text
stringlengths 4
2.78M
| meta
dict |
|---|---|
---
abstract: 'Living cells use phase separation and concentration gradients to organize chemical compartments in space. Here, we present a theoretical study of droplet dynamics in gradient systems. We derive the corresponding growth law of droplets and find that droplets exhibit a drift velocity and position dependent growth. As a consequence, the dissolution boundary moves through the system, thereby segregating droplets to one end. We show that for steep enough gradients, the ripening leads to a transient arrest that is induced by an narrowing of the droplet size distribution.'
address:
- '$^1$ Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany '
- '$^2$ Center for Advancing Electronics Dresden cfAED, Dresden, Germany'
- '$^3$ Department of Bioengineering, Imperial College London, South Kensington Campus, London SW7 2AZ, U.K.'
author:
- 'Christoph A. Weber$^{1,2}$, Chiu Fan Lee$^3$ and Frank Jülicher$^{1,2}$'
title: Droplet Ripening in Concentration Gradients
---
Introduction: Droplet ripening in concentration gradients in biology
====================================================================
Living cells have to organize many molecules in space and time in order to build compartments which can perform certain biological functions. The formation of these compartments is often regulated by spatially heterogenous distributions of molecular species. An example is the polarized distribution of polarity proteins in the course of asymmetric cell division [@brangwynne2011soft; @hyman2014liquid; @brangwynne2015polymer]. During asymmetric cell division, molecules of the cell cytoplasm are distributed unequally between both daughter cells [@cowanhyman2004asymmetric; @betschinger2004dare]. This can be studied in the first division of the fertilized egg of the roundworm *C.elegans*. RNA-protein aggregates called P-granules are segregated to the posterior side of the cell and are located in the posterior daughter cell after division. P-granules are liquid like droplets that form by phase separation from the cell cytoplasm [@Brangwynne_2009; @brangwynne2011soft; @hyman2014liquid; @brangwynne2015polymer]. The segregation and ripening of P-granule droplets toward the posterior is driven by a concentration gradient of the protein Mex-5 that regulates droplet dynamics [@Brangwynne_2009; @Lee_2013; @saha2016polar].
The ripening of drops guided by a concentration gradient of molecules that regulate phase separation fundamentally differs from classical Ostwald-ripening. In the case of Ostwald ripening, droplets are uniformly distributed throughout the system and the droplet size distribution broadens with time . If a concentration gradient of a regulator component is maintained, for example by sources and sinks [@saha2016polar], or [*via*]{} position-dependent reaction kinetics [@tenlen08; @griffin11], there is a broken symmetry generating a bias of droplet positions. Recently, droplet segregation in a concentration gradient has been discussed using a simplified model [@Lee_2013]. However, the dynamics of droplets ripening in a gradient of regulating molecules has not been explored (figure \[fig:Fig\_0\](a)).
![\[fig:Fig\_0\] (a) Schematic representation of the regulation of droplet (blue dot) formation by a regulator $R$ which can bind to droplet material $D$ to form the product ${\mathcal{R}}$:${{D}}$. (b) Illustration of droplet ripening in a gradient of regulator volume fraction $\phi_R$ (orange). At each time point a boundary (red dashed) divides the system into domains of growth and shrinkage. This boundary moves to the right (red arrow) leaving a region of dissolving drops behind. Droplets drift (black arrows) with velocity $V_\mathrm{d}$.](./Fig_1ab.jpg){width="80.00000%"}
In this paper we present a theoretical study of droplet ripening in a concentration gradient of a regulator that affects phase separation. Considering a simplified theory we extract generic physical features of droplet growth in the presence of concentration gradients. The generic features we study here are the spatially dependent, local equilibrium concentration and a spatially dependent actual concentration outside the droplets. If the distance between droplets is large, these features can be used to derive the generic laws of droplet ripening in concentrations gradients and thereby extend the classical theory for homogeneous systems [@Lifshitz_Slyozov_61; @wagner61]. Our central finding is that a regulator gradient leads to a drift velocity and a position dependent growth of drops (figure \[fig:Fig\_0\](b)). As a consequence, a dissolution boundary moves through the system, leaving droplets only in a region close to one boundary of the system. Using numerical calculations supported by analytic estimates, we study the growth dynamics of droplets in a gradient. We discover that, surprisingly, ripening is not always faster in the case of steeper regulator gradients. Instead, a transient arrest of ripening is observed that results from a narrowing of the droplet size distribution. Our work shows that a regulator gradient induces a novel and rich ripening dynamics in droplet systems.
Local regulation of phase separation
====================================
We use a simplified model to discuss two component phase separation that is influenced by a regulator. We consider a system consisting of a solvent ${{S}}$, droplet material ${{D}}$ and a regulator ${\mathcal{R}}$ In this model the regulator does not take part in demixing but influences phase separation of ${{D}}$ and ${{S}}$. We describe demixing by a $$\label{eq:FH_init}
f= {k_B T} \bigg[ \frac{\phi^T_{{D}}}{\nu_{{{D}}}} \ln{\phi^T_{{D}}}
+\frac{\phi_{{S}}}{\nu_{{{S}}}} \ln{\phi_{{S}}} \bigg]+ \mathcal{E} \, ,$$ where $k_B$ is the Boltzmann constant, $T$ is temperature and $\phi^T_{{D}}$ and $\phi_{{S}}$ denote the total volume fraction of droplet material and solvent, respectively, with $\phi^T_{{D}}+ \phi_{{S}}=1$. The molecular volumes $\nu_i$ connect volume fractions with concentrations $c_i$ by $\phi_i=\nu_i c_i$. The regulator influences phase separation by binding to droplet material, $${{D}}+{\mathcal{R}}\rightleftharpoons{\mathcal{R}}:{{D}}\, .$$ Here we consider the case where the bound state ${\mathcal{R}}$:${{D}}$ does not phase separate from the solvent. The total volume fraction of droplet material is given by the sum of contributions of bound and free molecules, $\phi^T_{{D}}=\phi_{{D}}+\phi_{{\mathcal{R}}:{{D}}} $. The binding process between regulator and droplet material can be described by mass action with the equilibrium binding constant $K_0=c_{{\mathcal{R}}:{{D}}}/(c_{{D}}c_{\mathcal{R}})$. Using the simplification $\nu_{{\mathcal{R}}:{{D}}}=\nu_{{D}}$, we write $K_0=\phi_{{\mathcal{R}}:{{D}}}/(\phi_{{D}}c_{\mathcal{R}})$. The interaction energy is given by $\mathcal{E}= {k_B T} \chi \, \phi_{{D}}\phi_{{S}}$, where $\chi$ is the interaction parameter. Expressing $\phi_{{D}}$ in terms of $\phi^T_{{D}}$ and considering a fast local equilibrium of the binding reaction we find
Spatial organization of phase separation
========================================
we consider a spatially inhomogeneous system that is locally at thermodynamic equilibrium such that at each position the local free energy is defined. Globally the system is maintained away from equilibrium by an imposed position dependent regulator gradient. For simplicity, we use a linear gradient along the $x$ direction, $\phi_{\mathcal{R}}(x)=\phi_0-m\cdot x$, with $x\in[0,L]$, where $L$ denotes the size of the system. We first look at a situation without droplets but with a possible spatial profile $\phi_{{D}}={\bar \Phi}(x)$ of droplet material. Since the spatial concentration profile of the regulator $\phi_{\mathcal{R}}(x)$ is imposed, the effective interaction parameter $\chi_\mathrm{eff}(x)$ becomes a function of $x$. As the droplet material is also distributed in space, the concentration at each position $x$ corresponds to a point in the phase diagram. The linear range $x\in[0,L]$ then maps onto a line that is indicated in the phase diagram in figure \[fig:Fig\_1\](a). Using the phase diagram, we can determine the position $x_\mathrm{d}$ of the dissolution boundary, which separates the region $x<x_\mathrm{d}$ where the fluid mixes, from the region $x>x_\mathrm{d}$ in which droplets can form. For $x>x_\mathrm{d}$, we can then determine the local equilibrium volume fraction ${{\Phi^\mathrm{in}_\mathrm{eq}}}(x)$ of the droplet material inside and ${{\Phi^\mathrm{out}_\mathrm{eq}}}(x)$ outside of a potential droplet, which depend on position. For $\nu_{{S}}\gg \nu_{{D}}$, ${{\Phi^\mathrm{in}_\mathrm{eq}}}$ is approximately constant along $x$. As we will see below, choosing this simple limit allows us to focus on the concentration field outside of the droplet. The spatial distribution of the regulator and droplet material imply a spatially dependent supersaturation defined as $$\label{eq:supersaturation}
\epsilon(x)=\frac{{\bar \Phi}(x)}{{{\Phi^\mathrm{out}_\mathrm{eq}}}(x)}-1 \, ,$$ which is positive for $x>x_\mathrm{d}$. In the absence of droplets, the concentration field ${\bar \Phi}(x)$ evolves in time satisfying a diffusion equation. If droplets are nucleated, their dynamics of growth or shrinkage is guided by the local supersaturation $\epsilon(x)$ as well as ${{\Phi^\mathrm{in}_\mathrm{eq}}}$ and ${{\Phi^\mathrm{out}_\mathrm{eq}}}(x)$. This droplet dynamics then in turn also influences the concentration field ${\bar \Phi}(x)$.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[fig:Fig\_1\] (a) Phase diagram. Interaction parameter $\chi_\mathrm{eff}$ as a function of the volume fraction of droplet material $\phi_{{D}}^T$. The binodal line (blue) and the critical point (triangle) are indicated. For a regulator concentration gradient, the positions $x$ in the system are mapped to a line (dashed/green line in the mixed/demixed region). At the position $x=x_\mathrm{d}$ this line crosses the binodal. (b) Equilibrium volume fractions outside and inside the droplet, ${{\Phi^\mathrm{out}_\mathrm{eq}}}$ and ${{\Phi^\mathrm{in}_\mathrm{eq}}}$, corresponding to the binodal line in (a), are shown as functions of position $x>x_\mathrm{d}$ for $K=500$. Parameters: $m=-3\cdot 10^{-3}$, $\phi_0=4\cdot 10^{-3}$, $\nu_{{S}}=10 \nu_{{D}}$.](./Fig_2a.pdf "fig:"){width="40.00000%"} ![\[fig:Fig\_1\] (a) Phase diagram. Interaction parameter $\chi_\mathrm{eff}$ as a function of the volume fraction of droplet material $\phi_{{D}}^T$. The binodal line (blue) and the critical point (triangle) are indicated. For a regulator concentration gradient, the positions $x$ in the system are mapped to a line (dashed/green line in the mixed/demixed region). At the position $x=x_\mathrm{d}$ this line crosses the binodal. (b) Equilibrium volume fractions outside and inside the droplet, ${{\Phi^\mathrm{out}_\mathrm{eq}}}$ and ${{\Phi^\mathrm{in}_\mathrm{eq}}}$, corresponding to the binodal line in (a), are shown as functions of position $x>x_\mathrm{d}$ for $K=500$. Parameters: $m=-3\cdot 10^{-3}$, $\phi_0=4\cdot 10^{-3}$, $\nu_{{S}}=10 \nu_{{D}}$.](./Fig_2b.pdf "fig:"){width="40.00000%"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Dynamics of a single drop in a concentration gradient
=====================================================
Here, $\ell_\mathrm{c}= 2\gamma \nu_{D}/(k_bT)$ is the capillary length, $\gamma$ denotes the surface tension of the droplet. The values of $\alpha$ and $\beta$ characterizing the far field together with the local concentration at the droplet surface, ${\phi^\mathrm{out}}(R,{\textcolor{black}{{\theta}}})$, then determine the local rates of growth or shrinkage of the drop. Deformations of the spherical shape of the droplet can be neglected if the surface tension is large and concentration gradients on the scale of the droplet are small. Furthermore, we focus, for simplicity, on the case where the Onsager cross coupling coefficient between the regulator and droplet material is negligible and we thus ignore how the spatial distribution of droplet material affects the maintained regulator gradient.
The solution to the diffusion equation (\[eq:diff\_equ\]) is of the form, $ {\phi^\mathrm{out}}(r,\theta) = \sum_{n=0}^\infty \left(A_n r^n +B_n r^{-n-1}\right) P_n (\cos \theta)$, where $P_n (\cos \theta)$ are the Legendre polynomials. Using the boundary conditions (\[eq:bc1\]) and (\[eq:bc2\]), we find $$\begin{aligned}
\nonumber
{\phi^\mathrm{out}}(r, \theta)&= \alpha \cos \theta \left( r - \frac{R^3}{r^2} \right)
+\beta \left( 1- \frac{R}{r}\right)
\\&+ \left({{\Phi^\mathrm{out}_\mathrm{eq}}}+ R \cos(\theta)\partial_x {{\Phi^\mathrm{out}_\mathrm{eq}}}\right) \left( 1+\frac{\ell_\mathrm{c}}{R}\right) \frac{R}{r} \, .
\label{eq:phi_quasi_static}\end{aligned}$$ a droplet at position $x_0$ can be expressed by a function ${\mathpzc R}(\theta, \varphi, t; x_0)$. The is $\partial_t {\mathpzc R}(\theta, \varphi,t; x_0)=v_\mathrm{n}(\theta, \varphi; x_0) $, where $v_\mathrm{n}=\vec n \cdot \vec{J}$ is the local velocity normal to the interface and $\vec n$ denotes a surface normal. Here, $\vec{J}=(\vec j^\mathrm{in}-\vec j^\mathrm{out})/(\Phi^\mathrm{in}_{{D}}-{{\Phi^\mathrm{out}_\mathrm{eq}}})$ is the [@Bray_Review_1994], and $\vec j^\mathrm{in}$ and $\vec j^\mathrm{out}$ denote the fluxes at the droplet surface inside and outside of the drop. Since the volume fraction inside the droplet is considered as constant and independent of the droplet position, $\vec j^\mathrm{in}=0$ and $\vec j^\mathrm{out}=-D \nabla {\phi^\mathrm{out}}$. In the limit of strong phase separation (${{\Phi^\mathrm{in}_\mathrm{eq}}}\gg {{\Phi^\mathrm{out}_\mathrm{eq}}}$) the growth velocity normal to the interface is $v_\mathrm{n}= (D/{{\Phi^\mathrm{in}_\mathrm{eq}}}) \partial_r {\phi^\mathrm{out}}|_{r=R} $.
Ripening of multiple drops in a regulator gradient
==================================================
$$\label{eq:LS_inhomogi}
\frac{\mathrm{d} }{\mathrm{d}t} R_i=
\frac{D}{R_i}\; \frac{{{\Phi^\mathrm{out}_\mathrm{eq}}}(x_i)}{{{\Phi^\mathrm{in}_\mathrm{eq}}}} \left[ \epsilon(x_i) - \frac{\ell_\mathrm{c}}{R_i}\right] \, .$$ The droplet drift velocity, $\mathrm{d} x_i/\mathrm{d}t= {V}_\mathrm{d}(x_i)$, is given by $$\label{eq:drifti}
\frac{\mathrm{d} x_i}{\mathrm{d}t}= \frac{D}{{{\Phi^\mathrm{in}_\mathrm{eq}}}} \left[ \partial_x {\bar \Phi}(x)\vert_{x_i} - \partial_x {{\Phi^\mathrm{out}_\mathrm{eq}}}(x)\vert_{x_i} \left(1 + \frac{\ell_\mathrm{c}}{R_i} \right) \right] \, .$$ If the distance between droplets is large relative to their size, droplets only interact [*via*]{} the concentration field $\bar \Phi(x,t)$ which represents the far field. It is governed by a diffusion equation including gain and loss terms associated with growth or shrinkage of drops: $$\label{eq:beta_eq}
\partial_t \bar \Phi(x,t) = D \frac{
\partial^2}{
\partial x^2} \bar \Phi(x,t) - \frac{4\pi {{\Phi^\mathrm{in}_\mathrm{eq}}}}{3L^3} \sum^{N}_{i=1}\delta(x_i-x)\frac{\mathrm{d} }{\mathrm{d} t} R^3_i(t)
\, .$$ For simplicity, in the above equation we consider a regulator gradient along the $x$ axis. Please note that equation (\[eq:beta\_eq\]) describes the effects of large scale spatial inhomogeneities on the ripening dynamics. Since large scale variations of $\bar \Phi(x,t)$ only build up along the $x$-directions, derivatives of $\bar \Phi$ along the $y$ and $z$ axes do not contribute.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![\[fig:Fig\_2\] Droplet ripening in concentration gradients. (a) Mean droplet radius $\langle R \rangle_x$ at position $x$ as a function of time for different $x$ as indicated. A spatial profile of equilibrium volume fraction of slope $s=0.5$ is imposed at time $t=L^2/D$ (quench). (b) Characteristic time $\tau_L$ required to segregate the volume of droplet material toward $x=L$, and dissolution time $\tau_\mathrm{D}$ required to reach $10$ droplets starting from $\mathcal{O}(10^4)$, as a function of quench slope $s$. The horizontal grey line indicated the value of $\tau_\mathrm{D}$ for classical Ostwald ripening ($s=0$). ](./Fig_3a.pdf "fig:"){width="40.00000%"} ![\[fig:Fig\_2\] Droplet ripening in concentration gradients. (a) Mean droplet radius $\langle R \rangle_x$ at position $x$ as a function of time for different $x$ as indicated. A spatial profile of equilibrium volume fraction of slope $s=0.5$ is imposed at time $t=L^2/D$ (quench). (b) Characteristic time $\tau_L$ required to segregate the volume of droplet material toward $x=L$, and dissolution time $\tau_\mathrm{D}$ required to reach $10$ droplets starting from $\mathcal{O}(10^4)$, as a function of quench slope $s$. The horizontal grey line indicated the value of $\tau_\mathrm{D}$ for classical Ostwald ripening ($s=0$). ](./Fig_3b.pdf "fig:"){width="40.00000%"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[fig:Fig\_3\] Narrowing droplet size distribution. (a) Mean radius $\langle R \rangle$ averaged over all drops in the system as a function of time for three quench slopes $s$. The onset of arrest is indicated (arrows). (b) Duration and time of onset of the arrest phase as a function of quench slope $s$. The vertical black line indicates the quench slope $s_\mathrm{c}=\{1-[3/2-\Phi_0/(2\bar \Phi(t=L^2/D))]^{-1}\}/L$ below which no arrest can occur. It can be calculated by the condition that the critical radius $R_\mathrm{c}$ at $x=L$ is reduced by at least a factor of $3/2$ during the quench such that the largest droplets in the distribution grow more slowly than smaller ones and the distribution narrows. For our numerical solutions $s_\mathrm{c}\approx0.017/L$, which is consistent with the emergence of the arrest along quench slope found in our numerical calculations. (c) Standard deviation $\delta R$ of the droplet radius distribution as a function of time for three different quench slopes $s$. The onset of arrest corresponds to a sudden narrowing of the distribution (arrows). (d) Rate of droplet growth $\mathrm{d} R/\mathrm{d}t$ as a function of droplet radius $R$ before (grey) and after (black) the spatial quench. The droplet radius distribution $p(R)$ at the moment of the quench is shown (blue). Narrowing of $p(R)$ occurs if droplet size exceeds the radius for which the growth rate is maximal (black dots). ](./Fig_4a.pdf "fig:"){width="40.00000%"} ![\[fig:Fig\_3\] Narrowing droplet size distribution. (a) Mean radius $\langle R \rangle$ averaged over all drops in the system as a function of time for three quench slopes $s$. The onset of arrest is indicated (arrows). (b) Duration and time of onset of the arrest phase as a function of quench slope $s$. The vertical black line indicates the quench slope $s_\mathrm{c}=\{1-[3/2-\Phi_0/(2\bar \Phi(t=L^2/D))]^{-1}\}/L$ below which no arrest can occur. It can be calculated by the condition that the critical radius $R_\mathrm{c}$ at $x=L$ is reduced by at least a factor of $3/2$ during the quench such that the largest droplets in the distribution grow more slowly than smaller ones and the distribution narrows. For our numerical solutions $s_\mathrm{c}\approx0.017/L$, which is consistent with the emergence of the arrest along quench slope found in our numerical calculations. (c) Standard deviation $\delta R$ of the droplet radius distribution as a function of time for three different quench slopes $s$. The onset of arrest corresponds to a sudden narrowing of the distribution (arrows). (d) Rate of droplet growth $\mathrm{d} R/\mathrm{d}t$ as a function of droplet radius $R$ before (grey) and after (black) the spatial quench. The droplet radius distribution $p(R)$ at the moment of the quench is shown (blue). Narrowing of $p(R)$ occurs if droplet size exceeds the radius for which the growth rate is maximal (black dots). ](./Fig_4b.pdf "fig:"){width="40.00000%"}
![\[fig:Fig\_3\] Narrowing droplet size distribution. (a) Mean radius $\langle R \rangle$ averaged over all drops in the system as a function of time for three quench slopes $s$. The onset of arrest is indicated (arrows). (b) Duration and time of onset of the arrest phase as a function of quench slope $s$. The vertical black line indicates the quench slope $s_\mathrm{c}=\{1-[3/2-\Phi_0/(2\bar \Phi(t=L^2/D))]^{-1}\}/L$ below which no arrest can occur. It can be calculated by the condition that the critical radius $R_\mathrm{c}$ at $x=L$ is reduced by at least a factor of $3/2$ during the quench such that the largest droplets in the distribution grow more slowly than smaller ones and the distribution narrows. For our numerical solutions $s_\mathrm{c}\approx0.017/L$, which is consistent with the emergence of the arrest along quench slope found in our numerical calculations. (c) Standard deviation $\delta R$ of the droplet radius distribution as a function of time for three different quench slopes $s$. The onset of arrest corresponds to a sudden narrowing of the distribution (arrows). (d) Rate of droplet growth $\mathrm{d} R/\mathrm{d}t$ as a function of droplet radius $R$ before (grey) and after (black) the spatial quench. The droplet radius distribution $p(R)$ at the moment of the quench is shown (blue). Narrowing of $p(R)$ occurs if droplet size exceeds the radius for which the growth rate is maximal (black dots). ](./Fig_4c.pdf "fig:"){width="40.00000%"} ![\[fig:Fig\_3\] Narrowing droplet size distribution. (a) Mean radius $\langle R \rangle$ averaged over all drops in the system as a function of time for three quench slopes $s$. The onset of arrest is indicated (arrows). (b) Duration and time of onset of the arrest phase as a function of quench slope $s$. The vertical black line indicates the quench slope $s_\mathrm{c}=\{1-[3/2-\Phi_0/(2\bar \Phi(t=L^2/D))]^{-1}\}/L$ below which no arrest can occur. It can be calculated by the condition that the critical radius $R_\mathrm{c}$ at $x=L$ is reduced by at least a factor of $3/2$ during the quench such that the largest droplets in the distribution grow more slowly than smaller ones and the distribution narrows. For our numerical solutions $s_\mathrm{c}\approx0.017/L$, which is consistent with the emergence of the arrest along quench slope found in our numerical calculations. (c) Standard deviation $\delta R$ of the droplet radius distribution as a function of time for three different quench slopes $s$. The onset of arrest corresponds to a sudden narrowing of the distribution (arrows). (d) Rate of droplet growth $\mathrm{d} R/\mathrm{d}t$ as a function of droplet radius $R$ before (grey) and after (black) the spatial quench. The droplet radius distribution $p(R)$ at the moment of the quench is shown (blue). Narrowing of $p(R)$ occurs if droplet size exceeds the radius for which the growth rate is maximal (black dots). ](./Fig_4d.pdf "fig:"){width="40.00000%"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
To study the ripening dynamics of droplets in a concentration gradient we solved the equations (\[eq:LS\_inhomogi\]) to (\[eq:beta\_eq\]) numerically. To access the late time regime of ripening we first initialize about $N=10^7$ drops with radii taken from the Lifschitz-Slyozov distribution [@Lifshitz_Slyozov_61; @wagner61] in a system of position independent equilibrium concentration $\Phi_0$, and fix the concetration inside $ {{\Phi^\mathrm{in}_\mathrm{eq}}}=1$. For $t\ge L^2/D$, we then spatially quench the system by imposing the spatially varying equilibrium concentration ${{\Phi^\mathrm{out}_\mathrm{eq}}}(x)= \Phi_0 (1- s\, x)$ [@footnote2], which we refer to as “spatial quench” in the following. In our numerical studies we find that droplets experience a non-uniform growth depending on the position and the stage of ripening (figure \[fig:Fig\_2\](a)). At the beginning, all drops grow in the region where the concentration ${\bar \Phi}(x)$ exceeds the local equilibrium concentration at the drop surface, ${{\Phi^\mathrm{out}_\mathrm{eq}}}(x) \left(1+\ell_\mathrm{c}/R\right)$, and shrink otherwise. The dissolution boundary at $x=x_\mathrm{d}$ obeys . It moves according to $$\frac{\mathrm{d} x_\mathrm{d\mathrm}}{\mathrm{d}t} = {\frac{\mathrm{d} \bar \Phi(x_\mathrm{d}(t))} {\mathrm{d}t} }
\bigg/{ \left.\frac{ \mathrm{d}{{\Phi^\mathrm{out}_\mathrm{eq}}}(x)}{\mathrm{d}x}\right|_{x=x_\mathrm{d}(t)} } \, .$$ For $\mathrm{d}{{\Phi^\mathrm{out}_\mathrm{eq}}}(x)/\mathrm{d}x <0$, the position of the dissolution boundary $x_\mathrm{d}$ moves to the right until it reaches the system boundary at $x= L$ (Supplemental video [@SM]). At long times, the volume fraction at all positions approaches the minimum of the equilibrium volume fraction, $\bar\Phi(x)\to {{\Phi^\mathrm{out}_\mathrm{eq}}}(L)$, and all droplets dissolve except at $x=L$ (figure \[fig:Fig\_2\](a), Supplemental video [@SM]).
The characteristic time $\tau_L$ of droplet segregation depends on the quench slope $s$. It decreases for increasing $s$ according $\tau_L\propto s^{-1}$ (figure \[fig:Fig\_2\](b)). In contrast the time of droplet dissolution, $\tau_\mathrm{D}$, defined as the time to reach 10 droplets, changes only weakly with the quench slope $s$ and can even increase (figure \[fig:Fig\_2\](b)). Interestingly, the droplet ripening exhibits periods of transient arrest, during which droplet number and size remain almost constant (figure \[fig:Fig\_3\](a)). These arrest phases govern the time of droplet dissolution for large quench slopes since they occur for sufficiently large quench slopes $s$. The duration of arrest is roughly constant as a function of $s$ and the onset of the arrest phase is delayed for decreasing $s$ [@footnote1] (figure \[fig:Fig\_3\](b)). In particular, the standard deviation of droplet radius exhibits a pronounced minimum when the arrest begins (figure \[fig:Fig\_3\](c)). After the arrest phase droplets undergo classical Ostwald ripening where time-dependence of $\langle R \rangle$ and $\delta R$ is consistent with $t^{1/3}$ (figures \[fig:Fig\_2\](a) and \[fig:Fig\_3\](a)). The effect of a narrowing droplet size distribution has also been observed in spatially homogeneous systems with constant influx of phase separation material [@clark_nanolett11; @Vollmer_size_focussing].
The narrowing of the droplet size distribution in a concentration gradient is fundamentally different from broadening of the droplet size distributions during classical Ostwald-ripening [@Lifshitz_Slyozov_61; @wagner61]. Ostwald ripening is characterized by a supersaturation that decreases with time, leading to an increase of the critical droplet radius $R_\mathrm{c}=\ell_\mathrm{c}/\epsilon(t)\propto t^{1/3}$. The droplet size distribution $p(R)$ has a universal shape and is nonzero only in the interval $[0, 3R_\mathrm{c}/2]$ (figure \[fig:Fig\_3\](d), ). The broadening of $p(R)$ follows from larger droplets growing at a larger rate $\mathrm{d}R/\mathrm{d}t$ than smaller droplets. Though $\mathrm{d}R/\mathrm{d}t$ has a maximum at $R=2R_\mathrm{c}$ and decreases for large $R$, no droplets exist larger than $3R_\mathrm{c}/2$.
This situation changes in the presence of a concentration gradient. The spatial quench reduces the local critical radius $R_\mathrm{c}(x\simeq L)=\ell_\mathrm{c}/\epsilon(x\simeq L)$ at the rightmost boundary $x\simeq L$ as compared to the critical radius before the quench (equation (\[eq:supersaturation\])). This quench also shifts the maximum of $\mathrm{d}R/\mathrm{d}t$ for droplets at $x\simeq L$ to smaller radii (black line in figure \[fig:Fig\_3\](d)) since the radius corresponding to the maximum occurs at $R=2 R_\mathrm{c}$. As a result, many droplets now exist after the spatial quench with large radii $R>2 R_\mathrm{c}(x\simeq L)$. These droplets grow more slowly than those at $R=2R_\mathrm{c}$ which leads to a narrowing of the size distribution $p(R)$ at $x\simeq L$. The critical radius $R_\mathrm{c} (x \simeq L)$ remains small because dissolution of droplets at $x<L$ leads to a diffusive flux toward $x\simeq L$ and thus keeps the volume fraction $\bar \Phi(L)$ at increased levels. These conditions hold longer if the spatial quench has a steeper slope. As a result the distribution narrows more for steeper quenches. When the critical radius catches up with the mean droplet size narrowing stops and the onset of arrest occurs. At this time droplets have almost equal size which slows down the exchange of material between droplets [*via*]{} Ostwald ripening, leading to a long phase of almost constant size and number of droplets (figure \[fig:Fig\_2\](a)). During this arrest phase, the droplet distribution broadens slowly.
Conclusion and Outlook
======================
We would like to thank Shambaditya Saha and Anthony A. Hyman for stimulating discussions.
References {#references .unnumbered}
==========
[10]{}
Brangwynne CP. Soft active aggregates: mechanics, dynamics and self-assembly of liquid-like intracellular protein bodies. Soft Matter. 2011;7(7):3052–3059.
Hyman AA, Weber CA, J[ü]{}licher F. Liquid-liquid phase separation in biology. Annual review of cell and developmental biology. 2014;30:39–58.
Brangwynne CP, Tompa P, Pappu RV. Polymer physics of intracellular phase transitions. Nature Physics. 2015;11(11):899–904.
Cowan CR, Hyman AA. Asymmetric cell division in C. elegans: cortical polarity and spindle positioning. Annu Rev Cell Dev Biol. 2004;20:427–453.
Betschinger J, Knoblich JA. Dare to be different: asymmetric cell division in Drosophila, C. elegans and vertebrates. Current biology. 2004;14(16):R674–R685.
Brangwynne CP, Eckmann CR, Courson DS, Rybarska A, Hoege C, Gharakhani J, et al. Germline P Granules Are Liquid Droplets That Localize by Controlled Dissolution/Condensation. Science. 2009;324(5935):1729–1732.
Lee CF, Brangwynne CP, Gharakhani J, Hyman AA, Jülicher F. Spatial Organization of the Cell Cytoplasm by Position-Dependent Phase Separation. Phys Rev Lett. 2013 Aug;111:088101.
Saha S, Weber CA, Nousch M, Adame-Arana O, Hoege C, Hein MY, et al. Polar Positioning of Phase-Separated Liquid Compartments in Cells Regulated by an mRNA Competition Mechanism. Cell. 2016;166(6):1572–1584.
Lifshitz IM, Slyozov VV. The kinetics of precipitation from supersaturated solid solutions. Journal of Physics and Chemistry of Solids. 1961;19(1?2):35 – 50.
Wagner C. Theorie der Alterung von Niederschlägen durch Umlösen (Ostwald-Reifung). Berichte der Bunsengesellschaft für physikalische Chemie. 1961;65(7):581–591.
Yao JH, Elder K, Guo H, Grant M. Theory and simulation of Ostwald ripening. Physical review B. 1993;47(21):14110.
Bray AJ. Theory of phase-ordering kinetics. Advances in Physics. 1994;43(3):357–459.
Tenlen J, Molk J, London N, Page B, Priess J. . Development. 2008;135(22):3665–3675.
Griffin E, Odde D, Seydoux G. . Cell. 2011;146(6):955–968.
Rubinstein M, Colby RH. Polymer physics. Oxford: OUP Oxford; 2003.
We checked that a quench of the form ${{\Phi^\mathrm{out}_\mathrm{eq}}}(x)= \Phi_0 [1- s (x-L/2)]$ leads to qualitatively similar results;.
See Supplemental Material for videos and more information at http://;.
The arrest phase is defined as the time interval during which $N(t)$ decreases more slowly then $t^{-1/2}$ For Ostwald ripening ($s=0$) $N(t)\sim t^{-1}$;.
Clark MD, Kumar SK, Owen JS, Chan EM. . Nano Letters. 2011 may;11(5):1976–1980.
Vollmer J, Papke A, Rohloff M. Ripening and focusing of aggregate size distributions with overall volume growth. Frontiers in Physics. 2014;2:18.
Alberti S, Hyman AA. Are aberrant phase transitions a driver of cellular aging? BioEssays. 2016;38(10):959–968.
Zwicker D, Hyman AA, Jülicher F. Suppression of Ostwald ripening in active emulsions. Phys Rev E. 2015 Jul;92:012317.
Zwicker D, Seyboldt R, Weber CA, Hyman AA, Jülicher F. Growth and division of active droplets provides a model for protocells. Nature Physics. 2016;10.1038/nphys3984:1745–2481.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Recently, there has been a surge of interest in using Rényi entropies as quantifiers of correlations in many-body quantum systems. However, it is well known that in general these entropies do not satisfy the strong subadditivity inequality, which is a central property ensuring the positivity of correlation measures. In fact, in many cases they do not even satisfy the weaker condition of subadditivity. In the present paper we shed light on this subject by providing a detailed survey of Rényi entropies for bosonic and fermionic Gaussian states. We show that for bosons the Rényi entropies always satisfy subadditivity, but not necessarily strong subadditivity. Conversely, for fermions both do not hold in general. We provide the precise intervals of the Rényi index $\alpha$ for which subadditivity and strong subadditivity are valid in each case.'
author:
- Giancarlo Camilo
- 'Gabriel T. Landi'
- Sebas Eliëns
bibliography:
- 'library.bib'
title: On the Strong Subadditivity of the Rényi entropies for bosonic and fermionic Gaussian states
---
\[sec:int\]Introduction
=======================
There is currently a large effort from the quantum physics community to further our understanding of the typical correlation patterns of many-body quantum states [@Huber2010; @Bennett2011; @Levi2013; @Schwaiger2015; @Goold2015a; @Girolami2017; @Calabrese2005]. Examples include the dynamics of quantum quenches [@calabrese_evolution_2005; @calabrese_entanglement_2007; @2011_Santos_PRL_107; @nezhadhaghighi_entanglement_2014; @alba_quench_2017; @alba_entanglement_2018; @alba_renyi_2017], quantum Markov chains [@Kato2016; @Renner2002; @Ibinson2008], Bell non-locality [@Chaves2016], among others. Remarkably, correlation patterns are also starting to become accessible to controlled quantum platforms. For instance, the correlations in a 8-site Bose-Hubbard model have been measured in [@Islam2015], while in Ref. [@Lanyon2017] the authors have implemented a type of matrix product state tomography for a trapped ion system. These efforts are helping to shed light on the important question of typicality of many body states [@Bianchi2018; @Eisert2010a; @Alba2017; @Calabrese2005; @Eisler2007; @Zurek2018; @Deutsch2018; @Srednicki1994; @Deutsch1991; @Kaufman2016; @Popescu2006; @White1992; @Schollwock2011; @DeChiara2006; @McCulloch2007], i.e., which are the typical sectors of the Hilbert space that are usually occupied by systems with well defined structures.
Central to this discussion, therefore, are the tools used to quantify correlations. Quite often, these are based on entropic quantities. In the simplest scenario, a system with density matrix $\rho_{AB}$ is divided into a bipartition $AB$. The amount of information shared between the two parts is then characterized by the Mutual Information (MI) defined as [@Nielsen] $$\label{MI}
\mathcal{I}(A\!:\!B) = S(\rho_A) + S(\rho_B) - S(\rho_{AB}) \geq 0\,,$$ where $S(\rho) = - \tr(\rho \ln \rho)$ is the von Neumann entropy of $\rho$ and $\rho_A = \tr_B \rho_{AB}$ and $\rho_B = \tr_A \rho_{AB}$ are the reduced density matrices of $A$ and $B$. The MI is always non-negative, a fact known as the *subadditivity* (SA) of the von Neumann entropy. Moreover, it is zero if and only if $A$ and $B$ are in a product state, i.e., $\rho_{AB}=\rho_A\otimes\rho_B$. This ensures that $\mathcal{I}(A\!:\!B)$ is a genuine measure of correlations. When $\rho_{AB}$ is a pure state we get $S(\rho_{AB}) = 0$ and $S(\rho_A) = S(\rho_B)$. In this case the MI becomes twice the entanglement entropy. Instead, for mixed states the MI provides a measure of the total amount of correlations (quantum and classical) between $A$ and $B$.
In addition to subadditivity, the von Neumann entropy also satisfies a more general inequality known as strong subadditivity (SSA) [@Nielsen]. Namely, given a tripartite state $\rho_{ABC}$, then
[rCl]{}\[SSA\] S(\_[AB]{}) + S(\_[BC]{}) && S(\_[ABC]{}) + S(\_B).
The proof of this inequality, which is rather easy in the classical setting, turns out to be much more complicated in the quantum scenario [@Lieb1973] (although for theories with a holographic dual it is remarkably simple [@Headrick:2007km]). The SSA is one of the most fundamental results in quantum information theory, being the bedrock behind a large number of important results, including quantification of multipartite correlations. Indeed, from it one is naturally led to define a quantity called the Conditional Mutual Information (CMI) [@Renner2002; @Ibinson2008; @Kato2016; @Fawzi; @Berta2018; @Wyner1978] $$\label{CMI}
\mathcal{I}(A\!:\!C|B) = S(\rho_{AB}) + S(\rho_{BC}) - S(\rho_{ABC}) - S(\rho_B) \geq 0\,,$$ which quantifies the amount of information shared between $A$ and $C$ *given* knowledge of $B$.
Despite the enormous success of the von Neumann entropy, in recent years there has been a surge of interest in alternative entropic quantifiers, particularly those given by the so-called Rényi entropies, defined as $$\label{renyi}
S_\alpha(\rho) = \frac{1}{1-\alpha} \ln \tr\rho^\alpha\,,$$ where $\alpha \in [0,\infty)$ is a continuous parameter. The von Neumann entropy is recovered for $\alpha=1$ (understood as the limit $\alpha\to1$) whereas for $\alpha = 2$ one gets the log of the purity of the state, $S_2(\rho) = - \ln \tr\rho^2$. The Rényi entropy satisfies several properties expected from an entropic quantifier, such as non-negativity (vanishing for pure states and positive otherwise) and additivity under tensor product. However, there are other desired properties that they generally violate. The most important is precisely the SSA [(\[SSA\])]{}, which is *not* satisfied for any $\alpha\ne1$. In fact, as shown in Ref. [@Matus2007], the Rényi entropies of a fixed order $\alpha$ in general satisfy no linear inequalities whatsoever (although a weak subadditivity relation mixing $S_\alpha$ and $S_0$ was proved in [@2002quant.ph..4093V] for all $\alpha$). As a result, if one naïvely introduces the Rényi Mutual Information (RMI) and Rényi Conditional Mutual Information (RCMI) as in [(\[MI\])]{} and [(\[CMI\])]{}, i.e., $$\begin{aligned}
\mathcal{I}_\alpha(A\!:\!B) &= S_\alpha(\rho_{A}) + S_\alpha(\rho_{B}) - S_\alpha(\rho_{AB}),\label{RMI}\\[0.3cm]
\mathcal{I}_\alpha(A\!:\!C|B) &= S_\alpha(\rho_{AB}) + S_\alpha(\rho_{BC}) - S_\alpha(\rho_{ABC}) - S_\alpha(\rho_B)\,,\label{RCMI}\end{aligned}$$ these quantities are not ensured to be non-negative and therefore become meaningless as correlation measures. Explicit violations of SSA were illustrated, for instance, in Ref. [@Adesso2003] for two qubits and in Ref. [@Kormos2017] in the quench dynamics of the transverse field Ising model. In spite of this, the quantities above were recently measured experimentally in Ref. [@Islam2015].
However, quite surprisingly, Adesso *et. al.* have recently found a specific situation that offers a curious exception to the problem mentioned above [@Adesso2012] (see also [@Adesso:2016ldo]). Namely, when restricted to Gaussian states of a many-body bosonic system the SSA turns out to be true for $\alpha = 2$. This result offered an interesting alternative for quantifying correlations using $\mathcal{I}_2$ in bosonic Gaussian states, which are of central interest, for instance, to quantum optics and continuous variable quantum computation. It has since led to more theoretical developments including results that have no counterpart for the Von Neumann entropy [@lami_log-determinant_2017; @lami_schur_2016].
Apart from this success story of the Rényi-2 entropy for bosonic Gaussian states not much is known about the Rényi entropies for Gaussian states in general. In particular, there is no information theory for fermionic Gaussian states based on Rényi entropies with $\alpha \neq 1$. It is worth stressing at this point that Gaussian states, both bosonic and fermionic, are quite prevalent in modern condensed matter physics, appearing in a multitude of paradigmatic models such as the transverse field Ising model [@1970_Barouch_PRA_2; @2008_Fagotti_PRA_78; @2011_Calabrese_PRL_106; @2013_Fagott_PRB_87] and models for topological phases [@2006_Kitaev_AP_321; @2007_Pachos_AP_322; @2007_Feng_PRL_98; @2009_Kells_PRB_80; @2015_Dubail_PRB_92]. For the transverse field Ising and similar models mappable to free fermions, Gaussian states naturally arise as steady states when the system is brought out of equilibrium by a quantum quench, in accordance with the logic of the generalized Gibbs ensemble, even when the initial state is not Gaussian [@2008_Cramer_PRL_100; @2013_Fagott_PRB_87; @2016_Gluza_PRL_117; @2018_Murthy_arxiv; @2018_Gluza_arxiv] (note the exception of massless relativistic bosons [@sotiriadis_memory-preserving_2016]). Moreover, in a recent work, the dynamics of the logarithmic negativity in such a quench scenario has been related to the RMI with $\alpha = 1/2$ [@Alba:2018hie]. A moments thought reveals that Gaussian states are also at play behind the scenes in certain mean-field approximations such as Hartree-Fock and Bogoliubov theory. With the ubiquitousness of Gaussian states and the continued progress in uncovering relations between observables and entropic quantities [@2010PhRvL.104o7201H; @Cornfeld:2018sac] in mind, we see that furthering our understanding of SA and SSA for Rényi entropies of Gaussian states is both a natural and timely question. This is precisely the goal of this work.
In this paper we set out to map the full range of values $\alpha$ for which the SA and SSA conditions are satisfied in the case of bosonic and fermionic Gaussian many-body states. We begin by discussing the covariance matrix approach for computing the Rényi entropies for Gaussian states of bosons and fermions in Sec. \[sec:entropies\]. We make an effort to emphasize as much as possible the similarity between both cases. From our development we then find the following results: for bosonic Gaussian states we prove that SA is satisfied for all $0 \leq \alpha < \infty$, while for fermions this only holds true for the interval $0 \leq \alpha \leq 2$ and we explicitly give a procedure to construct violations for $\alpha >2$ (Sec. \[sec:SA\]). Sec. \[sec:SSA\] is devoted to SSA, where we rely on numerical evidence to conjecture that for bosons SSA holds true in the domain $0\le\alpha \leq 2$ and we show explicit violations for $\alpha > 2$. For fermions, on the other hand, we suggest that SSA holds for $0\le \alpha \leq 1$, although explicit violations in this case are only found for $\alpha \gtrsim 1.3$. In Sec. \[sec:final\] we gather some concluding remarks.
\[sec:entropies\]Rényi entropy for Gaussian states
==================================================
In this section we discuss how to compute Rényi entropies for Gaussian states by considering first bosonic and then fermionic states
Bosonic systems
---------------
We consider a system of $N$ bosonic modes $a_1, \ldots, a_N$ satisfying $[a_i,a_j^\dagger] = \delta_{ij}$ and $[a_i, a_j] = 0$. We then define the quadrature operators $$\label{quadrature_bosons}
q_i = \frac{1}{\sqrt{2}} (a_i^\dagger + a_i),
\qquad
p_i = \frac{{\mathrm{i}}}{\sqrt{2}}(a_i^\dagger - a_i)$$ and collect them in the vector $\bm{X} = (q_1, p_1, \ldots, q_N, p_N)$. The canonical commutation relations in terms of $\bm{X}$ are then written as $$\label{algebra_bosons}
[X_I,X_J] = {\mathrm{i}}\, \Omega_{IJ},
\qquad
\Omega = \mathbb{I}_N \otimes ({\mathrm{i}}\sigma_y),$$ where $I,J=1,\ldots,2N$ and $\sigma_y$ is the Pauli matrix. The antisymmetric matrix $\Omega$ is the symplectic form of the bosonic algebra [@Simon1994; @Dutta1995].
Given a state $\rho$, we now define the $2N\times 2N$ Covariance Matrix (CM) associated to the operators $\bm{X}$ as [@PirlaRMP; @Adesso2014] $$\label{boson_CM}
\Gamma_{IJ} = \langle \{X_I,X_J\} \rangle,$$ where $\{, \}$ represents the anti-commutator and, for simplicity, we assume $\langle X_I \rangle = 0$ since local unitary transformations are not expected to affect the entanglement properties of the state. The algebra (\[algebra\_bosons\]) imposes that any physically reasonable CM must satisfy the following *bona fide* condition $$\label{bona_fide_boson}
\Gamma - {\mathrm{i}}\, \Omega \geq 0\,,$$ which can be viewed as the generalized Schrödinger-Robertson uncertainty relation.
It is well-known from Williamson’s theorem [@Simon1994; @Dutta1995] that any CM may be diagonalized by a symplectic transformation $M$ (i.e., $M \Omega M{^{\text{T}}}= \Omega$) so that [(\[algebra\_bosons\])]{} is preserved and $$\label{CMdiagonal_boson}
M \Gamma M{^{\text{T}}}= \text{diag}(\sigma_1, \sigma_1, \ldots, \sigma_L, \sigma_L)\,,$$ where the $\sigma_i \geq 1$ are called the symplectic eigenvalues of $\Gamma$ and correspond to the $N$ positive eigenvalues of ${\mathrm{i}}\Omega \Gamma$. It is important to stress that they are not the true eigenvalues of the CM (these are not preserved by $M$), even though their product $\prod_{i=1}^N\sigma_i^2=\det(M\Gamma M{^{\text{T}}})=\det(\Gamma)$ happens to be basis-independent. So far these facts hold for arbitrary density matrices. If we now assume that the state is Gaussian, i.e., fully characterized by its CM, then the density matrix in this diagonal basis may be written as a product of thermal oscillators $$\label{rho_diagonal_boson}
\rho = \prod\limits_{i=1}^N Z_i{^{-1}}e^{-\frac{1}{2} \beta_i (\tilde{p}_i^2 + \tilde{q}_i^2)}\,,$$ where $Z_i{^{-1}}=(1- e^{-\beta_i})$ is a normalization constant while $\tilde{q}_i$ and $\tilde{p}_i$ are related to the original quadrature operators $q_i,p_i$ by means of $\tilde{\bm{X}} = M \bm{X}$. Moreover, the local temperatures $\beta_i$ are uniquely determined by the symplectic eigenvalues according to $$\label{sigma_boson}
\sigma_i = \coth\left(\frac{\beta_i}{2}\right)\,.$$ From Eq. (\[rho\_diagonal\_boson\]) it is now straightforward to compute the corresponding Rényi-$\alpha$ entropy (\[renyi\]), which becomes
[rCl]{} \[renyi\_bosonic\_1\] S\_() &=& \_[i=1]{}\^N .
For future convenience, let us introduce the function $$\label{func_bosons}
f_\alpha^+(x) = \left(\frac{x+1}{2}\right)^\alpha - \left(\frac{x-1}{2}\right)^\alpha$$ and rewrite [(\[renyi\_bosonic\_1\])]{} as
[rCl]{} \[renyi\_bosonic\_2\] S\_() &=& \_[i=1]{}\^N f\_\^+(\_i).
From this one can already see that the case $\alpha = 2$ is rather special since $f_2^+(x)=x$, which allows expressing the Rényi entropy as the log determinant of the CM, namely $$\label{renyi_bosonic_alpha2}
S_2(\rho) = \frac{1}{2} \ln \det (\Gamma)\,.$$ The Rényi entropies satisfy a monotonicity property $S_{\alpha_1}(\rho) \geq S_{\alpha_2}(\rho)$ if $\alpha_1 \leq \alpha_2$. It is therefore useful to study the limits $\alpha \to 0,\infty$. This gives rise to the max-entropy $$S_0(\rho) = N \ln 0^+ + \sum_{i=1}^N \ln \beta_i$$ which diverges as $N\log \alpha$ for $\alpha \to 0^+$. This is not surprising as the max-entropy is formally equivalent to the log of the rank of $\rho$ and we are working with the infinite dimensional Hilbert space for bosons. However, in suitable linear combinations of entropies such as for the (conditional) mutual information, the divergences cancel and we get a meaningful, finite, limiting result. For pure states $\beta_i \to \infty$ and $S_0(\rho) = 0$. The min-entropy can be computed as $$\label{b_infty}
S_{\infty}(\rho) = \sum_{i=1}^N \ln (1+n_i)$$ in terms of the occupation numbers of the normal modes $n_i =(e^{\beta_i}-1)^{-1} = (\sigma_i -1)/2$. We will see that this can be used to bound the (conditional) mutual information.
Fermionic systems
-----------------
We now parallel the development above for the case of fermions. Consider a system of $N$ fermionic operators $c_i$ satisfying $\{c_i, c_j^\dagger\} = \delta_{ij}$ and $\{c_i,c_j\} = 0$. We define the set of majorana operators analogously to Eq. (\[quadrature\_bosons\]), as $$\gamma_{2i-1} = \frac{1}{\sqrt{2}}(c_i + c_i^\dagger)\,,
\qquad
\gamma_{2i} = \frac{{\mathrm{i}}}{\sqrt{2}} (c_i^\dagger - c_i),$$ which together form the analog of $\bm{X}$ and satisfy $$\label{algebra_fermions}
\{\gamma_I,\gamma_J\} = \delta_{IJ}\,.$$ The fermionic CM is then constructed similarly to Eq. (\[boson\_CM\]) as $$\Gamma_{IJ} = {\mathrm{i}}\, \langle[ \gamma_I, \gamma_J]\rangle.$$ Any valid fermionic covariance matrix must now satisfy the bona-fide relation [@2009PhRvA..79a2306K] $$\label{bona_fide_fermion}
{\mathrm{i}}\, \Gamma - 1 \leq 0,$$ which again parallels Eq. (\[bona\_fide\_boson\]). This can be equivalently stated as $\Gamma\Gamma^\dagger\le1$. The Gaussian state is pure iff $\Gamma^2=-1$.
The fermionic CM can always be put in block diagonal form by an orthogonal transformation $M$ (i.e. $M M{^{\text{T}}}= 1$) that preserves [(\[algebra\_fermions\])]{}, namely $$\label{CMdiagonal_fermion}
M \Gamma M{^{\text{T}}}= \bigoplus_{i=1}^N
\begin{pmatrix}0 &-\sigma_i\\ \sigma_i &0\end{pmatrix}\,,$$ where $\sigma_i\in[-1,1]$. Each block is then trivially diagonalized and the eigenvalues of $\Gamma$ are simply $\pm{\mathrm{i}}\sigma_i$. Unlike in the bosonic case [(\[CMdiagonal\_boson\])]{}, these are the true eigenvalues of $\Gamma$. This means in particular that any matrix function $f(\Gamma)$ is block-diagonalized by the same $M$, having $f(\pm{\mathrm{i}}\sigma_i)$ as its eigenvalues. A Gaussian fermionic state may then be written as $$\label{rho_diagonal_fermion}
\rho = \prod\limits_{j=1}^N Z_j{^{-1}}e^{- {\mathrm{i}}\,\beta_j \gamma_{2j-1} \gamma_{2j}},$$ where $Z_i=2\cosh \beta_i$ and $$\label{sigma_fermion}
\sigma_i = \tanh\left(\frac{\beta_i}{2}\right),$$ which is the analog of Eq. [(\[sigma\_boson\])]{}.
From this we once again can readily compute the Rényi entropy, which reads
[rCl]{} \[renyi\_fermionic\_1\] S\_() &=& \_[i=1]{}\^N .
By defining $$\label{func_fermions}
f_\alpha^-(x) = \left(\frac{1+x}{2}\right)^\alpha + \left(\frac{1-x}{2}\right)^\alpha\,,$$ we can write the Rényi entropy as
[rCl]{} \[renyi\_fermionic\_2\] S\_() &=& \_[i=1]{}\^N f\_\^-(\_i).
Together with Eq. [(\[renyi\_bosonic\_2\])]{} this gives a fully unified description of Rényi entropies for both bosonic or fermionic modes. However, an important difference with respect to bosons comes from the fact that here $\pm{\mathrm{i}}\sigma_i$ are the true eigenvalues of the CM. Namely, using $f_\alpha^-(x)=f_\alpha^-(-x)$ one can rewrite [(\[renyi\_fermionic\_2\])]{} as a log determinant for any $\alpha$,
[rCl]{} \[renyi\_fermionic\_3\] S\_() &=& f\_\^-().
The case $\alpha = 2$ once again allows a simple expression linear in the CM even though $f_2^-(x) = (1+x^2)/2$ is not a linear function. This is due to the peculiar structure of $\Gamma$, which implies $\det(1-\Gamma^2) = \big[\det(1+\Gamma)\big]^2$ and yields for the Rényi-2 entropy $$S_2(\rho) = N\ln(2) - \ln \det (1+\Gamma).$$ For fermions the max-entropy is simply $$\label{f_0}
S_0(\rho) = N\ln 2$$ as expected from the finite dimension of the Hilbert space, except for pure states for which we should put $S_0(\rho) =0$. The min-entropy can be expressed in exactly the same way as for bosons $$S_{\infty}(\rho) = \sum_{i=1}^N\ln (1+n_i)$$ in terms of the fermionic occupation numbers $n_i =(e^{\beta_i}+1)^{-1} =(1-\sigma_i)/2$ of the normal modes.
\[sec:SA\]Subadditivity
=======================
In this section we study the analog of the SA inequality [(\[MI\])]{} for the Rényi-$\alpha$ entropies of Gaussian states, which is equivalent to non-negativity of the Rényi mutual information [(\[RMI\])]{}. We show that it holds for any $\alpha$ in the bosonic case, while for fermions it holds in the window $\alpha\in[0,2]$.
Bosons
------
Let us consider a bipartition $A\cup B$ of a $N$-boson system in the Gaussian state $\rho_{AB}$. The corresponding CM can be parametrized in the block form $$\label{GammaAB}
\Gamma_{AB} =
\begin{pmatrix}
\Gamma_A & \chi_{AB} \\[0.2cm]
\chi_{AB}^T & \Gamma_B
\end{pmatrix}\,.$$ The reduced states $\rho_A,\rho_B$ are also Gaussian, being fully characterized by the reduced CM’s $\Gamma_A,\Gamma_B$. We denote the set of symplectic eigenvalues $\sigma_i^{AB}$ of the full system by $\{d_i\}$ and collect the symplectic eigenvalues $\sigma_i^A$ and $\sigma_i^B$ of the subsystems into a single set of elements $\{c_i\}$. Both sets are assumed to be organized in non-decreasing order. Finding the necessary and sufficient conditions under which the symplectic spectra $\{c_i\}$ and $\{d_i\}$ are mutually consistent defines the Gaussian version of the so called *quantum marginal problem*. These conditions have been found in [@2008CMaPh.280..263E] and amount to the following chain of inequalities
\[Eisert\] $$\begin{aligned}
\sum_{j=1}^k c_j &\ge \sum_{j=1}^k d_j\,,\qquad k=1,\ldots,N\label{Eisertsum}\\
c_n-\sum_{j=1}^{N-1}c_j &\le d_n-\sum_{j=1}^{N-1}d_j\label{Eisertdifference}\,.\end{aligned}$$
With the conventions above, the Rényi mutual information [(\[RMI\])]{} associated to the Rényi entropies [(\[renyi\_bosonic\_2\])]{} can be written as $$\begin{aligned}
\mathcal{I}_\alpha(A:B) = \sum_{j=1}^N\left[g^+_\alpha(c_j)-g^+_\alpha(d_j)\right]\,,\end{aligned}$$ where $g^+_\alpha(x)\equiv\tfrac{1}{1-\alpha}\log f_\alpha^+(x)$. The non-negativity of $I_\alpha$ then follows straightforwardly from the fact that, for any $\alpha\ge0$ ($\alpha\ne1$), $g^+_\alpha(x)$ is a positive, monotonically increasing, concave function of $x$ in the domain $x\ge1$ (See Appendix \[app\]). Namely, $$\begin{aligned}
\label{SA_proof_bosonic}
\mathcal{I}_\alpha &\ge\sum_{j=1}^N {g^+_\alpha}'(c_j)(c_j-d_j)\notag\\
&= \sum_{k=1}^{N-1}\!\big[{g^+_\alpha}'(c_k)-{g^+_\alpha}'(c_{k+1})\big]\!\sum_{j=1}^k\!(c_j-d_j) + {g^+_\alpha}'(c_N)\!\sum_{j=1}^N\!(c_j-d_j)\notag\\
&\ge0\,.\end{aligned}$$ where the inequality in the first line uses concavity of $g^+_\alpha$, the second line is a convenient rewriting using Abel’s partial summation formula, and the last inequality holds since each term in the previous expression is ensured to be non-negative by [(\[Eisertsum\])]{} together with monotonicity and concavity of $g^+_\alpha$. In other words, [(\[SA\_proof\_bosonic\])]{} shows rather remarkably the subadditivity of all the quantum Rényi entropies in the particular class of Gaussian states. It is interesting to note that the second constraint [(\[Eisertdifference\])]{} plays no role in the proof.
Fermions
--------
Now consider a bipartite $N$-fermion Gaussian state $\rho_{AB}$ with associated fermionic CM $$\label{GammaABfermion}
\Gamma_{AB} =
\begin{pmatrix}
\Gamma_A & \chi_{AB} \\[0.2cm]
-\chi_{AB}^T & \Gamma_B
\end{pmatrix}\,.$$ The positive eigenvalues ${\mathrm{i}}\Gamma_{AB}$ by will be denoted by $\{y_i\}$, and we combine the positive eigenvalues ${\mathrm{i}}\Gamma_A,{\mathrm{i}}\Gamma_B$ into the set $\{x_i\}$. The ordering in the fermionic case is assumed to be non-increasing. The Sing-Thomson theorem [@sing1976; @thompson1977; @thompson1979] then implies that $$\begin{aligned}
\sum_{j=1}^k x_j &\leq \sum_{j=1}^ky_j,\qquad k =1 ,\ldots, N,\\
\sum_{j=1}^{N-1}x_j - x_N &\leq \sum_{j=1}^{N-1} y_j - y_N\,.\end{aligned}$$ Let us define $g^-_{\alpha}(x) = (1-\alpha)^{-1}\log f^-_{\alpha}(x)$. Then we can write $$\mathcal{I}_{\alpha}(A\!:\!B) = \sum_{j=1}^N\left[g^-_{\alpha}(x_j) - g^-_{\alpha}(y_j)\right].$$ The function $g^-_{\alpha}$ is monotonically *decreasing* and concave for the interval $\alpha \in [0,2]$. Hence, for these values of $\alpha$ we can use the sequence of steps identical to Eq. to prove SA for fermionic Gaussian states and $0 \leq \alpha \leq 2$.
For $\alpha>2$ we have checked numerically for $2$ and $3$ modes that SA is violated. Indeed, for $2$ modes it is straightforward to construct a prototypical violating CM in this range. For instance, $$\begin{aligned}
\label{fermion_2mode_violation}
\Gamma_{AB} =
\begin{pmatrix}
0 &-\lambda &-\xi &0 \\
\lambda &0 &0 &-\xi \\
\xi &0 &0 &-\lambda\\
0 &\xi &\lambda &0
\end{pmatrix}\,\end{aligned}$$ has $\sigma_{\pm} = |\lambda \pm \xi|$ and satisfies the bona fide condition [(\[bona\_fide\_fermion\])]{} as long as $|\lambda\pm \xi|\le 1$ with $|\lambda|\le1$ (for simplicity one can take both to be positive). Assuming small $\xi$ (for illustration purposes only – this is not needed) it follows that $$\begin{aligned}
I_{\alpha}(A:B) &= 2g^-_{\alpha}(\lambda) - g^-_{\alpha}(\lambda+\xi) - g^-_{\alpha}(\lambda-\xi)\\
&= -{g^-_{\alpha}}''(\lambda)\,\xi^2 + \mathcal{O}(\xi^4)\,,\end{aligned}$$ which for any $\alpha>2$ can be made to violate SA since in this case it is always possible to find a $\lambda$ for which ${g^-_\alpha}''(\lambda)$ is positive (recall that $g^-_\alpha(x)$ is no longer concave on the domain $x\in[0,1]$). A simple calculation shows that this correlation matrix $\Gamma_{AB}$ is realized by a thermal state of the Hamiltonian $$H = (\beta_+ + \beta_-)\left[c^\dagger_1 c_1 + c^\dagger_2 c_2\right] + {\mathrm{i}}(\beta_+ - \beta_-)\left[c^\dagger_1 c_2 - c^\dagger_2 c_1\right]\,,$$ where $\beta_\pm=2\arctan\sigma_\pm$ and the temperature defines the unit of energy. In other words, the state $\rho\sim e^{-H}$ with the Hamiltonian above leads precisely to the SA-violating CM [(\[fermion\_2mode\_violation\])]{}.
\[sec:SSA\]Strong subadditivity
===============================
In this section we take a step further over Sec. \[sec:SA\] and present a complete survey of the regimes of validity of SSA inequality [(\[SSA\])]{} for the Rényi-$\alpha$ entropies of Gaussian states. That is, we map the full range of values of $\alpha$ for which the Rényi conditional mutual information [(\[RCMI\])]{} is ensured to be non-negative (SSA satisfied) and those where it is not (SSA violated). This is done both for bosons and fermions by numerically generating a large number of bona fide CMs and using them to find explicit violations of SSA for some $\alpha$. For bosons, we find strong evidence that SSA holds for all $\alpha\in[0,2]$ while in the fermionic case we find no violations in the interval $\alpha\in[0,\alpha_\text{max}]$ with $\alpha_\text{max}\approx 1.3$.
Bosons
------
Consider a tripartite system in the state $\rho_{ABC}$ and let us parametrize its CM in block form as $$\Gamma_{ABC} = \begin{pmatrix}
\Gamma_A & \chi_{AB} & \chi_{AC} \\[0.2cm]
\chi_{AB}^T & \Gamma_B & \chi_{BC} \\[0.2cm]
\chi_{AC}^T & \chi_{BC}^T & \Gamma_C
\end{pmatrix}.$$ Since the reduced density matrix is still Gaussian, the corresponding covariance matrix may be simply obtained by discarding the blocks one is tracing over. For instance, the CM $\Gamma_{AB}$ associated with $\rho_{AB}=\Tr_C\,\rho_{ABC}$ will be the one in [(\[GammaAB\])]{}.
Here we focus only on the cases where the full state $\rho_{ABC}$ is mixed, since for pure states the SSA follows from SA using the property that $S(\rho_A)=S(\rho_{\bar{A}})$ for any subsystem $A$ and its complement $\bar{A}$ [@Nielsen].
We start by reviewing the special case of $\alpha=2$ studied in [@Adesso2012], the only one (apart from the trivial von Neumann case $\alpha=1)$ for which SSA is known to be satisfied. From [(\[renyi\_bosonic\_alpha2\])]{} it is straightforward to write the corresponding RCMI [(\[RCMI\])]{} as $$\label{CMI_Gaussian}
\mathcal{I}_2(A\!:\!C|B) = \frac{1}{2} \ln \frac{\det(\Gamma_{AB}) \det(\Gamma_{BC}) }{\det(\Gamma_{ABC})
\det(\Gamma_{B})}\ge0\,.$$ The non-negativity follows immediately from the Hadamard-Fischer determinant inequality relating the minors of the symmetric positive-semidefinite matrix $\Gamma_{ABC}$.
For $\alpha\ne2$ we need to deal with the generic expression [(\[renyi\_bosonic\_1\])]{} involving particular functions of the symplectic eigenvalues. This is rather non-trivial since the only known inequalities relating the symplectic eigenvalues of the CM and those of its reduced CMs are the ones in [(\[Eisert\])]{}. We do this numerically by generating a huge number of bona fide CMs of three- and four-mode Gaussian states and use them to scan for violations of SSA by computing the RCMI [(\[RCMI\])]{} for different values of the index $\alpha$. Figure \[SSA\_violations\] shows a scatter plot of the result. It provides clear evidence that SSA is violated for all $\alpha>2$, while no violation is found for $0\le\alpha\le2$. We conjecture that this result holds true in general. Less extensive searches for SSA violations by bona fide CMs with up to six modes have not given any reason to believe that violations will be found for higher numbers of modes, but we presently do not have a proof. We hope to come back to this in future work.
Let us conclude this subsection with the statement that if $\mathcal{I}_{\infty}(A\!:\!B|C)<0$, we find that $\mathcal{I}_{\alpha}(A\!:\!B|C)<0$ for all $\alpha > \alpha_*$ with $$\alpha_* =1 + \frac{S_{\infty}(\rho_{AB}) + S_{\infty}(\rho_{BC})}{|\mathcal{I}_{\infty}(A\!:\!B|C)|}.$$ Hence a SSA violation for $S_{\infty}(\rho)$ implies SSA violation for any finite $\alpha$ [^1].
![Violation of strong subadditivity (SSA) of bosonic Rényi entropies for three-mode Gaussian states. The plots were created by randomly generating $\sim200k$ bona fide correlation matrices and computing for different $\alpha$ the Rényi conditional mutual information $\mathcal{I}_{\alpha}(A\!:\!C|B)$, where $A$, $B$, and $C$ correspond to three single-mode subsystems.[]{data-label="SSA_violations"}](bosonsSSA){width="45.00000%"}
Fermions
--------
The story for fermions is quite similar to that of bosons. We consider a tripartite system in a state given by the block-diagonal CM $$\Gamma_{ABC} = \begin{pmatrix}
\Gamma_A & \chi_{AB} & \chi_{AC} \\[0.2cm]
-\chi_{AB}^T & \Gamma_B & \chi_{BC} \\[0.2cm]
-\chi_{AC}^T & -\chi_{BC}^T & \Gamma_C
\end{pmatrix}$$ and its corresponding reductions to subsystems $AB,BC,A,C$. Once more we can restrict attention only to mixed states.
Even though here all the Rényi entropies can be written in the determinant form [(\[renyi\_fermionic\_3\])]{}, no claim can be made based on the Hadamard-Fisher determinant inequality since the fermionic CM is not positive semi-definite. In particular, unlike the case of bosons, not even the second Rényi entropy is guaranteed to be strongly subadditive.
One again has to resort to numerics and deal with the generic expression [(\[renyi\_bosonic\_2\])]{}. We generate a large number of random bona fide CMs of three- and four-mode states and compute [(\[RCMI\])]{} looking for SSA violations as the Rényi index $\alpha$ is varied. The results appear in Figure \[SSA\_violations\_fermions\]. We find no violations of SSA in the region $\alpha\in[0,\alpha_\text{max}]$ with $\alpha_\text{max}\approx1.3$, while many violating counterexamples are found beyond this window. The limiting value $\alpha_\text{max}$ above which violations occur is a bit surprising. The most reasonable possibility is that the limiting value is actually $\alpha = 1$, but violations for $1< \alpha <\alpha_\text{max}$ are either very hard to find by random sampling or only possible for larger numbers of modes. We generated similarly exhaustive numbers of bona fide CMs with up to twelve modes for fermions, in an attempt to find such violations with $\alpha< \alpha_\text{max}$, but without result. Again, we lack a formal proof for SSA to hold for $\alpha \in [0,1]$, but we conjecture it to be true.
![Violation of strong subadditivity (SSA) of fermions for three-mode Gaussian states. The plots were created by randomly generating $\sim300k$ bona fide correlation matrices and computing for different $\alpha$ the Rényi conditional mutual information $\mathcal{I}_{\alpha}(A\!:\!C|B),0)$ for single-mode subsystems $A$, $B$, and $C$.[]{data-label="SSA_violations_fermions"}](fermionsSSA){width="45.00000%"}
\[sec:final\]Final remarks
==========================
We have studied the Rényi entropies of bosonic and fermionic Gaussian states using the covariance matrix approach. A special effort has been made to clarify as much as possible the technical similarities between the bosonic and fermionic calculations. As our main result, we have obtained a complete map of the regimes of validity of the strong subaddivity (SSA) and subadditivity (SA) inequalities as a function of the Rényi index $\alpha$. We prove that SA holds for all $\alpha\ge0$ in the case of bosons and for $\alpha\in[0,2]$ in the case of fermions. The proofs rely only on concavity properties of the entropy functions together with a set of inequalities relating the (symplectic) eigenvalues of the full correlation matrix $\Gamma_{AB}$ and those of its bipartitions $\Gamma_{A,B}$. The situation becomes more complicated for the SSA, for which it was necessary to resort to numerics. We provided strong numerical evidence that SSA is satisfied for $\alpha\in[0,2]$ and violated for $\alpha>2$ in the case of bosons, while for fermions we conjecture $\alpha\in[0,1]$ to be free of violations even though explicit violations are only found for $\alpha \geq \alpha_\text{max}\approx1.3$.
Our calculations for the SA for fermions put on firmer grounds the results reported in [@Kormos2017], where it was shown that for temperature-driven quenches in the Ising model the Rényi mutual information in the resulting non-equilibrium steady state can become negative for $\alpha>2$ while it is definitely positive for $\alpha<2$. It also sheds light on the recent results of [@Alba:2018hie], which showed that at late times after a quench in integrable theories the logarithmic negativity becomes proportional to the Rényi mutual information with $\alpha=\frac{1}{2}$. Our results guarantee that this object is always non-negative for both free bosons and free fermions, which strengthen the case for it as a good entanglement quantifier.
As discussed in the introduction, our main goal with these results was to clarify the possible ranges of validity in which Rényi-based correlation quantifiers can be employed. This is particularly important in light of the fact that some Rényi entropies (particularly the Rényi-2) naturally appear in analytical, numerical and even experimental approaches. For instance, in Ref. [@Islam2015] the authors experimentally implemented a method to measure the Rényi-2 entropy in a bosonic system, from which they construct the corresponding Rényi-2 mutual information. Their system, however, are generally in non-Gaussian states so that the positivity of the Rényi-2 mutual information is not guaranteed.
Notwithstanding, it is our hope that by continuing with this approach one may be able to map out these ranges of validity for different classes of states. For instance, a natural candidate would be tensor networks with well defined structures, such as matrix product states.
An obvious continuation of this work is to prove the conjectured domains of validity of the SSA for the Rényi-$\alpha$ entropies. The proof is likely to involve tools other than the ones appearing in the SA proof (in particular, a $\alpha$-dependent property of the entropy functions $g^\pm_\alpha$ that restricts the proof to the range $\alpha\in[0,2]$ for bosons and $\alpha\in[0,1]$ for fermions). One can also use inspiration from standard operator-based approaches to similar proofs (as opposed to the present one based on eigenvalues), such as the one used in [@Adesso:2016ldo; @2010arXiv1007.4626A] or the Schur complement techniques introduced in [@lami_log-determinant_2017; @lami_schur_2016]. We hope to report on this in the near future.
Acknowledgements {#acknowledgements .unnumbered}
=================
We are grateful to Diego P. Pires for helpful discussions and to Gerardo Adesso and Ludovico Lami for useful correspondence. GTL acknowledges the International Institute of Physics, where part of this work was developed, for both the hospitality and the financial support. GTL also acknowledges the funding from the University of São Paulo, the São Paulo Research Foundation FAPESP (grant numbers 2016/08721-7 and 2017/20725-0), and the Brazilian funding agency CNPq (grant number INCT-IQ 246569/2014-0). GC and SE acknowledge financial support from the Brazilian ministries MEC and MCTIC.
Concavity properties of the functions $g^\pm_\alpha$ {#app}
====================================================
This appendix is devoted to study the concavity properties of the entropy functions $$\begin{aligned}
g_\alpha^\pm(x) = \frac{\pm1}{\alpha-1}\log f^\pm_\alpha(x)\,,\quad f^\pm_\alpha(x) = \frac{[x+1]^\alpha}{2^{\alpha}}\mp\frac{[\pm (x-1)]^\alpha}{2^{\alpha}},\end{aligned}$$ where upper signs correspond to bosons and lower signs to fermions. It will be convenient to introduce the shorthand notation $x_1=x+1$ and $x_2=\pm(x-1)$ so that both cases can be treated in a unified way as $f^\pm_\alpha(x) = 2^{-\alpha}(x_1^\alpha\mp x_2^\alpha)$. Recall that in the bosonic case the domain is $x\ge1$, meaning that $x_1\ge2$ and $x_2\ge0$; for fermions, on the other hand, the domain is $x\in[-1,1]$ but since the function is even one can focus only on the subdomain $x\in[0,1]$ so that $x_{1}\in[1,2],x_{2}\in[0,1]$. It is then straightforward to write the second derivative of $g^\pm_\alpha$ as $$\begin{aligned}
\label{gpp}
&\partial_x^2 g^\pm_\alpha(x) = \frac{\pm1}{\alpha-1}\frac{f^\pm_\alpha \,\partial_x^2 f^\pm_{\alpha} - (\partial_x f^\pm_{\alpha})^2}{(f^\pm_\alpha)^2}\notag\\
&= \frac{\alpha(x_1x_2)^{\alpha-1}}{(\alpha-1)(f^\pm_\alpha)^2}\left[\mp\left(y^{\alpha-1}+y^{1-\alpha}\right)+(1-\alpha)(y+y^{-1})\pm2\alpha\right]\,,\end{aligned}$$ where we introduced $y=\frac{x_1}{x_2}\ge1$. In other to prove concavity of $g_\alpha^\pm$, we have to show that $\partial_x^2 g_\alpha^\pm\le0$ for every $x$ in the domain.
We focus first on bosons. Since the prefactor in [(\[gpp\])]{} is negative for $0\le\alpha<1$ and positive for $\alpha>1$, the task becomes to show that the term in the square brackets is non-negative in the former case and non-positive in the latter. Both results follow trivially from the Bernoulli inequalities $y^\mu \le(1-\mu)+\mu y$ (for $y>0$ and $0\le\mu\le1$) and $y^\mu \ge(1-\mu)+\mu y$ (for $y>0$ and $\mu\ge1$ or $\mu\le0$) with $\mu=\alpha$. This proves that $g^+_\alpha(x)$ is concave for all $\alpha\ge0$.
Now we move to fermions. For $0\le\alpha<1$, the concavity of $g^-_\alpha$ is a direct consequence of the concavity of $f^-_\alpha$ (i.e., it is preserved by the log), namely the fact that $\partial_x^2 f^-_\alpha=\alpha(\alpha-1)f^-_{\alpha-2}\le0$. For $\alpha>1$, we first notice that the prefactor in [(\[gpp\])]{} is positive and hence what remains is to show that the term inside the square brackets is non-positive. Clearly this is not going to happen for all $\alpha$ since the positive $y$-dependent piece can easily overcome the negative contributions for large enough $\alpha$. The limiting value for which this is avoided turns out to be $\alpha=2$. Indeed, the non-positivity of the square brackets for $\alpha\in[1,2]$ follows straightforwardly from the first Bernoulli inequality above with $\mu=\alpha-1$. In other words, $g^-_\alpha(x)$ is concave for $0\le\alpha\le2$.
[^1]: Using the bound $[(\sigma +1)/2 ]^{\alpha-1} \leq f^-_{\alpha}(\sigma) \leq [(\sigma +1)/2 ]^{\alpha}$ we can bound the RCMI as $S_{\infty}(\rho_{AB})\leq(\alpha-1)[\mathcal{I}_{\alpha}(A\!:\!B\!|\!C) - \mathcal{I}_{\infty}(A\!:\!B|\!C))] \leq S_{\infty}(\rho_{AB}) + S_{\infty}(\rho_{BC})$ from which the result follows.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present the discovery of 3 transiting planets from the WASP survey, two hot-Jupiters: WASP-177b ($\sim$0.5 M$_{\rm Jup}$, $\sim$1.6 R$_{\rm Jup}$) in a 3.07-d orbit of a $V = 12.6$ K2 star, WASP-183b ($\sim$0.5 M$_{\rm Jup}$, $\sim$1.5 R$_{\rm Jup}$) in a 4.11-d orbit of a $V = 12.8$ G9/K0 star; and one hot-Saturn planet WASP-181b ($\sim$0.3 M$_{\rm Jup}$, $\sim$1.2 R$_{\rm Jup}$) in a 4.52-d orbit of a $V = 12.9$ G2 star. Each planet is close to the upper bound of mass-radius space and has a scaled semi-major axis, $a/R_{*}$, between 9.6 and 12.1. These lie in the transition between systems that tend to be in orbits that are well aligned with their host-star’s spin and those that show a higher dispersion.'
author:
- |
Oliver D. Turner,$^{1}$[^1] D.R. Anderson,$^{2}$ K. Barkaoui,$^{3,4}$, F. Bouchy,$^{1}$ Z. Benkhaldoun$^{4}$ D.J.A. Brown,$^{5,6}$ A. Burdanov,$^{3}$ A. Collier Cameron,$^{7}$ E. Ducrot,$^{3}$ M. Gillon,$^{3}$ C. Hellier,$^{2}$ E. Jehin,$^{3}$ M. Lendl,$^{8,1}$ P.F.L. Maxted,$^{2}$ L.D. Nielsen,$^{1}$ F. Pepe,$^{1}$ D. Pollacco,$^{5,6}$ F.J. Pozuelos,$^{3}$ D. Queloz,$^{1,9}$ D. Ségransan,$^{1}$ B. Smalley,$^{2}$ A.H.M.J. Triaud,$^{10}$ S. Udry$^{1}$, and R.G. West$^{5,6}$\
$^{1}$Observatoire de Genève, Université de Genève, 51 Chemin des Maillettes, 1290 Sauverny, Switzerland\
$^{2}$Astrophysics Group, Keele University, Staffordshire ST5 5BG, UK\
$^{3}$Space sciences, Technologies and Astrophysics Research (STAR) Institute, Université de Liège, Liège 1, Belgium\
$^{4}$Oukaimeden Observatory, High Energy Physics and Astrophysics Laboratory, Cadi Ayyad University, Marrakech, Morocco\
$^{5}$Department of Physics, University of Warwick, Coventry CV4 7AL, UK\
$^{6}$Centre for Exoplanets and Habitability, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK\
$^{7}$SUPA, School of Physics and Astronomy, University of St. Andrews, North Haugh, Fife KY16 9SS, UK\
$^{8}$Space Research Institute, Austrian Academy of Sciences, Schmiedlstr. 6, A-8042 Graz, Austria\
$^{9}$Cavendish Laboratory, J J Thomson Avenue, Cambridge CB3 0HE, UK\
$^{10}$School of Physics & Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK\
bibliography:
- 'w177-w181-w183\_bib.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: 'Three Hot-Jupiters on the upper edge of the mass-radius distribution: WASP-177, WASP-181 and WASP-183'
---
\[firstpage\]
planets and satellites: detection – planets and satellites: individual: WASP-177b – planets and satellites: individual: WASP-181b – planets and satellites: individual: WASP-183b
Introduction
============
Since the beginning of the project the Wide Angle Search for Planets (WASP; @2006PASP..118.1407P) survey has discovered nearly 190 transiting, close-in, giant exoplanets. As they transit their host stars their bulk properties, mass and radius, can be determined relatively easily. Their transits allow for deeper characterisation that has led to the discovery of multiple chemical and molecular species in their atmospheres and the observation of planetary winds [@2016ApJ...817..106B].
Close-in exoplanets can also provide information on the formation and migration mechanisms of solar systems. It is expected that hot-Jupiter exoplanets initially form much further from their stars than where we detect them today. Therefore some mechanism must cause this migration. There are two proposed pathways, high eccentricity migration or disk migration. In the former some mechanism e.g. Kozai cycles [@2003ApJ...589..605W; @2013apf..book.....A] or planet-planet scattering [@1996Sci...274..954R; @1996Natur.384..619W], forces the cold Jupiter into a highly eccentric orbit which then is tidally circularised via interaction with the star. During this kind of migration it is possible for the planet orbital axis to become mis-aligned with the stellar spin axis [@2007ApJ...669.1298F]. In the latter mechanism the planet loses angular momentum via interaction with the stellar disk during formation and migrates inward [@1980ApJ...241..425G]. This is expected to preserve the initial spin-orbit alignment [@2009ApJ...705.1575M], though work is being done to investigate the production of mis-aligned planets due to inclined protoplanetary discs [@2017MNRAS.471.2334X].
The alignment between the stellar rotation axis and planet orbit can be investigated with the Rossiter-M$^{\rm c}$Laughlin (RM) technique (, etc.). These observations have shown a general trend for systems orbiting cool stars (with $T_{\rm eff} < 6250$K; @2012ApJ...757...18A [@2015ApJ...800L...9A]) to be more well aligned than systems orbiting hotter stars. Tides are also expected to play a role. In cool star systems, those with smaller scaled semi-major axes, $a/R_{*}$, tend to be more often well aligned than those with larger $a/R_{*}$. Though this picture is far from clear as there seems to be evidence for the hot/cool alignment disparity holding even for systems with large separations or low mass planets meaning tidal effects should be minimal (@2015ApJ...801....3M) casting tidal realignment into doubt (see also the discussion of @2017AJ....153..205D).
In this paper we present the discovery of three systems at the upper edge of the mass-radius envelop of hot-giants that could be useful probes of tidal re-alignment.
Observations
============
Each of these planets was initially flagged as a candidate in data taken with both WASP arrays located at Roque de los Muchachos Observatory on La Palma and at the South African Astronomical Observatory (SAAO). The data were searched for periodic signals using a BLS method as per [@2006MNRAS.373..799C; @2007MNRAS.380.1230C]. The survey itself is described in more detail by [@2006PASP..118.1407P].
In order to confirm the planetary nature of the signals radial velocity (RV) data were obtained with the CORALIE spectrograph on the 1.2-m Swiss telescope at La Silla, Chile . Additional photometry was acquired using EulerCam (, also on the 1.2-m Swiss) and the two 0.6-m TRAPPIST telescopes , based at La Silla and Oukaimeden Observatory in Morocco [@Gillon2017Natur; @Barkaoui2018].
Due to the low masses of WASP-181b and WASP-183b, we also acquired HARPS data[^2]. These observations are summarised in Table \[tab:obs\]. The TRAPPIST data from 2018-08-13 contain a meridian flip at BJD = 2458344.5639. During analysis the data were partitioned at this point and modeled as two datasets.
Figures \[fig:W177-phot\], \[fig:W181-phot\] and \[fig:W183-phot\] show the phase folded discovery and follow-up data. The RVs exhibit signals in phase with those found in the transit data and are consistent with companion objects of planetary mass. We checked for correlation between the RV variation and the bisector spans, see Fig.\[fig:bisectors\]. We find no strong correlation and so further exclude the possibility that these objects are transit mimics.
![As for Fig.\[fig:W177-phot\] for the WASP-181 system. CORALIE data in bottom figure are small (red) while HARPS data are larger (blue) symbols.[]{data-label="fig:W181-phot"}](wasp-181-rv-phot-nobi_span-labels){width="\columnwidth"}
![As for Fig.\[fig:W181-phot\] for the WASP-183 system.[]{data-label="fig:W183-phot"}](wasp-183-rv-phot-nobi_span-labels){width="\columnwidth"}
![Radial velocity measurements plotted against line bisector spans. There is no strong correlation between the two, thus ruling out transit mimics. Solid lines are the linear best fit to the data. The dotted lines show the 1$\sigma$ uncertainty limits on the fit. []{data-label="fig:bisectors"}](bisector-w177-w181-w183-col){width="\columnwidth"}
Analysis
========
Stellar Parameters
------------------
To obtain the stellar parameters effective temperature, $T_{\rm eff}$, metallicity, \[Fe/H\], and surface gravity, $\log g$, we followed the method of using iSpec . To do this we corrected each spectrum for the computed RV shift, cleaned them of cosmic ray strikes and convolved them to a spectral resolution, $R$, of $47\,000$. Then, ignoring areas typically affected by telluric lines we used the synthetic spectral fitting technique to derive the stellar parameters. Via iSpec we used SPECTRUM [@1994AJ....107..742G] as the radiative transfer code with atomic data from VALD [@2011BaltA..20..503K], a line selection based on a $R\sim 47\,000$ solar spectrum [@2016csss.confE..22B; @2017hsa9.conf..334B] and the MARCS model atmospheres in the wavelength range 480- to 680-nm. We increased the uncertainties in these parameters by adding the dispersion found by analysing the [*Gaia*]{} benchmark stars with iSpec as per .
We determined the stellar density from and initial fit to the lightcurves and then used it along with the $T_{\rm eff}$ and metallicity to determine stellar masses, for later use in the joint analysis, and the stellar ages with the Bayesian stellar evolution code BAGEMASS . The resulting parameters are presented in the top part of Table \[tab:all\_results\] and the corresponding isochrons/evolutionary tracks are shown in Fig \[fig:tracks\].
![Isochrones (solid/blue) and evolution tracks (dot-dashed/red) output by BAGEMASS for each of the planets we present with the corresponding isochrone age and mass (labelled).[]{data-label="fig:tracks"}](a_MLT1pt5){width="\columnwidth"}
BAGEMASS uses an MCMC with a densely sampled grid of stellar models to compute stellar masses and ages. There are three usable grids with differing mixing length parameters, $\alpha_{\rm MLT}$, and helium enhancement. The default values of these are $\alpha_{\rm MLT} = 1.78$ and no He-enhancement. We used the BAGEMASS default parameters to model WASP-177 and WASP-181 but found that they did not fit WASP-183 very well. This is likely because WASP-183 is among the $\sim 3\%$ of the K-dwarf population that are larger than models would predict [@2013ApJ...776...87S]. To account for this we follow the method of in the case of Qatar-2 and use a the grid provided by BAGEMASS with $\alpha_{\rm MLT} = 1.5$. This results in a much improved fit to the observed density and temperature. We find that the resulting mass estimate is unaffected.
System parameters
-----------------
To determine the system parameters we modeled the discovery and follow-up data together using the most recent version of the Markov-Chain Monte Carlo (MCMC) code described in detail in [@2007MNRAS.380.1230C] and . We modeled the transit lightcurves using the models of [@2002ApJ...580L.171M] with the 4 parameter limb darkening law of .
In brief, the models were initialised using the period, $P$, epoch, $T_{0}$, transit depth, $(R_{p}/R_{s})^{2}$, transit duration, $T_{14}$, and impact parameter, $b$, output by the BLS search of each discovery lightcurve. The spectroscopic stellar effective temperature, $T_{\rm eff}$, and metallicity, \[Fe/H\], were used initially to estimate the stellar mass using the updated Torres mass calibration by [@2011MNRAS.417.2166S]. To explore the effect of limb-darkening we extracted tables of limb- darkening parameters in each photometric band used for each star. They were extracted for a range of effective temperatures while keeping the stellar metallicity and surface gravity constant. The values used were perturbed during the MCMC via $T_{\rm L-D}$, the ‘limb-darkening temperature’, which has a mean and standard deviation corresponding to the spectroscopic $T_{\rm eff}$ and its uncertainty.
At each step of the MCMC each of these values are perturbed and the models are re-fit. These new proposed parameters are then accepted if the $\chi^2$ of the fit is better or accepted with a probability proportional to $\exp(-\Delta\chi^2)$ if the $\chi^2$ of the fit is worse.
In the final MCMCs, in place of using the Torres relation to determine a mass, we provided the value given by BAGEMASS. The code then drew values at each step from a Gaussian with a mean and standard deviation given by the value and its uncertainty respectively. Due to the lack of good quality follow-up photometry we imposed a similar prior on the radius of the star WASP-183 using the Gaia parallax. Lacking a complete, good quality follow-up lightcurve can lead to a poor determination of, $\Delta F$, $T_{14}$ and $b$ which we use to calculate the $R_{*}/a$. This in turn results in a poorer determination of $R_{*}$, $R_{p}$ and other parameters that depend upon them.
In this way we also explored models allowing for eccentric orbits and the potential for linear drifts in the RVs. There was no strong evidence supporting either scenario so we present the system solutions corresponding to circular orbits [@2012MNRAS.422.1988A] with no trends due to unseen companions. The parameters derived by these fits can be found in the lower part of Table \[tab:all\_results\].
Rotational modulation
---------------------
We checked the WASP lightcurves of the three stars for rotational modulation that could be caused by star spots using the method described by [@2011PASP..123..547M]. The transits were fit with a simple model and removed. We performed the search over 16384 frequencies ranging from 0 to 1 cycles/day. Due to the limited lifetime and variable distribution of star spots this modulation is not expected to be coherent over long periods of time. As such, we modeled each season of data from each camera individually. WASP-181 and WASP-183 show no significant modulation, with an upper limit on the amplitude of 2- and 3-mmag respectively.
However, WASP-177 was found to exhibit modulation consistent with a rotational period, $P_{\rm rot} = 14.86 \pm 0.14$ days and amplitude of $5 \pm 1$ mmag. The results of this analysis for each camera and season of data is shown in Table \[tab:rot\_mod\]. Fig. \[fig:rot-mod\] shows the periodograms of the fits and the discovery lightcurves phase-folded on the corresponding period of modulation. Three of the datasets exhibit $P_{\rm rot} \sim 7$-days while the other two exhibit $P_{\rm rot} \sim 14$-days. We interpret the $\sim 7$-day signals as a harmonic of the longer $\sim 14$-day signal as it is more easy for multiple active regions to produce a $\sim 7$-day signal when the true period is $\sim 14$-days than vice versa. Using this rotational period and our value for the stellar radius we find a stellar rotational velocity of, $v_{*} = 2.9 \pm 0.2$ km/s. When compared to the projected equatorial spin velocity we find a stellar inclination to our line of sight of $38 \pm 25\degr$ which suggests that WASP-177b could be quite mis-aligned.
![[*Left*]{}: Periodograms of the WASP lightcurves of WASP-177. Each is labeled with the corresponding camera ID, dates of the observation period (in JD-2450000) and period of the most significant signal. Horizontal lines indicate false-alarm probability levels of 0.1, 0.01 and 0.001. [*Right*]{}: Lightcurves folded on the most significant detected period.[]{data-label="fig:rot-mod"}](wasp177_swlomb_mod){width="\columnwidth"}
Discussion
==========
Our joint analysis shows that in this ensemble we have two large sub-Jupiter mass planets: WASP-177b ($\sim$0.5 M$_{\rm Jup}$, $\sim$1.6 R$_{\rm Jup}$) and WASP-183b ($\sim$0.5 M$_{\rm Jup}$, $\sim$1.5 R$_{\rm Jup}$) orbiting old stars. The third planet, WASP-181b, is a large Saturn mass planet ($\sim$0.3 M$_{\rm Jup}$, $\sim$1.2 R$_{\rm Jup}$) . According to the analysis with BAGEMASS, WASP-177 and WASP-183 are both at the latter end of the main sequence explaining their slightly larger radii for stars of their spectral class; a $9.7 \pm 3.9$ Gyr K2 and $14.9 \pm 1.7$ Gyr G9/K0 respectively. WASP-183 is particularly noteworthy as its advanced age makes it one of the oldest stars known to host a transiting planet (see Fig. \[fig:age\]). Though, WASP-183 appears to be subject to the K-dwarf radius anomaly, making this determination less clear. Meanwhile, WASP-181 is a relatively young, standard example of a G2 star.
We compared the stellar radii derived from our MCMC to those we can calculate using the Gaia DR2 parallaxes , with the correction from [@2018ApJ...862...61S], and stellar angular radii from the infra-red flux fitting method (IRFM) these radii, with reddening accounted for by the use of dust maps [@2011ApJ...737..103S]. We find good agreement and present a summary in table \[tab:radii\_comp\].
![Age distribution for known exoplanet hosts with published uncertainties (grey) and planets presented in this paper (see legend). WASP-183 appears to be particularly old amongst planet hosts. However, we note it is unphysically old and so caution that this determination may be in part due to the K-dwarf radius anomaly. (Data from exoplanet.eu.)[]{data-label="fig:age"}](w177w181w183-ages){width="\columnwidth"}
![Mass-radius distribution for transiting planets. planets with masses determined to better than 10% precision are plotted in blue, otherwise the symbols are gray. WASP-177b, WASP-181b and WASP-183b have been plotted with their error bars. Each is close to the upper most part of the distribution. WASP-177b is in an area particularly sparsely populated by planets with well determined masses. (Prepared using data collated the TEPCat.)[]{data-label="fig:mass-radius"}](Mass-Rad-w177w181w183_zoom){width="\columnwidth"}
All three planets occupy the upper edge of the mass-radius distribution, seen in Fig \[fig:mass-radius\]. WASP-181b is amongst the group of the largest planets for an object of its mass. While its mass is not as well determined as the other two, further HARPS observations will help to refine this. WASP-177b and WASP-183b do lie above the bulk of the distribution, especially when compared to other objects with mass determinations of 10% precision or better. However, it is difficult to say how exceptional they are as a precise radius determination has proven difficult for them both. The transit of WASP-177b is grazing and the transit of WASP-183b, in addition to being grazing, lacks a full high precision follow-up lightcurve to refine the transit shape. We anticipate that [*TESS*]{} observations could soon solve the latter problem;the long cadence data would capture roughly 24 in transit points with a predicted precision from the [*ticgen*]{} tool of better than 1000 ppm in each 30-minute observation.
We used the the values derived for planet equilibrium temperature, $T_{\rm eq}$, and surface gravity, $g$, along with Boltzmann’s constant, $k$, and the atmospheric mean molecular mass, $\mu$, to estimate the scale heights, $H$, of these planets as:
$$H = \frac{k T_{\rm eq}}{g \mu}$$
assuming an isothermal, hydrogen dominated atmosphere. The resulting scale heights were; $790 \pm 320$ km, $770 \pm 200$ km, $696 \pm 464$ km for WASP-177b, WASP-181b and WASP-183b respectively. These translate to transit depth variations of just under 300 ppm for WASP-177 and WASP-181 and $\sim 300$ ppm for WASP-183. If we account for the K-band flux and scale in the same way as , we get atmospheric signals of; 70, 41 and 60. In reality, we can expect this metric to be an over estimate of detectability for WASP-177b and WASP-183b as the grazing nature of their transits reduces the impact of the atmospheric signal further. For comparison we used the same metric on other planets with atmospheric detections: water has been detected in the atmospheres of both WASP-12b (@2015ApJ...814...66K; signal $\sim 93$) and WASP-43b (@2014ApJ...793L..27K; signal $\sim 74$); titanium oxide has been detected in the atmosphere of WASP-19b (@2017Natur.549..238S; signal $\sim 83$); sodium and potassium have both been detected in the atmosphere of WASP-103b (; signal $\sim 37$). While not ideal targets, this suggests such detections may be possible.
Investigation into any eccentricity or long-period massive companions in these systems has not yielded anything convincing. All of the orbits are circular, with the $2\sigma$ upper limits quoted in Table \[tab:all\_results\]. As for long term trends, WASP-177 shows the possibility of a very low significance ($\sim 1.5 \sigma$) drift with $\delta\gamma/\delta t$ of $(-2.4 \pm 1.6)\times 10^{-5}$ km/s/d. Neither WASP-181 nor WASP-183 show significant drifts with $\delta\gamma/\delta t$ of $(1.2 \pm 4.0)\times 10^{-5}$ km/s/d and $(-1.9 \pm 5.1)\times 10^{-5}$ km/s/d respectively.
![Distribution of planets with measured spin-orbit angles with cool host stars. WASP-177, WASP-181 and WASP-183 are all cool stars by this definition and the planets lie in the region where mis-alignment is often said to become more common. WASP-177 shows signs of being misaligned and so may be an interesting diagnostic in this region.[]{data-label="fig:spin-orbit"}](Spin-orbit-w177w181w183-cool){width="\columnwidth"}
Finally, these systems do present interesting targets for the investigation of the observed spin-orbit mis-alignment distribution (@2012ApJ...757...18A [@2015ApJ...800L...9A]). All of the stellar hosts fall into the “cool” regime of [@2012ApJ...757...18A] and despite their short periods have scaled semi-major axes, $a/R_{*}$, above 8. They are therefore above the empirical boundary noted by [@2017AJ....153..205D] as the transition region where systems with cooler stars show more tendency to be mis-aligned. Since the study in 2017 the number of systems with obliquity measurements has increased. Most of the cool-star systems with $a/R_{*}$ above 8 are well aligned, see Fig \[fig:spin-orbit\].
We estimated the alignment time-scale for each system using Eq.4 of [@2012ApJ...757...18A] as was done for WASP-117 . These time-scales, along with the mass of the convective zone, $M_{\rm cz}$, are shown in Tab \[tab:alignment\]. In each case, the time-scale for realignment is much longer than the ages of the systems. Therefore, we would expect the initial state of alignment of the systems to have been preserved. We have estimated the inclination of WASP-177 to be $38 \pm 25\degr$ and so may expect it to join only 12 systems with $a/R_{*}$ < 15 that show mis-alignment this makes it a potentially important diagnostic in determining the factors that cause or preserve mis-alignment.
We calculate that the amplitude of the RM effect will be greatest for WASP-181 at $\sim 50$ ms$^{-1}$. The effect should also be detectable for WASP-177 and WASP-183 despite their more grazing transits, with an amplitude of $\sim 10$ ms$^{-1}$.
Conclusions
===========
We have presented the discovery of 3 transiting exoplanets from the WASP survey; WASP-177b ($\sim$0.5 M$_{\rm Jup}$, $\sim$1.6 R$_{\rm Jup}$), WASP-181b ($\sim$0.3 M$_{\rm Jup}$, $\sim$1.2 R$_{\rm Jup}$), and WASP-183b ($\sim$0.5 M$_{\rm Jup}$, $\sim$1.5 R$_{\rm Jup}$). They all occupy the upper region of the mass-radius distribution for hot gas-giant planets but do not present exceptional targets for transmission spectroscopy. However, regarding the investigation of system spin-orbit alignment they do occupy an under investigated range of $a/R_{*}$ and so could act as good probes of tidal realignment time-scales.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank the Swiss National Science Foundation (SNSF) and the Geneva University for their continuous support to our planet search programs. This work has been in particular carried out in the frame of the National Centre for Competence in Research ‘PlanetS’ supported by the Swiss National Science Foundation (SNSF). WASP-South is hosted by the South African Astronomical Observatory and we are grateful for their ongoing support and assistance. Funding for WASP comes from consortium universities and from the UK’s Science and Technology Facilities Council. TRAPPIST is funded by the Belgian Fund for Scientific Research (Fond National de la Recherche Scientifique, FNRS) under the grant FRFC 2.5.594.09.F, with the participation of the Swiss National Science Fundation (SNF). MG is a F.R.S.-FNRS Senior Research Associate. The research leading to these results has received funding from the European Research Council under the FP/2007-2013 ERC Grant Agreement 336480, from the ARC grant for Concerted Research Actions financed by the Wallonia-Brussels Federation, from the Balzan Foundation, and a grant from the Erasmus+ International Credit Mobility programme (K Barkaoui). We thank our anonymous reviewer for their comments which helped improve the clarity of the paper.
We include the data we used in this paper as online material. Examples of the tables are show here.
Online Data
===========
\[lastpage\]
[^1]: E-mail: oliver.turner@unige.ch
[^2]: These observations were made as part of the programs Anderson:0100.C-0847(A) and Nielsen:0102.C-0414(A).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
A vector field $X$ on a manifold $M$ with possibly nonempty boundary is [*inward*]{} if it generates a unique local semiflow $\Phi^X$. A compact relatively open set $K$ in the zero set ${\ensuremath{{\msf Z ( X)}}}$ is a [*block*]{}. The Poincaré-Hopf index is extended to blocks $K\subset {\ensuremath{{\msf Z ( X)}}}$ where $X$ is inward and $K$ may meet $\p M$. A block with nonzero index is [ *essential*]{}.
Let $X, Y$ be inward $C^1$ vector fields on surface $M$ such that $[X, Y]\wedge X=0$ and let $K$ be an essential block of zeros for $X$. Among the main results are that $Y$ has a zero in $K$ if $X$ and $Y$ are analytic, or $Y$ is $C^2$ and $\Phi^Y$ preserves area. Applications are made to actions of Lie algebras and groups.
author:
- |
Morris W. Hirsch[This research was supported in part by National Science Foundation grant DMS-9802182.]{}\
Mathematics Department\
University of Wisconsin at Madison\
University of California at Berkeley
title: '**Common zeros of inward vector fields on surfaces**'
---
\#1[\[\#1\]]{}
Introduction
============
Let $M$ be an $n$-dimensional manifold with boundary $\p M$ and $X$ a vector field on $M$, whose value at $p$ is denoted by $X_p$. The [ *zero set*]{} of $X$ is $${\ensuremath{{\msf Z ( X)}}}:= \{p\in M\co X_p=0\}.$$ The set of common zeros for a set ${\ensuremath{{\mathfrak {s}}}}$ of vector fields is $$\textstyle
{\ensuremath{{\msf Z ( {\ensuremath{{\mathfrak {s}}}})}}} :=\bigcap_{X\in {\ensuremath{{\mathfrak {s}}}}}{\ensuremath{{\msf Z ( X)}}}.$$
A [*block*]{} for $X$ is a compact, relatively open subset $K\subset
{\ensuremath{{\msf Z ( X)}}}$. This means $K$ lies in a precompact open set $U\subset M$ whose topological boundary $\msf{bd}(U)$ contains no zeros of $X$. We say that $U$ is [*isolating*]{} for $(X,K)$, and for $X$ when $K:={\ensuremath{{\msf Z ( X)}}}
\cap U$. When $M$ is compact, ${\ensuremath{{\msf Z ( X)}}}$ is a block for $X$ with $M$ as an isolating neighborhood.
If $p\in M\verb=\=\p M$ is an isolated zero of $X$, the [ *index of $X$ at $p$*]{}, denoted by $\msf i_p X$, is the degree of the map of the unit $(n-1)$-sphere $$\S{n-1}\to\S {n-1}, \quad z\mapsto \frac{\hat X(z)}{\|\hat X (z)\|},$$ where $\hat X$ is the representative of $X$ in an arbitrary coordinate system centered at $p$. When $U$ is isolating for $(X, K)$ and disjoint from ${\ensuremath{{\msf Z ( X)}}} \cap \p M$, the [*Poincaré-Hopf index*]{} of $X$ at $K$ is $$\msf i^{\rm PH}_K (X) :=\sum_p\msf i_pY, \quad (p\in {\ensuremath{{\msf Z ( Y)}}} \cap U)$$ where $Y$ is any vector field on $M$ such that ${\ensuremath{{\msf Z ( Y)}}} \cap \ov U$ is finite, and there is a homotopy of vector fields $\left\{X^t\right\}_{0\le t\le 1}$ from $X^0=X$ to $X^1=Y$ such that $\bigcup_{t}]{\ensuremath{{\msf Z ( X^t)}}} \cap U$ is compact.[^1]
The block $K$ is [*essential*]{} for $X$ if $\msf i^{\rm PH}_K (X)\ne 0$.
Christian Bonatti [@Bonatti92] proved the following remarkable result:
Assume ${\ensuremath{{\mathsf {dim}\,}}}M\le 4$ and $\p M=\empty$. If $X,Y$ are analytic vector fields on $M$ such that $[X, Y]= 0$, then ${\ensuremath{{\msf Z ( Y)}}}$ meets every essential block of zeros for $X$.[^2]
Our main results, Theorems \[th:main\] and \[th:mainB\], reach similar conclusions for surfaces $M$ which may have nonsmooth boundaries, and certain pairs of vector fields that generate local semiflows on $M$, including cases where the fields are not analytic and do not not commute. Applications are made to actions of Lie algebras and Lie groups.
Next we define terms (postponing some details), state the main theorems and apply them to Lie actions. After sections on dynamics and index functions, the main results are proved in Section \[sec:mainproofs\].
Terminology
-----------
$\ZZ$ denotes the integers, $\Np$ the positive integers, $\RR$ the real numbers, and $\Rp$ the closed half line $[0,\infty)$. Maps are continuous, and manifolds are real, smooth and metrizable, unless otherwise noted. The set of fixed points of a map $f$ is $\Fix f$.
The following assumptions are always in force:
$\tilde M$ is an analytic $n$-manifold with empty boundary. $M\subset \tilde M$ is a connected topological $n$-manifold.[^3]
We call $M$ an [*analytic manifold*]{} when $\p M$ is an analytic submanifold of $\tilde M$. The tangent vector bundle of $\tilde M$ is $T\tilde M$, whose fibre over $p$ is the vector space $T_p\tilde M$. The restriction of $T\tilde M$ to a subset $S\subset \tilde M$ is the vector bundle $T_S\tilde M$. We set $TM:=T_M\tilde M$.
A map $f$ sending a set $S\subset M$ into a smooth manifold $N$ is called $C^r$ if it extends to a map $\tilde f$, defined on an open subset of $\tilde M$, that is $C^r$ in the usual sense. Here $r\in\Np\cup \{\infty, \omega\}$, and $C^\omega$ means analytic. If $f$ is $C^1$ and $S$ has dense interior in $M$, the tangent map $T\tilde f\co T\tilde M\to T N$ restricts to a bundle map $Tf\co T_SM
\to TN$ determined by $f$.
A [*vector field*]{} on $S$ is a section $X\co S\to T_S M$, whose value at $p$ is denoted by $X_p$. The set of these vector fields is a linear space ${\mcal V}(S)$. The linear subspaces ${\mcal V}^r (S)$ and ${\mcal V}^{\msf{L}}(M)$, comprising $C^r$ and locally Lipschitz fields respectively, are given the compact-open topology (uniform convergence on compact sets).
$X$ and $Y$ always denote vector fields on $M$. When $X$ is $C^r$, $\tilde X$ denotes an extension of $X$ to a $C^r$ vector field on an open set $W\subset \tilde M$.
The [*Lie bracket*]{} of $X, Y \in {\mcal V}^1 (M)$ is the restriction to $M$ of $[\tilde X, \tilde Y]$. This operation makes ${\mcal V}^\omega (M)$ and ${\mcal V}^\infty (M)$ into Lie algebras.
$X\wedge Y$ denotes the tensor field of exterior 2-vectors $p\mapsto
X_p\wedge Y_p \in \Lam^2(T_pM)$. Evidently $X\wedge Y=0$ iff $X_p$ and $Y_p$ are linearly dependent at all $p\in M$.
#### Inward vector fields
A tangent vector to $M$ at $p$ is [*inward*]{} if it is the tangent at $p$ to a smooth curve in $M$ through $p$. The set of inward vectors at $p$ is $T^{\msf {in}}_pM$. A vector field $X$ is [*inward*]{} if $X(M)\subset T^{\msf {in}} (M)$, and there is a unique local semiflow $\Phi^X:=\left\{\Phi^X_t\right\}_{t\in \Rp}$ on $M$ whose trajectories are the maximally defined solutions to the initial value problems
$$\ode y t = X(y), \quad y(0)=p, \qquad p\in M, \quad t \ge 0.$$
The set of inward vector fields is ${\mcal V}_{\msf {in}} (M)$. When $\p M$ is a $C^1$ submanifold of $\tilde M$, it can be shown that $X$ is inward iff $X (M)\subset T^{\msf {in}} (M)$.
Define $${\mcal V}^r_{\msf {in}} (M):= {\mcal V}_{\msf {in}} (M)\cap{\mcal V}^r (M), \qquad
{\mcal V}^{\msf{L}}_{\msf {in}}(M):= {\mcal V}_{\msf {in}} (M)\cap {\mcal V}^{\msf{L}}(M).$$ Proposition \[th:convex\] shows these sets are convex cones in ${\mcal V}(M)$.
#### The vector field index and essential blocks of zeros
Let $K$ be a block of zeros for $X\in {\mcal V}_{\msf{in}} (M)$, and $U\subset M$ an isolating neighborhood for $(X, K)$. The [*vector field index*]{} $$\msf i_K (X):=\msf i (X, U)\in \ZZ$$ is defined in Section \[sec:index\] as the fixed point index of the map $\Phi^X_t|U\co U\to M$, for any $t >0$ so small that the compact set $\ov U$ lies in the domain of $\Phi^X_t$.
The block $K$ is [*essential*]{} (for $X$) when $\msf i_K (X)\ne 0$. A version of the Poincaré-Hopf theorem implies $K$ is essential if it is an attractor for $\Phi^X$ and has nonzero Euler characteristic $\chi
(K)$.
Statement of results
--------------------
In the next two theorems, besides Hypothesis \[th:hypmain\] we assume:
[ ]{}
- $M$ and $\tilde M$ are surfaces,
- $X$ and $Y$ are $C^1$ inward vector fields on $M$,
- $K\subset M$ is an essential block of zeros for $X$,
- $[X, Y]\wedge X=0$.
The last condition has the following dynamical significance (Proposition \[th:wedge\]):
- [*$\Phi^Y$ permutes integral curves of $X$.*]{}
This implies:
- [*if $q=\Phi^Y_t (p)$ then $ X_q=\lam\cdot T\Phi^Y_t(X_p)$ for some $ \lam >0$,*]{}
- [*${\ensuremath{{\msf Z ( X)}}}$ is positively invariant under $\Phi^Y$.*]{} (See Definition \[th:posinv\].)
A [*cycle*]{} for $Y$, or a $Y$-[*cycle*]{}, is a periodic orbit of $\Phi^Y$ that is not a fixed point.
Assume [*Hypothesis \[th:hyp2\]*]{}. Each of the following conditions implies ${\ensuremath{{\msf Z ( Y)}}}\cap
K\ne\varnothing$:
(a)
: $X$ and $Y$ are analytic.
(b)
: Every neighborhood of $K$ contains an open neighborhood whose boundary is a nonempty union of finitely many $Y$-cycles.
When $[X, Y]=0$ this extends Bonatti’s Theorem to surfaces with nonempty boundaries. The case $[X, Y]= cX, \, c\in\RR$ yields applications to actions of Lie algebras and Lie groups.
In his pioneering paper [@Lima64], E. Lima constructs vector fields $X, Y$ on the closed disk $\D 2$, tangent to $\p \D 2$ and generating unique flows, such that $[X, Y]= X$ and ${\ensuremath{{\msf Z ( X)}}}\cap{\ensuremath{{\msf Z ( Y)}}}=\empty$ (see Remark \[th:remlima\]). Such fields can be $C^\infty$ (M. Belliart & I. Liousse [@BL96], F.-J Turiel [@Turiel03]). The unique block of zeros for $X$ is ${\ensuremath{{\msf Z ( X)}}}=\p \D 2$, which is essential because $\chi (\D
2)\ne\varnothing$, but ${\ensuremath{{\msf Z ( Y)}}}$ is a point in the interior of $\D 2$. This shows that the conclusion of Theorem \[th:main\](a) can fail when $X$ and $Y$ are not analytic. The flows of $X$ and $Y$ generate an effective, fixed-point free action by a connected, solvable nonabelian Lie group.
A local semiflow on a surface $M$ [*preserves area*]{} if it preserves a Borel measure on $M$ that is positive and finite on nonempty precompact sets.
Assume [*Hypothesis \[th:hyp2\]*]{}. If $\Phi^Y$ preserves area, each of the following conditions implies ${\ensuremath{{\msf Z ( Y)}}} \cap K\ne\varnothing$:
(i)
: $K$ contains a $Y$-cycle,
(ii)
: $Y$ is $C^2$,
(iii)
: $K$ has a planar neighborhood in $M$.
For $X\in {\mcal V}^{\omega}_{\msf {in}} (M)$ define $$\msf W(X):=\{Y\in{\mcal V}^{\omega}_{\msf {in}} (M) \co[X, Y]\wedge X=0\},$$ which is the set of inward analytic vector fields on $M$ whose local semiflows permute integral curves of $X$. Propositions \[th:wedge\] and \[th:convex\] imply $\msf W(X)$ is a convex cone that is closed under Lie brackets, and a subalgebra of ${\mcal V}^{\omega}(M)$ if $M$ is an analytic manifold without boundary.
Assume Hypothesis \[th:hyp2\] holds. If $X$ is analytic and $\p M$ is an analytic subset of $\tilde M$, then $ {\ensuremath{{\msf Z ( \msf W (X))}}}\cap K\ne\varnothing$.
#### Actions of Lie algebras and Lie groups
Let $M$ be an analytic surface and $\gg$ a Lie algebra (perhaps infinite dimensional) of analytic vector fields on $M$ that are tangent to $\p M$. Assume $X\in \gg$ spans a nontrivial ideal. Then:
(a)
: ${\ensuremath{{\msf Z ( \gg)}}}$ meets every essential block of zeros for $X$.
(b)
: If $M$ is compact and $\chi (M)\ne 0$, then ${\ensuremath{{\msf Z ( \gg)}}}\ne\varnothing$.
Note that $\gg$ has $1$-dimensional ideal if its center is nontrivial, or $\gg$ is finite dimensional and supersoluble (Jacobson [@Jacobson62 Ch[. ]{}2, Th[. ]{}14]). A finite dimensional solvable Lie algebra of vector fields on a surface has derived length $\le 3$ (Epstein & Thurston [@ET79]). Plante [@Plante88] points out that ${\mcal V}^\omega (\R 2)$ contains infinite-dimensional subalgebras, such as the Lie algebra of quadratic vector fields. An [*action*]{} of a group $G$ on a manifold $M$ is a homomorphism $\alpha\co g\to g^\alpha$ from $G$ to the homeomorphism group of $P$, such that the corresponding [*evaluation map*]{} $$\msf{ev_\alpha}\co G\times P\to P, \quad (g,p)\mapsto g^\alpha (p)$$ is continuous. When $\msf{ev_\alpha}$ is analytic, $\alpha$ is an [*analytic action*]{}.
Assume $M$ is a compact analytic surface and $G$ is a connected Lie group having a one-dimensional normal subgroup. If $\chi (M)\ne 0$, every effective analytic action of $G$ on $M$ has a fixed point.
For supersoluble $G$ this is due to Hirsch & Weinstein [@HW00].
Background on group actions
---------------------------
The literature on actions of connected Lie groups $G$ include the following notable results:
If $G$ is solvable (respectively, nilpotent) and acts effectively on an $n$-dimensional manifold, the derived length of $G$ is $\le n+1$ (respectively, $\le n$)
In the next two propositions $M$ denotes a compact connected surface.
Assume $G$ acts on $M$ without fixed points.
(i)
: If $G$ is nilpotent, $\chi (M)\ne 0$ [*(Plante [@Plante86]).*]{}
(ii)
: If the action is analytic, $\chi (M)\ge 0$ [*(Turiel [@Turiel03], Hirsch [@Hirsch03]).*]{}
Let ${\rm Aff}_+ (\R m)$ denote the group of orientation-preserving affine homeomorphisms of $\R m$.
(a)
: If $\chi (M) < 0$ and $G$ acts effectively on $M$ without fixed points, then $G$ has a quotient isomorphic to ${\rm Aff}_+ (\R 1)$ [ *(Belliart [@Belliart97]).*]{}
(b)
: ${\rm Aff}_+(\R 1)$ has effective fixed-point free actions on $M$ [*(Plante [@Plante86])*]{}.
(c)
: ${\rm Aff}_+ (\R 2)$ has effective analytic actions on $M$ [*(Turiel [@Turiel03])*]{}.
For related results see the references above, also Belliart [@Belliart97], Hirsch [@Hirsch2010; @Hirsch2011], Molino & Turiel [@MolinoTuriel86; @MolinoTuriel88], Plante [@Plante91], Thurston [@Thurston74], Turiel [@Turiel89]. Transitive effective surface actions are classified in Mostow’s thesis [@Mostow50], with a useful summary in Belliart [@Belliart97].
Dynamics
========
Let $\Psi:=\{\Psi_t\}_{t\in \msf T}$ denote a local flow ($\msf
T=\RR)$ or a local semiflow ($\msf T=\Rp)$ on a topological space $S$. Each $\Psi_t$ is a homeomorphism from an open set $\mcal D
(\Psi_t)\subset S$ onto a set $\mcal R (\Psi_t)\subset S$, such that:
- $\Psi_t (p)$ is continuous in $(t,p)$,
- $\Psi_0$ is the identity map of $S$,
- if $0 \le |s| \le |t|$ and $|st|\ge 0$ then $\mcal D
(\Psi^s)\supset \mcal D (\Psi^t)$,
- $ \Psi_t (\Psi_s (p)) = \Psi_{t+s} (p)$.
We adopt the convention that notation of the form “$\Psi_t (x)$” presumes $x\in\mcal D (\Psi_t)$.
A set $L\subset S$ is [*positively invariant*]{} under $\Psi$ provided $\Psi_t$ maps $L\cap \mcal D
(\Psi_t)$ into $L$ for all $t\ge 0$, and [*invariant*]{} when $L\subset \mcal D (\Psi_t)\cap\mcal R (\Psi_t)$ for all $t \in \msf
T$. When $\Psi$ is generated by a vector field $Y$ we use the analogous terms “positively $Y$-invariant” and “$Y$-invariant.”
Let $M, \tilde M$ be as in Hypothesis \[th:hypmain\]. When $\Psi$ is a local semiflow on $M$, the theorem on invariance of domain shows that $\mcal R (\Psi_t)$ is open in $M$ when $\Phi$ is a local flow, and also when $\mcal D (\Psi_t)\cap\p M=\empty$. This implies $M\setminus \p M$ is positively invariant under every local semiflow on $M$.
Assume $X, Y\in {\mcal V}_{\msf{in}} (M)$ and $[X, Y]\wedge X=0$.
(a)
: If $\Phi^Y_t (p)=q$ then $T_p\Phi^Y_t \co X_p\mapsto
cX_q,\, c >0$.
(b)
: $\Phi^Y_t$ sends integral curves of $X$ to integral curves of $X$.
(c)
: ${\ensuremath{{\msf Z ( X)}}}$ is $Y$-invariant.
Let $\tilde X$ and $\tilde Y\in{\mcal V}^1 (\tilde M)$ be extensions of $X$ and $Y$, respectively. It suffices to prove:
#### (\*)
[*If $p\in M,\ t\ge 0$ and $p(t):= \Phi_t^{\tilde X} (p)$, then the linear map $$T_p\Phi_t^{ \tilde Y}\co T_p \tilde M\to T_{p(t)}\tilde M$$ sends $X_p$ to a positive scalar multiple of $X_{p(t)}$.*]{}
By continuity it suffices to prove this when $Y_p\ne 0$, $X_p\ne 0$ and $|t|$ sufficiently small. Working in flowbox coordinates $(u_j)$ for $\tilde Y$ in a neighborhood of $p$, we assume $\tilde M$ is an open set in $ \R n$, $\tilde Y=\pde{~}{u_1}$, and $\tilde X$ has no zeros. Because $[X, Y]\wedge X=0$, there is a unique continuous map $f\co
M\to \RR$ such that $[X, Y]=fX$. Since $\tilde Y$ is a constant vector field, the vector-valued function $t\mapsto X_{p(t)}$ satisfies $$\ode {X_{p(t)}} t = -f(p(t))\cdot X_{p(t)},$$ whose solution is $$X_{p (t)}= e^{-\int_0^t f(s)ds}\cdot X_{p(0)}.$$ This implies (\*).
The following fact is somewhat surprising because $T^{\msf{in}}_pM$ need not be convex in $T_pM$:
${\mcal V}^{\msf{L}}_{\msf{in}}(M)$ is a convex cone in ${\mcal V}(M)$.
As ${\mcal V}^{\msf{L}}(M)$ is a convex cone in ${\mcal V}(M)$, it suffices to show that ${\mcal V}^{\msf{L}}_{\msf{in}} (M)$ is closed under addition. Let $X, Y\in {\mcal V}^{\msf{L}}(M)$. We need to prove: $$\label{eq:ppM}
\mbox{\em If $p\in \p M$ there exists $\eps >0$ such that $0\le t\le \eps
\implies \Phi_t^{ X + Y} (p)\in M$.}$$ This is easily reduced to a local result, hence we assume $M$ is relatively open in the closed halfplane $\RR \times [0,\infty)$ and $X, Y$ are Lipschitz vector rields on $M$. Let $\tilde X, \tilde Y$ be extensions of $X, Y$ to Lipschitz vector fields on an open neighborhood $\tilde M\subset \R 2$ of $M$ ( Johnson [*et al. *]{}[@Lindenstrauss86]). Denote the local flows of $\tilde
X, \, \tilde Y, \, \tilde X+ \tilde Y$ respectively by $\{f_t\},
\{g_t\}, \{h_t\}, \ (t\in \RR)$. We use a special case of Nelson [@Nelson70 Th.1, Sec.4]:
\[th:nelson\] For every $p\in \tilde M$ there exists $\eps >0$ and a neighborhood $W\subset
\tilde M$ of $p$ such that $$h_t (x)=\lim_{k\to\infty}\left(
f_{t/k}\circ g_{t/k}\right)^k (x)$$ uniformly for $x\in W$ and $|t|<\eps$.
Because $X$ and $Y$ are inward, $M$ is positively invariant under the local semiflows $\{f_t\}_{t\ge 0}$ and $\{ g_t\}_{t\ge 0}$. Therefore $$0\le t\le \eps \implies \left(f_{t/k}\circ g_{t/k}\right)^k \in M,
\qquad (k\in \Np).$$ As $M$ is relatively closed in $\tilde M$, Proposition \[th:nelson\] implies $h_t (x)\in M$ for $0\le t\le \eps$, which yields (\[eq:ppM\]).
Examination of the proof yields:
If $L$ is a closed subset of a smooth manifold $N$, the set of locally Lipschitz vector fields on $N$ for which $L$ is positively invariant is a convex cone.
Is ${\mcal V}_{\msf{in}}(M)$ is a convex cone in ${\mcal V}(M)$?
Index functions
===============
We review properties of the fixed point index $I (f)$ defined by the late Professor Albrecht Dold ([@Dold65; @Dold72]). Using it we define an [*equilibrium index*]{} $I_K (\Phi)$ for local semiflows, and a [*vector field index*]{} $\msf i_K (X)$ for inward vector fields.
The fixed point index for maps
------------------------------
#### Dold’s Hypothesis:
- $V$ is an open set in a topological space $S$.
- $f\co V \to
S$ is a continuous map with compact fixed point set $\Fix
f\subset V$.
- $V$ is a Euclidean neighborhood retract (ENR).[^4]
On the class of maps satisfying these conditions, Dold constructs an integer-valued [*fixed point index*]{} denoted here by $I (f)$, uniquely characterized by the following five properties (see [@Dold72 VII.5.17, Ex.5\*]):
(FP1)
: $I(f)=I(f|V_0)$ if $V_0\subset V$ is an open neighborhood of $\Fix f$.
(FP2)
: $I(f)=\begin{cases}
& 0 \ \mbox{if $\Fix f =\empty$,}\\
& 1 \ \mbox{if $f$ is constant.}
\end{cases}
$
(FP3)
: $I(f)=\sum_{i=1}^m I(f| V_i)$ if $V$ is the disjoint union of $m$ open sets $V_i$.
(FP4)
: $I (f\times g)=I (f)\cdot I (g)$.
(FP5)
: $I(f_0)=I(f_1)$ if there is a homotopy $f_t\co V\to S,\, (0\le t \le 1)$ such that $\bigcup_t\Fix{f_t}$ is compact.
These correspond to (5.5.11) — (5.5.15) in [@Dold72 Chap.VII]. In addition:
(FP6)
: If $f$ is $C^1$ and $\Fix f$ is an isolated fixed point $p$, then $$I(f)= (-1)^\nu$$ where $\nu$ is the number of eigenvalues $\lam$ of $f$ such that $\lam
>1$, ignoring multiplicities ([@Dold72 VII.5.17, Ex. 3]).
<!-- -->
(FP7)
: If $S$ is an ENR and $f\co S\to S$ is homotopic to the identity map, then $$I(f)= \chi (S).$$ See Dold [@Dold72 VII.6.22].
If $g$ is sufficiently close to $f$ in the compact open topology, then $\Fix g$ is compact and $I(g)=I(f)$.
We can assume $\rho\co W\to V$ is a retraction, where $W\subset \R m$ is an open set containing $V$. For $g$ sufficiently close to $f$ the following hold: $W$ contains the line segment (or point) spanned by $\{f(p), g(p)\}$ for every $p\in V$, and the maps $$f_t\co (t,p)\mapsto \rho((1-t)f(p) + t g(p)),\qquad (0\le t\le 1,\quad p\in V)$$ constitute a homotopy in $V$ from $f_0=f$ to $f_1=g$ with $\bigcup_t\Fix{f_t}$ is compact. Therefore the conclusion follows from (FP5).
The equilibrium index for local semiflows
-----------------------------------------
Let $\Phi:=\{\Phi_t\}_{t\ge 0}$ be a local semiflow in a topological space $\mcal C$, with [*equilibrium set*]{} $$\mcal E (\Phi):={\textstyle\bigcap}_{t\ge 0}\Fix{\Phi_t}.
$$ $K\subset \mcal E (\Phi)$ is a [*block*]{} if $K$ is compact and has an open, precompact ENR neighborhood $V\subset \mcal C$ such that $\ov
V\cap\mcal E (\Phi)\subset V$. Such a $V$ is an [ *isolating neighborhood*]{} for $K$. With these assumptions on $\Phi$ and $V$ we have:
There exists $\tau >0$ such that the following hold when $0 <t \le \tau$:
(a)
: $\Fix {\Phi_t}\cap V$ is compact,
(b)
: $I (\Phi_t|V)=I (\Phi_\tau|V)$.
If (a) fails, there are convergent sequences $\{t_k\}$ in $[0,\infty)$ and $\{p_k\}$ in $V$ such that $$t_k\searrow 0, \quad
p_k\in \Fix{\Phi_{t_k}}\cap V, \quad p_k\to q\in \msf{bd} (V).$$ Joint continuity of $(t,x)\mapsto \Phi_t (x)$ yields the contradiction $q\in \mcal E (\Phi) \cap \msf{bd} (V)$. Assertion (b) is a consequence of (a) and (FP5).
It follows that the fixed point index $I
(\Phi_\tau|V)$ depends only on $\Phi$ and $K$, and is the same for all isolating neighborhoods $V$ of $K$.
Let $\tau>0$ be as in Lemma \[th:tau\](b). We call $I(\Phi_\tau|V)$ the [*equilibrium index*]{} of $\Phi$ in $V$, and at $K$, denoted by $\msf i(\Phi,V)$ and $\msf i_K (\Phi)$.
The vector field index for inward vector fields
-----------------------------------------------
In the rest of this section the manifolds $\tilde M$ and $M\subset
\tilde M$ are as in Hypothesis \[th:hypmain\], $K$ is a block of zeros for $X$ for $X\in{\mcal V}_{\msf{in}} (M)$, and $U$ an isolating neighborhood for $(X, K)$. Then $K$ is also a block of equilibria for the local semiflow $\Phi^X$, and the equilibrium index $\msf i (\Phi^X,U)$ is defined in Definition \[th:defeqindex\].
The [*vector field index of $X$ in $U$*]{} (and [*at $K$*]{}) is $$\begin{split}
\msf i (X, U)=\msf i_K (X) & := \msf i (\Phi^X, U).
\end{split}$$ $K$ is [*essential*]{} (for $X$) if $\msf i_K (X)\ne 0$, and [ *inessential*]{} otherwise.
Two vector fields $X_j\in {\mcal V}_{\msf{in}} (M_j), \,j=1,2$ have [ *isomorphic germs*]{} at $K_j\subset M_j$ provided there are open neighborhoods $U_j\subset
M_j$ of $X_j$ and a homeomorphism $U_1\approx U_2$ conjugating $\Phi^{X_1}|U_1
$ to $\Phi^{X_2}|U_2$.
The vector field index has the following properties:
(VF1)
: $i_K (X)=\msf i_{K'} (X')$ if $K'$ is a block of zeros for $X'\in {\mcal V}_{\msf{in}} (M')$ and the germs of $X$ at $K$ and $X'$ at $K'$ are isomomorphic.
(VF2)
: If $\msf i (X, U)\ne 0$ then ${\ensuremath{{\msf Z ( X)}}}\cap
U\ne\varnothing$.
(VF3)
: $\msf i (X, U)=\sum_{j=1}^m \msf i (X, U_j)$ provided $U$ is the union of disjoint open sets $U_1,\dots,U_m$.
(VF4)
: If $Y$ is sufficiently close to $X$ in ${\mcal V}_{\msf{in}}
(M)$, then $U$ is isolating for $Y$ and $\msf i (X, U)=\msf i
(Y, U)$.
(VF5)
: $\msf i_K (X)$ equals the Poincaré-Hopf index $\msf i^{\rm PH}_K (X)$ provided $K\cap\p M=\empty$.
[ ]{}
[(VF1): ]{} A consequence of (FP1).
[(VF2): ]{} Follows from (FP2).
[(VF3): ]{} Follows from (FP3).
[(VF4): ]{} Use Lemma \[th:gnearf\].
[(VF5): ]{} Since $X$ can be approximated by locally $C^\infty$ vector fields transverse to the zero section, using compactness of $\ov U$ and (VF4) we assume ${\ensuremath{{\msf Z ( X)}}} \cap U$ is finite set of hyperbolic equilibria. By (FP3) we assume ${\ensuremath{{\msf Z ( X)}}} \cap
U$ is a hyperbolic equilibrium $p$. In this case the index of $X$ at $p$ is $(-1)^\nu$ where $\nu$ is the number of positive eigenvalues of $dX_p$ (ignoring multiplicity). The conclusion follows from (FP6) and Definitions \[th:defeqindex\], \[th:defindex\].
If $M$ is compact, $\msf i (X, M)= \chi (M)$.
Follows from (FP7).
Assume $X, Y\in {\mcal V}_{\msf {in}} (M)$ and $U\subset M$ is isolating for $X$. Then $U$ is isolating for $Y$, and $$\msf i (X, U) =\msf i
(Y, U),$$ provided one of the following holds:
(i)
: $Y|\msf{bd}(U)$ is sufficiently close to $X|\msf {bd} (U)$,
(ii)
: $Y|\msf{bd}(U)$ is nonsingularly homotopic to $X|\msf{bd}(U)$.
Both (i) and (ii) imply $U$ is isolating for $Y$. Consider the homotopy $$Z^t:= (1-t)X + tY,\qquad (0\le t\le 1).$$ When (i) holds each vector field $Z^t|{\ensuremath{{\msf {bd} (U)}}}$ is nonsingular, implying (ii). In addition, $Z^t$ is inward by Proposition \[th:convex\], and Lemma \[th:tau\] yields $\tau >0$ such that $$0<t\le \tau \implies
\msf i (X, U)=I(\Phi^X_t|U), \quad \msf i (Y,
U)=I(\Phi^Y_t|U).$$ By Lemma \[th:gnearf\], each $t\in [0,1]$ has an open neighborhood $J_t\subset[0,1]$ such that $$s\in J_t\implies I (\Phi^{X^s}_\tau|U) =
I (\Phi^{X^t}_\tau|U).$$ Covering $[0,1]$ with sets $J_{t_1},\dots, J_{t_\nu}$ and inducting on $\nu\in \Np$ shows that $$I (\Phi^X_\tau|U) = I (\Phi^Y_\tau|U),$$ which by Definition \[th:defindex\] implies the conclusion.
#### The index as an obstruction
The following algebraic calculation of the index is included for completeness, but not used. Assume $M$ is oriented and $U$ is an isolating neighborhood for a block $K\subset{\ensuremath{{\msf Z ( X)}}}$. Let $V\subset U$ be a compact smooth $n$-manifold with the orientation induced from $M$, such that $K\subset V\verb=\= \p V$. The primary obstruction to extending $X|\p V$ to a nonsingular section of $TV$ is the [ *relative Euler class*]{} $${\bf e}_{(X, V)}\in H^n (V, \p V)$$ Let $${\bf v}\in H_n (V, \p V)$$ be the homology class corresponding to the induced orientation of $V$. Denote by $$H^n (V, \p V) \times H_n (V,\p V)\to\ZZ,\quad ({\bf c}, {\bf u}) \to
\langle {\bf c}, {\bf u}\rangle,$$ the Kronecker Index pairing, induced by evaluating cocycles on cycles. Unwinding definitions leads to:
With $M, X, K, V$ are as above, $$\msf i_K (X)= \langle {\bf v}, {\bf e}_{(X, V)} \rangle.$$
When $M$ is nonorientable the same formula holds provided the coefficients for $H^n (V, \p V)$ and $ H_n (V,\p V)$ are twisted by the orientation sheaf of $V$.
Stability of essential blocks
------------------------------
An immediate consequence of Propositions \[th:homsections\](i) and property (VF2) of \[th:index\] is:
If a block $K$ is essential for $X$, and $Y\in {\mcal V}_{\msf{in}} (M)$ is sufficiently close to $X$, then every neighborhood of $K$ contains an essential block for $Y$.
Thus essential blocks are stable under perturbations of the vector field. It is easy to see that a block is stable if it contains a stable block. For example, the block $\{-1, 1\}$ for $X= (x^2-1)\pd x$ on $\RR$ is stable, but inessential. But the following result (not used) means that a block can be perturbed away iff every subblock is inessential:
Assume $\p M$ is a smooth submanifold of $\tilde M$ and every block for $X$ in $U$ is inessential. Then $ X=\lim_{k\to\infty}X^k $ where $$X=\lim_{k\to\infty}X^k, \quad X^k\in {\mcal V}^{\msf L}_{\msf{in}} (M), \quad
{\ensuremath{{\msf Z ( X^k)}}}\cap U =\empty, \qquad (k\in \Np)$$ and $X^k$ coincides with $X$ outside $U$.
Fix a Riemannian metric on $M$. For every $\eps >0$ choose an isolating neighborhood $W:=W(\eps)\subset U$ of $K$ having only finitely many components, and such that $$\|X_p\|<\eps, \qquad (p\in W).$$ Thus $X (W)$ lies in the bundle $T^\eps W$ whose fibre over $p$ is the open disk of radius $\eps$ in $T_p W$. Smoothness of $\p M$ enables an approximation $Y^\eps\in {\mcal V}^{\msf
L}_{\msf{in}}$ to $X$ such that $Y^\eps
(W)\subset T^\eps W$ and ${\ensuremath{{\msf Z ( Y^\eps)}}}\cap \ov U$ is finite. By Proposition \[th:homsections\](ii) and the hypothesisw we choose the approximation close enough so that for each component $W_j$ of $W$: $$\msf i (Y^\eps, W_j) =0.$$ Standard deformation techniques (compare Hirsch [@Hirsch76 Th. 5.2.10]) permit pairwise cancellation in each $W_j$ of the zeros of $Y^\eps$, without changing $Y^\eps$ near $\msf{bd} (W_j)$. This yields a vector field $X^{\eps}\in {\mcal V}^{\msf L}_{\msf{in}} (M)$ coinciding with $X$ in a neighborhood of $M\verb=\= W$ and nonsingular in $W$, and such that $$\|X^\eps_p - X_p\| < 2\eps, \qquad (p\in M)$$ The sequence $\{X^{1/k}\}_{k\in\Np}$ has the required properties.
Plante [@Plante91] discusses index functions for abelian Lie algebras of vector fields on closed surfaces.
Proofs of the main theorems
============================
Proof of Theorem \[th:main\]
----------------------------
We recall the hypothesis:
- $\tilde M$ is an analytic surface with empty boundary, $M\subset \tilde M$ is a connected topological surface embedded in $\tilde M$.
- $X$ and $Y$ are inward $C^1$ vector fields on $M$.
- $[X, Y]\wedge X=0$.
- $K$ is an essential block of zeros for $X$.
Let $A, B$ be vector fields on a set $S\subset \tilde M$. The [*dependency set*]{} of $A$ and $B$ is $$\msf D (A, B):=\{p\in S\co A_p\wedge B_p=0\}$$
Evidently $$\msf D (X, Y)= \msf D (\tilde X, \tilde Y)\cap M.$$ Proposition \[th:wedge\] implies $ \msf D (\tilde X, \tilde Y)$ is $\tilde
Y$-invariant and $\msf D (X, Y)$ is positively $Y$-invariant.
Then $\msf D (\tilde X, \tilde Y)$ and its subset ${\ensuremath{{\msf Z ( \tilde X)}}}$ are analytic sets in $\tilde M$, hence $\tilde M$ is a simplicial complex with subcomplexes $\msf D (\tilde X, \tilde Y)$ and ${\ensuremath{{\msf Z ( \tilde
X)}}}$ by S. [Ł]{}ojasiewicz’s triangulation theorem [@Lo64].
We assume
- ${\ensuremath{{\mathsf {dim}\,}}}{\ensuremath{{\msf Z ( \tilde Y)}}} < 2$,
as otherwise $Y=0$ and the conclusion is trivial. We also assume
- [*every component of $K$ has dimension $1$*]{}
because isolated points of $K$ lie in ${\ensuremath{{\msf Z ( Y)}}}$ and $K=M$ by analyticity if some component of $K$ is $2$-dimensional, and either of these conditions imply the conclusion. Thus $\Psi^{\tilde Y}$ restricts to a semiflow on the $1$-dimensional complex $ D (\tilde X, \tilde Y)$ having ${\ensuremath{{\msf Z ( X)}}}$ and $\msf D (X, Y)$ as positively invariant subcomplexes.
Let $J\subset K$ be any component. $J$ is a compact, connected, triangulable space of dimension $\le 1$ which is positively $Y$-invariant. From the topology of $J$ we see that ${\ensuremath{{\msf Z ( Y)}}}$ meets $J$ and therefore $K$, unless $$\label{eq:jordan}
\mbox {\em $J$ is a Jordan curve on which $\Phi^Y$ acts
transitively.}$$ Henceforth (\[eq:jordan\]) is assumed.
Let $L\subset \msf D (X, Y)$ be the component containing $J$. The set $Q:=J\cap \ov{L\setminus J}$ is positively $Y$-invariant, whence (\[eq:jordan\]) implies $Q=J$ or $Q=\empty$. Therefore one of the following holds:
(D1)
: $J\subset \Int_M\msf D (X, Y)$,
(D2)
: $J$ is a component of $\msf D (X, Y)$.
Assume (D1) and suppose [*per contra*]{} that ${\ensuremath{{\msf Z ( Y)}}}\cap K
=\varnothing$. Then $ \msf D (X, Y)$ contains the compact closure of an open set $U$ that is isolating for $(X,K)$. We choose $U$ so that each component $C$ of the topological boundary $\msf{bd}(U)$ is a Jordan curve or a compact arc. It suffices by Proposition \[th:homsections\](ii) to prove for each $C$: $$\label{eq:scyc}
\mbox{\em the vector fields $X|C$ and $Y|C$ are nonsingularly homotopic.}$$ Since this holds when $C$ is an arc, we assume $C$ is a Jordan curve. Fix a Riemannian metric on $M$ and define $$\hat X_p=\frac{1}{\|X_p\|}X_p, \quad \hat Y_p=\frac{1}{\|Y_p\|}Y_p,
\qquad (p\in C).$$ These unit vector fields are nonsingularly homotopic to $X|C$ and $Y|C$ respectively, and the assumption $C\subset \msf D (X, Y)$ implies $\hat X=\hat Y$ or $\hat X= -\hat Y$. In the first case there is nothing more to prove. In the second case $\hat X$ and $\hat Y$ are antipodal sections of the unit circle bundle $\eta$ associated to $T_CM$. As the identity and antipodal maps of the circle are homotopic through rotations, (\[eq:scyc\]) is proved.
Now assume (D2). There is an isolating neighborhood $U$ for $X$ such that $$\label{eq:UD}
U\cap \msf D (X, Y) = K.$$ If $0<\eps <1$ the field $X^\eps:= (1-\eps X)+\eps
Y$ belongs to ${\mcal V}^{\msf L}_{\msf{in}}(M)$ (Proposition \[th:convex\]), and has a zero $p\in U$ provided $\eps$ is sufficiently small (Proposition \[th:ess\]). In that case $X_p$ and $Y_p$ are linearly dependent, therefore $p\in K$ by (\[eq:UD\]), whence $Y_p=0$.
[*Case (b): Every neighborhood of $K$ contains an open neighborhood $W$ whose boundary consist of finitely many $Y$-cycles.*]{} It suffices to prove that ${\ensuremath{{\msf Z ( Y)}}}\cap \ov W \ne\varnothing$ if $W$ is isolating for $(X, K)$. Given such a $W$, let $C$ be a component of $\msf{bd}(W)$. By Proposition \[th:wedge\](a), $X_p$ and $Y_p$ are linearly dependent at all points of $C$, or at no point of $C$. In the first case $X|C$ and $Y|C$ are nonsingularly homotopic, as in the proof of (\[eq:scyc\]). In the second case they are nonsingularly homotopic by the restriction to $C$ of the path of vector fields $(1-t)X+tY$, $0\le t\le 1$. It follows that $X|\msf{bd}(W)$ and $Y|\msf{bd}(W)$ are nonsingularly homotopic. Now Proposition \[th:homsections\](ii) implies $$\msf i(Y, W)=\msf i (X, W),$$ which is nonzero because $K$ is essential for $X$. Hence either ${\ensuremath{{\msf Z ( Y)}}}$ meets $\msf {bd} (W)$, or $W$ is isolating for $(Y, K)$ and ${\ensuremath{{\msf Z ( Y)}}}\cap W\ne\varnothing$. This shows that ${\ensuremath{{\msf Z ( Y)}}}$ meets $\ov W$.
It is interesting to see where the proof Theorem \[th:main\] breaks down in Lima’s counterexample to a nonanalytic version (see Example \[th:exlima\]). Lima starts from the planar vector fields $$X^1:= \pd x,\qquad Y^1:=x \pd x + y \pd y$$ satisfying $[X^1, Y^1]= X^1$ and transfers them to the open disk by an analytic diffeomormophism $f\co \R 2 \approx \Int \D 2$. This is done in such a way that the push-forwards of $X^1$ and $Y^1$ extend to continuous vector fields $X, Y$ on $M:=\D 2$ satisfying $[X, Y]= Y$, with ${\ensuremath{{\msf Z ( X)}}}=K= \p \D 2$ and $\msf i_K (X) =1$, while ${\ensuremath{{\msf Z ( Y)}}}$ is a singleton in the interior of $\D 2$. This can be done so that $X$ and $Y$ are $C^\infty$ (see [@BL96]) and therefore generate unique local semiflows. The dependency set $\msf D (X, Y)$ is $R\cup\p \D
2$, where $R$ is the $\Phi^X$-orbit of $z$, a topological line in $\Int\, \D 2$ that spirals toward the boundary in both direction. $\msf D (X, Y)$ is not triangulable because it is connected but not path connected. It is easily seen that neither (D1) nor (D2) holds.
Proof of Theorem \[th:mainB\]
-----------------------------
Here $K$ is essential for $X$ and $\Phi^Y$ preserves area. Suppose [*per contra*]{} $$\label{eq:zyk}
{\ensuremath{{\msf Z ( Y)}}}\cap K=\empty.$$ We can assume $K$ contains a $Y$-cycle $\gam$, for (\[eq:zyk\]) implies every minimal set for $\Phi^Y$ in $K$ is a cycle: This follows from the Poincaré-Bendixson theorem (Hartman [@Hartman64]) when $K$ has a planar neighborhood, and from the Schwartz-Sacksteder Theorem [@Sacksteder65; @Schwartz63] when $Y$ is $C^2$.
Let $J\subset M$ be a half-open arc with endpoint $p\in \gam$ and otherwise topologically transverse to $Y$ orbits (Whitney [@Whitney33]). For any sufficiently small half-open subarc $J_0\subset J$ with endpoint $p$, there is a first-return Poincaré map $f\co J_0\hookrightarrow J$ obtained by following trajectories. By the area-preserving hypothesis and Fubini’s Theorem, $f$ is the identity map of $J_0$. Therefore $\gam$ has a neighborhood $U\subset
M$, homeomorphic to a cylinder or a Möbius band, filled with $Y$-cycles. Theorem \[th:main\](b) implies $K$ is inessential for $X$, contradicting to the hypothesis.
Proof of Theorem \[th:mainC\]
-----------------------------
$K$ is an essential block for $X\in{\mcal V}^{\omega}_{\msf{in}}(M)$ and $\p M$ is an analytic set in $\tilde M$. We can assume $\msf W
(X)\ne\varnothing$ (see Definition \[th:defW\]). Our goal is to prove $${\ensuremath{{\msf Z ( \msf W
(X))}}}\cap K\ne\varnothing.$$ The main step is to show that the set $$\mcal P (K):=\{\ttt\subset \msf W (X)\co {\ensuremath{{\msf Z ( \ttt)}}} \cap
K\ne\varnothing\}$$ is inductively ordered by inclusion. Note that $\mcal P (K)$ is nonempty because it contains the singleton $\{Y\}$. In fact Theorem \[th:main\] states: $$\label{eq:yddx}
Y\in \msf W (X)\implies \{Y\}\in \mcal P (K).$$ We rely on a consequence of Proposition \[th:wedge\]: $$\label{eq:kddx}
\mbox{\em $K$ is positively invariant under $\msf W (X)$}$$ The assumption on $\p M$ implies $M$ is semianalytic as a subset of $\tilde M$, and this implies $K$ is also semianalytic. Therefore $K$, being compact, has only finitely many components and one of them is essential for $X$ by Proposition \[th:index\], (VF3). Therefore we can assume $K$ is a connected semianalytic set of $\tilde M$. Now $K$ is the intersection of $M$ with the component of ${\ensuremath{{\msf Z ( \tilde)}}} X$ that contains $K$, which is an analytic set. This implies ${\ensuremath{{\mathsf {dim}\,}}}K \le
1$, and we assume ${\ensuremath{{\mathsf {dim}\,}}}K=1$, as otherwise $K$ is finite and contained in ${\ensuremath{{\msf Z ( \msf W (X))}}}$ by (\[eq:kddx\]).
The set $K_{sing}\subset K$ where $K$ is not locally an analytic $1$-manifold is finite and positively invariant under $\msf W
(X)$ As this implies $K_{sing}\subset{\ensuremath{{\msf Z ( \msf W (X))}}}\cap K$, we can assume $K_{sing}=\empty$, which under current assumptions means: $$\mbox{\em $K$ is an analytic submanifold of $\tilde M$
diffeomorphic to a circle.}$$ We can also assume:
(K)
: [*$K\subset L$ if $K\cap L\ne\varnothing$ and $L$ is positively $\msf W (X)$-invariant semianalytic set in $\tilde M$.*]{}
For if $Y\in \msf W (S)$ then $K\cap L$, being nonempty, finite and positively invariant under every $Y$, is necessarily contained in ${\ensuremath{{\msf Z ( Y)}}}$.
From (K) we infer $$\label{eq:hhK}
\ttt\in \mcal P (K) \implies K\subset {\ensuremath{{\msf Z ( \ttt)}}}.$$ Consequently $\mcal P (K)$ is inductively ordered by inclusion. By Zorn’s lemma there is a maximal element $\mm\in\mcal P (K)$, and (\[eq:hhK\]) implies $$\label{eq:kmm}
K\subset {\ensuremath{{\msf Z ( \mm)}}}.$$ To prove every $Y\in \gg$ lies in $\mm$, let $\nn_Y\subset\gg$ be the smallest ideal containing $Y$ and $\mm$. Theorem \[th:main\] implies ${\ensuremath{{\msf Z ( Y)}}}\cap
K\ne\varnothing$, whence $${\ensuremath{{\msf Z ( \nn_Y)}}}\cap K= {\ensuremath{{\msf Z ( Y)}}}\cap {\ensuremath{{\msf Z ( \mm)}}} \cap K \ne\varnothing$$ by (\[eq:kmm\]). Property (K) shows that $\nn_Y\in \mcal
H$, so $\nn_Y=\mm$ by maximality of $\mm$.
Proof of Theorem \[th:liealg\]
-------------------------------
The theorem states:
[*Let $M$ be an analytic surface and $\gg$ a Lie algebra of analytic vector fields on $M$ that are tangent to $\p M$. If $X\in \gg$ spans a one-dimensional ideal, then:*]{}
(a)
: [*${\ensuremath{{\msf Z ( \gg)}}}$ meets every essential block $K$ of zeros for $X$,*]{}
(b)
: [*if $M$ is compact and $\chi (M)\ne 0$ then ${\ensuremath{{\msf Z ( \gg)}}}\ne\varnothing$.*]{}
The hypotheses imply $\gg\subset \msf W (X)$, because if $Y\in \gg$ then $[X, Y]=cX, c\in \RR$. Therefore (a) follows from Theorem \[th:mainC\]. Conclusion (b) is a consequence, because its assumptions imply the block ${\ensuremath{{\msf Z ( X)}}}$ is essential for $X$ (Proposition \[th:ichi2\]).
Proof of Theorem \[th:liegroup\]
--------------------------------
An effective analytic action $\alpha$ of $G$ on $M$ induces an isomorphism $\phi$ mapping the Lie algebra $\gg_0$ of $G$ isomorphically onto a subalgebra $\gg\subset {\mcal V}^{\omega}(M)$. Let $X^0\in\gg_0$ span the Lie algebra of a one-dimensional normal subgroup of $G$. Then $\phi
(X^0)$ spans a $1$-dimensional ideal in $\gg$, hence Theorem \[th:liealg\] implies ${\ensuremath{{\msf Z ( \gg)}}}\ne\varnothing$. The conclusion follows because ${\ensuremath{{\msf Z ( \gg)}}}=\Fix {\alpha(G)}$.
[99]{} M. Belliart, [*Actions sans points fixes sur les surfaces compactes,*]{} Math. Zeit. [**225**]{} (1997) 453–465
M. Belliart & I. Liousse, [*Actions affines sur les surfaces,*]{} Publications IRMA, Universite de Lille, [**38**]{} (1996) exposé X
C. Bonatti, [*Champs de vecteurs analytiques commutants, en dimension $3$ ou $4$: existence de zéros communs,*]{} Bol. Soc. Brasil. Mat. (N.S.) [**22**]{} (1992) 215–247
A. Dold, [*Fixed point index and fixed point theorem for Euclidean neighborhood retracts,*]{} Topology [**4**]{} (1965) 1––8
A. Dold, “Lectures on Algebraic Topology,” Die Grundlehren der matematischen Wissenschaften Bd. 52, second edition. Springer, New York (1972)
D. Epstein & W. Thurston, [*Transformation groups and natural bundles*]{}, Proc. London Math. Soc. [**38**]{} (1979) 219–236
M. Freedman, [*The topology of 4-dimensional manifolds,*]{} J. Diff. Geometry [**17**]{} (1982) 357–453
S. Friedl, I. Hambleton, P. Melvin, P. Teichner, [*Non-smoothable four-manifolds with infinite cyclic fundamental group,*]{} Int. Math. Res. Not. 2007, Vol. 2007, article ID rnm031, 20 pages, doi:10.1093/imrn/rnm031
P. Hartman, “Ordinary Differential Equations,” John Wiley & Sons, New York (1964)
M. Hirsch, “Differential Topology,” Springer-Verlag, New York, 1976
M. Hirsch, manuscript (2003)
M. Hirsch, [*Actions of Lie groups and Lie algebras on manifolds,*]{} “A Celebration of the Mathematical Legacy of Raoul Bott,” 69-78. Centre de Recherches Mathématiques, U. de Montréal. Proceedings & Lecture Notes Vol. 50 (P. R. Kotiuga, ed.), Amer. Math. Soc. Providence RI (2010)
M. Hirsch, [*Smooth actions of Lie groups and Lie algebras on manifolds,*]{} J. Fixed Point Th. App. [**10**]{} (2011) 219–232
M. Hirsch & A. Weinstein, [*Fixed points of analytic actions of supersoluble Lie groups on compact surfaces,*]{} Ergod. Th. Dyn. Sys. [**21**]{} (2001) 1783–1787
H. Hopf, [*Vektorfelder in Mannifaltigkeiten*]{}, Math. Annalen [**95**]{} (1925), 340–367
N. Jacobson, “Lie Algebras,” John Wiley & Sons, New York (1962)
R. Kirby, Personal communication (2012)
E. Lima, [*Common singularities of commuting vector fields on $2$-manifolds*]{}, Comment. Math. Helv. [**39**]{} (1964) 97–110
W. Johnson, J. Lindenstrauss, G. Schechtman, [*Extensions of Lipschitz maps into Banach spaces,*]{} Israel J. Math. [**54**]{} (1986) 129-–138
S. [Ł]{}ojasiewicz, [*Triangulation of semi-analytic sets*]{}, Ann. Scuola Norm. Sup. Pisa (3) [**18**]{} (1964) 449–474
P. Molino & F.-J. Turiel, [ *Une observation sur les actions de ${\bf R}^p$ sur les variétés compactes de caractéristique non nulle,*]{} Comment. Math. Helv. [**61**]{} (1986), 370–-375
P. Molino & F.-J. Turiel, [*Dimension des orbites d’une action de ${\bf R}^p$ sur une variété compacte,*]{} Comment. Math. Helv. [**63**]{} (1988) 253–-258
G. Mostow, [*The extensibility of local Lie groups of transformations and groups on surfaces*]{}, Ann. of Math. [**52**]{} (1950) 606–636. E. Nelson, “Topics in Dynamics I: Flows,” Princeton University Press (1970)
J. Plante, [*Fixed points of Lie group actions on surfaces,*]{} Erg. Th. Dyn. Sys. [**6**]{} (1986) 149–161
J. Plante, [*Lie algebras of vector fields which vanish at a point,*]{} J. London Math. Soc (2) [**38**]{} (1988) 379–384
J. Plante, [*Elementary zeros of Lie algebras of vector fields,*]{} Topology [**30**]{} (1991) 215–222
H. Poincaré, [*Sur les courbes définies par une équation différentielle,*]{} J. Math. Pures Appl. [**1**]{} (1885) 167–244
A. J. Schwartz, [*Generalization of a Poincaré-Bendixson theorem to closed two dimensional manifolds,*]{} Amer. J. Math. [**85**]{} (1963) 453-458
R. Sacksteder, [*Foliations and pseudogroups,*]{} Amer. J. Math. [**87**]{} (1965) 79–102
W. Thurston, [*A generalization of the Reeb stability theorem*]{}, Topology [**13**]{} (1974) 347-352
F.-J. Turiel, [*An elementary proof of a Lima’s theorem for surfaces*]{}, Publ. Mat. [**3**]{} (1989) 555–557
F.-J. Turiel, [*Analytic actions on compact surfaces and fixed points,*]{} Manuscripta Mathematica [**110**]{} (2003) 195–201 F.-J. Turiel, personal communication (2006)
H. Whitney, [*Regular families of curves*]{}, Ann. Math. [**34**]{} (1933) 244–270
[^1]: The Poincaré-Hopf index goes back to Poincaré [@Poincare85] and Hopf [@Hopf25]. It is usually defined only when $M$ is compact and $K={\ensuremath{{\msf Z ( X)}}}$. The more general definition above is adapted from Bonatti [@Bonatti92].
[^2]: Bonatti assumes ${\ensuremath{{\mathsf {dim}\,}}}M = 3$ or $4$, but the conclusion for ${\ensuremath{{\mathsf {dim}\,}}}M
\le 2$ follows easily: If ${\ensuremath{{\mathsf {dim}\,}}}M=2$, identify $M$ with $M\times\{0\}\subset M\times \RR$ and apply Bonatti’s theorem to the vector fields $\left(X, x\pde{~} {x}\right)$ and $\left(Y, x\pde{~} {x}\right)$ on $M\times
\RR$. For ${\ensuremath{{\mathsf {dim}\,}}}M=1$ there is a simple direct proof.
[^3]: The only role of $\tilde M$ is to permit a simple definition of smooth maps on $M$. Its global topology is never used, and in any discussion $\tilde M$ can be replaced by any smaller open neighborhood of $M$. If $n>2$ then $M$ might not be smoothable, as shown by a construction due to Kirby [@Kirby12]: Let $P$ be a nonsmoothable closed $4$-manifold (Freedman [@Freedman82]). Let $D\subset P$ be a compact $4$-disk. Then $M:=P\setminus \Int D$ is not smoothable, for otherwise $\p M$ would be diffeomorphic to $\S 3$ and $P$ could be smoothed by gluing $\D
4$ to $M$. Define $\tilde M$ as the connected sum of $P$ with a nontrivial $\S
2$-bundle over $\S 2$. Then $\tilde M$ is smoothable and contains $M$ (compare Friedl [*et al. *]{}[@Friedl07]).
[^4]: This means $V$ is homeomorphic to a retract of an open subset of some Euclidean space. Polyhedra and connected metrizable manifolds are ENRs.
| {
"pile_set_name": "ArXiv"
} |
=-1cm =-0.7cm
[.7ex]{} [.7ex]{}
0.05in
0.05in [[**Marco Battaglia**]{}]{} 0.05in [*[CERN, Geneva, Switzerland]{}* ]{} 0.15in [**Abstract**]{}
The problems of the origin of mass and of electro-weak symmetry breaking are central to the programme of research in particle physics, at present and in the coming decades. This paper reviews the potential of high energy, high luminosity $e^+e^-$ linear colliders in exploring the Higgs sector, to extend and complement the data which will become available from hadron colliders. The accuracy of measurements of the Higgs boson properties will not only probe the validity of the Higgs mechanism but also provide sensitivity to New Physics beyond the Standard Model.
[Invited Talk at the 10$^{th}$ International Conference on Supersymmetry\
and Unification of Fundamental Interactions\
June 17-23, 2002, DESY, Hamburg]{}
[*This paper is dedicated to Laura Alidori*]{}
0.15in
Prologue
========
Explaining the origin of mass is one of the great scientific quests of our time. The Standard Model (SM), successfully tested to an unprecedented level of accuracy by the LEP, Tevatron and SLC experiments, and now also by the $B$-factories, addresses this question with the Higgs mechanism [@Higgs].
In this paper, I review the anticipated potential of $e^+e^-$ linear colliders (LC) in probing the nature of the Higgs sector. As the title suggests, such studies might be compared to the efforts of explorers and cartographers, from the first sightings of new lands to the systematic charting of their coastline profiles. The first manifestation of the Higgs mechanism through the Higgs sector is the existence of at least one Higgs boson, denoted with $H^0$. The observation of a new spin-0 particle would represent a first sign that the Higgs mechanism of mass generation is realised in Nature. Results of direct searches and indirect constraints, from precise electro-weak data, tell us that the Higgs boson is heavier than 114 GeV and possibly lighter than about 195 GeV [@ichep]. We expect the Higgs boson to be first sighted by the Tevatron or the [Lhc]{}, the CERN hadron collider, which will determine its mass and perform a first survey of its basic properties. Similarly to the first sighting of Hispaniola in 1492, such a discovery will represent a major breakthrough and will bring to a successful completion an intense program of experimental searches and phenomenological speculations lasting since several decades. But after the observation of a new particle with properties compatible with those expected for the Higgs boson, a significant experimental and theoretical effort will be needed to verify that the newly-discovered particle is indeed the boson of the scalar field responsible for the electro-weak symmetry breaking and the generation of mass. An $e^+e^-$ linear collider with its well defined energy, known identity of the initial state partons, opportunity to control their helicities and a detectors which provides highly accurate information on the event properties, will promote the Higgs studies into the domain of precision physics.
$e^+e^-$ collisions from beyond LEP-2 to\
the multi-TeV scale
=========================================
The [Lep-2]{} collider at CERN has set the highest centre-of-mass energy, reached so far in $e^+e^-$ collisions, at $\sqrt{s}$=209 GeV. A significant further increase in both the beam energy and the luminosity has been the aim of several decades of LC designs and R&D, to advance the energy frontier in $e^+e^-$ physics. These developments have now matured to the point where we can contemplate construction of a linear collider with initial energy in the 500 GeV range and a credible upgrade path to $\sim1$ TeV. The [Tesla]{} project [@Brinkmann:2001qn] adopts super-conducting (SC) cavities which offer high luminosity with rather relaxed alignment requirements. As a result of a successful R&D program, gradients in excess of 23 MV/m, necessary to reach $\sqrt{s}$=500 GeV have been demonstrated. An alternative approach is taken by the [Nlc]{} [@nlc] and [Jlc]{} projects which adopt X-band (11 GHz) warm cavities, evolving the concept of the [Slc]{} at SLAC, the only linear collider operated so far.
The LC energy can be later upgraded by three different strategies (see Figure \[fig:1\]). The first is to increase the energy at the expense of the luminosity, where the limit is set by the gradient and the available power. The second is to increase the accelerating gradient and/or adiabatically extend the active linac by adding extra accelerating structure. This scheme was successfully adopted at [Lep]{} in raising the collision energy by more than a factor of two. Here the limit is set by the site length, the achieved gradient and the RF power. Four super-conducting nine-cell cavity prototypes have been conditioned at 35 MV/m.
This gradient would allow to push the [Tesla]{} $\sqrt{s}$ energy up to 800 GeV, with a luminosity of $5.8 \times 10^{34}$ cm$^{-2}$s$^{-1}$. The [NLC]{} project aims at achieving 1 TeV, with luminosity of 3.4$\times 10^{34}$cm$^{-2}$s$^{-1}$, by doubling the number of components and, possibly, increasing the gradient as well.
Beyond these energies, the extensions of the SC and X-band technology are more speculative. In order to attain collisions at energies in excess of 1 TeV, with high luminosity, significantly higher gradients are necessary. Also, the number of active elements in the linac must be kept low enough to ensure reliable operation. The two-beam acceleration scheme, presently being developed within the [Clic]{} study [@clic] at CERN, suggests a unique opportunity to extend the physics at $e^+e^-$ colliders to constituent energies of the order of the LHC energy frontier, and beyond. The recent test of a 30 GHz tungsten iris structure, successfully reaching an accelerating gradient of 150 MV/m [@ctf], for a 16 ns pulse, is highly encouraging for the continuation of these studies.
The Neutral Higgs Boson Profile
===============================
Outlining the Higgs boson profile, through the determination of its mass, width, quantum numbers, couplings to gauge bosons and fermions and the reconstruction of the Higgs potential, stands as a greatly challenging programme. The spin, parity and charge-conjugation quantum numbers $J^{PC}$ of Higgs bosons can be determined at the LC in a model-independent way. Already the observation of either $\gamma \gamma \rightarrow H$ production or $H \rightarrow \gamma\gamma$ decay sets $J \ne 1$ and $C=+$. The angular dependence $\frac{d \sigma_{ZH}}{d \theta} \propto \sin^2 \theta$ and the rise of the Higgs-strahlung cross section $\sigma_{ZH} \propto~\beta
\sim~\sqrt{s-(M_H+M_Z)^2}$ allows to determine $J^P = 0^+$ and distinguish the SM Higgs from a $CP$-odd $0^{-+}$ state $A^0$, or a $CP$-violating mixture of the two. But the LC has a unique potential for verifying that the Higgs boson does its job of providing gauge bosons, quarks and leptons with their masses. This can be obtained by testing the relation $g_{HXX} \propto {M_X}$ between Yukawa couplings, $g_{HXX}$, and the corresponding particle masses, $M_X$, precisely enough. It is important to ensure that the LC sensitivity extends over a wide range of Higgs boson masses and that a significant accuracy is achieved for all particle species. Here, the LC adds the precision which establishes the key elements of the Higgs mechanism, as discussed in this section and summarised in Table 1.
Couplings to Gauge Bosons
-------------------------
The determination of the cross-section for the $e^+e^- \to H^0Z^0$ Higgs-strahlung process measures the Higgs coupling to the $Z^0$ boson and it is also a key input to extract absolute branching fractions from the experimental determination of products of production cross sections and decay rates. At the LC, the use of the di-lepton recoil mass from the $Z^0 \rightarrow \ell^+ \ell^-$ decay (see Figure \[fig:hz\]) provides a model-independent method which critically depends on the momentum resolution. An high precision central tracker and the addition of the beam-spot constraint guarantees $\Delta p/p \le 5 \times 10^{-5} p$ (GeV/c), which is $\simeq$12 times better than that obtained with the [Lep]{} detectors. At the nominal luminosity expected at $\sqrt{s}$=350 GeV, of the order of 24000 Higgs bosons would be observable in one year of operation ($=10^7$ s), of which 4000 in a model-independent way, if $M_H$=120 GeV. A relative accuracy on the $e^+e^- \to HZ$ cross section of 2.4-3.0% can be achieved with 0.5 ab$^{-1}$ of data at $\sqrt{s}$ = 350 GeV, assuming 120 GeV$< M_H <$160 GeV [@hzxs].
Since the recoil mass analysis is independent on the decay mode, it is also sensitive to non-standard decay modes, such as $H \rightarrow {\mathrm{invisible}}$. Several theoretical models introduce an invisible $H$ decay width (possibly SUSY decays $\chi^0 \chi^0$, but also signatures of Radion-Higgs mixing or so-called Stealth models). Such invisible Higgs decays may be problematic at hadron colliders but are detectable both indirectly, by subtracting from the total decay width the sum of visible decay modes, and directly, by analysing the system recoiling against the $Z^0$ in the $e^+e^- \rightarrow HZ$ process, at the LC. The direct method generally provides with a higher accuracy, corresponding to a determination of BR($H \rightarrow {\mathrm{invisible}}$) to better than 5%, so long as the invisible yield exceeds 10% of the Higgs decay width [@invisible].
-- --
-- --
The $WW$-fusion reaction, $e^+e^- \rightarrow WW \nu \bar \nu \to H \nu \bar \nu$, measures the $H^0$ coupling to the $W^{\pm}$ boson. A template analysis is based on $b$-tagged hadronic events at $\sqrt{s}$ = 350 GeV with large missing mass and missing energy. The main background is due to the Higgs-strahlung process, when $Z^0 \to \nu \bar{\nu}$. It is possible to extract $\sigma_{H \nu\nu}$ from a $\chi^2$ fit to the missing mass distribution, which efficiently discriminates between the two contributions [@wwxs]. Overlapping accelerator-induced $\gamma \gamma \to {\mathrm{hadrons}}$ background can be suppressed by an impact parameter analysis of the forward produced particles, if the tracking resolution is small compared to the bunch length [@Battaglia:1999ux]. A relative accuracy of 2.6% is obtained for $M_H$=120 GeV, which becomes 10%, for $M_H$=150 GeV. However, since the branching fraction for the $H^0 \to W^*W$ decay increases sharply in this mass interval, the accuracy on the Higgs coupling to $W$ boson can be extracted with a small and constant uncertainties, when combining the results for production and decay processes, involving the same $HWW$ coupling.
Couplings to Fermions
---------------------
Measuring the Higgs couplings to quarks and leptons precisely is one of the main aims of Higgs studies at the LC. The requirements of such analyses have driven the concept of the innermost vertex detector and the design of the interaction region. The issue here is to measure the charged particle trajectories accurately enough to distinguish the decays $H^0 \to b \bar{b}$ from $H^0 \to c \bar{c}$, and these from $H^0 \to gg$, by reconstructing the signature decay patterns of heavy flavour hadrons. Several independent studies have been performed which indicate that the BR($H^0 \to b \bar{b}$) can be measured to better than 3%, BR($H^0 \to c \bar{c}$) to about 9-19% and BR($H^0 \to g g$) to 6-10%, if the Higgs boson is light [@Battaglia:1999re; @brientbr; @Potter:2001ap]. The spread in the estimated accuracy for $c \bar{c}$ and $gg$ results is attributed in terms of the different simulation and data analyses methods adopted which are currently under study. It is interesting to observe that with these experimental accuracies, the test of the coupling scaling with the fermion masses will be dominated by the present uncertainties on the latter. The case of the top quark is of special interests as it is the only fermion with an ${\cal O}(1)$ Yukawa coupling to the SM Higgs boson. This coupling can be measured through a determination of the cross section $e^+e^- \to t \bar t H^0$ [@Dittmaier:1998dz].
Tests of the mass generation mechanism in the lepton sector are also possible by studying the decays $H^0 \to \tau^+ \tau^-$ and $\mu^+ \mu^-$. The $\tau$ Yukawa coupling can be measured to 2.5-5.0% using $\tau$ identification based on multiplicity and kinematics for 120 GeV$< M_H <$140 GeV [@Battaglia:1999re; @brientbr]. More recently, it has been shown that also the rare decay $H^0 \rightarrow \mu\mu$ is observable at TeV-class LC and quite accurately measurable at a multi-TeV LC [@Battaglia:2001vf]. At 3 TeV, the relative accuracy on the muon Yukawa coupling is 3.5-10% for 120 GeV$< M_H <$150 GeV. This would allow to test the $g_{H\mu\mu}/g_{H\tau\tau}$ coupling ratio to a 5-8% accuracy at a multi-TeV LC, which must be compared to the 3-4% accuracy expected from combining the Muon Collider and TeV-class LC data for 120$<M_H<$140 GeV.
Higgs Potential
---------------
A most distinctive feature of the Higgs mechanism is the shape of the Higgs potential, $V(\Phi^* \Phi) = \lambda (\Phi^*\Phi - \frac{1}{2}v^2)^2$. In the SM, the triple Higgs coupling, $g_{HHH}$, is related to the Higgs mass, $M_H$, through the relation $g_{HHH} = \frac{3}{2} \frac{M_H^2}{v}$, where $v$=246 GeV. By determining $g_{HHH}$, in the double Higgs production processes $e^+e^- \to H H Z$ and $e^+e^- \to H H \nu\nu$ [@Djouadi:1999gv], the above relation can be tested at the LC. These measurements are made difficult by the tiny production cross sections and the dilution due to diagrams leading to double Higgs production, but not sensitive to the triple Higgs vertex. A LC operating at $\sqrt{s}$ = 500 GeV can measure the $HHZ$ production cross section to about 15% accuracy, if the Higgs boson mass is 120 GeV, corresponding to a fractional accuracy of 23% on $g_{HHH}$ [@Castanier:2001sf]. Improvements can be obtained both by performing the analysis at very high energy and by introducing observables sensitive to the presence of the triple Higgs vertex (see Figure \[fig:hhh\]) [@Battaglia:2001nn]. On the contrary, the quartic Higgs coupling remains elusive, due to the smallness of the relevant triple Higgs production cross sections.
-- --
-- --
----------------- --------- -------------------------------
$M_H$ $\delta(X)/X$
(GeV) LC-500 $|$ LC-3000
0.5 ab$^{-1}$ $|$ 5 ab$^{-1}$
$M_H$ 120-180 (3-5) $\times 10^{-4}$
$\Gamma_{tot}$ 120-140 0.04-0.06
$g_{HWW}$ 120-160 0.01-0.03
$g_{HZZ}$ 120-160 0.01-0.02
$g_{Htt}$ 120-140 0.02-0.06
$g_{Hbb}$ 120-160 0.01-0.03
$g_{Hcc}$ 120-140 0.03-0.10
$g_{H\tau\tau}$ 120-140 0.03-0.05
$g_{H\mu\mu}$ 120-140 0.15 $|$ 0.04-0.06
[CP]{} test 120 0.03
$g_{HHH}$ 120-180 0.20 - – $|$ 0.07-0.09
----------------- --------- -------------------------------
: *Summary of the accuracies on the determination of the Higgs boson profile at the LC. Results are given for a 350-500 GeV LC with ${\cal{L}}$=0.5 ab$^{-1}$. Further improvements expected from a 3 TeV LC are also shown for some of the measurements.*
\[tab:summary\]
-- --
-- --
--------------------------------- ------- ---------------------------------
$M_H$ $\delta X/X$
GeV LC-500/800 $|$ LC-3000
0.5/1 ab$^{-1}$ $|$ 5 ab$^{-1}$
$M_H$ 240 $9 \times 10^{-4}$
$\Gamma_H$ 240 0.12
$\sigma(e^+e^- \rightarrow HZ)$ 240 0.04
BR($H \rightarrow ZZ$) 240 0.10
BR($H \rightarrow WW$) 240 0.07
BR($H \rightarrow b \bar b$) 200 0.16 $|$ 0.04
BR($H \rightarrow b \bar b$) 220 0.27 $|$ 0.05
--------------------------------- ------- ---------------------------------
: *Summary of the accuracies on the determination of a heavy Higgs boson profile at the LC. Results are given for a 500-800 GeV LC with ${\cal{L}}$=0.5 and 1.0 ab$^{-1}$. Further improvements expected from a 3 TeV LC are also shown for some of the measurements.*
\[tab:heavy\]
What if the Higgs is heavier ?
------------------------------
Precision electro-weak data indicate that a SM-like Higgs boson should be lighter than $\simeq$195 GeV. However, scenarios exists where the Higgs boson is heavier as a result of New Physics affecting the electro-weak observables. It is therefore important to assess the LC sensitivity to heavier bosons. Analyses have considered the $HZ \rightarrow\ell^+\ell^-$, $q \bar q$ recoil mass at 500 GeV and the $H \nu \bar \nu$ process at 800 GeV. In order to extract $M_H$, $\Gamma_H$ and $\sigma$ a fit to the recoil mass spectrum can be performed while the $H \rightarrow WW$ and $ZZ$ branching fractions can be measured from the jet-jet mass in $HZ$ [@heavy]. The couplings to fermions are still accessible through $H \rightarrow b \bar b$ studied in $H \nu \bar \nu$. Results are summarised in Table \[tab:heavy\].
An Extended Higgs Sector
========================
Despite its successful tests, the Standard Model must be embedded into a more fundamental theory valid at higher energies, in order to explain its many parameters and cure remaining problems, such as the stabilisation of the Higgs potential. Supersymmetry is the simplest perturbative explanation for low-scale electroweak symmetry breaking, and, in its minimal realization (MSSM), predicts that the lightest Higgs boson must be lighter than about 130 GeV. In the MSSM, as in more general 2HDM extensions of the SM, the Higgs sector consists of two doublets, generating five physical Higgs states: $h^0$, $H^0$, $A^0$ and $H^{\pm}$. The $h^0$ and $H^0$ states are CP even while the $A^0$ is CP odd. The masses of the CP-odd Higgs boson, $M_A$, and the ratio of the vacuum expectation values of the two doublets $\tan \beta = v_2/v_1$ are free parameters.
-- --
-- --
The study of the lightest neutral MSSM Higgs boson, $h^0$, follows closely that of the SM $H$ discussed above and those results remain in general valid. The mass and coupling patterns of the other bosons vary with the model parameters. However, in the decoupling limit the $H^{\pm}$, $H^0$ and $A^0$ bosons are expected to be heavy and to decay predominantly into quarks of the third generation. Establishing their existence and determining their masses and main decay modes, through their pair production $e^+e^- \to H^0 A^0$ and $H^+H^-$ would represent a decisive step in the understanding of the Higgs sector, which will most probably require extended LC operations at $\sqrt{s} \ge$ 500 GeV.
Indirect Sensitivity
--------------------
The precision study of the Higgs couplings to fermions and gauge bosons may already reveal its SM or Supersymmetric nature, before the direct observation of heavier bosons. In fact, in the SM we expect the ratios of Yukawa couplings to be proportional to those of the particle masses. On the contrary, in Supersymmetry, couplings to up-like and down-like fermions are shifted w.r.t. their SM predictions as $\frac{BR(h \rightarrow f_u \bar{f_u})}{BR(h \rightarrow f_d \bar{f_d})}
\propto \frac{1}{tan^2 \alpha \tan^2 \beta} \simeq
\frac{(M^2_h - M^2_A)^2}{(M^2_Z + M^2_A)^2}$. This can be exploited to distinguish the SM $H^0$ from the Supersymmetric $h^0$ with precise determination of its couplings (see Figure \[fig:hfitter\]). If they would result to be incompatible with those predicted by the SM, informations can be extracted on the value of the $A^0$ mass, $M_A$, which is a fundamental parameter in Supersymmetry. Several analyses of the indirect sensitivity to the MSSM provided by the accurate determination of the Higgs couplings have been conducted as a function of $M_A$. Results show consistently that for $M_A<$650 GeV, the MSSM $h^0$ can be distinguished from the SM $H^0$, mostly from its $b \bar{b}$, $c \bar{c}$ and $WW$ couplings (see Figure \[fig:matb\]). Furthermore, SUSY sbottom-gluino and stop-higgsino loops may shift the effective $b$-quark mass in the $hbb$ couplings: $\Delta m_b \propto \mu ~M_{\tilde{g}}~\tan \beta~f(M_{\tilde{b_1}}, M_{\tilde{b_2}},
M_{\tilde{g}})$. This becomes important in specific regions of the parameter space, at large $\tan \beta$ and small $M_A$ values, where the $b \bar{b}$ coupling can be dramatically suppressed. These effects can be accurately surveyed at the LC.
Direct Sensitivity
------------------
If the heavy Higgs bosons are above pair-production threshold, the $e^+e^- \to H^0$, $A^0 \to b \bar b$, $e^+e^- \to H^+ \rightarrow t \bar b$ or $H^+ \to W^+ h^0$, $h^0 \to b \bar b$ processes will provide with very distinctive, yet challenging, multi-jet final states with multiple $b$-quark jets, which must be efficiently identified and reconstructed. Example analyses have shown that an accuracy of about 0.3% on the boson masses and of $\simeq 10\%$ on the product $\sigma \times BR$ can be obtained at the LC [@hpm; @ha].
In addition, the $\gamma \gamma \to A^0$ and $H^0$ process at the $\gamma \gamma$ collider, is characterised by a sizable cross section which may probe the heavier part of the Higgs spectrum, beyond the $e^+e^-$ reach. Finally, a scan of the $A^0$ and $H^0$ thresholds at the $\gamma \gamma$ collider can in principle resolve a moderate $A^0 - H^0$ mass splitting [@ggha; @Muhlleitner:2001kw], which could not otherwise be observed at other colliders.
CP violation
------------
Extensions of the SM may introduce new sources of CP violation, through additional physical phases whose effects can be searched for in the Higgs sector. Supersymmetric one-loop contributions can lead to differences in the decay rates of $H^+ \to t \bar{b}$ and $H^- \to \bar{t} b$, in the MSSM with complex parameters [@Christova:2002ke]. This CP asymmetry is expressed as $\delta CP = \frac{\Gamma(H^- \rightarrow b\bar{t})-\Gamma(H^+ \rightarrow t\bar{b})}
{\Gamma(H^- \rightarrow b\bar{t})+\Gamma(H^+ \rightarrow t\bar{b})}$ and it can amount to up to $\simeq 15$%. As the leading contributions come from loops with $\tilde{t}$, $\tilde{b}$ and $\tilde{g}$, $\delta^{CP}$ is sensitive to these parameters. With the expected statistics of $e^+e^- \to H^+H^- \to t \bar{b} \bar{t} b$ at $\sqrt{s}$=3 TeV and assuming realistic charge tagging performances, a 3 $\sigma$ effect for ${\cal{L}}$=5 ab$^{-1}$ would be observed for an asymmetry $|\delta^{CP}|$=0.10.
NMSSM
-----
While the MSSM has been the main model for surveying the SUSY Higgs phenomenology so far, Supersymmetry may be realised in a non-minimal scenario. The introduction of an additional Higgs singlet has been proposed as a natural explanation of the value of the $\mu$ term in the MSSM Higgs potential. The resulting NMSSM Higgs sector has seven physical Higgs bosons and six free parameters.
-- --
-- --
It has been shown that the prospects for Higgs discovery at the [Lhc]{} are not undermined in this scenario, provided that the full integrated luminosity of ${\cal{L}}$=300 fb$^{-1}$ is considered [@Ellwanger:2001iw]. This would lead to an interesting phenomenology with two scalar Higgs bosons possibly within reach of a TeV-class LC, one light pseudo-scalar and four heavy bosons almost degenerate in mass (see Figure \[fig:nmssm\]) [@miller].
The Higgs Boson and the Radion\
in scenarios with Extra-Dimensions
==================================
The hierarchy problem, originating from the mismatch between the electroweak scale, defined by the Higgs field vacuum expectation value $v$ = 246 GeV and the Planck scale, has motivated the introduction of models with hidden extra dimensions. In the original construction of the Randall-Sundrum formulation [@Randall:1999vf], the SM particles live on a brane, while gravity expands on a second, parallel brane and in the bulk. This scenario introduces a new particle, the Radion, which represents the quantum excitation of the brane separation. By mixing with the Higgs field, the Radion modifies the Higgs couplings to SM particles and thus its decay branching fractions BR($H \rightarrow f \bar{f}$) [@Hewett:2002nk; @Dominici:2002jv]. Examples are shown in Figure \[fig:radion\] for $M_H$=125 GeV. The typical accuracies for the branching fraction determination at the LC make these shifts significant enough to reveal the radion effects over a large fraction of the parameter space, including regions where evidence of the radions and of the extra-dimensions could not be obtained directly.
-- --
-- --
Conclusions
===========
The present theory of fundamental interactions and the experimental evidence strongly suggest that the electro-weak symmetry is broken via the Higgs mechanisms, while particles acquire their mass through their interaction with the scalar Higgs field. The Higgs boson is heavier than 114 GeV and possibly lighter than about 210 GeV. Within this scenario we expect the [Lhc]{} to discover such Higgs boson and a high energy, high luminosity $e^+e^-$ linear collider to accurately determine its properties. These precisions can be obtained within a realistic run plan scenario. Charting the Higgs profile will not only clarify if the Higgs mechanism is indeed responsible for electro-weak symmetry breaking and mass generation. It will also elucidate the nature of the Higgs particle. As the discovery of Hispaniola deceived Columbus on the essence of his achievement, the Higgs sector may have more and different properties than those expected in the SM or even considered in this paper. Higgs physics at the LC may thus provide the first clues on New Physics beyond the SM and possibly a new world of particles, as predicted in Supersymmetry. While most of the mapping of the lighter Higgs profile can be optimally performed at a LC with centre-of-mass energies in the range 300-500 GeV, there are measurements which will require higher energies. An upgrade of the centre-of-mass energy to $\simeq$ 1 TeV and a second stage multi-TeV LC should complete the measurement of the Higgs properties with the needed accuracy and extend the sensitivity to heavier Higgs bosons, beyond the [Lhc]{} reach, depending on the nature of the Higgs sector. In all cases, linear colliders add crucial information to previous data and to the data that the [Lhc]{} will obtain.
[*It is a pleasure to thank the organisers for their invitation and for a most pleasant Conference. I am grateful to Albert De Roeck, Klaus Desch, Daniele Dominici, Jack Gunion, Joanne Hewett, Tom Rizzo, Ian Wilson and Peter Zerwas for suggestions and discussion.*]{}
[99]{}
P.W. Higgs, [*Phys. Rev. Lett.*]{} [**12**]{} (1964) 132; [*idem*]{}, [*Phys. Rev.*]{} [**145**]{} (1966) 1156; F. Englert and R. Brout, [*Phys. Rev. Lett.*]{} [**13**]{} (1964) 321; G.S. Guralnik, C.R. Hagen and T.W. Kibble, [*Phys. Rev. Lett.*]{} [**13**]{} (1964) 585.
M. Grünewald, to appear in the Proc. of the 31$^{st}$ Int. Conf. on High Energy Physics, Amsterdam, July 2002 and \[arXiv:hep-ex/0210003\].
, R. Brinkmann, K. Flottmann, J. Rossbach, P. Schmuser, N. Walker and H. Weise (editors), DESY-01-011 and ECFA-2001-209.
, NLC Collaboration, in [*Proc. of the APS/DPF/DPB Summer Study on the Future of Particle Physics (Snowmass 2001)*]{} ed. N. Graf, SLAC-R-571 [*Prepared for APS / DPF / DPB Summer Study on the Future of Particle Physics (Snowmass 2001), Snowmass, Colorado, 30 Jun - 21 Jul 2001*]{}.
M. Battaglia, I. Hinchliffe, J. Jaros and J. Wells, in [*Proc. of the APS/DPF/DPB Summer Study on the Future of Particle Physics (Snowmass 2001)* ]{} ed. N. Graf, \[arXiv:hep-ex/0201018\].
, The CLIC Study Team, G. Guignard (editor), CERN-2000-008.
H.Braun, S. Doebert, I. Syratchev, M. Taborelli, I. Wilson and W. Wuensch, to appear in [*Proc. of the XXI International Linac Conference*]{}, August 19-23 2002, Gyeongiu (Kyongiu), Korea.
P. Garcia [*et al.*]{}, LC-PHSM-2001-054; T. Abe [*et al.*]{}, \[arXiv:hep-ex/0106056\].
M. Schumacher, LC Note in preparation.
T. Abe [*et al.*]{} \[American Linear Collider Working Group Collaboration\], in [*Proc. of the APS/DPF/DPB Summer Study on the Future of Particle Physics (Snowmass 2001)* ]{} ed. N. Graf, SLAC-R-570 [*Resource book for Snowmass 2001, 30 Jun - 21 Jul 2001, Snowmass, Colorado*]{}.
K. Desch and N. Meyer, LC-PHSM-2001-025.
M. Battaglia and D. Schulte, \[arXiv:hep-ex/0011085\]. M. Battaglia, in Proc. of 4th Int. Workshop on Linear Colliders (LCWS 99) and \[arXiv:hep-ph/9910271\].
J.C. Brient, Note LC-PHSM-2002-003.
C. T. Potter, J. E. Brau and M. Iwasaki, in [*Proc. of the APS/DPF/DPB Summer Study on the Future of Particle Physics (Snowmass 2001)* ]{} ed. N. Graf and [Proc. of the 6$^{th}$ Int. Linear Collider Workshop (LCWS2002)]{}, Jeju Island (Korea), 2002.
S. Dittmaier, M. Kramer, Y. Liao, M. Spira and P. M. Zerwas, Phys. Lett. B [**441**]{} (1998) 383 \[arXiv:hep-ph/9808433\]. M. Battaglia and A. De Roeck, in [*Proc. of the APS/DPF/DPB Summer Study on the Future of Particle Physics (Snowmass 2001)* ]{} ed. N. Graf, \[arXiv:hep-ph/0111307\].
A. Djouadi, W. Kilian, M. Muhlleitner and P. M. Zerwas, Eur. Phys. J. C [**10**]{} (1999) 27 \[arXiv:hep-ph/9903229\]. C. Castanier, P. Gay, P. Lutz and J. Orloff, \[arXiv:hep-ex/0101028\].
M. Battaglia, E. Boos and W. M. Yao, in [*Proc. of the APS/DPF/DPB Summer Study on the Future of Particle Physics (Snowmass 2001)* ]{} ed. N. Graf, \[arXiv:hep-ph/0111276\].
K. Desch and M. Battaglia, LC-PHSM-2001-053.
K. Desch and N. Meyer, LC Note in preparation.
M. Battaglia A. Ferrari and A. Kiiskinen, in [*Proc. of the APS/DPF/DPB Summer Study on the Future of Particle Physics (Snowmass 2001)* ]{} ed. N. Graf, \[arXiv:hep-ph/0111307\].
A. Andreazza and C. Troncon, DESY-123-E, 417.
M. Müehlleitner, in [*Proc. of the 5$^{th}$ Int. Linear Collider Workshop (LCWS 2000)*]{} A. Para and H.E. Fisk (editors), Melville, AIP, (2001).
M. M. Muhlleitner, M. Kramer, M. Spira and P. M. Zerwas, Phys. Lett. B [**508**]{} (2001) 311 \[arXiv:hep-ph/0101083\].
E. Christova, H. Eberl, W. Majerotto and S. Kraml, Nucl. Phys. B [**639**]{} (2002) 263 \[arXiv:hep-ph/0205227\]. D.J. Miller, in these proceedings.
U. Ellwanger, J. F. Gunion and C. Hugonie, \[arXiv:hep-ph/0111179\]. L. Randall and R. Sundrum, Phys. Rev. Lett. [**83**]{} (1999) 4690 \[arXiv:hep-th/9906064\] and Phys. Rev. Lett. [**83**]{} (1999) 3370 \[arXiv:hep-ph/9905221\].
J. L. Hewett and T. G. Rizzo, \[arXiv:hep-ph/0202155\]. D. Dominici, B. Grzadkowski, J. F. Gunion and M. Toharia, \[arXiv:hep-ph/0206192\].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The dual concepts of coverings and packings are well studied in graph theory. Coverings of graphs with balls of radius one and packings of vertices with pairwise distances at least two are the well-known concepts of domination and independence, respectively. In 2001, Erwin introduced *broadcast domination* in graphs, a covering problem using balls of various radii where the cost of a ball is its radius. The minimum cost of a dominating broadcast in a graph $G$ is denoted by ${\ensuremath{\gamma_b}}(G)$. The dual (in the sense of linear programming) of broadcast domination is *multipacking*: a multipacking is a set $P \subseteq V(G)$ such that for any vertex $v$ and any positive integer $r$, the ball of radius $r$ around $v$ contains at most $r$ vertices of $P$. The maximum size of a multipacking in a graph $G$ is denoted by ${\ensuremath{\mathrm{mp}}}(G)$. Naturally, ${\ensuremath{\mathrm{mp}}}(G) \leq {\ensuremath{\gamma_b}}(G)$. Hartnell and Mynhardt proved that (whenever ${\ensuremath{\mathrm{mp}}}(G)\geq 2$). In this paper, we show that . Moreover, we conjecture that this can be improved to ${\ensuremath{\gamma_b}}(G) \leq 2{\ensuremath{\mathrm{mp}}}(G)$ (which would be sharp).'
author:
- 'L. Beaudou[^1]'
- 'R. C. Brewster[^2]'
- 'F. Foucaud'
title: |
Broadcast domination and multipacking:\
bounds and the integrality gap
---
Introduction {#introduction .unnumbered}
============
The dual concepts of coverings and packings are well studied in graph theory. Coverings of graphs with balls of radius one and packings of vertices with pairwise distances at least two are the well-known concepts of domination and independence respectively. Typically we are interested in minimum (cost) coverings and maximum packings. Natural questions to ask are for what graph do these dual problems have equal (integer) values, and in the case they are not equal, can we bound the difference between the two values? The second question is the focus of this paper.
The particular covering problem we study is broadcast domination. Let $G=(V,E)$ be a graph. Define the *ball of radius $r$ around $v$* by $N_r(v) = \{ u : d(u,v) \leq r \}$. A *dominating broadcast* of $G$ is a collection of balls $N_{r_1}(v_1), N_{r_2}(v_2), \dots, N_{r_t}(v_t)$ (each $r_i > 0$) such that $\bigcup_{i=1}^t N_{r_i}(v_i) = V$. Alternatively, a dominating broadcast is a function $f: V \to \mathbb{N}$ such that for any vertex $u \in V$, there is a vertex $v \in V$ with $f(v)$ positive and $\mathrm{dist}(u,v) \leq f(v)$. (The ball around $v$ with radius $f(v)$ belongs to the covering.) The *cost* of a dominating broadcast $f$ is $\sum_{v \in V} f(v)$ and the minimum cost of a dominating broadcast in $G$, its *broadcast number*, is denoted by ${\ensuremath{\gamma_b}}(G)$.[^3]
When broadcast domination is formulated as an integer linear program, its dual problem is *multipacking* [@Brewster2013; @Teshima2012]. A multipacking in a graph $G$ is a subset $P$ of its vertices such that for any positive integer $r$ and any vertex $v$ in $V$, the ball of radius $r$ centered at $v$ contains at most $r$ vertices of $P$. The maximum size of a multipacking of $G$, its *multipacking number*, is denoted by ${\ensuremath{\mathrm{mp}}}(G)$. Broadcast domination was introduced by Erwin [@Erwin2001; @Erwin2004] in his doctoral thesis in 2001. Multipacking was then defined in Teshima’s Master’s Thesis [@Teshima2012] in 2012, see also [@Brewster2013] (and [@Brewster2017; @hartnell_2014; @Yang2015] for subsequent studies). As we have already mentioned, this work fits into the general study of coverings and packings, which has a rich history in Graph Theory: Cornuéjols wrote a monograph on the topic [@Cornuejols2001].
In early work, Meir and Moon [@MeirMoon1975] studied various coverings and packings in trees, providing several inequalities relating the size of a minimum covering and a maximum packing. Giving such inequalities connecting the parameters ${\ensuremath{\gamma_b}}$ and ${\ensuremath{\mathrm{mp}}}$ is the focus of our work. Since broadcast domination and multipacking are dual problems, we know that for any graph $G$, $${\ensuremath{\mathrm{mp}}}(G) \leq {\ensuremath{\gamma_b}}(G).$$
This bound is tight, in particular for strongly chordal graphs, see [@Farber84; @Lubiw87; @Teshima2012]. (In a recent companion work we prove equality for grids [@Beaudou2018].) A natural question comes to mind. How far apart can these two parameters be? Hartnell and Mynhardt [@hartnell_2014] gave a family of graphs $(G_k)_{k \in
\mathbb{N}}$ for which the difference between both parameters is $k$. In other words, the difference can be arbitrarily large. Nonetheless, they proved that for any graph $G$ with ${\ensuremath{\mathrm{mp}}}(G)\geq 2$, $${\ensuremath{\gamma_b}}(G) \leq 3 {\ensuremath{\mathrm{mp}}}(G) - 2$$ and asked [@hartnell_2014 Section 5] whether the factor $3$ can be improved. Answering their question in the affirmative, our main result is the following.
\[thm:bounding\] Let $G$ be a graph. Then, $${\ensuremath{\gamma_b}}(G) \leq 2 {\ensuremath{\mathrm{mp}}}(G) + 3.$$
Moreover, we conjecture that the additive constant in the bound of Theorem \[thm:bounding\] can be removed.
\[conj:fac2\] For any graph $G$, ${\ensuremath{\gamma_b}}(G) \leq 2 {\ensuremath{\mathrm{mp}}}(G)$.
In Section \[sec:bound\], we prove Theorem \[thm:bounding\]. In Section \[sec:discussion\], we show that Conjecture \[conj:fac2\] holds for all graphs with multipacking number at most $4$. We conclude the paper with some discussions in Section \[sec:remarks\].
Proof of Theorem \[thm:bounding\] {#sec:bound}
=================================
We want to bound the broadcast number of a graph by a function of its multipacking number. We first state a key counting result which is used throughout the remainder of this paper.
For any two relative integers $a$ and $b$ such that $a \leq b$, $\llbracket a, b\rrbracket$ denotes the set $\mathbb{Z} \cap [a,b]$.
\[lem:path\] Let $G$ be a graph, $k$ be a positive integer and $(u_0,\ldots,u_{3k})$ be an isometric path in $G$. Let be the set of every third vertex on this path. Then, for any positive integer $r$ and any ball $B$ of radius $r$ in $G$, $$|B \cap P| \leq \left\lceil \frac{2r+1}{3} \right\rceil.$$
Let $B$ be a ball of radius $r$ in $G$, then any two vertices in $B$ are at distance at most $2r$. Since the path $(u_0,\ldots,u_{3k})$ is isometric the intersection of the path and $B$ is included in a subpath of length $2r$. This subpath contains at most $2r+1$ vertices and only one third of those vertices can be in $P$.
Any positive integer $r$ is greater than or equal to $\left\lceil
\frac{2r+1}{3} \right\rceil$. Thus, Lemma \[lem:path\] ensures that $P$ is a valid multipacking of size $k+1$. We have the following (see also [@dun_al_2006]):
For any graph $G$, $${\ensuremath{\mathrm{mp}}}(G) \geq \left\lceil\frac{{\text{diam}}(G)+1}{3}\right\rceil.$$
Building on this idea, we have the following result.
\[thm:main\] Given a graph $G$ and two positive integers $k$ and $k'$ such that , if there are four vertices $x,y,u$ and $v$ in $G$ such that $$d_G(x,u) = d_G(x,v) = 3k \text{, } d_G(u,v) = 6k \text{ and }d_G(x,y) = 3k
+ 3k',$$ then $${\ensuremath{\mathrm{mp}}}(G) \geq 2k + k'.$$
Let $(u_{-3k},\ldots,u_0,\ldots,u_{3k})$ be the vertices of an isometric path from $u$ to $v$ going through $x$. Note that $u =
u_{-3k}$, $x = u_0$ and $v = u_{3k}$. We shall select every third vertex of this isometric path and let $P_1$ be the set $\{u_{3i}
| i \in \llbracket -k, k \rrbracket\}$.
We thus have already selected $2k+1$ vertices. In order to complete our goal, we need $k'-1$ additional vertices. Let $(x_0,\ldots,x_{3k + 3k'})$ be the vertices of an isometric path from $x$ to $y$. Note that $x =
x_0$ and $y = x_{3k+3k'}$. We shall select every third vertex on this isometric path starting at $x_{3k+6}$. Formally, we let $P_2$ be the set $\{ x_{3k+3(i+2)} | i \in \llbracket 0,k'-2
\rrbracket\}$. Finally, we let $P$ be the union of $P_1$ and $P_2$. An illustration of this is displayed in Figure \[fig:firstscheme\].
\(u) at (-3,0) ; (v) at (3,0) ; (x) at (0,0) ; (y) at (0,-5) ; (x3k6) at (0,-3.4) ; (x3k) at (0,-2.6) ; (x3k3) at (0,-3) ; (um3) at (-.4,0) ; (u3) at (.4,0) ; (x3) at (0,-.4) ; (u) – (v); (x) – (x3) – (x3k) – (x3k3) – (y); at ($(x) + (0,.2)$) [$x = x_0 = u_0$]{}; at ($(u) + (0,.2)$) [$u = u_{-3k}$]{}; at ($(v) + (0,.2)$) [$v = u_{3k}$]{}; at ($(y) + (0,-.2)$) [$y = x_{3k+3k'}$]{}; at ($(x3) + (0,0)$) [$x_3$]{}; at (x3k) [$x_{3k}$]{}; at ($(x3k6)+(.1,0)$) [$x_{3k+6}$]{}; at (x3k3) [$x_{3k+3}$]{}; at (-.1,-4.2) [$P_2$]{}; at (-1.5,-0.2) [$P_1$]{}; (-3.1,.1) rectangle (3.1,-.1); (-.1,-3.3) rectangle (.1,-5.1);
Since every vertex of $P_2$ is at distance at least $3k + 6$ from $x$, while every vertex of $P_1$ is at distance at most $3k$ from $x$, we infer that $P_1$ and $P_2$ are disjoint. Thus $|P| =
2k+k'$. We shall now prove that $P$ is a valid multipacking.
Let $r$ be an integer between 1 and $|P| - 1$, and let $B$ be a ball of radius $r$ in $G$ (we do not care about the center of the ball). If this ball $B$ intersects only $P_1$ or only $P_2$, then we know by Lemma \[lem:path\] that it cannot contain more than $r$ vertices of $P$. We may then consider that the ball $B$ intersects both $P_1$ and $P_2$. Let $l$ denote the greatest integer $i$ such that $x_{3k+3(i+2)}$ is in $B$ and in $P_2$. Let us name this vertex $z$. From this, we may say that $$\label{eq:P2}
|B \cap P_2| \leq l + 1$$
Before ending this preamble, we state an easy inequality. For every integer $n$, $$\label{eq:mod3}
\left\lceil\frac{n}{3}\right\rceil \leq \frac{n}{3} + \frac{2}{3}$$
We now split the remainder of the proof into two cases.
#### Case 1: $3(l+2) \leq r$.
In this case, we just use Lemma \[lem:path\] for $P_1$. We have $$|B \cap P_1 | \leq \left\lceil \frac{2r + 1}{3} \right\rceil,$$ and by Inequality , this quantity is bounded above by $\frac{2r+1}{3} + \frac{2}{3}$. We obtain with Inequality ,
&&|B P| & l+1 + + &&\
&& & l+2 + &&\
&& & + &&\
&& & r.&&
Therefore, the ball $B$ contains at most $r$ vertices of $P$, as required.
#### Case 2: $3(l+2) > r$.
Here we need some more insight. Recall that $l + 2 $ cannot exceed $k'$ and that $k' \leq k$. Thus $$\begin{aligned}
r & < 3(l+2) \\
& < 2k' + l +2\\
& < 2k + l + 2,
\end{aligned}$$ and since $r$ is an integer, we get $$\label{eq:nice}
r \leq 2k + l + 1.$$
We also note that any vertex $u_i$ for $|i| \leq 3k + 3(l+2) -
(2r+1)$ is at distance at least $2r+1$ from $z$. By the triangle inequality $d(z,u_i) \geq d(z,x)-d(u_i,x)$, where $d(z,x)=3k + 3(l+2)$, and $d(u_i,x) = |i|$. Since the ball $B$ has radius $r$, no such vertex can be in $B$. Since we assumed that $B$ intersects $P_1$, not all the vertices of the $uv$-path are excluded from $B$. This means that $$\label{eq:nonzero}
3k > 3k + 3(l+2) - (2r+1).$$ We partition the vertices of $P_1$ into three sets: $U_L, U_M, U_R$. The vertex $u_i$ belongs to: $U_L$ if $i < -3k - 3(l+2) + 2(r+1)$; $U_M$ if $|i| \leq 3k + 3(l+2) - (2r+1)$; and $U_R$ if $i > 3k + 3(l+2) - (2r+1)$. See Figure \[fig:case21\]. The distance from $u = u_{-3k}$ to the first vertex (smallest positive index) in $U_R$ is then $6k + 3(l+2) - (2r+1) + 1$. We compare this distance with $2r+1$.
##### Case 2.1: $6k + 3(l+2) - (2r+1) + 1 \geq 2r+1$.
We match $U_L$ with $U_R$ so that each pair is at distance at least $2r+1$ (match $u_{-3k}$ with the first vertex in $U_R$ and so on, as pictured in Figure \[fig:case21\]). Therefore the ball $B$ contains at most one vertex of each matched pair. In other words, $B$ contains at most $\lceil |U_R|/3 \rceil$ vertices from $U_L \cup U_R$, and so $$|B \cap P_1| \leq \left\lceil \frac{3k - (3k + 3(l+2) -
2r) + 1}{3} \right\rceil.$$ By using Inequality again, $$\begin{aligned}
|B \cap P| & \leq l+1 + \left\lceil \frac{2r+1}{3} \right\rceil - (l+2)\\
& \leq r.
\end{aligned}$$ Therefore, the ball $B$ contains at most $r$ vertices of $P$, as required.
##### Case 2.2: $6k + 3(l+2) - (2r+1) + 1 < 2r+1$.
We partition each of $U_L$ and $U_R$ as shown in Figure \[fig:case22\]. The vertices that are distance at least $2r+1$ from a vertex in $U_L \cup U_R$ are the sets $U'_L$ and $U'_R$, and those that are close to all other vertices are $U''_L$ and $U''_R$. We can match pairs of vertices $U'_L \cup U'_R$. This allows us to say that the extremities of $P_1$ will contribute at most $\left\lceil \frac{6k - (2r+1) + 1}{3}
\right\rceil$ which equals $2k + \lceil\frac{-2r}{3}\rceil$. Using again Inequality , this is bounded above by $2k -
\frac{2r}{3} + \frac{2}{3}$.
For any integer $i$ between $6k + 3(l+2) - (2r+1) + 1$ and $2r$, vertices $u_{-i}$ and $u_{i}$ belong to $U''_L$ and $U''_R$ respectively. Such vertices may be in $B$. Since $P_1$ contains every third vertex on these two subpaths, this amounts to at most $$2 \left\lceil\frac{2r - 6k - 3(l+2) + (2r+1)}{3}\right\rceil$$ such vertices. This quantity is equal to $$2\left\lceil \frac{4r+1}{3} \right\rceil -4k - 2(l+2),$$ which in turn, using Inequality is bounded above by $$\frac{8r}{3} + 2 -4k -2(l+2).$$
By putting everything together, we derive that
&& |B P| & (l+1) + (2k - + ) + ( +2 -4k - 2(l+2)) &&\
&&& 2r - 2k - l - .&&
But since $|B \cap P|$ is an integer, we may rewrite this last inequality as
&& |B P| & r + (r - 2k - l - 1) &&\
&&& r. &&
Thus, $|B \cap P|$ cannot exceed $r$ and the ball $B$ contains at most $r$ vertices of $P$, as required. This concludes the proof of Theorem \[thm:main\].
Theorem \[thm:main\] allows us to give a lower bound on the size of a maximum multipacking in a graph in terms of its diameter and radius.
\[coro:diam-rad\] For any graph $G$ of diameter $d$ and radius r, $${\ensuremath{\mathrm{mp}}}(G) \geq \frac{d}{6} + \frac{r}{3} - \frac{3}{2}.$$
We just pick the integer $k$ such that $d$ can be expressed as $6k + \alpha$ where $\alpha$ is in $\llbracket 0,5 \rrbracket$ and the integer $k'$ such that $r$ can be expressed as $3k +
3k'+\beta$ where $\beta$ is in $\llbracket 0,2\rrbracket$.
We must have two vertices at distance $6k$ in $G$. On a shortest path of length $6k$, the middle vertex has some vertex at distance $3k+3k'$. We can then apply Theorem \[thm:main\]. $$\begin{aligned}
{\ensuremath{\mathrm{mp}}}(G) &\geq 2k + k'\\
& \geq \frac{1}{3}(d - \alpha) + \frac{1}{3} \left(r - \beta - \frac{1}{2}(d - \alpha)\right)\\
& \geq \frac{d}{6} + \frac{r}{3} - \frac{9}{6}.\qedhere
\end{aligned}$$
We can now finalize the proof of our main theorem.
Since the diameter of a graph is always greater than or equal to its radius, we conclude from Corollary \[coro:diam-rad\] that $$\frac{{\text{rad}}(G)-3}{2} \leq {\ensuremath{\mathrm{mp}}}(G) \leq {\ensuremath{\gamma_b}}(G) \leq {\text{rad}}(G).$$ Hence, for any graph $G$, $${\ensuremath{\gamma_b}}(G) \leq 2 {\ensuremath{\mathrm{mp}}}(G) + 3,$$ proving Theorem \[thm:bounding\].
Note that in our proof, we chose the length of the long path to be a multiple of $6$ for the reading to be smooth. We think that the same ideas implemented with more care would work for multiples of $3$. This might slightly improve the additive constant in our bound, but we believe that it would not be enough to prove Conjecture \[conj:fac2\] (while adding too much complexity to the proof).
Proving Conjecture \[conj:fac2\] when ${\ensuremath{\mathrm{mp}}}(G)\leq 4$ {#sec:discussion}
===========================================================================
The following collection of results shows that Conjecture \[conj:fac2\] holds for graphs whose multipacking number is at most $4$.
\[lemma-distances\] Let $G$ be a graph and $P$ a subset of vertices of $G$. If, for every subset $U$ of at least two vertices of $P$, there exist two vertices of $U$ that are at distance at least $2|U|-1$, then $P$ is a multipacking of $G$.
We prove the contrapositive. Let $G$ be a graph and $P$ a subset of its vertices which is not a multipacking. Then there is a ball $B$ of radius $r$ which contains $r+1$ vertices of $P$.
Let $U$ be the set $B \cap P$, then $U$ has size at least $r+1$. Moreover, any two vertices in $U$ are at distance at most $2r$ which is stricly smaller than $2|U|-1$.
\[prop:mp=3\] Let $G$ be a graph. If ${\ensuremath{\mathrm{mp}}}(G)=3$, then ${\ensuremath{\gamma_b}}(G)\leq 6$.
We prove the contrapositive again. Let $G$ be a graph with broadcast number at least 7. Then, the eccentricity of any vertex is at least 7 (otherwise we could cover the whole graph by broadcasting with power 6 from a single vertex).
Let $x$ be any vertex of $G$. There must be a vertex $y$ at distance 7 from $x$. Let $u$ be any vertex at distance 3 from $x$ and on a shortest path from $x$ to $y$. Then $u$ is at distance 4 from $y$. But $u$ has also eccentricity at least 7. So there is a vertex $v$ at distance 7 from $u$. By the triangle inequality, $v$ is at distance at least 4 from $x$ and at least 3 from $y$. Therefore the set $\{u,v,x,y\}$ satisfies the condition of Lemma \[lemma-distances\] and the multipacking number of $G$ is at least 4 (and so it is not equal to 3).
The following proposition improves Theorem \[thm:bounding\] for graphs $G$ with ${\ensuremath{\mathrm{mp}}}(G) \leq 6$ and shows that Conjecture \[conj:fac2\] holds when ${\ensuremath{\mathrm{mp}}}(G) = 4$.
\[prop:mp=4\] Let $G$ be a graph. If ${\ensuremath{\mathrm{mp}}}(G)\geq 4$, then ${\ensuremath{\gamma_b}}(G)\leq 3{\ensuremath{\mathrm{mp}}}(G)-4$.
For a contradiction, let $G$ be a counterexample, that is a graph with multipacking number $p$ at least 4 while ${\ensuremath{\gamma_b}}(G)\geq
3p-3$. Then, the eccentricity of any vertex of $G$ is at least $3p-3$ (otherwise we could broadcast at distance $3p-4$ from a single vertex). Let $x$ be a vertex of $G$ and let $V_i$ denote the set of vertices at distance exactly $i$ of $x$. By our previous remark, $V_{3p-3}$ is non-empty. Let $y$ be a vertex in $V_{3p-3}$ and consider a shortest path $P_{xy}$ from $x$ to $y$ in $G$. Let $v_0=x$, and for $1\leq i\leq p-1$, let $v_i$ be the vertex on $P_{xy}$ belonging to $V_{3i}$ (thus $v_{p-1}=y$).
Now, since ${\ensuremath{\gamma_b}}(G)\geq 3p-3$, there must be a vertex $u$ at distance at least $3p-3$ of $v_{p-2}$ (otherwise we could broadcast from that single vertex). Note that the triangle inequality ensures that the distance between $u$ and $v_i$ is at least $3+3i$ for $i$ between $0$ and $p-2$. The distance from $u$ to $v_{p-1}$ is at least $3p-6$ which is at least 6 since $p$ is at least 4. Consider the set $P=\{u,v_0,\ldots, v_{p-1}\}$. We claim that $P$ is a multipacking of $G$ of size $p+1$, which is a contradiction.
Let $B$ be a ball of radius $r$. Since $P_{xy}$ is an isometric path, Lemma \[lem:path\] ensures us that $B$ contains at most $$\left\lceil \frac{2r+1}{3} \right\rceil$$ vertices from $P \cap P_{xy}$ which is smaller than $r$. When $B$ does not include $u$, the ball is satisfied. For balls that contain vertex $u$, the maximum size of $P \cap B$ is $$\left\lceil \frac{2r+1}{3} \right\rceil + 1.$$ Whenever $r$ is 4 or more, this quantity does not exceed $r$. So every ball with radius $4$ or more is satisfied. We still need to check balls of radius 1,2, and 3 which contain $u$.
- Balls of radius 1 are easy to check since every vertex of $P_{xy}$ is at distance at least 3 from $u$.
- For balls of radius 2, it is enough to check that there is only one vertex at distance 4 or less from $u$ in $P \cap P_{xy}$.
- For balls of radius 3, there is only one way to select $u$ and three vertices in $P \cap P_{xy}$ within distance 6 from $u$. We should take $v_0, v_1$ and $v_{p-1}$. But since $v_0$ and $v_{p-1}$ are at distance $3p-3$ from each other, they cannot appear simultaneously in a ball of radius 3 (since $p$ is at least 4, $3p-3$ is at least 9).
Therefore $P$ is a multipacking of size $p+1$, which is a contradiction.
Let $G$ be a graph. If ${\ensuremath{\mathrm{mp}}}(G)\leq 4$, then ${\ensuremath{\gamma_b}}(G)\leq 2{\ensuremath{\mathrm{mp}}}(G)$.
When ${\ensuremath{\mathrm{mp}}}(G)\leq 2$, this is shown in [@hartnell_2014]. The case ${\ensuremath{\mathrm{mp}}}(G)=3$ is implied by Proposition \[prop:mp=3\], and the case ${\ensuremath{\mathrm{mp}}}(G)=4$ follows from Proposition \[prop:mp=4\].
Concluding remarks {#sec:remarks}
==================
We conclude the paper with some remarks.
The optimality of Conjecture \[conj:fac2\]
------------------------------------------
We know a few examples of connected graphs $G$ which achieve the conjectured bound, that is, ${\ensuremath{\gamma_b}}(G)=2{\ensuremath{\mathrm{mp}}}(G)$. For example, one can easily check that $C_4$ and $C_5$ have multipacking number $1$ and broadcast number $2$. In Figure \[fig:twoFour\], we depict three examples having multipacking number $2$ and broadcast number $4$. By making disjoint unions of these graphs, we can build further extremal graphs with arbitrary multipacking number. However, if we only consider connected graphs, we do not even know an example with multipacking number $3$ and broadcast number $6$. Hartnell and Mynhardt [@hartnell_2014] constructed an infinite family of connected graphs $G$ with ${\ensuremath{\gamma_b}}(G)=\tfrac{4}{3}{\ensuremath{\mathrm{mp}}}(G)$, but we do not know any construction with a higher ratio. Are there arbitrarily large connected graphs that reach the bound of Conjecture \[conj:fac2\]?
An approximation algorithm
--------------------------
The computational complexity of broadcast domination has been extensively studied, see for example [@Dabney2009; @HeggernesLokshtanov2006] and references of [@Brewster2013; @Teshima2012; @Yang2015]. It is particularly interesting to note that, unlike most other natural covering problems, broadcast domination is solvable in polynomial (sextic) time [@HeggernesLokshtanov2006]. It is not known whether this is also the case for multipacking, but a cubic-time algorithm exists for strongly chordal graphs [@Brewster2017; @Yang2015], as well as a linear-time algorithm for trees [@Brewster2013; @Brewster2017; @Yang2015]. We note that our proof of Theorem \[thm:bounding\], being constructive, implies the existence of a $(2+o(1))$-factor approximation algorithm for the multipacking problem.
There is a polynomial-time algorithm that, given a graph $G$, constructs a multipacking of $G$ of size at least $\frac{{\ensuremath{\mathrm{mp}}}(G)-3}{2}$.
To construct the multipacking, one first needs to compute the radius $r$ and diameter $d$ of the graph $G$. Then, as described in the proof of Corollary \[coro:diam-rad\], we compute $\alpha$ and $k$, and find the four vertices $x$, $y$, $u$, $v$ and the two isometric paths $P_1$ and $P_2$ described in Theorem \[thm:main\]. Finally, we proceed as in the proof of Theorem \[thm:main\], that is, we essentially select every third vertex of these two paths to obtain the multipacking $P$. All distances and paths can be computed in polynomial time using classic methods. By Corollary \[coro:diam-rad\], $P$ has size at least $\frac{{\text{rad}}(G)-3}{2}$. Since ${\ensuremath{\mathrm{mp}}}(G)\leq {\text{rad}}(G)$, the approximation factor follows.
[00]{} L. Beaudou and R. C. Brewster, On the multipacking number of grid graphs, manuscript. arXiv e-prints:1803.09639.
R. C. Brewster, C. M. Mynhardt and L. Teshima, New bounds for the broadcast domination number of a graph, *Central European Journal of Mathematics*, [**11**]{} (2013), 1334–1343.
R. C. Brewster, G. MacGillivray and F. Yang, Broadcast domination and multipacking in strongly chordal graphs, submitted.
G. Cornuéjols. [*Combinatorial Optimization: packing and covering.*]{} CBMS-NSF regional conference series in applied mathematics, vol. 74. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2001.
J. Dabney, B. C. Dean, S. T. Hedetniemi, A linear-time algorithm for broadcast domination in a tree, *Networks*, **53** (2009), 160–169.
J. E. Dunbar, D. J. Erwin, T. W. Haynes, S. M. Hedetniemi and S. T. Hedetniemi, Broadcasts in graphs, *Discrete Applied Mathematics*, **154** (2006), 59–75.
D. J. Erwin, *Cost domination in graphs*, PhD Thesis, Department of Mathematics, Western Michigan University, 2001.
D. J. Erwin, Dominating broadcasts in graphs, *Bulletin of the ICA*, [**42**]{} (2004), 89–105.
M. Farber, Domination, Independent Domination, and Duality in Strongly Chordal Graphs, *Discrete Applied Mathematics*, [**7**]{} (1984), 115–130.
B. L. Hartnell and C. M. Mynhardt, On the difference between broadcast and multipacking numbers of graphs, *Utilitas Mathematica*, **94** (2014), 19–29.
P. Heggernes and D. Lokshtanov, Optimal broadcast domination in polynomial time, *Discrete Mathematics*, [**306**]{} (2006), 3267–3280.
A. Lubiw, Doubly Lexical Orderings of Matrices, *SIAM Journal on Computing*, [**16**]{} (1987), 854–879.
A. Meir and John W. Moon, Relations between packing and covering numbers of a tree, *Pacific Journal of Mathematics*, [**61**]{} (1975), 225–233.
L. Teshima, [*Broadcasts and multipackings in graphs*]{}, Master’s Thesis, Department of Mathematics and Statistics, University of Victoria, 2012.
F. Yang, [*New results on broadcast domination and multipacking*]{}, Master’s Thesis, Department of Mathematics and Statistics, University of Victoria, 2015.
[^1]: LIMOS, Université Clermont Auvergne, Aubière (France). E-mails: laurent.beaudou@uca.fr, florent.foucaud@gmail.com
[^2]: Department of Mathematics and Statistics, Thompson Rivers University, Kamloops (Canada). E-mail: rbrewster@tru.ca
[^3]: One may consider the cost to be any function of the powers (for example the sum of the squares), see e.g. [@HeggernesLokshtanov2006]. We shall stick to the classical convention of linear cost.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
Z. C. Tu and Z. C. Ou-Yang\
*Institute of Theoretical Physics, The Chinese Academy of Sciences*\
*P.O.Box 2735 Beijing 100080, China*
title: 'Structures, Symmetries, Mechanics and Motors of carbon nanotubes'
---
ABSTRACT {#abstract .unnumbered}
========
The structures and symmetries of single-walled carbon nanotubes (SWNTs) are introduced in detail. The physical properties of SWNTs induced by their symmetries can be described by tensors in mathematical point of view. It is found that there are 2, 4, and 5 different parameters in the second, third, and fourth rank tensors representing electronic conductivity (or static polarizability), the second order nonlinear polarizability, and elastic constants of SWNTs, respectively. The values of elastic constants obtained from tight-binding method imply that SWNTs might be very weakly anisotropic in mechanical properties. The further study on the mechanical properties shows that the elastic shell theory in the macroscopic scale can be applied to carbon nanotubes (CNTs) in the mesoscopic scale, as a result, SWNTs can be regarded as an isotropic material with Poisson ratio, effective thickness, and Young’s modulus being $\nu=0.34$, $h=0.75$Å, $Y=4.70$TPa, respectively, while the Young’s moduli of multi-walled carbon nanotubes (MWNTs) are apparent functions of the number of layers, $N$, varying from 4.70TPa to 1.04TPa for $N=1$ to $\infty$. Based on the chirality of CNTs, it is predicted that a new kind of molecular motor driven by alternating voltage can be constructed from double walled carbon nanotubes (DWNTs).
INTRODUCTION {#introduction .unnumbered}
============
Carbon is the core element to construction of organic matters and has always attracted much attention up to now. Many decades ago, people only knew two kinds of crystals consisting of carbon: graphite with layer structure and diamond with tetrahedral shape. The situation was changed in 1985 when Kroto *et al.* synthesized bucky ball—a football-like molecule consisting of 60 carbon atoms [@Kroto] which marked the beginning of carbon times. After that, Iijima synthesized MWNTs in 1991 [@Iijima1] and SWNTs in 1993 [@Iijima2]. Simply speaking, a SWNT can be regarded as a graphitic sheet with hexagonal lattices that was wrapped up into a seamless cylinder with diameter in nanometer scale and length from tens of nanometers even to several micrometers if we ignore its two end caps, while a MWNT consists of a series of coaxial SWNTs with layer distance about 3.4Å[@Saito].
SWNTs have many unique properties. Viewed from the chirality, some of them are chiral but others are achiral. Viewed from the electronic properties, some of them are metallic but others are semiconductive. Moreover, their electronic properties depend sensitively on their chirality [@Mintmire; @Saito2; @Hamada; @Tans]. The conductivity of metallic SWNT does not satisfy Ohm’s law because the electron transport in it is ballistic [@Kong; @Liang; @White]. Otherwise, theoretical [@bi; @x; @e; @jp] and experimental [@a] studies have suggested that SWNTs also possess many novel mechanical properties, in particular high stiffness and axial strength, which are insensitive to the tube diameters and chirality. MWNTs have the similar mechanical properties to SWNTs [@mf; @mmj; @ew].
In purely theoretical point of view, we should consider the following two facts:
\(i) As quasi-one-dimensional structures with periodic boundary conditions, SWNTs might show anisotropic physical properties which may depend on the tube diameters and chirality. Generally speaking, the physical properties of crystals can be represented by tensors [@Nyejf]. For example, the electronic conductivity can be expressed by second-rank tensor and the elastic constants can be described as fourth-rank tensor. These tensors can be derived from the structures and symmetries of SWNTs. Therefore fully discussing structures and symmetries of SWNTs is one of the topic in this chapter.
\(ii) The SWNT is a single layer of carbon atoms. What is the thickness of the layer? It is a widely controversial question. Some researchers take 3.4 Å, the layer distance of bulk graphite, as the thickness of SWNT [@jp; @a]. Others define an effective thickness (about 0.7Å) by admitting the validity of elastic shell theory in nanometer scale [@bi; @x]. The present authors have proved that the elastic shell theory can indeed be applied to SWNTs [@tzcnt1] which supports the latter standpoint. Because the two standpoints have little effect on the mechanical results in many cases, the controversy is still being discussed [@Tabar; @Pantano; @Chandra; @Ogata; @Pantano2; @Natsuki; @SunCQ]. Therefore it is necessary to point out when the two standpoints will give different results.
In the applied point of view, the present authors have predicted a molecular motor constructed from a DWNT driven by temperature variation [@tzcnt2] which induces to molecular dynamics simulations by Dendzik *et al.* [@Dendzik]. Using molecular dynamics simulations, Kang *et al.* recently predicted a carbon-nanotube motor driven by fluidic gas [@KangJW2]. In this chapter, a conceptual motor of DWNT driven by alternating voltage will be proposed based on previous work [@tzcnt2].
STRUCTURES, SYMMETRIES AND THEIR INDUCING PHYSICAL PROPERTIES OF SWNTS {#structures-symmetries-and-their-inducing-physical-properties-of-swnts .unnumbered}
======================================================================
To describe the SWNT, some characteristic vectors require introducing. As shown in Fig.\[fig1\], the chiral vector ${\bf
C}_{h}$, which defines the relative location of two sites, is specified by a pair of integers $(n, m)$ which is called the index of the SWNT and relates ${\bf C}_{h}$ to two unit vectors ${\bf
a}_{1}$ and ${\bf a}_{2}$ of graphite (${\bf C}_{h}=n{\bf
a}_{1}+m{\bf a}_{2}$). The chiral angle $\theta_0$ defines the angle between $\mathbf{a}_1$ and $\mathbf{C}_h$. For $(n, m)$ nanotube, $\theta_0=\arccos\left[\frac{2n+m}{2\sqrt{n^2+m^2+nm}}\right]$. The translational vector ${\bf T}$ corresponds to the first lattice point of 2D graphitic sheet through which the line normal to the chiral vector ${\bf C}_{h}$ passes. The unit cell of the SWNT is the rectangle defined by vectors ${\bf C}_{h}$ and ${\bf
T}$, while vectors ${\bf a}_{1}$ and ${\bf a}_{2}$ define the area of the unit cell of 2D graphite. The number $N$ of hexagons per unit cell of SWNT is obtained as a function of $n$ and $m$ as $N=2(n^2+m^2+nm)/d_R$ which is larger than 8 for SWNTs in practice, where $d_R$ is the greatest common divisor of ($2m+n$) and ($2n+m$). There are $2N$ carbon atoms in each unit cell of SWNT because every hexagon contains two atoms. To denote the $2N$ atoms, we use a symmetry vector ${\bf R}$ to generate coordinates of carbon atoms in the nanotube and it is defined as the site vector having the smallest component in the direction of ${\bf
C}_h$. From a geometric standpoint, vector ${\bf R}$ consists of a rotation around the nanotube axis by an angle $\Psi=2\pi/N$ combined with a translation $\tau$ in the direction of ${\bf T}$; therefore, ${\bf R}$ can be denoted by ${\bf R}=(\Psi|\tau)$. Using the symmetry vector ${\bf R}$, we can divide the $2N$ carbon atoms in the unit cell of SWNT into two classes [@tzcnt3]: one includes $N$ atoms whose site vectors satisfy $$\label{sitea}
{\bf A}_l=l{\bf R}-[l{\bf R}\cdot{\bf T}/{\bf T}^2]{\bf T} \quad
(l=0,1,2,\cdots,N-1),$$ another includes the remainder $N$ atoms whose site vectors satisfy $$\begin{aligned}
\label{siteb}&&{\bf B}_l=l{\bf R}+{\bf
B}_0-[(l{\bf R}+{\bf B}_0)\cdot{\bf T}/{\bf T}^2]{\bf T}\nonumber
\\&&\quad -[(l{\bf R}+{\bf B}_0)\cdot{\bf C}_h/{\bf C}_h^2]{\bf C}_h
\quad (l=0,1,\cdots,N-1),\end{aligned}$$ where ${\bf B}_0\equiv\left (\Psi_0|\tau_0\right)=\left (\left.\frac{2\pi
r_0\cos(\theta_0-\frac{\pi}{6})}{|\mathbf{C}_h|}\right|r_0\cos(\theta_0-\frac{\pi}{6})\right)$ represents one of the nearest neighbor atoms to [**A**]{}$_0$ and $r_0$ is the carbon-carbon bond length.
![\[fig1\]The unrolled honeycomb lattice of a SWNT. By rolling up the sheet along the chiral vector ${\bf C}_h$, that is, such that the point $A_0$ coincides with the point corresponding to vector ${\bf C}_h$, a nanotube is formed. The vectors ${\bf
a}_{1}$ and ${\bf a}_{2}$ are the real space unit vectors of the hexagonal lattice. The translational vector ${\bf T}$ is perpendicular to ${\bf C}_h$ and runs in the direction of the tube axis. The vector ${\bf R}$ is the symmetry vector. $A_0$, $B_0$ and $A_l, B_l (l=1,2,\cdots,N)$ are used to denote the sites of carbon atoms.](graphite.eps){width="7cm"}
![\[fig2\] The coordinates of SWNT.](coordnt.eps){width="5cm"}
If we introduce cylindrical coordinate system $(r, \theta, z)$ whose $z$-axis is the tube axis parallel to vector $\mathbf{T}$. Its $r\theta$-plane is perpendicular to $z$-axis and contains atom $A_0$ in the nanotube. $r$ the distance from some point to $z$-axis, and $\theta$ the angle rotating around $z$-axis from an axis which is vertical to $z$-axis and passes through atom $A_0$ in the tube to the point. In this coordinate system, we can express Eqs.(\[sitea\]) and (\[siteb\]) as [@tzcnt33]: $$\label{siteaa}
{\bf A}_l=\{\rho,l\Psi,l\tau-[l\tau/T]T\}\quad
(l=0,1,2,\cdots,N-1) ,$$ and $$\begin{aligned}
\label{sitebb} {\bf
B}_l&=&\left\{\rho,l\Psi+\Psi_0-2\pi\left[\frac{l\Psi+\Psi_0}{2\pi}\right],l\tau+\tau_0-\left[\frac{l\tau+\tau_0}{T}\right]T\right\}\nonumber\\
&&\quad (l=0,1,2,\cdots,N-1),\end{aligned}$$ where $\rho=\frac{|\mathbf{C}_h|}{2\pi}$. In Eqs.(\[sitea\])-(\[sitebb\]), the symbol $[\cdots]$ denotes the largest integer smaller than $\cdots$, e.g., $[7.3]=7$.
In the following contents of this section, we will derive the general forms of the second, third, and fourth rank tensors by fully considering the symmetries of SWNTs. As shown in Fig.\[fig2\], $oxyz$ is the initial coordinate system whose $z$-axis is the tube axis and $x$-axis passes through atom $A_0$. If we use ${\bf R}$ to act $l$ times on the initial system, we can get the coordinate system $O_l x_l y_l z_l$. Obviously, the transformation from the bases $\hat{x}, \hat{y}, \hat{z}$ of the initial system to the new bases $\hat{x}_l, \hat{y}_l, \hat{z}_l$ can be expressed as, $$\label{trans}
\left ( \begin{array}{c}
\hat{x}_{l}\\
\hat{y}_{l}\\
\hat{z}_{l}
\end{array} \right )=(a_{ij})
\left ( \begin{array}{c} \hat{x}\\ \hat{y}\\ \hat{z}\\
\end{array} \right )
,$$ where $(a_{ij})$ is the matrix with elements $a_{11}=a_{22}=\cos l\psi$, $a_{12}=\sin l\psi$, $a_{21}=-\sin
l\psi$, $a_{33}=1$, $a_{13}=a_{31}=a_{23}=a_{32}=0$.
Above all, let us consider the second-rank tensor ${\bf S}$. $s_{ij}$ and $s_{ij}^{(l)}$ $(i,j=1,2,3; l=1,2,\cdots,N)$ denote its components in the coordinate system $Oxyz$ and $O_l x_l y_l
z_l$. The transformation law of the components is $s_{ij}^{(l)}=a_{im}a_{jk}s_{mk}$ [@Nyejf], where the Einstein summation convention is used. But the symmetry of SWNTs requires $s_{ij}^{(l)}=s_{ij}$. From this condition and the transformation law we derive out $s_{11}=s_{22}$, $s_{12}=-s_{21}$, $s_{13}=s_{31}=0$, $s_{23}=s_{32}=0$. Especially, there are only 2 different nonzero parameters in symmetric 2nd-rank tensors which represent the electronic conductivity or static polarizabilities: $s_{11}=s_{22}\neq 0$, $s_{33}\neq 0$.
Next, let us deal with the third-rank tensor ${\bf D}$ whose components are denoted by $d_{ijk}$ and $d_{ijk}^{(l)}$ $(i,j,k=1,2,3)$ in the coordinate system $Oxyz$ and $O_l x_l y_l
z_l$, respectively. From the transformation law of the components $d_{ijk}^{(l)}=a_{iq}a_{ju}a_{kv}d_{quv}$ [@Nyejf] and the symmetry of SWNTs, we can obtain the non-vanishing components of ${\bf D}$: $d_{113}=d_{223}$, $d_{123}=-d_{213}$, $d_{131}=d_{232}$, $d_{132}=-d_{231}$, $d_{311}=d_{322}$, $d_{312}=-d_{321}$, $d_{333}$. If some physical property requires $d_{ijk}=d_{ikj}$ (e.g. the second order polarization effect), then there are only 4 different nonzero parameters in its components: $d_{123}=-d_{213}=d_{132}=-d_{231}\equiv d_1/2$, $d_{113}=d_{223}=d_{131}=d_{232}\equiv d_2/2$, $d_{311}=d_{322}\equiv d_3$, $d_{333}\equiv d_4$. Thus the relation $P_i^{NL}=d_{ijk}E_jE_k$ between the nonlinear polarization and field can be expressed in matrix form, $$\label{nonpolar}
\left\{ \begin{array}{l}
P_x^{NL}=d_1 E_yE_z+d_2 E_xE_z\\
P_y^{NL}=d_2 E_yE_z-d_1 E_xE_z\\
P_z^{NL}=d_3 (E_x^2+E_y^2)+d_4 E_z^2
\end{array} \right.
.$$
Similarly, we can obtain the nonzero components of the fourth-rank tensor ${\bf C}$ from the transformation law $c_{ijkm}^{(l)}=a_{iq}a_{ju}a_{kv}a_{mw}c_{quvw} (i,j,k,m=1,2,3)$ [@Nyejf] and the symmetry of SWNTs: $c_{1111}=c_{2222}$, $c_{1112}=-c_{2221}$, $c_{1121}=-c_{2212}$, $c_{1122}=c_{2211}$, $c_{1133}=c_{2233}$, $c_{1211}=-c_{2122}$, $c_{1212}=c_{2121}$, $c_{1221}=c_{2112}=c_{2222}-c_{2121}-c_{2211}$, $c_{2111}=-c_{1222}=c_{2122}+c_{2212}+c_{2221}$, $c_{1233}=-c_{2133}$, $c_{1313}=c_{2323}$, $c_{1323}=-c_{2313}$, $c_{1331}=c_{2332}$, $c_{1332}=-c_{2331}$, $c_{3113}=c_{3223}$, $c_{3123}=-c_{3213}$, $c_{3131}=c_{3232}$, $c_{3132}=-c_{3321}$, $c_{3311}=c_{3322}$, $c_{3312}=-c_{3321}$, $c_{3333}$. If we consider the elastic property, the elastic constants can be expressed by the fourth-rank tensor whose components satisfy $c_{ijkm}=c_{ijmk}=c_{jikm}=c_{kmij}$[@Nyejf]. Thus there are only 5 different non-vanishing parameters ($c_1, c_2, c_3, c_4,
c_5$) in its components and $c_{1111}=c_{2222}\equiv c_1$, $c_{1133}=c_{3311}=c_{2233}=c_{3322}\equiv c_2$, $c_{3333}=c_3$, $c_{1313}=c_{2323}=c_{1331}=c_{2332}=c_{3113}=c_{3223}=c_{3131}=c_{3232}\equiv
c_4$, $c_{1122}=c_{2211}\equiv c_5$, $c_{1212}=c_{2121}=c_{1221}=c_{2112}=(c_1-c_5)/2$. The stress-strain relation ${\bm \sigma}={\bf C}{\bm \varepsilon}$ can be expressed by the matrix notations, $$\label{c2}
\left(\begin{array}{c} \sigma_{xx} \\ \sigma_{yy} \\ \sigma_{zz}
\\ \sigma_{yz} \\ \sigma_{xz} \\ \sigma_{xy}
\end{array} \right)=\left(\begin{array}{cccccc}
c_1 & c_5 & c_2 & & & \\
c_5 & c_1 & c_2 & & & \\
c_2 & c_2 & c_3 & & & \\
& & & c_4& & \\
& & & &c_4 & \\
& & & & &\frac{(c_1-c_5)}{2}
\end{array} \right) \left( \begin{array}{c}
\varepsilon_{xx} \\ \varepsilon_{yy} \\ \varepsilon_{zz} \\
\gamma_{yz} \\ \gamma_{xz} \\ \gamma_{xy}
\end{array} \right)
,$$ where $\gamma_{xy}=2\varepsilon_{xy}$, $\gamma_{xz}=2\varepsilon_{xz}$,$\gamma_{yz}=2\varepsilon_{yz}$ [@ld]. The axial Young’s modulus $Y_z$ defined as the stress/strain ratio when the tube is axially strained and Poisson ratio $\nu_z$ defined as the ratio of the reduction in radial dimension to the axial elongation can be expressed as $Y_z=c_3-2c_2^2/(c_1+c_5)$ and $\nu_z=c_2/(c_1+c_5)$, respectively.
Obviously, the numbers of different parameters in the expressions [**S**]{}, [**D**]{}, [**C**]{} of SWNTs are more than that in isotropic materials (see also Table \[number\]) if the different parameters are independent, which implies that SWNTs might possess anisotropic physical properties. All parameters are functions of $n$, $m$, which reveals the physical properties depend on the chirality and diameters of SWNTs to some extent.
[|cccc|]{}Tensors & [**S**]{}(2nd-rank) &[**D**]{}(3rd-rank)& [**C**]{}(4th-rank)\
Isotropic materials& 1 & 0 & 2\
carbon nanotubes & 2 & 4 & 5\
As examples, we will give the forms of second and fourth-rank order tensors—the static polarizabilities and elastic constants of SWNTs, respectively.
It is well known that the relation between the polarization [**P**]{} and external electric field [**E**]{} is ${\bf P}={\bm
\alpha}\bf{E}$[@Landau2], where $\bm{\alpha}$ is the static polarizability, a second-rank tensor. From above discussions, we known $\bm{\alpha}$ can be expressed as matrix form, $$\label{polar}
\bm{\alpha}=\left(\begin{array}{ccc}
\alpha_{xx} & & \\
&\alpha_{yy} & \\
& & \alpha_{zz}
\end{array} \right)
,$$ with $\alpha_{xx}=\alpha_{yy}$. Benedict [*et al.*]{}[@Benedict] have studied the polarizabilities and their results are shown in Table \[lxt\], where we have changed their values to polarizabilities per atom. From Table \[lxt\] we find that the polarizabilities of SWNTs are sensitive to the tube indexes $(n,m)$, particularly, the value of $\alpha_{zz}$ is extremely large when $n-m$ is a multiple of three, which corresponds metallic tubes.
[|cccc|]{} $(n,m)$ & $\rho$(Å) &$\alpha_{zz}$(Å$^3$/atom) & $\alpha_{xx}$(Å$^3$/atom)\
(9,0)& 3.57 & & 1.05\
(10,0) & 3.94 & 18.62 & 1.10\
(11,0) & 4.33 & 17.04 & 1.20\
(12,0) & 4.73 & & 1.23\
(13,0) & 5.12 & 23.98 & 1.29\
(4,4) & 2.73 & & 0.92\
(5,5) & 3.41 & & 1.02\
(6,6) & 4.10 & & 1.13\
(4,2) & 2.09 & 9.87 & 0.84\
(5,2) & 2.46 & & 0.89\
Otherwise, we calculate the elastic constants of SWNTs through the tight binding method [@x] with considering the curvature and bond-length change effects. The results are listed in Table \[const\] which suggests that the elastic properties of SWNTs slightly depend on the tube indexes $(n,m)$. We also give the corresponding axial Young’s moduli and Poisson ratios of single-walled carbon nanotubes with different indexes. Moreover, we find $(c_1+c_5)-c_4\approx c_2$, $c_3\approx 2(c_1+c_5)$ and $(c_1+c_5)\approx 4c_2$, i,e., there might be only two independent parameters in the elastic constants, which implies that the mechanical anisotropy of SWNTs is so weakly that we can regard them as approximately isotropic materials (Remark: the isotropic materials have two independent elastic constants, see also Table \[number\]).
[|ccccccc|]{} (n,m) & $c_1+c_5$ & $c_2$ & $c_3$ & $c_4$ & $Y_z$ & $\nu_z$\
(6,0) & 29.04 & 7.03 & 56.97 & 22.72 & 53.56 & 0.24\
(8,0) & 29.34 & 7.05 & 57.68 & 23.43 & 54.29 & 0.24\
(10,0) & 29.50 & 7.07 & 57.92 & 23.79 & 54.53 & 0.24\
(50,0) & 29.51 & 7.08 & 58.97 & 24.36 & 55.57 & 0.24\
(6,6) & 29.99 & 7.08 & 56.97 & 24.09 & 53.63 & 0.24\
(8,8) & 29.76 & 7.08 & 57.92 & 24.22 & 54.55 & 0.24\
(10,10) & 29.66 & 7.08 & 58.33 & 24.28 & 54.96 & 0.24\
(50,50) & 29.51 & 7.08 & 59.00 & 24.38 & 55.60 & 0.24\
(6,4) & 30.21 & 7.07 & 56.15 & 23.13 & 52.83 & 0.23\
(7,3) & 30.00 & 7.07 & 56.60 & 22.53 & 53.27 & 0.24\
(8,2) & 29.64 & 7.07 & 57.44 & 22.62 & 54.07 & 0.24\
It is necessary to discuss the meanings of strains and stresses in nanometer scale. Strains are geometric quantities so that their definitions in macroscopic theory of elasticity still hold for SWNTs. But we must redefine stresses because they are not well-defined quantities for SWNTs. Given strains, we can calculate the energy variation of the SWNTs due to the strains through quantum mechanics in principle. The stresses are defined as the partial derivatives of energy variation with respect to the strains. In fact, stresses are not necessary concepts. We can directly determine the elastic constants by strains and the corresponding energy variation.
Otherwise, we do not separate $c_1$ and $c_5$ in Table \[const\]. Up to now, we do not know how to separate them. A possibility is that we need not do that when we discuss the mechanical properties of SWNTs.
MECHANICAL PROPERTIES OF CNTS {#mechanical-properties-of-cnts .unnumbered}
=============================
In this section, we will continue to discuss the mechanical properties of CNTs in detail.
We start from the concise formula proposed by Lenosky [*et al.*]{} in 1992 to describe the deformation energy of a single layer of curved graphite[@Lenosky] $$\begin{aligned}
\label{leno}
E^g&=&(\epsilon_{0}/2) \sum_{(ij)}\left(r_{ij}-r_{0}\right)
^2+\epsilon_{1} \sum_{i}(\sum_{(j)}{\bf u}_{ij})^2 \nonumber \\
&+& \epsilon_{2} \sum_{(ij)}\left(1-{\bf n}_{i} \cdot {\bf n}_{j}
\right)+\epsilon_{3} \sum_{(ij)}\left({\bf n}_{i} \cdot {\bf
u}_{ij} \right) \left({\bf n}_{j} \cdot {\bf u}_{ji} \right)
.\end{aligned}$$ The first two terms are the contributions of bond length and bond angle changes to the energy. The last two terms are the contributions of the $\pi$-electron resonance. In the first term, $r_{0}=1.42$ Å is the initial bond length of planar graphite, and $r_{ij}$ is the bond length between atoms $i$ and $j$ after the deformations. In the remaining terms, ${\bf
u}_{ij}$ is a unit vector pointing from atom $i$ to its neighbor $j$, and ${\bf n}_{i}$ is the unit vector normal to the plane determined by the three neighbors of atom $i$. The summation ${\sum_{(j)}}$ is taken over the three nearest neighbor $j$ atoms to $i$ atom, and ${\sum_{(ij)}}$ taken over all the nearest neighbor atoms. The parameters $(\epsilon_{1},\epsilon_{2},\epsilon_{3})=(0.96,1.29,0.05)$eV were determined by Lenosky [*et al.*]{} [@Lenosky] through local density approximation. The value of $\epsilon_{0}$ was not given by Lenosky [*et al.*]{}, but given by Zhou [*et al.*]{} [@zhoux] $\epsilon_{0}=57{\rm eV/\AA^2}$ from the force-constant method.
In 1997, Ou-Yang *et al.* [@oy1997] reduced Eq.(\[leno\]) into a continuum limit form without taking the bond length change into account and obtained the curvature elastic energy of a SWNT $$\label{oy1}
E^{(s)}=\int \left[\frac{1}{2} k_{c} (2H)^2+\bar{k}_{1} K \right]
dA ,$$ where the bending elastic constant $$\label{oy2}
k_{c}=(18\epsilon_{1}+24\epsilon_{2}+9\epsilon_{3})r_{0}^2/(32
\Omega)=1.17{\rm eV}$$ with $\Omega=2.62$ Å$^2$ being the occupied area per atom, and $$\label{oy3}
\bar{k}_{1}/k_{c}=-(8\epsilon_{2}+3\epsilon_{3})/(6\epsilon_{1}+8\epsilon_{2}+3\epsilon_{3})=-0.645
.$$ In Eq.(\[oy1\]), $H$, $K$, and $dA$ are mean curvature, Gaussian curvature, and area element of the SWNTs surface, respectively.
In 2002, we obtained the total free energy [@tzcnt1] of a strained SWNT with in-plane strain ${\bf
\varepsilon}_i=\left(\begin{array}{cc} \varepsilon_{x} &
\varepsilon_{xy} \\ \varepsilon_{xy} & \varepsilon_{y} \end{array}
\right)$ at the $i$-atom site, where $\varepsilon_{x}$, $\varepsilon_{y}$, and $\varepsilon_{xy}$ are the axial, circumferential, and shear strains, respectively. The total free energy contains two parts: one is the curvature energy expressed as Eq.(\[oy1\]); another is the deformation energy [@tzcnt1] $$\label{tzc20021}
E_{d}=\int \left[\frac{1}{2} k_{d} (2J)^2+\bar{k}_{2} Q \right] dA
,$$ where $2J=\varepsilon_{x}+\varepsilon_{y}$ and $Q=\varepsilon_{x} \varepsilon_{y}-\varepsilon_{xy}^2$, are respectively named “mean” and “Gaussian” strains, and $$\label{tzc20022}
k_{d}=9\left(\epsilon_{0}
r_{0}^2+\epsilon_{1}\right)/(16\Omega)=24.88{\rm eV/\AA}^2
,$$ $$\label{tzc20023}
\bar{k}_{2}=-3\left(\epsilon_{0}
r_{0}^2+3\epsilon_{1}\right)/(8\Omega)=-0.678k_d.$$ The value of $\bar{k}_{2}/k_d$ is so excellently close to the value of $\bar{k}_{1}/k_{c}$ shown in Eq.(\[oy3\]) that we can regard that they are, in fact, equal to each other. We assume both $\bar{k}_{2}/k_{d}$ and $\bar{k}_{1}/k_{c}$ are equal to their average value, $$\label{tzc200266}
\bar{k}_{1}/k_{c}=\bar{k}_{2}/k_{d}=-0.66 .$$ This is the key relation that allows to describe the deformations of SWNT with classic elastic theory. Thus, the deformation energy of a SWNT, the sum of Eqs.(\[oy1\]) and (\[tzc20021\]) $$\label{tzc20024} E_{d}^{(s)}=\int \left[\frac{1}{2} k_{c}
(2H)^2+\bar{k}_{1} K \right] dA+\int \left[\frac{1}{2} k_{d}
(2J)^2+\bar{k}_{2} Q \right] dA$$ can be expressed as the form of the classic shell theory[@ld]: $$\begin{aligned}
\label{bi19971}
E_{c}&=&\frac{1}{2}\int D\left[ (2H)^2-2(1-\nu)K \right] dA
\nonumber \\ &+&\frac{1}{2}\int
\frac{C}{1-\nu^2}\left[(2J)^2-2(1-\nu)Q \right] dA ,\end{aligned}$$ where $D=(1/12)Yh^3/(1-\nu ^2)$ and $C=Yh$ are bending rigidity and in-plane stiffness of shell. $\nu$ is the Poisson ratio and $h$ is the thickness of shell. Comparing Eq.(\[tzc20024\]) with Eq.(\[bi19971\]), we have $$\label{tzcc2002}
\left \{ \begin{array}{l}
(1/12)Yh^3/(1-\nu^2)=k_c\\
Yh/(1-\nu^2)=k_d\\
1-\nu=-\bar{k}_{1}/k_{c}=-\bar{k}_{2}/k_{d}
\end{array} \right..$$ From above equations we obtain the Poisson ratio, effective wall thickness, and Young’s modulus of SWNTs are $\nu=0.34$, $h=0.75$Å and, $Y=4.70$TPa, respectively. Our numerical results are close to those given by Yakobson [*et al.*]{} [@bi].
Through above discussion, we can declare that: (i) Eqs.(\[tzc200266\]), (\[tzc20024\]) and (\[bi19971\]) imply that elastic shell theory in macroscopic scale can be applied to the SWNT in mesoscopic scale provided that its radius are not too small. (ii) SWNT can be regard as being made from isotropic materials with $Y=4.70$TPa and $\nu=0.34$. Its effective thickness can be well-defined as $h=0.75$Å.
We now turn to discuss the axial Young’s modulus of MWNT. A MWNT can be thought of as a series of coaxial SWNTs with layer distance $d=3.4{\rm \AA}$.
Due to the deformation energy of SWNT, we can write the deformation energy of MWNT as [@tzcnt1] $$\begin{aligned}
E^{(m)}&=&\sum_{l=1}^N\int \left[\frac{1}{2} k_{d}
(2J)^2+\bar{k}_{2} Q \right] dA\nonumber\\&+&\sum_{l=1}^N \pi
k_{c} L/ \rho_{l} +\sum_{l=1}^{N-1}(\Delta E_{coh}/d) \pi L
(\rho_{l+1}^2-\rho_l^2) ,\label{oy19974}\end{aligned}$$ where $\rho_{l}$ is the radius of the $l$-th layer from inner one, $N$ is the layer number of MWNT, and $\Delta E_{coh}=-2.04 {\rm
eV/nm^2}$ [@Girifalco] being the interlayer cohesive energy per area of planar graphite. $L$ is the length of MWNT. The second term in Eq.(\[oy19974\]) expresses the summation of curvature energies on all layers given in Eq.(\[oy1\]), and the third term represents the total interlayer cohesive energy which actually arises from the relatively weaker Van der Waals’ interactions. On this account, we can reasonably believe that the axial strain $\varepsilon_{x}$ and circumferential strain $\varepsilon_{y}$ still satisfy $\varepsilon_{y}=-\nu \varepsilon_{x}$ for every layer of SWNT in the MWNT when uniform stresses apply along axial direction. Thus Eq.(\[oy19974\]) becomes $$E^{(m)}=\frac{k_{d}}{2}(1-\nu^2)\varepsilon_{x}^2 \sum_{l=1}^N
2\pi \rho_l L+\sum_{l=1}^N \pi k_{c} L/ \rho_{l}
+\sum_{l=1}^{N-1}\Delta E_{coh} \pi L (\rho_{l+1}+\rho_l)
.\label{oy19975}$$
The axial Young’s modulus of the MWNT $Y_m$ is defined as $$\label{tzc20026}
Y_m(N)=\frac{1}{V}\frac{\partial^2 E^{(m)}}{\partial
\varepsilon_{x}^2} ,$$ where $V=\pi L
[(\rho_N+h/2)^2-(\rho_1-h/2)^2]$ is the volume of MWNT. If considering $\sum_{l=1}^N 2\pi \rho_l L=(\rho_1+\rho_N)N\pi L$, from Eqs.(\[oy19975\]) and (\[tzc20026\]) we have [@tzcnt1] $$\label{tzc20027}
Y_m(N)=\frac{N}{N-1+h/d}\frac{h}{d}Y ,$$ where $Y$ and $h$ are the Young’s modulus and effective wall thickness of SWNTs. Obviously, $Y_m=Y=4.70$TPa if $N$=1, which corresponds to the result of SWNTs, and $Y_m=Yh/d=1.04$TPa if $N \gg 1$, which is just the Young’s modulus of bulk graphite. The layer number dependence of MWNT’s Young’s modulus can be used to discuss mechanical properties of nanotube/polymer composites [@LauKT; @LauShi; @XiaoT].
Now let us discuss the mechanical stabilities of SWNTs and MWNTs. In above discussion, we have prove that elastic shell theory can be applied to SWNTs. Thus we can directly use the classical results in elastic shell theory [@wujk; @Pogorelov].
The critical pressure for SWNT by axial stress on both free ends is $$\label{critpswnta}
p_{csa}=\frac{1}{\sqrt{3(1-\nu^2)}}\frac{Yh}{\rho}.$$ The critical pressure for SWNT (with two free ends) by radial stress is $$\label{critpswntr}
p_{csr}=\frac{Yh^3}{4(1-\nu^2)\rho^3}.$$ The critical moment for SWMT by uniform bending moment is $$M_{cs}=\frac{2\sqrt{2}\pi Y\rho
h^2}{9\sqrt{1-\nu^2}}.$$ The critical torsion for SWMT by uniform torsion is $$T_{cs}=\frac{\sqrt{2}\pi Y\sqrt{\rho
h^5}}{3(1-\nu^2)^{3/4}}.$$ In above four equations, $\nu=0.34$, $h=0.75$Å, $Y=4.70$TPa, and $\rho$ is the radius of SWNT.
The stability of MWNT is determined by its weakest layer of SWNT. Thus we have the critical pressure for MWNT by axial stress on both free ends $$\label{critpmwnta}
p_{cma}=\frac{1}{\sqrt{3(1-\nu^2)}}\frac{Yh}{\rho_o}$$ and the critical pressure for MWNT (with two free ends) by radial stress $$\label{critpmwntr}
p_{cmr}=\frac{Yh^3}{4(1-\nu^2)\rho_o^3},$$ where $\nu=0.34$, $h=0.75$Å, $Y=4.70$TPa, and $\rho_o$ is the outmost radius of MWNT.
Eqs.(\[critpswnta\])–(\[critpmwntr\]) are valid provided that the elastic shell theory can be applied to SWNTs. Adopting the *ad hoc* convention $h=3.4$Å—the layer distance of bulk graphite as the thickness of SWNTs will give different values of these equations. Thus the crucial experiments to test above relations will judge whether we should take the *ad hoc* convention or effective thickness of SWNTs.
MOTORS OF DWNTS {#motors-of-dwnts .unnumbered}
===============
With the development of nanotechnology, especially the discovery of carbon nanotubes, people are putting their dream of manufacturing nanodevices [@Feynman1; @Drexler] into practice. In this section, we will discuss the potential application of CNTs. First, we specifically introduce a molecular motor constructed from a DWNT driven by temperature variation due to the different chirality of DWNT proposed in Ref.[@tzcnt2]. As shown in Fig.\[dwnteps\], inner tube’s index is (8, 4) with a length long enough to be regarded as infinite while outer tube is a (14, 8) tube with just a single unit cell [@footno1]. Obviously, they are both chiral nanotubes and their layer distance is about 3.4 Å. If we prohibit the motion of outer tube in the direction of nanotube axis, it will be proved that, in a thermal bath, outer tube exhibits a directional rotation when the temperature of the bath varies with time. Thus it could serve as a thermal ratchet.
![ A double-walled carbon nanotube with inner tube’s index being (8, 4) and outer tube’s index being (14, 8). $z$-axis is the tube axis parallel to vector $\mathbf{T}$. $x$-axis is perpendicular to $z$ passes through one of carbon atoms in inner tube and $y$-axis perpendicular to the $xz$-plane. The triangles represent the fixed devices of outer tube. There is no obviously relative motion along radial direction between inner and outer tubes at low temperature. If we forbid the motion of outer tube in the direction of $z$-axis, only the rotation of outer tube around inner tube is permitted.[]{data-label="dwnteps"}](dwnteps.eps){width="7cm"}
To see this, we first select an orthogonal coordinate system shown in Fig.\[dwnteps\] whose $z$-axis is the tube axis and $x$-axis passes through one of carbon atoms in the inner tube. We fix the inner tube and forbid the $z$-directional motion of the outer tube. The rotation angle of outer tube around the inner one is denoted by $\theta$.
We take the interaction between atoms in outer and inner tube as the Lennard-Jones potential $u(r_{ij})=4\epsilon[(\sigma/r_{ij})^{12}-(\sigma/r_{ij})^6]$, where $r_{ij}$ is the distance between atom $i$ in inner tube and atom $j$ in outer tube, $\epsilon=28$ K, and $\sigma=3.4$Å[@hir]. We calculate the potential $V(\theta)$ when outer tube rotates around inner tube with angle $\theta$ and plot it in Fig.\[potentialeps\].
![ The potentials $V(\theta)$ between outer and inner tubes when outer tube rotates around inner tube. $\theta$ is the rotating angle. Here we have set $V(0)=0$. The squares are the numerical results which can be well fitted by $V(\theta)=15.7-0.6\cos
4\theta-2.2\sin4\theta-12.7\cos8\theta-6\sin8\theta-1.7\cos12\theta+10.8\sin
12\theta$ (solid curve).[]{data-label="potentialeps"}](potential.eps){width="6.8cm"}
If putting our system in a thermal bath full of He gas whose temperature varies with time, We can show that outer tube will exhibit a directional rotation. Let us consider the overdamped case when the Langevin equation for outer tube is expressed $$\label{Langevin}\eta\dot{\theta}=-V'(\theta) +\xi(t),$$ where $\eta$ is the rotating viscosity coefficient, and dot and prime indicate, respectively, differentiations with respect to time $t$ and angle $\theta$. $\xi(t)$ is thermal noise which satisfies $\langle \xi(t)\rangle=0$ and the fluctuation-dissipation relation $\langle\xi(t)\xi(s)\rangle=2\eta
T(t)\delta(t-s)$, where $T(t)$ is temperature and the Boltzmann factor is set to 1. The Fokker-Planck equation corresponding to Eq.(\[Langevin\]) is [@Reimann]: $$\begin{aligned}
\label{fplanck}
\frac{\partial P(\theta,t)}{\partial
t}=\frac{\partial}{\partial\theta}\left[\frac{V'(\theta)P(\theta,t)}{\eta}\right]+\frac{T(t)}{\eta}\frac{\partial^2P(\theta,t)}{\partial\theta^2},\end{aligned}$$ where $P(\theta,t)$ represents the probability of finding the outer tube at angle $\theta$ and time $t$ which satisfies $P(\theta+\pi/2,t)=P(\theta,t)$. If the period of temperature variation is $\mathcal{T}$, we arrive at the average angular velocity in the long-time limit [@Reimann] $$\label{current}
\langle\dot{\theta}\rangle=\lim_{t\rightarrow
\infty}\frac{1}{\mathcal{T}}\int_t^{t+\mathcal{T}}dt\int_0^{\pi/2}d\theta\left[-\frac{V'(\theta)P(\theta,t)}{\eta}\right].$$
Here we take the periodical varying temperature $T(t)=\bar{T}[1+A\sin(2\pi t/\mathcal{T})]$ with $\bar{T}=50$ K and $|A|\ll 1$. Changing variables $D=\eta/\bar{T}$, $t=D\tau$, $U(\theta)=V(\theta)/\bar{T}$, $\mathcal{T}=D\mathcal{J}$ and $\tilde{P}(\theta,\tau)=P(\theta,D\tau)$, We arrive at the dimensionless equations of Eqs.(\[fplanck\]) and (\[current\]) $$\begin{aligned}
&&\frac{\partial \tilde{P}}{\partial
\tau}=\frac{\partial}{\partial\theta}[U'(\theta)\tilde{P}]+(1+A\sin\frac{2\pi\tau}{\mathcal{J}})\frac{\partial^2\tilde{P}}{\partial\theta^2},\label{fplanck2}\\
&&\langle \frac{d\theta}{d\tau}\rangle=\lim_{\tau\rightarrow
\infty}\frac{1}{\mathcal{J}}\int_{\tau}^{\tau+\mathcal{J}}d\tau\int_0^{\pi/2}d\theta\left[-U'(\theta)\tilde{P}\right].\label{current2}\end{aligned}$$
We numerically solve Eq.(\[fplanck2\]) and calculate Eq.(\[current2\]) with $A=0.01$ and different $\mathcal{J}$. The curve in Fig.\[averagevel\] shows the relation between the average dimensionless angular velocity $\langle\frac{d\theta}{d\tau}\rangle$ and the dimensionless period $\mathcal{J}$ of temperature variation. We find that $\langle\frac{d\theta}{d\tau}\rangle\simeq 0$ for very small and large $\mathcal{J}$, and $\langle\frac{d\theta}{d\tau}\rangle\neq
0$ for the middle values of $\mathcal{J}$, which implies that outer tube has an evident directional rotation in this period range.
![The average dimensionless angular velocity $\langle
d\theta/d\tau\rangle$ of outer tube rotating around inner tube in thermal bath whose temperature changes with dimensionless period $\mathcal{J}$. The minus sign means rotation around $z$-axis is the left-handed, and vice versa. []{data-label="averagevel"}](averagevel.eps){width="7cm"}
In above discussion, we have conceptually constructed a temperature ratchet—a kind of molecular motor driven by temperature variations which satisfies two necessary conditions required by the second law of thermodynamics [@Feynman2]: One is breaking spatial inversion symmetry (through different chirality of two SWNTs in DWNT); Another is breaking of thermal equilibrium (through temperature variations). But, in fact, this motor is not convenient because it is hard to control temperature variations. Our dream is construct an electric motor driven by alternating voltage discussed as below.
Although no one believe SWNTs are piezoelectric materials, an exceptionally large axial deformation in SWNTs induced by applied electrostatic field along tube axis is demonstrated using Hartree-Fock and density functional calculations by Guo *et al.* [@Guowl]. Additionally, their calculations reveal that the bond-elongation of SWNTs is linearly dependent on the axial field when the field is not strong enough. These results shed a light on our dream of making the electric motor. Now, we will qualitatively construct it. Put the device shown in Fig.\[dwnteps\] between two electrodes and apply alternating voltage $U_A(t)=U_0\sin(\omega t)$ on the electrodes, where $U_0$ and $\omega$ are constants. We expect that the field due to alternating voltage also induces bond-elongation of the DWNT. Thus the layer distance will change, which will change the interaction between layers. At least for small $U_0$, the lowest order Taylor Series of interaction between layers might be written as $\tilde{V}(\theta,t)\approx V(\theta)+\delta(\theta) |\sin\omega
t|$ with $\delta(\theta)$ being a small quantity. If putting our system in a thermal bath full of He gas with constant temperature $T$, we have the Langevin equation of outer tube in the overdamped case [@Reimann]: $$\label{elctromotor}\eta\dot{\theta}=-[V'(\theta)+\delta'(\theta)
|\sin\omega t|] +\xi(t),$$ where $\xi(t)$ is thermal noise which satisfies $\langle \xi(t)\rangle=0$ and the fluctuation-dissipation relation $\langle\xi(t)\xi(s)\rangle=2\eta
T\delta(t-s)$. Consequently, the thermal equilibrium is broken by the shaking potential $\tilde{V}(\theta,t)$, and we may construct a fluctuating potential ratchet proposed in Ref.[@Reimann; @RDAstumian].
CONCLUSION {#conclusion .unnumbered}
==========
In this chapter, we introduce the structures, symmetries and their inducing physical properties of SWNTs in detail. We discuss the mechanics of nanotubes and explain why we must define the effective thickness of SWNTs. We also qualitatively construct an electric motor driven by alternating voltage. We expect some experiments which can put our electric motor into practice in near future.
[99]{} Kroto, H. *et al.* *Nature* 1985, *318*, 162-163. Iijima, S. *Nature* 1991, *354*, 56-58. Iijima, S.; Ichihashi, T. *Nature* 1993, *363*, 603-605. Saito, R.; Dresselhaus, M. S.; Dresselhaus, G. [*Physical Properties of Carbon Nanotubes*]{}; Imperial College Press: London, 1998; pp 35-48. Mintmire, J. W.; Dunlap, B. I.; White, C. T. [*Phys. Rev. Lett.* ]{}1992, *68*, 631-634. Hamada, N.; Sawada, S. I.; Oshiyama, A. [*Phys. Rev. Lett.* ]{}1992, *68*, 1579-1581. Saito, R.; Fujita, M.; Dresselhaus, G.; Dresselhaus, M. S. [*Appl. Phys. Lett.*]{} 1992, *60*, 2204-2206. Tans, S. J. *et al.* *Nature* 1997, *386*, 474-477. Kong, J. *et al.* [*Phys. Rev. Lett.* ]{}2001, *87*, 106801. Liang, W. J. *et al.* *Nature* 2001, *411*, 665-669. White, C. T.; Todorov, T. N. *Nature* 1998, *393*, 240-242. Yakobson, B. I.; Brabec, C. J.; Bernholc, J. [*Phys. Rev. Lett.* ]{}1996, *76*, 2511-2514. Zhou, X.; Zhou, J. J.; Ou-Yang, Z. C. [*Phys. Rev. B* ]{}2000, *62*, 13692-13696. Hernández, E. *et al.* [*Phys. Rev. Lett.* ]{}1998, *80*, 4502-4505. Lu, J. P. [*Phys. Rev. Lett.* ]{}1997, *79*, 1297-1300. Krishnan, A. *et al.* [*Phys. Rev. B* ]{}1998, *58*, 14013-14019. Yu, M. F. *et al.* *Science* 2000, *287*, 637-640. Treacy, M. M. J.; Ebbesen, T. W.; Gibson, J. M. *Nature* 1996, *381*, 678-680. Wong, E. W.; Sheehan, P. E.; Lieber, C. M. *Science* 1997, *277* 1971-1975. Nye, J. F. [*Physical Properties of Crystals*]{}; Clarendon Press: Oxford, 1985; pp 33-148. Tu, Z. C.; Ou-Yang, Z. C. [*Phys. Rev. B* ]{}2002, *65*, 233407. Rafii-Tabar, H. *Phys. Rep.* 2004, *390*, 235-452. Pantano, A.; Parks, D. M.; Boyce, M. C. *J. Mech. Phys. Solids* 2004, *52* 789-821. Chandra, N. *et al.* [*Phys. Rev. B* ]{}2004, *69*, 94101. Ogata, S. *et al.* [*Phys. Rev. B* ]{}2003, *68*, 165409. Pantano, A. *et al.* [*Phys. Rev. Lett.* ]{}2003, *91*, 145504. Natsuki, T.; Tantrakarn, K.; Endo, M. *Appl. Phys. A-Mater.* 2004, *79*, 117-124. Sun, C. Q. *et al.* *J Phys Chem B* 2003, *107*, 7544-7546. Tu, Z. C.; Ou-Yang Z. C. *J. Phys.: Condens. Matter* 2004, *16*, 1287-1292. Dendzik, Z. *et al.* *Poster at E-MRS Fall Meeting* 2004, Symposium H, Poland. Kang, J. W. *et al.* *J. Korean Phys. Soc.* 2004, *45*, 573-576. Tu, Z. C.; Ou-Yang Z. C. [*Phys. Rev. B* ]{}2003, *68*, 153403. Tu, Z. C.; Ou-Yang Z. C. *J. Phys.: Condens. Matter* 2003, *15*, 6759-6771. Landau, L. D.; Lifshiz, E. M. [*Elasticity Theory*]{}; Pergamon: Oxford, 1986. Landau, L. D.; Lifshiz, E. M. [*Electrodynamics of Continuous Media*]{}; Pergamon: Oxford, 1984. Benedict, L. X.; Louie, S. G.; Cohen, M. L. [*Phys. Rev. B* ]{}1995, *52*, 8541-8549. Lenosky, T. *et al.* *Nature* 1992 *355*, 333-335. Zhou, X.; Zhou, J. J.; Ou-Yang, Z. C. *Physica B* 2001, *304*, 86-90. Ou-Yang, Z. C.; Su, Z. B.; Wang, C. L. [*Phys. Rev. Lett.* ]{}1997, *78*, 4055-4058. Girifalco, L. A.; Lad, R. A. *J. Chem. Phys.* 1956, *25*, 693. Lau, K. T. *Chem. Phys. Lett.* 2003 *370*, 399-405. Lau, K. T.; Shi, S. Q. *Carbon* 2002, *40*, 2965-2968. Xiao, T.; Liao, K. *Compos. Part. B-Eng.* 2004, *35* 211-217. Wu, J. K.; Su, X. Y. *Stabilities of Elastic Systems*; Science Press: Beijing, 1994. Pogorelov, A. V. *Bendings of surfaces and stability of shells*; AMS: Providence, 1989. Feynman, R. P. [*J. Microelectromechanical Systems*]{} 1992, *1*, 60. Drexler, K. E. [*Nanosystems: Molecular Machinery, Manufacturing and Computation*]{}; Wiley: New York, 1992. If we consider several unit cells, the final qualitative results sitll hold. Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. [*Molecular Theory of Gases and Liquids*]{}; John Wiley & Sons: New York, 1954. Reimann, P. [*Phys. Rep.*]{} 2002, *361*, 57-265. Feynman, R. P.; Leighton, R. B.; Sands, M. [*The Feynman Lectures on Physics*]{}; Addison-Wesley: Reading, 1966; *Vol 1*, Chapter 46. Guo, W.; Guo, F. [*Phys. Rev. Lett.* ]{}2003, *91*, 115501. Astumian, R. D.; Bier, M. [*Phys. Rev. Lett.* ]{}1994, *72*, 1766-1769.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We prove the results announced in \[KSV\] modulo one general fact on Voevodsky motives that does not exist in the published literature. Namely, we assume that the functor of motivic vanishing cycles commutes with the Hodge and l-adic realizations.'
author:
- Vadim Vologodsky
title: 'On the N-integrality of instanton numbers'
---
This paper is a result of a joint work with Maxim Kontsevich and Albert Schwarz. However, they decided not to sign it in the capacity of authors.
Let $\pi: X \to C$ be a family of Calabi-Yau n-folds over a smooth curve and let $a\in \overline C - C$ be a maximal degeneracy point of $\pi$. We assume that the pair $( \pi: X \to C,
a)$ is defined over $\mathbb{Z}$. It is predicted that the power series expansion for the canonical coordinate on $C$ at the point $a$ has integral coefficients (\[M\]). This is a higher-dimensional generalization of the classical fact that the Fourier coefficients of the j-invariant are integers. This conjecture was checked in a number of cases in \[LY\]. Another related conjecture says the instanton numbers $n_d$, defined from the Picard-Fuchs equation, are integers (see [*loc. cit.*]{}). We do not know how to prove these conjectures. However, in the present paper we indicate a proof of a weaker statement, namely, that these numbers belong to the subring $\mathbb{Z}[N^{-1}]\subset \mathbb{Q}$ [^1], where $N$ is an explicitly defined integer. [^2] There are two main ingredients in our proof. The first one is [*the Frobenius action*]{} on the p-adic de Rham cohomology. It easy to see, that under our assumptions, both the coefficients of power series expansion for the canonical coordinates and the instanton numbers are rational. Thus, to prove the integrality statement, it will suffice to show that for almost every prime $p$ they are p-adic integers. To do this we look at the relative de Rham cohomology of our family over p-adic numbers. Then the Frobenius symmetry or, more precisely, the existence of the Fontaine-Laffaille structure on the cohomology bundle imply certain strong integrality properties of the parallel sections (i.e. solutions to the Picard-Fuchs equation ). [^3]
The second ingredient is the motivic vanishing cycles. Assume, for the purpose of Introduction, that $dim H^n(X/C)=n+1$. Then the limit Hodge structure of the variation $H^n(X/C)$ is a mixed Hodge-Tate structure. Consider the corresponding [*period matrix*]{} $(a_{ij})$. This is a matrix with highly transcendental complex coefficients. On the other hand, we consider the limit Fontaine-Laffaille module and look at the corresponding [*Frobenius action*]{} $(b_{ij})$. This is a matrix with p-adic coefficients. To complete the proof of the integrality statement we need to establish a certain relation between the superdiagonal entries of the two matrices. Namely, we have to show, that $$\label{cperiod}
a_{m\,m+1}= (2\pi i)^{m-1} log\, c$$ $$b_{m\,m+1}= \pm p^{m-1}
log\, c^{1-p}$$ for some [*rational*]{} number $c$. [^4] Standard conjectures on motives imply the existence of a mixed Artin-Tate motive $T$ over $\mathbb{Q}$ whose Hodge and p-adic realizations are the limit Hodge and Fontaine-Laffaille structure correspondingly. This yields a certain explicit relation between [*all*]{} the coefficients of matrices $(a_{ij})$ and $(b_{ij})$ and, in particular, formula (\[cperiod\]). Unfortunately, the motivic conjectures needed to justify this argument are very far from being proved. However, we construct in Section 4 a 1-motive $M_{t,n}(X_{\mathbb{Q}})$ that should be thought of as the maximal 1-motive quotient of $T^*$ and then use it to prove (\[cperiod\]).
Still, at one point we have to rely on a general fact that has not yet appeared in the published literature. Namely, we have to assume the compatibility of Ayoub’s motivic vanishing cycles functor with the Hodge and l-adic realization functors. Although not published the required compatibility is known to experts \[A2\], \[BOV\] and hopefully this piece of a general theory will be written in a matter of time.
The paper is organized as follows. Section 1 contains statements of the results. In Section 2 we recall some well known facts on vector bundles with logarithmic connection, variations of Hodge structure, and give an interpretation of the canonical coordinate as an extension class of certain variations of Hodge structure. In Section 3 we define, using p-adic Hodge Theory, a p-adic analog of the canonical coordinate and Yukawa coupling (for 1-parameter families of Calabi-Yau varieties over $\mathbb{Z}_p$) and prove, using Dwork’s Lemma, the (p-adic) integrality statements for these objects. Finally, in the last section (the most technical one) we show for families over $\mathbb{Z}$ that the two constructions (complex and p-adic) give the same functions. [^5] To do this we give a third geometric definition (i.e. which makes sense over any ground field) of the canonical coordinate. The construction is based on the notion of motivic nearby cycles (due to Ayoub \[A1\]) and uses the language of Voevodsky motives.
[**Acknowledgments.**]{} Besides Maxim Kontsevich and Albert Schwarz whom many ideas in the present paper are due to, I wish to express my gratitude to Joseph Ayoub, who drew my attention to the paper \[BK\]. Special thanks go to the referee for his generous help in turning a raw draft into a paper. In particular, the referee pointed out that the compatibility of the motivic Albanese functor with the Hodge realization, that plays an important role in our argument, is not proved in the published literature. At the end the author wrote a separate paper (\[Vol\]) with a proof of the required compatibility.\
I acknowledge the stimulating atmosphere of IHES and MPIM in Bonn where parts of this work were done.\
This research was partially supported by NSF grant DMS-0401164.
Terminology and statements of the results
=========================================
[**1.1. Definition of the canonical coordinate.** ]{} The following construction is due to Morrison \[M\]. Let $\pi: X \to C$ be a family of Calabi-Yau varieties of dimension $n$ over a smooth curve over $\mathbb{C}$. This means that locally on $C$ the relative canonical bundle $\Omega^n_{X/C}$ is trivial. Assume that $C$ is embedded into a larger smooth curve $\overline C
\supset C $ and $a \in \overline C - C$ is a boundary point. The point $a$ is called a maximal degeneracy point if the monodromy operator $M: H_n(X_{a'}, \mathbb{Q}) \to H_n(X_{a'}, \mathbb{Q})$, corresponding to a small loop around $a$ is unipotent and $(M -
Id)^n \ne 0$.
[**Remark.**]{} It is known that for any smooth proper family $\pi: X \to C$ with a unipotent monodromy, $(M - Id)^{n+1} = 0$.
From now on we will assume that $a$ is a maximal degeneracy point. Set $N_B= log\, M$. The following remarkable result was derived by Morrison from the very basic properties of the limit Hodge structure [^6].
(\[M\], Lemma 1.) \[M\] $dim_{\mathbb{Q}} Im\, N_B^n =1$ and $dim_{\mathbb{Q}} Im\, N_B^{n-1} =2$.
Denote by $ {\cal T}_{\mathbb{Z}}$ the local system over a punctured neighborhood $D^*$ of $a$, whose fiber over a point $a'\in C$ is $Im( H_n(X_{a'}, \mathbb{Z}) \to H_n(X_{a'}, \mathbb{Q})) $. Let $\delta_1, \delta_2$ be a basis for $Im\, N_B^{n-1} \cap ({\cal
T}_{\mathbb{Z}})_{a'} $ such that $\delta_1 $ generates $Im\, N_B^{n} \cap ({\cal T}_{\mathbb{Z}})_{a'} $ and such that $N_B(\delta_2)$ is a positive multiple of $\delta_1$. We may view $\delta_1$ as a section of ${\cal
T}_{\mathbb{Z}} $ over $D^*$ and $\delta_2$ as a section of the quotient ${\cal T}_{\mathbb{Z}}/\mathbb{Z}\delta _1 $. Choose a non-vanishing section $\omega $ of $\pi_*\Omega^n_{X/C}$ over $D^*$. We then see that $$\label{cc}
q= exp(2\pi i \frac{\int _{\delta _2} \omega}{ \int _{\delta _1} \omega})$$ is a well defined function on a punctured neighborhood $ D^*$ and that $q$ does not depend on the choice of $\delta_i$ and $\omega$ we made. Moreover, the function $q$ extends to $a$ and $ord_a q = k$, where $k$ is defined from the equation $N_B(\delta_2)= k \delta
_1$. [^7] We shall say the Betti monodromy of the family $X\to C$ is small if $k=1$. In this case, the function $q$ is called the canonical (local) coordinate on $\overline C$.
[**1.2. Yukawa function.**]{} Denote by $${\cal H}= ({\cal H}_{\mathbb{Z}}= Im( R^n \pi_* \mathbb{Z} \to R^n \pi_* \mathbb{Q}),\, {\cal F}^{n} \subset {\cal F}^{n-1}
\subset \cdots \subset {\cal F}^0 = {\cal H}_{\mathbb{Z}} \otimes
{\cal O}_{D^*})$$ the variation of Hodge structure associated to $\pi: X\to C$. The Kodaira-Spencer operator $$\overline \nabla: {\cal F}^p/{\cal F}^{p+1} \to {\cal F}^{p-1}/{\cal F}^p \otimes \Omega^1_{D^*}$$ extends to a homomorphism of graded algebras $$S^{\cdot}T_{D^*} \to End_{{\cal O}_{D^*}}(\bigoplus_ p {\cal F}^p/{\cal F}^{p+1}).$$ Specializing, we get a morphism $$\label{ks}
\kappa : S^{n}T_{D^*} \to Hom _{{\cal O}_{D^*}} ({\cal F}^n, {\cal
F}^0/{\cal F}^1)\simeq ({\cal F}^n \otimes {\cal F}^n)^*.$$ The line bundle ${\cal F}^n \otimes {\cal F}^n$ is naturally trivialized over $D^*$. To see this, denote by $\omega \in {\cal F}^n$ the differential form such that $$\int _{\delta _1} \omega = (2\pi i)^n.$$ The section $\omega \otimes \omega \in {\cal F}^n \otimes {\cal
F}^n$ defines the desired trivialization. [^8] Define the Yukawa function on $D^*$ to be $$Y = \kappa((q\frac{d}{dq})^n)\cdot (\omega \otimes \omega).$$ One can check that $Y$ extends to $D$.[^9]
[**1.3. Statement of main results.**]{} Let $S= spec \,
\mathbb{Z}[N^{-1}]$ be an open subscheme of $spec \, \mathbb{Z}, \;
$ $\overline C_S$ a smooth curve over $S$, and let $a: S
\hookrightarrow \overline C_S$ be a section. Denote by $t$ a local coordinate on a open neighborhood of $a$ such that $t(a)=0$. Let $\pi: X_S \to C_S = \overline C_S- a$ be a smooth proper family of Calabi-Yau schemes. We will make the following assumptions:\
i) $a_{\mathbb{C}} $ is the maximal degeneracy point of the complex family $X_{\mathbb{C}}\to C_{\mathbb{C}}$\
ii) $\pi: X_S \to C_S $ extends to a semi-stable morphism $\overline{\pi}: \overline X_S \to \overline C_S$ [^10]\
iii) All primes $p\leq dim \, X_{\mathbb{C}}$ are invertible on $\mathbb{Z}[N^{-1}]$\
iv) The Betti monodromy of the family $X_{\mathbb{C}}\to
C_{\mathbb{C}}$ is small ( see 1.1 ).
\[th1\] Assume that $q'(0)$ is a rational number . Then $$q(t)\in (\mathbb{Z}[N^{-1}]((t)))^* .$$
We shall see in Section [**4.5**]{} that, for any family $X_{\mathbb{Q}} \to C_{\mathbb{Q}} $ over $\mathbb{Q}$ with a maximal degeneracy point at $a\in \overline
C_{\mathbb{Q}}(\mathbb{Q})$, $q'(t)^r\in \mathbb{Q}((t))$, for some integer $r$.
For the next result we assume that $dim \, X_{\mathbb{C}} = 4$ (i.e. $X_{\mathbb{C}}\to C_{\mathbb{C}}$ is a family of threefolds) and [^11] $$\label{one}
rk \, H^3_{DR}(X_{\mathbb{C}}/C_{\mathbb{C}}) = 4.$$ We also assume that $q'(0)$ is a rational number.
\[th2\] One has $$\label{instanton}
Y(q)= n_0+ \sum^{\infty}_{d=1}n_d d^3 \frac{q^d}{1-q^d},$$ where $n_i \in \mathbb{Z}[N^{-1}]$. [^12]
Hodge Theory
============
[**2.1. Logarithmic De Rham cohomology.**]{} We will need the following construction from \[Ste\]. Let $\overline{\pi}: \overline
X_S \to \overline C_S$ be a semi-stable morphism.
$$\def\normalbaselines{\baselineskip20pt
\lineskip3pt \lineskiplimit3pt}
\def\mapright#1{\smash{
\mathop{\to}\limits^{#1}}}
\def\mapdown#1{\Big\downarrow\rlap
{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\begin{matrix}
Y_S= \overline X_S \times _{\overline C_S} S & \stackrel{}{\hookrightarrow}
& \overline X_S & \hookleftarrow & X_S \cr
\mapdown{} & &\mapdown{\overline{\pi}} & & \mapdown{\pi} \cr
S & \stackrel{a}{\hookrightarrow}
& \overline C_S & \hookleftarrow & C_S
\end{matrix}$$ We then consider the relative logarithmic De Rham complex $(\Omega^*_{\overline X_S/\overline C_S}(log \,Y_S), d)$ on $\overline X_S$ defined as follows. Let $\Omega^i_{\overline
X_S}(log \,Y_S)$ be the sheaf of differential forms with logarithmic singularities along $Y_S$, and let $\Omega^*_{\overline
X_S/\overline C_S}(log \,Y_S)$ be the quotient of the sheaf of algebras $\Omega^*_{\overline X_S}(log \,Y_S)$ by the ideal generated by $\overline {\pi}^* \eta, \, \, \eta \in
\Omega^1_{\overline C_S}(log \, S)$. One immediately sees that $\Omega^*_{\overline X_S/\overline C_S}(log \,Y_S)$ is a locally free sheaf of ${\cal O}_{\overline X_S}$-modules and that the exterior differential $d: \Omega^i_{\overline X_S}(log \,Y_S)\to
\Omega^{i+1}_{\overline X_S}(log \,Y_S)$ descends to $\Omega^*_{\overline X_S/\overline C_S}(log \,Y_S)$ . We then define the logarithmic De Rham cohomology by $$H^i_{log}(\overline X_S/\overline C_S)= R^i\overline{\pi}_*(\Omega^*_{\overline X_S/\overline C_S}(log \,Y_S), d).$$ $H^i_{log}(\overline X_S/\overline C_S)$ is a coherent sheaf on $\overline C_S$ equipped with a logarithmic connection: $$\nabla: H^i_{log}(\overline X_S/\overline C_S)\to H^i_{log}(\overline X_S/\overline C_S) \otimes \Omega^1_{\overline C_S}(log \, S).$$ Assume that all primes $p\leq n= dim \, _{C_S} X_S$ are invertible in $\mathbb{Z}[N^{-1}]$. Then\
i) the coherent sheaf $H^i_{log}(\overline X_S/\overline C_S)$ is locally isomorphic to a direct sum of the sheaves ${\cal
O}_{\overline C_S}$, ${\cal O}_{\overline C_S}/p^e$. In particular, the quotient of $H^i_{log}(\overline X_S/\overline C_S)$ modulo torsion is locally free.\
ii) the Hodge spectral sequence (i.e. the spectral sequence associated to the stupid filtration on $(\Omega^*_{\overline
X_S/\overline C_S}(log \,Y_S), d)$) degenerates in the first term. Moreover the induced filtration $\overline{{\cal F}}^{\cdot}\subset
H^i_{log}(\overline X_S/\overline C_S)$ splits (in the category of ${\cal O}_{\overline C_S}$-modules) locally on $\overline C_S$. In particular, $R^0\overline{\pi}_*(\Omega^i_{\overline X_S/\overline
C_S}(log \,Y_S))$ is a locally free ${\cal O}_{ \overline C_S}$-module.\
iii) the residue of the connection $$N_{DR}= Res \, \nabla: a^* H^i_{log}(\overline X_S/\overline C_S) \to a^* H^i_{log}(\overline X_S/\overline C_S)$$ is nilpotent. In particular, $H^i_{log}(\overline
X_{\mathbb{Q}}/\overline C_{\mathbb{Q}})$ is the Deligne canonical extension of the vector bundle $ H^i_{DR}(X_{\mathbb{Q}}/ C_{\mathbb{Q}})$ equipped with the Gauss-Manin connection.\
iv) there is a canonical pairing $$\label{derhampairing}
<\cdot ,\cdot >_{DR}: H^i_{log}(\overline X_S/\overline C_S)\otimes
H^{2n -i}_{log}(\overline X_S/\overline C_S) \to
H^{2n}_{log}(\overline X_S/\overline C_S)\simeq {\cal O}_{\overline C_S}$$ The induced pairing on the quotient of $H^*_{log}(\overline X_S/\overline C_S)$ modulo torsion is perfect.\
Over $\mathbb{C}$ these facts are proven in \[Ste\]; the integral version is contained in \[Fa\] (Theorems 2.1 and 6.2).
The De Rham isomorphism $$H^i_{DR}(X_{\mathbb{C}}/ C_{\mathbb{C}}) \simeq
R^i \pi^{an}_{\mathbb{C}*} \mathbb{Z} \otimes_{\mathbb{Z}} {\cal
O}_{C_{\mathbb{C}}}$$ and a choice of a local coordinate $t$ on $\overline C_{\mathbb{C}}$ yield an integral structure (of a topological nature) on the vector space $a_{\mathbb{C}}^*
H^i_{log}(\overline X_{\mathbb{C}}/\overline C_{\mathbb{C}})$: $$\label{derhamv}
a_{\mathbb{C}}^* H^i_{log}(\overline X_{\mathbb{C}}/\overline
C_{\mathbb{C}})\simeq \Psi_t^{an, un}(R^i \pi^{an}_{\mathbb{C}*}
\mathbb{Z}) \otimes _{\mathbb{Z}} \mathbb{C}$$ To see this, let $Log^{\infty}= {\cal O}_{\mathbb{C}^*}[log\, t]$ be the universal unipotent local system on $\mathbb{C}^*$ and let $\overline{Log}^{\infty}={\cal O}_{\mathbb{C}}[log\, t] $ be the Deligne extension of $Log^{\infty}$ to $\mathbb{C}$. Let us view $Log^{\infty}$ as a subsheaf of the direct image $(exp)_* {\cal
O_{\mathbb{C}}}$ of the structure sheaf on the universal cover $exp:
\mathbb{C} \to \mathbb{C}^*$. Define a $\mathbb{Z}$-lattice $Log^{\infty}_{\mathbb{Z}} \subset Log^{\infty}$ to be $(exp)_*
\mathbb{Z} \cap Log^{\infty}$. We have then $$Log^{\infty}_{\mathbb{Z}}\otimes {\cal O}_{\mathbb{C}^*} \simeq Log^{\infty}.$$ Let $a_{\mathbb{C}} \in D \subset \overline C_{\mathbb{C}}(\mathbb{C})$ be a disk such that the map $t: D \hookrightarrow \mathbb{C}$ defined by the coordinate is an embedding. Given a $\mathbb{Z}$-local system ${\cal H}_{\mathbb{Z}}$ over $D^*$ we define the space of unipotent vanishing cycles by $$\Psi_t^{an, un}({\cal H}_{\mathbb{Z}}):= H^0(D^*, {\cal H}_{\mathbb{Z}}\otimes
(t|_{D^*})^* Log^{\infty}_{\mathbb{Z}} ).
\footnote{This definition is borrowed from [B].}$$ Assume that ${\cal H}_{\mathbb{Z}}$ is unipotent and denote by $\overline {\cal
H}$ the Deligne extension of ${\cal H}={\cal H}_{\mathbb{Z}}\otimes
{\cal O}_{D^*}$ to $D$. We shall define a canonical isomorphism: $$\label{BS}
a_{\mathbb{C}}^* \overline {\cal H} \simeq \Psi_t^{an, un}( {\cal
H}_{\mathbb{Z}}) \otimes _{\mathbb{Z}} \mathbb{C}.$$ This will induce (\[derhamv\]). To construct (\[BS\]) observe that for any vector bundle $E$ over $D$ with a logarithmic nilpotent connection (i.e. a logarithmic connection such that $N_{DR}$ is nilpotent) we have $$\label{ncon}
(E|_{D^*})^{\nabla}\stackrel{\simeq}{\longleftarrow}
E^{\nabla}\stackrel{\simeq}{\longrightarrow}
(a_{\mathbb{C}}^* E)^{N_{DR}=0}.$$ We apply (\[ncon\]) to the ind-object $\overline {\cal H} \otimes
t^* (\overline{Log}^{\infty})$ and use a canonical isomorphism $$(a_{\mathbb{C}}^* \overline {\cal H} \otimes a_{\mathbb{C}}^* t^* (\overline{Log}^{\infty}))^{N_{DR}=0}
\simeq a_{\mathbb{C}}^* \overline {\cal H} ,$$ which takes an element $ v\otimes log^k t \in (a_{\mathbb{C}}^* \overline {\cal H}
\otimes a_{\mathbb{C}}^* t^* (\overline{Log}^{\infty}))^{N_{DR}=0}
$ to $v$ if $k=0$ and to $0$ otherwise.
Denote by $$N_B:
\Psi_t^{an, un}(R^i \pi^{an}_{\mathbb{C}*} \mathbb{Z}) \otimes
_{\mathbb{Z}} \mathbb{Q} \to \Psi_t^{an, un}(R^i
\pi^{an}_{\mathbb{C}*} \mathbb{Z}) \otimes _{\mathbb{Z}} \mathbb{Q}$$ the logarithm of the monodromy operator and by $$< \cdot, \cdot>_B: \Psi_t^{an, un}(R^i \pi^{an}_{\mathbb{C}*} \mathbb{Z})\otimes
\Psi_t^{an, un}(R^{2n-i} \pi^{an}_{\mathbb{C}*} \mathbb{Z}) \to
\mathbb{Z}$$ the pairing induced by the Poincare duality. We then have $$N_B= -2\pi i N_{DR} \, , \, \, < \cdot, \cdot>_B= (2\pi i)^n < \cdot, \cdot>_{DR}.$$
[**2.2. A variation of mixed Hodge structure.**]{} Let $\pi_{\mathbb{C}}: X_{\mathbb{C}} \to C_{\mathbb{C}}$ be a smooth family of Calabi-Yau schemes with a maximal degeneracy point at $a_{\mathbb{C}}\in \overline C_{\mathbb{C}}$ and let $\overline \pi_{\mathbb{C}}: \overline X_{\mathbb{C}} \to \overline C_{\mathbb{C}}$ be a semi-stable morphism which extends $\pi_{\mathbb{C}}$. Set $$\overline {\cal H}= H^n_{log}(\overline X_{\mathbb{C}}/\overline C_{\mathbb{C}})\, , \,
{\cal H}_{\mathbb{Z}}= Im( R^n \pi_{\mathbb{C} *} \mathbb{Z} \to R^n
\pi_{\mathbb{C} *} \mathbb{Q}).$$ Denote by $W_{\cdot}\Psi_t^{an,un}( {\cal H}_{\mathbb{Z}})\subset \Psi_t^{an,
un}( {\cal H}_{\mathbb{Z}}) $ the monodromy filtration. [^13] We then consider the limit Hodge structure $$\Psi_t^{Hodge , un}( {\cal H})= (W_{\cdot}\Psi_t^{an, un}( {\cal H}_{\mathbb{Z}})\subset
\Psi_t^{an, un}( {\cal H}_{\mathbb{Z}})\, ,\,
a_{\mathbb{C}}^*\overline{{\cal F}}^{\cdot}\subset a_{\mathbb{C}}^*
\overline {\cal H} ).$$ Since $\overline{{\cal F}}^{n+1}= 0$, we have $$W_{-1}\Psi_t^{an, un}( {\cal H}_{\mathbb{Z}}) =0 \, , \, W_{2n} \Psi_t^{an, un}( {\cal
H}_{\mathbb{Z}})= \Psi_t^{an, un}( {\cal H}_{\mathbb{Z}})$$ $$Im
\, N^n_B = W_{0}\Psi_t^{an, un}( {\cal H}_{\mathbb{Z}})\otimes
\mathbb{Q}.$$ Furthermore, since $ rk\, \overline{{\cal F}}^n =rk \,
\overline{{\cal F}}^0/\overline{{\cal F}}^1 = 1$ and $Im\, N_B^n\ne
0$ we must have $$rk \, W_0 \Psi_t^{an, un}( {\cal H}_{\mathbb{Z}}) = rk \, W_1 \Psi_t^{an, un}(
{\cal H}_{\mathbb{Z}})=1.$$ It follows that the map $$N_B^{n-1}:
\Psi_t^{an, un}( {\cal H}_{\mathbb{Z}}) \to Im \, N^{n-1}_B/ Im \,
N^n_B$$ factors through the quotient $\Psi_t^{an, un}( {\cal
H}_{\mathbb{Z}})/ W_{2n-1} \Psi_t^{an, un}( {\cal H}_{\mathbb{Z}})$ of rank 1. In particular, $dim \, Im \, N^{n-1}_B = 2$.
Note that for any monodromy invariant lattice $P_{\mathbb{Z}}
\subset \Psi_t^{an, un}( {\cal H}_{\mathbb{Z}})$ there is a unique local system ${\cal P}_{\mathbb{Z}}\subset {\cal H}_{\mathbb{Z}}$ over a punctured disk $D^*\subset C_{\mathbb{C}}$ such that $\Psi_t^{an, un}({\cal P}_{\mathbb{Z}})=P_{\mathbb{Z}}$. We apply this remark to $L_{\mathbb{Z}}= Im \, N^{n-1}_B \cap \Psi_t^{an,
un}( {\cal H}_{\mathbb{Z}})$ and to $W_{\cdot} \Psi_t^{an,un}( {\cal
H}_{\mathbb{Z}})$. Call the corresponding local systems by ${\cal
L}_{\mathbb{Z}}$ and ${\cal W}_{\cdot} {\cal H}_{\mathbb{Z}}$. We claim that $${\cal L}^{Hodge}= ( {\cal W}_{\cdot} {\cal L}_{\mathbb{Z}} \, , \, {\cal F}^{\cdot}{\cal L}),$$ where ${\cal W}_{-1} {\cal L}_{\mathbb{Z}}=0$, ${\cal W}_0 {\cal
L}_{\mathbb{Z}}
= {\cal W}_1 {\cal L}_{\mathbb{Z}}=
{\cal W}_{0} {\cal H}_{\mathbb{Z}} $, ${\cal W}_2 {\cal
L}_{\mathbb{Z}} = {\cal L}_{ \mathbb{Z}}$ and ${\cal F}^{2}{\cal
L}=0$, ${\cal F}^{1}{\cal L} =
{\cal F}^{1}\cap {\cal L}$, ${\cal F}^{0}{\cal L} = {\cal
L}= {\cal L}_{ \mathbb{Z}} \otimes {\cal O}_{D^*}$, is an admissible variation of mixed Hodge structure over a sufficiently small punctured disk $D^*$.[^14] Indeed, over a small disk we have ${\cal W}_{0} \overline {\cal H} \oplus \overline{\cal F}^1 =
\overline {\cal H} $. The claim follows.
Thus we get a class $$[{\cal L}^{Hodge}]\in Ext^1_{VMHS}({\cal L}^{Hodge}/{\cal W}_{0}{\cal
L}^{Hodge}, {\cal W}_{0}{\cal L}^{Hodge} )$$ $$\simeq Ext^1_{VMHS}(
\mathbb{Z}(-1), \mathbb{Z}(0)) \otimes Hom_{\mathbb{Z}} ( L_{\mathbb
{Z}}/ W_{0} \Psi_t^{an,un}( {\cal H}_{\mathbb{Z}}), W_{0}
\Psi_t^{an,un}( {\cal H}_{\mathbb{Z}})).$$ Define $$\label{ccc}
q_{\mathbb C} \in Ext^1_{VMHS}( \mathbb{Z}(-1), \mathbb{Z}(0))
\otimes \mathbb{Q}$$ to be the composition of $[{\cal L}^{Hodge}]$ with $$N_B^{-1}:W_{0}
\Psi_t^{an,un}( {\cal H}_{\mathbb{Z}}) \otimes \mathbb{Q} \to
L_{\mathbb{Z}}/ W_{0} \Psi_t^{an,un}( {\cal H}_{\mathbb{Z}}) \otimes \mathbb{Q}.$$ If the monodromy is small i.e. that the map $N_B: L_{\mathbb{Z}}/ W_{0} \Psi_t^{an,un}( {\cal H}_{\mathbb{Z}})
\to W_{0} \Psi_t^{an,un}( {\cal H}_{\mathbb{Z}}) $ is an isomorphism, the class $q_{\mathbb C}$ lifts canonically to $$\tilde q_{\mathbb C}\in Ext^1_{VMHS}( \mathbb{Z}(-1),
\mathbb{Z}(0)).$$ Recall that the group $Ext^1_{VMHS}(
\mathbb{Z}(-1), \mathbb{Z}(0))$ of admissible extensions is canonically identified with the group of invertible functions on $D^*$ with a regular singularity at the origin. The following lemma immediately follows from the construction.
The class $\tilde q_{\mathbb{C}}$ is equal to the canonical coordinate $q$.
We shall compute the logarithmic derivative $$q_{\mathbb{C}}\in (\mathbb{C}((t)))^* \otimes \mathbb{Q}
\stackrel{d\, log}{\longrightarrow} \mathbb{C}[[t]] \frac {dt}{t}.$$ Let $e^0$ be a nonzero parallel section of ${\cal W}_0\overline
{\cal L}$. Then there exists a unique section $e^1$ of $
{\cal F}^{1}\overline{\cal L}$ such that the projection of $e^1$ to $\overline{\cal L}/{\cal W}_0 \overline{\cal L}$ is parallel and $$- \frac{1}{2\pi i} N_B(e^1( a_{\mathbb{C}}))= N_{DR}(e^1(
a_{\mathbb{C}}))= e^0( a_{\mathbb{C}}).$$ We have then $$\label{deq}
\nabla e^1 = e^0 \otimes d\, log q_{\mathbb{C}} .$$ Assume that we are in the situation of 1.3. It follows then that $$log q_{\mathbb{C}}\in (1+ t \mathbb{Q}[[t]] ) \frac{dt}{t}.$$ Indeed, we can normalize $e^0$ such that $e^0( a_{\mathbb{Q}})\in
a_{\mathbb{Q}}^*H^n_{log}(\overline X_{\mathbb{Q}}/\overline
C_{\mathbb{Q}})$. Then $e^0 , e^1 \in H^n_{log}(\overline
X_{\mathbb{Q}}/\overline C_{\mathbb{Q}})$, and we are done by (\[deq\]).
p-adic Hodge Theory
===================
[**3.1. Fontaine-Laffaille modules.**]{} Fontaine-Laffaile modules over a scheme is a p-adic analog of variations of Hodge structure. Below we recall this notion in the special case of torsion free modules over a punctured disk. This is sufficient for our applications. [^15] The general definition can be found in \[Fa\].
Let $(D, a)$ be a formal disk over $\mathbb{Z}_p$ with a point $a: spec\,
\mathbb{Z}_p \hookrightarrow D$. We view $(D, a)$ as a logarithmic scheme (see \[Il\]). A logarithmic morphism $G: (D,a) \to (D', a')$ is morphism such that the scheme theoretical preimage of the section $a'$ is supported on $a$ i.e. $G^*(t)= t^{\prime n} f(t')$, where $t$ and $t'$ are coordinates on $D$ and $D'$ respectively, such that $t(a)= t'(a')=0$, and $f$ is an invertible function on $D'$. Denote by $\Omega ^1 (log)$ the space of 1-forms on $D$ with logarithmic singularities at $a$.
Let $\overline{\cal E}$ be a vector bundle over $D$ with a logarithmic connection $\nabla : \overline{\cal E} \to
\overline{\cal E} \otimes \Omega ^1 (log)$. For any logarithmic morphism $G: (D', a') \to (D, a)$, $G^*\overline{\cal E}$ is endowed with the induced logarithmic connection. Moreover, if two logarithmic morphisms $G$ and $G'$ are equal modulo $p$ we have a canonical parallel isomorphism $$\label{crystal}
\theta: G^*\overline{\cal E} \simeq G^{\prime *}\overline{\cal E}.$$ In coordinates $\theta: \overline{\cal E}\otimes _G \mathbb{Z}_p[[t']] \to \overline{\cal E}\otimes _{G'} \mathbb{Z}_p[[t']] $ is given by Taylor’s formula $$\theta (e\otimes 1) = \sum_{i=0}^{\infty} (\nabla_{\delta})^i e
\otimes \frac{(log(G^*(t)/G^{\prime *}(t)))^i}{i!},$$ where $\delta
= td/dt$ is the vector field. One readily checks that $\frac{(log(G^*(t)/G^{\prime *}(t)))^i}{i!} \in \mathbb{Z}_p[[t]] $ and that the series converges.
Let $\tilde F: (D, a)\to (D, a)$ be a logarithmic lifting of the Frobenius morphism (i.e. $\tilde F^*(t)= t^p (1 +p h(t) )$, where $h(t) \in \mathbb{Z}_p[[t]]$ ). A (torsion free) Fontaine-Laffaille module over the logarithmic disk $(D, a)$ amounts to the following data:\
i) a vector bundle $\overline{\cal E}$ over $D$ with a filtration by sub-bundles $$0= \overline{\cal F}^{p-1}\subset \overline {\cal F}^{p-2} \subset \cdots \subset \overline {\cal F}^0 = \overline{\cal E},$$ ii) a logarithmic connection $\nabla : \overline{\cal E} \to
\overline{\cal E} \otimes \Omega ^1 (log)$ satisfying the Griffiths transversality condition: $\nabla(\overline {\cal F}^i)\subset \overline {\cal F}^{i-1} \otimes \Omega ^1 (log)$,\
iii) a parallel morphism (“Frobenius”) $$\phi: \tilde F^*\overline{\cal E} \to \overline{\cal E}$$ with the following properties $$\phi(\tilde F^* \overline {\cal
F}^i)\subset p^i \overline{\cal E}$$ and $$\sum _i p^{-i}\phi (\tilde F^*\overline {\cal F}^i)= \overline{\cal E}.$$
[**Remark.**]{} The definition we gave above depends on the choice of a lifting $\tilde F$. Still, the categories corresponding to different liftings are canonically equivalent. To see this let $\tilde F'$ be another logarithmic lifting. By (\[crystal\]) there is a canonical parallel isomorphism $$\theta: \tilde F^*
\overline{\cal E} \simeq \tilde F^{\prime *}\overline{\cal E}.$$ The functor, that provides the equivalence, takes $(\overline{\cal E}, \overline {\cal F}^i , \nabla) $ to the same objects and sends $\phi$ to $\phi \theta ^{-1}$. The Griffiths transversality condition implies that $$\theta ^{-1} \tilde F^{\prime *} \overline {\cal F}^k
\subset \sum_{i\geq 0} \frac{p^i}{i!} \tilde F^* \overline {\cal
F}^{k-i} \subset \tilde F^* \overline{\cal E}.$$ Thus, thanks to the assumption on the range of the Hodge filtration ( $\overline
{\cal F}^{p-1}=0$ and $\overline {\cal F}^0=\overline{\cal E}$), $(\overline{\cal E}, \overline {\cal F}^i , \nabla, \phi \theta ^{-1}) $ satisfies the requirements in iii). Denote the category of Fontaine-Laffaille modules over $(D, a)$ by $MF_{[0,p-2]}(D^*)$. A similar construction is used to define the pullback functor $$\label{pullback}
G^*:MF_{[0,p-2]}(D^*)\to MF_{[0,p-2]}(D^{'*}),$$ for a logarithmic morphism $G: (D', a') \to (D, a)$.
[**3.2. Dwork’s Lemma.**]{} Denote by $\mathbb{Z}_p(-k)$, ($k\geq 0$), the constant variation: $\overline{\cal E}= {\cal O}$, $\overline{\cal F}^k=\overline{\cal E}$, $\overline{\cal
F}^{k+1}=0$, $\phi = p^k Id$. Let $ {\cal O}(D^*)$ be the space of functions on the punctured disk (i.e. $ {\cal O}(D^*)=
\mathbb{Z}_p((t))$).
The group $Ext^1_{MF_{[0,p-2]}(D^*)}( \mathbb{Z}_p(-1),
\mathbb{Z}_p(0))$ is canonically isomorphic (i.e. the isomorphism does not depend on the lifting $\tilde F$) to p-adic completion of the group $ {\cal O}^*(D^*)$: $$\label{ext}
\hat {\cal O}^*(D^*):=\underset \longleftarrow \lim {\cal
O}^*(D^*)/({\cal O}^*(D^*)) ^{p^i} \stackrel{\sim}{\to} Ext^1(
\mathbb{Z}_p(-1), \mathbb{Z}_p(0)).$$
Let $t$ be a coordinate on $D$, and let $\tilde F:D\to D $ send $t$ to $t^p$. Consider an extension $(\overline{\cal E}, \overline {\cal F}^i, \phi)$: $$0\to \mathbb{Z}_p(0) \to \overline{\cal E} \to \mathbb{Z}_p(-1) \to 0$$ Note that $\overline {\cal F}^0=\overline{\cal E}, \overline {\cal
F}^2=0$, and that last map in the exact sequence defines an isomorphism $\overline {\cal F}^1\simeq {\cal O}_{D}$. Let $e_1 \in
\overline {\cal F}^1$ be the preimage of $1$ under the above isomorphism, and let $e_0\in \overline{\cal E}$ be the image of $1$ under the first map in the exact sequence. Then $e_0, e_1$ form a basis for $\overline{\cal E}$. We have: $$\nabla e_0=0, \nabla e_1 = e_0 \otimes \omega$$ $$\phi (\tilde F^* (e_0)) = e_0, \phi( \tilde F^*( e_1))= p e_1 + p h e_0 ,$$ for some $\omega \in \Omega ^1 (log )$ and $h\in {\cal O}(D)$. Since $\phi$ is parallel, $\phi \nabla = \nabla \phi$. This amounts to the following equation $$1/p\tilde F^*\omega - \omega = dh .$$ Thus the set of extensions is in a bijection with the set of pairs $(\omega, h)$ satisfying the above equation. One can easily see that the above bijection is compatible with the group structure [^16] Define a homomorphism $$\label{reg}
{\cal O}^*(D^*) \to Ext^1_{MF_{[0,p-2]}(D^*)}( \mathbb{Z}_p(-1), \mathbb{Z}_p(0)).$$ sending an invertible function $q$ to the pair $( d\,log q,
\frac{1}{p} log \frac{\tilde F^*q}{q^p})$. One readily sees that (\[reg\]) extends to the p-adic completion of ${\cal O}^*(D^*)$. This is the map in (\[ext\]). The injectivity of (\[ext\]) is clear from the definition and the surjectivity is the content of the Dwork’s lemma [^17]. Let us check that (\[ext\]) is independent of the choice of the coordinate. Indeed, let $t'$ be another coordinate and let $\tilde
F'$ be the corresponding lifting of the Frobenius. The isomorphism (\[crystal\]): $$\theta: \tilde F ^*\overline{\cal E} \simeq \tilde F ^{\prime *}\overline{\cal E}$$ takes $\tilde F^*( e_0)$ to $\tilde F^{\prime *}( e_0)$ and $\tilde
F^* (e_1)$ to $$\sum_{i=0}^{\infty}\frac{(log(\tilde F^*(t)/\tilde F^{\prime
*}(t)))^i}{i!}\,
\tilde F^{\prime *}((\nabla_{\delta})^i e_1)
=\tilde F^{\prime *} e_1 + log \frac{\tilde F^* (q)}{\tilde F^{\prime
*}(q)}\, e_0 . \footnote{The last equality follows from the multiplicative version of Taylor's formula
$f(e^b a) = (exp(b \delta )(f))(a)= f(a)+ \delta f (a) b + \frac{\delta ^2 f(a)}{2!} b^2 +\cdots
$.}$$ The claim follows.
[**3.3. Limit Fontaine-Laffaille module.**]{} Let $t$ be a coordinate, $\tilde F$ the corresponding lifting of the Frobenius, and let $(\overline{\cal E}, \overline{\cal F}^{\cdot}, \nabla , \phi) \in
MF_{[0, p-2]}(D^*)$ be a Fontaine-Laffaille module. We define $$\Psi^{FL}_t((\overline{\cal E}, \overline{\cal F}^{\cdot}, \nabla , \phi))=
(E= a^*\overline{\cal E}, F^{\cdot}= a^*\overline{\cal F}^{\cdot},
\phi_a).$$ $\Psi^{FL}_t((\overline{\cal E}, \overline{\cal
F}^{\cdot}, \nabla , \phi))$ is a Fontaine-Laffaille module over the point. The residue of $\nabla$ is a morphism of Fontaine-Laffaille modules: $$N_{DR}= Res \, \nabla : \Psi^{FL}_t((\overline{\cal E}, \overline{\cal F}^{\cdot}, \nabla ,
\phi))\to \Psi^{FL}_t((\overline{\cal E}, \overline{\cal F}^{\cdot},
\nabla , \phi))(-1).$$ In particular, $$\label{rel}
N_{DR} \phi_a =p \phi_a N_{DR}.$$
[**Remark.**]{} The functor $
\Psi^{FL}_t$ depends on the choice of a coordinate $t$. If $t'= bt +\cdots$ , $b\in \mathbb{Z}_p^*$ is another coordinate we have $$\Psi^{FL}_{t'}((\overline{\cal E}, \overline{\cal F}^{\cdot}, \nabla ,
\phi \theta^{-1}))\simeq (E, F^{\cdot}, \phi_a exp(N_{DR} log\,
b^{p-1} )).$$ In particular, $ \Psi^{FL}_t$ does not get changed if we replace $t$ by $t'$ with the same derivative.
[**3.4. p-adic canonical coordinate.**]{} Let $\overline{\pi}:
\overline X\to D$ be a proper semi-stable morphism. For any $k<p-1$, Faltings constructed in \[Fa\] a Fontaine-Laffaille structure on the logarithmic De Rham cohomology $H_{log}^k(\overline X/D)$. In the rest of this section we assume that $n:= dim_ D \overline X <p-1 $, and let $\overline{\cal E}: = H_{log}^{n} (\overline X/D)/
\textstyle{p-torsion} \in MF_{[0, p-2]}(D^*)$. The cup product $H_{log}^{n}(\overline X/D) \otimes H_{log}^{n}(\overline X/D) \to H_{log}^{2 n}(\overline X/D)
\simeq \mathbb{Z}_p(-n) $ induces a perfect paring $$\label{paring}
<\cdot, \cdot>_{DR}: \overline{\cal E}\otimes \overline{\cal E} \to
\mathbb{Z}_p(-n).$$ In particular, $$\label{reltwo}
<\phi(\tilde F^* v) , \phi (\tilde F^*u)>_{DR}= p^n< v , u>_{DR}.$$ Assume that $X$ is a Calabi-Yau scheme over $D^*$ and that $a$ is the maximal degeneracy point. That means, by definition, that $dim \, F^n \otimes \mathbb{Q} =1$ and the operator $N_{DR}^n: E\to
E$ is not equal to $0$. Assume, in addition, that $\overline{\pi}:
\overline X \to D$ extends to a semi-stable scheme over a curve. We have then $$\label{pmor}
rk\, Im\, N_{DR}^n =1 \, , \, rk \, Im \, N_{DR}^{n-1} =2.$$ This follows from Lemma \[M\] and “the Lefschetz principle”.
\[Fr\] The Frobenius operator $\phi_a$ restricted to $Im\, N_{DR}^n$ is equal to $\pm Id$.
The lemma follows immediately from (\[rel\]) and (\[reltwo\]).
The above lemma implies the existence of a parallel section of $\overline{\cal E}$. Namely we have the following result.
\[fs\] Let $\overline{\cal E}$ be a vector bundle over $D$ with a logarithmic connection and $\phi: \tilde F ^* \overline{\cal E} \to \overline{\cal E}$ be a parallel morphism. For any element $w\in E$ such that $\phi_a (w)=\pm w$ there exists a unique parallel section $s $ of $\overline{\cal E}$ with $s(a)=w$. The section $s$ satisfies the property $\phi(\tilde F^* s) =\pm s$.
The uniqueness part is clear. To prove the existence we start with any section $s'$ of $\overline{\cal E}$ with $s'(a)=w$ and consider the sections $s'_k= (\phi \tilde F^*)^{2k}(s')$. It is easy to see that $\nabla s'_k \in p^{2k} \overline{\cal E} \otimes \Omega ^1
(log)$ and that $s'_{k}(a)= w$. This implies that the limit $$s= \underset \longrightarrow \lim s'_{k}$$ exists and satisfies all the required properties.
The nilpotent operator $N_{DR}: E\to E $ gives rise to a canonical filtration $W_0= W_1 \subset W_2 \subset \cdots \subset W_{2n}=E$ by Fontaine-Laffaille submodules. It is a unique filtration with torsion free quotients $W_{i+1}/W_i$ such that $W_{\cdot}\otimes
\mathbb{Q}_p$ is the monodromy filtration on $E\otimes
\mathbb{Q}_p$. The Frobenius $\phi$ preserves the filtration $W_{\cdot}$ and $N_{DR}(W_i)\subset W_{i-2}$. Let $L^{FL}: = Im\,
N_{DR}^{n-1} \otimes \mathbb{Q}_p\cap E$. This is a Fontaine-Laffaille submodule of $E$. It follows from (\[Fr\]) that the eigenvalue of $\phi$ on $W_0$ (resp. $L^{FL}/W_0$ ) is equal to $\pm 1$ (resp. $\pm p$). Lemma (\[fs\]) implies that the inclusion $W_0\hookrightarrow E$ extends uniquely to a parallel morphism ${\cal W}_0:= W_0 \otimes _{\mathbb{Z}_p} {\cal O}_D \hookrightarrow
\overline{\cal E}$. Note that the projection ${\cal W}_0
\hookrightarrow \overline{\cal E} \to \overline{\cal
F}^0/\overline{\cal F}^1$ is an isomorphism. Thus the Frobenius $\phi :\tilde F^* (\overline{\cal E}/{\cal W}_0) \to \overline{\cal
E}/{\cal W}_0 $ is divisible by $p$. Applying (\[fs\]) again to $\frac{\phi}{p}: \tilde F^*(\overline{\cal E}/{\cal W}_0) \to
\overline{\cal E}/{\cal W}_0$ we conclude that the inclusion $L^{FL}/ W_0\hookrightarrow E/W_0$ extends uniquely to a parallel morphism $L^{FL} / W_0 \otimes _{\mathbb{Z}_p }{\cal O}_D
\hookrightarrow \overline{\cal E}/ {\cal W}_0$. Finally, let ${\cal
L}^{FL}\subset \overline{\cal E}$ be the preimage of $L^{FL}/ W_0
\otimes _{\mathbb{Z}_p} {\cal O}_D$ in $\overline{\cal E}$. By construction, ${\cal L}^{FL}$ is a unique Fontaine-Laffaille submodule of $\overline{\cal E}$ with $\Psi^{FL}_t({\cal L}^{FL})=
L^{FL}$. Thus we get a canonical class $$[{\cal L}^{FL}]\in
Ext^1_{MF_{[0, p-2]}(D^*)}( {\cal L}^{FL}/{\cal W}_0 , {\cal
W}_0)\simeq$$ $$Ext^1_{MF_{[0, p-2]}(D^*)}( \mathbb{Z}_p(-1) ,
\mathbb{Z}_p(0))\otimes Hom_{\mathbb{Z}_p}(L^{FL}/W_0, W_0).$$ Composing this with $N_{DR}^{-1}\in Hom( W_0 \otimes \mathbb{Q}_p,
L^{FL}/ W_0 \otimes \mathbb{Q}_p )$ we get the “p-adic canonical coordinate”: $$\label{pcc}
q_{\mathbb{Z}_p}\in Ext^1_{MF_{[0, p-2]}(D^*)}( \mathbb{Z}_p(-1) ,
\mathbb{Z}_p(0)) \otimes _{\mathbb{Z}_p} {\mathbb{Q}_p}\simeq \hat
{\cal O}^*(D^*) \otimes _{\mathbb{Z}_p} {\mathbb{Q}_p}.$$ Observe that the order $$ord:\, \hat {\cal O}^*(D^*) \otimes _{\mathbb{Z}_p} {\mathbb{Q}_p} \to \mathbb{Q}_p$$ of $q_{\mathbb{Z}_p}$ is equal to $1$. In particular, $q_{\mathbb{Z}_p}\in {\cal O}^*(D) \otimes _{\mathbb{Z}}
{\mathbb{Q}}$.
Let $e^0$ be a nonzero parallel section of ${\cal W}_0\otimes
\mathbb{Q}_p$ and let $e^1$ be a section of $(\overline{\cal F}^1
\cap {\cal L}^{FL})\otimes \mathbb{Q}_p $ whose projection to $({\cal L}^{FL}/{\cal W}_0)\otimes \mathbb{Q}_p$ is parallel and such that $N_{DR}(e^1(a))= e^0(a)$. We then have $$\label{pdeq}
\nabla e^1 = e^0 \otimes d\, log q_{\mathbb{Z}_p} .$$
We shall [*the p-adic monodromy is small*]{}, if the operator $N_{DR}: L^{FL}/W_0 \to W_0 $ is an isomorphism. If this is the case, one has $$\label{pint}
q_{\mathbb{Z}_p}\in {\cal O}^*(D^*)/\mu_{p-1} \subset \hat {\cal
O}^*(D^*) \otimes \mathbb{Q}_p .$$
[**3.5. p-adic Yukawa map.**]{} In this subsection we assume that the p-adic monodromy is small. Denote by $q\in {\cal O}_{D}$ the p-adic canonical coordinate (defined up to a (p-1)th root of unity). Let $$\label{ks}
S^{n}T_{D, log} \to Hom _{{\cal O}_D} (\overline{\cal F}^n,
\overline{\cal F}^0/\overline{\cal F}^1)\simeq (\overline{\cal F}^n
\otimes \overline{\cal F}^n)^*$$ be the Kodaira-Spenser morphism. Here $T_{D, log}$ denotes the sheaf dual to $\Omega^1(log)$ i.e. the sheaf of vector fields on $D$ vanishing at $a$. Choose a generator $e^0 $ of ${\cal W}_0^{\nabla}$ and let $e_0\in \overline{\cal F}^n$ be a section with $(e^0, e_0)=
1$ [^18]. Applying (\[ks\]) to $(q \frac{d}{dq})^{\otimes ^n} $ and pairing the result with $e_0\otimes e_0 $ we obtain the p-adic Yukawa function $Y \in {\cal
O}_D$. Observe that $Y(q)$ is well defined up to multiplication by a constant in $\mathbb{Z}_p^*$.
\[kont\] Assume that $n=3$ and that $rk\, E = 4$. Then $$Y(q)= n_0 + \sum^{\infty}_{d=1}n_d d^3 \frac{q^d}{1-q^d},$$ where $n_d \in \mathbb{Z}_p$.
We shall use the following elementary result:
\[KSV\] (\[KSV\]. Lemma 2.) Assume that a formal power series $Y(q)\in \mathbb{Z}_p[[q]]$ is written in the form $$Y(q)= \sum^{\infty}_{d=1}n_d d^3 \frac{q^d}{1-q^d}.$$ Then $n_d \in \mathbb{Z}_p$ if and only if $Y(q)- Y(q^p)=
\delta^3(\psi(q))$, for some $\psi(q)\in \mathbb{Z}_p[[q]]$. Here $\delta= q\frac{d}{dq}$.
\[ht\] The monodromy filtration $W_0= W_1 \subset W_2= W_3 \subset W_4=W_5 \subset W_6= E$, extends to a filtration ${\cal W}_i \subset \overline{\cal E} $ by Fontaine-Laffaille submodules such that either ${\cal W}_{2i}/{\cal
W}_{2i-2} \simeq \mathbb{Z}_p(-i)$, for all $0\leq i \leq 3$, or ${\cal W}_{2i}/{\cal W}_{2i-2} \simeq \epsilon \mathbb{Z}_p(-i)$. Here $\epsilon \mathbb{Z}_p(-i)$ denotes the constant Fontaine-Laffaille module with $\overline{\cal F}^i={\cal O}_D$, $\overline{\cal F}^{i+1}=0$, $\phi = -p^i Id$.
Our assumptions imply that $rk\, W_{2i}/W_{2i-2}= 1$, for $0\leq i
\leq 3$. Thus, by Lemma \[Fr\] and (\[rel\]), the operator $\phi$ acts on $W_{2i}/W_{2i-2}$ as $\pm p^i Id$.
We prove by induction on $i$ that $W_{2i}\subset E$ extends to a subbundle ${\cal W}_{2i}$ of $\overline{\cal E}$ preserved by the connection and that $\overline{\cal F}^{i+1}\oplus {\cal W}_{2i}=
\overline{\cal E}$. Indeed, for $i=-1$, there is nothing to prove. Assume that we know the result for $i=k$. Then $(\overline{\cal
E}/{\cal W}_{2k}=(\overline{\cal F}^{k+1}+{\cal W}_{2k})/ {\cal
W}_{2k}\supset \cdots \supset (\overline{\cal F}^{3}+{\cal W}_{2k})/
{\cal W}_{2k}, \phi, \nabla) $ is a Fontaine-Laffaille module. Applying Lemma \[fs\] to $\overline{\cal E}/{\cal W}_{2k} \otimes
\mathbb{Z}_p(k+1)$ we see that $W_{2k+2}/W_{2k}\subset E/W_{2k}$ extends to a subbundle of ${\cal W}_{2k+2}/{\cal W}_{2k}\subset
\overline{\cal E}/{\cal W}_{2k}$. It remains to show that ${\cal
W}_{2k+2}/{\cal W}_{2k}\oplus (\overline{\cal F}^{k+2}+{\cal
W}_{2k})/ {\cal W}_{2k}= \overline{\cal E}/{\cal W}_{2k}$. We will be done if we prove that this is true over the closed point of $D$ i.e. $(W_{2k+2}/{W}_{2k}\oplus (F^{k+2}+ W_{2k})/ {W}_{2k})\otimes
\mathbb{F}_p = (E/W_{2k})\otimes \mathbb{F}_p$. Indeed, the operator $p^{-k-1} \phi$ induces an action on $ (E/W_{2k})\otimes
\mathbb{F}_p$ which is $0$ on $ ((F^{k+2}+ W_{2k})/ {W}_{2k})\otimes \mathbb{F}_p$ and invertible on $(W_{2k+2}/{W}_{2k})\otimes \mathbb{F}_p$. The claim follows.
For the rest of the proof we assume that $\phi$ acts on $W_0$ as $+Id$. The other alternative is considered in a similar way.
By the definition of the canonical coordinate $q$ we can find sections $e^0\in {\cal W}_0$, $e^1\in \overline{\cal F}^1\cap {\cal
W}_2$ such that $$\nabla_\delta e^0 = 0, \, \phi e^0= e^0, \, \nabla_\delta e^1 =
e^0, \, \phi e^1 = p e^1,$$ and such that $e^0, e^1$ generate ${\cal W}_2$. Next, it follows from Lemma \[ht\] that there exist unique $e_1 \in \overline{\cal
F}^2\cap {\cal W}_4$, $e_0 \in \overline{\cal F}^3 $ such that $$(e^0, e_0)= 1, \, (e^1, e_1)= -1$$ Observe that $e^i, e_i$ generate $\overline{\cal E}$. Thanks to the self-duality condition (\[paring\]) we have $$\nabla_{\delta} e_1 = Y(q)e^1 , \, \nabla_{\delta} e_0= e_1$$ $$\phi e_1 = p^2(e_1 + m_{23}(q) e^1 + m_{13}(q) e^0), \, \phi e_0=
p^3(e_0 - m_{13}(q) e^1 + m_{14}(q) e^0),$$ where $Y(q)$ is the Yukawa function. Finally, the relation $\nabla_{\delta} \phi = p \phi \nabla_\delta $ amounts to $$Y(q)- Y(q^p)= \delta (m_{23}), \, m_{23}= -\delta(m_{13}), \, \delta (m_{14})= 2 m_{13}.$$ Thus $$Y(q^p) - Y(q) = \frac{1}{2} \delta ^3 m_{14},$$ and we are done by Lemma \[KSV\].
Comparison
==========
[**4.1. Plan of the proofs of Theorems \[th1\] and \[th2\].**]{}
Let $\overline \pi: \overline X_S \to \overline C_S$ be a semi-stable morphism satisfying the conditions i) - iii) from Section 1.3. Denote by $q_{\mathbb{C}}\in (\mathbb{C}((t)))^*
\otimes _{\mathbb{Z}} {\mathbb{Q}}$ the complex canonical coordinate (\[ccc\]) and by $q_{\mathbb{Z}_p} \in (\mathbb{Z}_p((t)))^*
\otimes _{\mathbb{Z}} {\mathbb{Q}}$ the p-adic one (\[pcc\]).
\[dif\] a) For every prime prime $p$ such that $(p,N)=1$, we have $$q_{\mathbb{C} }= q_{\mathbb{Z}_p} \in ({\mathbb{Q}}((t)))^* \otimes \mathbb{Q}.$$ b) Assume that the Betti monodromy of the family $\overline X_S \to
\overline C_S$ is small (see 1.1 ). Then, for every prime $p$ with $(p,N)=1$, the p-adic monodromy is also small (see 3.4 ).\
c) Let $\omega $ be a nonvanishing section of the line bundle $\overline {\cal F}^n= \overline \pi_* \Omega_{\overline
X_S/\overline C_S}^n(log \, Y_S)$ over an open neighborhood of the subscheme $a: S \hookrightarrow \overline C_S $. Then $$(\frac{1}{(2\pi i)^n} \int
_{\delta _1} \omega)^2 \in (\mathbb{Z}[N^{-1}][[t]])^*.$$
In the remaining part of this section we complete the proofs of Theorems \[th1\] and \[th2\] assuming Proposition \[dif\]. A proof of the proposition (which is the hardest technical part of the argument) is given in Sections 4.2-4.5.
[**Proof of Theorem \[th1\].**]{} Since $q'(0)\in \mathbb{Q}^*$ and $ d\, log \, q (t) \in \mathbb{Q}[[t]] \frac{dt}{t}$ the coefficients of $q(t)$ are rational numbers. On the other hand, parts a) and b) of Proposition \[dif\] together with formula (\[pint\]) show that, for every prime $p$ such that $(p,N)=1$, $$q(t)\in (\mathbb{Z}_p((t)))^* \cap (\mathbb{Q}((t)))^* \subset
(\mathbb{Q}_p((t)))^*.$$ This completes the proof.
[**Proof of Theorem \[th2\].**]{} Let $\omega \times \omega$ be a local section of $\overline \pi_* \Omega_{\overline
X_{\mathbb{C}}/\overline C_{\mathbb{C}}}^n(log \, Y_{\mathbb{C}})
\otimes \overline \pi_* \Omega_{\overline X_{\mathbb{C}}/\overline
C_{\mathbb{C}}}^n(log \, Y_{\mathbb{C}})$ defined by the equation $$\frac{1}{(2\pi i)^n} \int
_{\delta _1} \omega =1.$$ Part c) of Proposition \[dif\] shows that $\omega \times \omega$ yields a nonvanishing section of $
\overline \pi_* \Omega_{\overline X_S/\overline C_S}^n(log \,
Y_S)\otimes \overline \pi_* \Omega_{\overline X_S/\overline
C_S}^n(log \, Y_S)$ over the formal neighborhood $D_S$. This together with Theorem \[th1\] imply that the coefficients of the Yukawa function $Y(q)$ are rational numbers and so are the instanton numbers $n_d$. It also follows that $Y(q)$ coincides (up to a constant factor in $\mathbb{Z}_p^*$) with the p-adic Yukawa function from [**3.5**]{}. Thus by Proposition \[kont\] the numbers $n_d$ are p-adic integers. This completes the proof.
[**4.2. Recollections on p-adic Comparison Theorem.**]{} Recall from \[FL\] that there is an exact tensor fully faithful functor $$U: MF_{[0,p-2]} \to Rep(\Gamma)$$ from the category $MF_{[0,p-2]}$ of Fontaine-Laffaille modules over $spec \, \mathbb{Z}_p$ to the category $Rep(\Gamma)$ of finitely generated $\mathbb{Z}_p$-modules equipped with an action of the Galois group $\Gamma =Gal(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$. We will use the following properties of $U$:
1\) $U$ takes a finite (as a plain abelian group) Fontaine-Laffaille module to a $\Gamma $-module of the same finite order.
2\) $U(\mathbb{Z}_p(i))= \mathbb{Z}_p(i)$ and the induced morphism $$\underset \longleftarrow \lim \mathbb{Z}_p ^* /( \mathbb{Z}_p ^* ) ^{p^i} \simeq
Ext^1_{MF_{[0,p-2]}}(\mathbb{Z}_p(-1), \mathbb{Z}_p(0)) \stackrel{U}{\hookrightarrow}
Ext^1_{Rep(\Gamma)}(\mathbb{Z}_p(-1), \mathbb{Z}_p(0))$$ $$\stackrel{Kummer}{\simeq}
\underset \longleftarrow \lim \mathbb{Q}_p ^* /( \mathbb{Q}_p ^* ) ^{p^i}$$ is identity.
3)Let $\overline {\pi}: \overline X_{\mathbb{Z}_p} \to \overline
C_{\mathbb{Z}_p} $ be a proper semi-stable (relative to $\mathbb{Z}_p$ ) scheme. Assume that $dim _{ \overline C_{\mathbb{Z}_p}} \overline X_{\mathbb{Z}_p}
\leq p-2$. Then there is a canonical isomorphism: $$U(\Psi^{FL}_t(H^k_{log}( \overline X_{\mathbb{Z}_p}/\overline C_{\mathbb{Z}_p})) ) \stackrel{\simeq}{\longrightarrow}
\Psi_{t}^{et} (R^k \pi_{\mathbb{Q}_p *}^{et} \mathbb{Z}_p)$$ Here $\pi_{\mathbb{Q}_p}$ denotes the projection $X_{\mathbb{Q}_p} \to C_{\mathbb{Q}_p}$ and $\Psi^{et}_{t}: Sh^{et}(C_{\mathbb{Q}_p}) \to
Sh^{et}(a_{\mathbb{Q}_p})= Rep(\Gamma) $ is the etale vanishing cycles functor. Moreover, we have the following commutative diagram
$$\def\normalbaselines{\baselineskip20pt
\lineskip3pt \lineskiplimit3pt}
\def\mapright#1{\smash{
\mathop{\to}\limits^{#1}}}
\def\mapdown#1{\Big\downarrow\rlap
{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\begin{matrix}
U(\Psi^{FL}_t(H^k_{log}( \overline X_{\mathbb{Z}_p}/\overline C_{\mathbb{Z}_p})) ) & \stackrel{\simeq}{\longrightarrow}
& \Psi_{t}^{et} (R^k \pi_{\mathbb{Q}_p *}^{et} \mathbb{Z}_p) \cr
\mapdown{N_{DR}} & &\mapdown{N_{et}} \cr
U(\Psi^{FL}_{t}(H^k_{log}( \overline X_{\mathbb{Z}_p}/\overline C_{\mathbb{Z}_p}))_a )\otimes \mathbb{Z}_p(-1) & \stackrel{\simeq}{\longrightarrow}
& \Psi_{t}^{et} (R^k \pi_{\mathbb{Q}_p *}^{et} \mathbb{Z}_p) \otimes \mathbb{Z}_p(-1)
\end{matrix}$$ This follows from the main Comparison Theorem in \[Fa\].
[**4.3. 1-motives, the motivic Albanese functor $LAlb$.**]{} The main references here are \[D3\] and \[BK\]. Let $k$ be a field of characteristic $0$. Fix an algebraic closure $\overline k \supset
k$. A 1-motive over $k$ is a triple $$M = (\Lambda, G, \Lambda \stackrel{u}{\longrightarrow} G(\overline{k})),$$ where $\Lambda$ is a free abelian group of finite rank equipped with an action of the Galois group $Gal(\overline k/k)$ that factors through a finite quotient, $G$ is an semi-abelian variety over $k$ i.e. an extension $$0\to T \to G \to A\to 0$$ of an abelian variety by a torus, and $u$ is a homomorphism of the Galois modules. We shall denote by ${\cal M}_1(k)$ the additive category of 1-motives [^19].
Every 1-motive is equipped with a canonical (weight) filtration: $$W_{-2}M = (0, T) \subset W_{-1}M = (0, G) \subset W_0 M= M.$$ Thus $W_0 M/W_{-1} M = (\Lambda, 0)$ and $W_{-1} M/W_{-2} M = (0,
A)$. The category ${\cal M}_1(k; \mathbb{Q}):= {\cal M}_1(k) \otimes
\mathbb{Q} $ is abelian (\[BK\], Proposition 1.1.5) and any morphism in ${\cal M}_1(k; \mathbb{Q})$ is strictly compatible with the weight filtration.
Set $\mathbb{Z}(0) = (\mathbb{Z}, 0)$ and $\mathbb{Z}(1) = (0,
\mathbb{G}_m)$. The same 1-motives $\mathbb{Z}(i)$ ($i=0,1$) but viewed as objects of ${\cal M}_1(k; \mathbb{Q})$ are denoted by $\mathbb{Q}(i)$. We have $$\label{tateext}
Ext^1_{{\cal M}_1(k)}(\mathbb{Z}(0), \mathbb{Z}(1)) \simeq k^*, \,
Ext^1_{{\cal M}_1(k; \mathbb{Q})}(\mathbb{Q}(0), \mathbb{Q}(1)) \simeq k^* \otimes \mathbb{Q}.$$ For any prime $p$, we have the etale realizations functors (\[D3\], 10.1.5): $$T_{\mathbb{Z}_p}^{et}: {\cal M}_1(k) \to Rep_{\mathbb{Z}_p}(Gal(\overline k/k)),$$ $$T_{\mathbb{Q}_p}^{et}: {\cal M}_1(k; \mathbb{Q}) \to
Rep_{\mathbb{Q}_p}(Gal(\overline k/k)).$$ We also set $$T_{\mathbb{Q}_p}^{*et}(M)= Hom_{\mathbb{Q}_p}( T_{\mathbb{Q}_p}^{et}(M), \mathbb{Q}_p)\in
Rep_{\mathbb{Q}_p}(Gal(\overline k/k)).$$
If $k=\mathbb{C}$ the category ${\cal M}_1(\mathbb{C})$ is equivalent to the category of torsion free polarizable mixed Hodge structures of type $\{(0,0),(0,-1),(-1, 0),(-1,-1)\}$ (\[D3\], 10.1.3): $$\label{DeRham}
T^{Hodge}: {\cal M}_1(\mathbb{C})
\stackrel{\simeq}{\longrightarrow} MHS_1$$ For $k\subset \mathbb{C}$, $M\in {\cal M}_1(k)$, $T^{Hodge}(M\times _k spec \, \mathbb{C})= (W_{\cdot}\subset
V_{\mathbb{Z}}, F^{\cdot}\subset V_{\mathbb{C}})$ there is a functorial isomorphism of $\mathbb{Z}_p$-modules $$\label{be}
V_{\mathbb{Z}}\otimes \mathbb{Z}_p \simeq T_{\mathbb{Z}_p}^{et}(M).$$ Abusing notation, we shall also denote by $T^{Hodge}$ the equivalence $${\cal M}_1(\mathbb{C};
\mathbb{Q})\stackrel{\simeq}{\longrightarrow} MHS_1^{\mathbb Q}=
MHS_1\otimes \mathbb{Q}$$ induced by (\[DeRham\]) and the corresponding equivalence of the derived categories $$D^b({\cal M}_1(\mathbb{C};
\mathbb{Q}))\stackrel{\simeq}{\longrightarrow} D^b(MHS_1^{\mathbb
Q}).$$ Let $$D^b({\cal M}_1(k; \mathbb{Q})) \stackrel{{\cal S}}{\hookrightarrow} DM^{eff}_{gm}(k; \mathbb{Q})$$ be embedding of the bounded derived category of 1-motives into the triangulated category of Voevodsky motives (\[O\]). By \[BK\] ${\cal S}$ has a left adjoint functor: $$LAlb: DM^{eff}_{gm}(k; \mathbb{Q}) \to D^b({\cal M}_1(k; \mathbb{Q})).$$ Denote by $MHS^{ \mathbb{Q}}$ the category of mixed polarizable Hodge structures over ${\mathbb Q}$ and by $MHS_{eff}^{\mathbb{Q}}$ the full subcategory of $MHS^{ \mathbb{Q}}$, whose objects are mixed Hodge structures $(W_{\cdot}\subset V_{\mathbb{Q}}, F^{\cdot}\subset
V_{\mathbb{C}})$ with $ F^1=0$. It is proven in \[Vol\] that embedding of the derived categories $$\overline {\cal S}: D^b(MHS_1^{\mathbb Q}) \to D^b(MHS_{eff}^{ \mathbb{Q}})$$ admits a $t$-exact left adjoint functor [^20] $$\overline {LAlb}: D^b(MHS_{eff}^{\mathbb{Q}}) \to D^b(MHS_1^{\mathbb Q})$$ and that $$T^{Hodge}\circ LAlb \simeq \overline{LAlb} \circ R^{Hodge}:
DM^{eff}_{gm}(\mathbb{C}; \mathbb{Q}) \to D^b(MHS_1^{\mathbb Q}).$$ Here $$R^{Hodge}: DM^{eff}_{gm}(\mathbb{C} ; \mathbb{Q}) \to D^b(MHS^{\mathbb
Q}_{eff})$$ is the homological Hodge realization functor (i.e. $R^{Hodge}(M)= R_{Hodge}(M)^*$, where $R_{Hodge}$ is Huber’s cohomological realization (\[Hu1\], \[Hu2\]).)
[**4.4. Motivic vanishing cycles.**]{} Let $X_k
\stackrel{\pi}{\longrightarrow} C_k $ be a smooth proper scheme over a punctured curve $ C_k \hookrightarrow \overline C_k
\stackrel{a}{\hookleftarrow} spec \, k$ over a field $k\subset
\mathbb{C}$. Fix a local coordinate $t$ at $a$ and an integer $m\geq 0$. Denote by $$H^m(X_{\mathbb{C}}/C_{\mathbb{C}})= ( R^m\pi_*
\mathbb{Q},F^{\cdot}\subset
H^m_{DR}(X_{\mathbb{C}}/C_{\mathbb{C}}))$$ the variation of Hodge structure associated to the family $X_{\mathbb{C}}
\stackrel{\pi_{\mathbb{C}}}{\longrightarrow} C_{\mathbb{C}} $ and by $ H_m(X_{\mathbb{C}}/C_{\mathbb{C}})$ the dual variation. Let $\Psi_t^{Hodge, un}( H_m(X_{\mathbb{C}}/C_{\mathbb{C}}))$ be the unipotent limiting mixed Hodge structure. The Hodge structure $$\overline {LAlb} (\Psi_t^{Hodge, un}(
H_m(X_{\mathbb{C}}/C_{\mathbb{C}})))$$ can be viewed as a 1-motive over $\mathbb{C}$. In this subsection we explain how this 1-motive canonically descends to a 1-motive $$M_{t,m}(X_k)= (\Lambda, G,
\Lambda \stackrel{u}{\longrightarrow} G(\overline{k}))\in {\cal
M}_1(k; \mathbb{Q})$$ over $k$. In addition, $M_{t,m}(X_k)$ comes equipped with a “monodromy" homomorphism $N: \Lambda \otimes
\mathbb{Q} \to T_* \otimes \mathbb{Q} $ of $Gal(\overline
k/k)$-modules. Here $T_*$ denotes the group of cocharacters of the torus $T$: $T_*= \underline {Hom}(\mathbb{G}_m, T)(\overline k)$. Equivalently, $N$ can be viewed as a morphism of 1-motives: $$N: (W_0 M_{t,m}(X_k) /W_{-1} M_{t,m}(X_k))(1) \to W_{-2} M_{t,m}(X_k).$$ The main properties of $M_{t,m}(X_k)$ are the following.
1\) There is a natural isomorphism: $$\label{c1}
\overline {LAlb} (\Psi_t ^{Hodge, un}(
H_m(X_{\mathbb{C}}/C_{\mathbb{C}})))\simeq T^{Hodge}(
M_{t,m}(X_k))\footnote{Abusing notation, we denote by $T^{Hodge}$
the composition of functors ${\cal M}_1(k; \mathbb{Q})\to {\cal
M}_1(\mathbb{C}; \mathbb{Q})
\stackrel{T^{Hodge}}{\longrightarrow}MHS_1^{\mathbb Q}$.}$$ compatible with the monodromy action.
2\) There is a natural morphism $Gal(\overline k/k)$-modules $$\label{c2}
\alpha: T^{*et}_{ \mathbb{Q}_p}(M_{t,m}(X_k)) \to \Psi_t^{et,
un}(R^m \pi_*^{et}\mathbb{Q}_p)$$ where $$\Psi_t^{et, un}: Sh^{et}(C_k) \to Sh^{et}(spec\, k)\to Rep_{\mathbb{Q}_p}(Gal(\overline k/k))$$ denotes the functor of unipotent vanishing cycles (see \[B\]). The morphism $\alpha $ commutes with the monodromy action. 3) If $k'\supset k$ is any field extension, there is a natural isomorphism $$\label{c3}
M_{t,m}(X_k \times _k spec\, k')\simeq M_{t,m}(X_k)\times _k spec\,
k'$$ compatible in the obvious way with (\[c2\]).
4\) Assume that $k\subset \mathbb{C}$. Set $$T^{Hodge}( M_{t,m}(X_k))= (W_{\cdot}\subset V_{\mathbb{Q}}, F^{\cdot}\subset V_{\mathbb{C}}).$$ The following diagram is commutative.
$$\def\normalbaselines{\baselineskip20pt
\lineskip3pt \lineskiplimit3pt}
\def\mapright#1{\smash{
\mathop{\to}\limits^{#1}}}
\def\mapdown#1{\Big\downarrow\rlap
{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\begin{matrix}
V^*_{\mathbb{Q}}\otimes \mathbb{Q}_p & \stackrel{(\ref{c1})}{\longrightarrow}
& \Psi_t^{an, un}(R^m \pi_*^{an}\mathbb{Q})\otimes \mathbb{Q}_p \cr
\mapdown{(\ref{be})} & &\mapdown{} \cr
T^{*et}_{ \mathbb{Q}_p}(M_{t,m}(X_k)) & \stackrel{\alpha}{\longrightarrow}
& \Psi_t^{et, un}(R^m \pi_*^{et}\mathbb{Q}_p)
\end{matrix}$$
The above properties of $M_{t,m}(X_k)$ are sufficient for our applications. We shall indicate a conceptual construction of $M_{t,m}(X_k)$ based on the theory of Voevodsky’s motives. Unfortunately, the construction relies on the following general fact that is not explained in the published literature.
Let $$\Psi^{mot, un}_t: DM^{eff}_{gm}(\eta; \mathbb{Q}) \to DM^{eff}_{gm}(k; \mathbb{Q})$$ the functor of (unipotent) motivic vanishing cycles from the triangulated category of motives over the generic point $\eta \in C_{k}$ to the category of motives over $a$, and let $$N: \Psi^{mot, un}_t (1) \to \Psi^{mot, un}_t$$ be the monodromy operator (see \[A1\]). The fact, we will need, is that the formation $(\Psi^{mot, un}_t, N)$ commutes with the etale and Hodge realizations \[Hu1\], \[Hu2\]:
$$\def\normalbaselines{\baselineskip20pt
\lineskip3pt \lineskiplimit3pt}
\def\mapright#1{\smash{
\mathop{\to}\limits^{#1}}}
\def\mapdown#1{\Big\downarrow\rlap
{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\begin{matrix}
DM^{eff}_{gm}(\eta; \mathbb{Q}) & \stackrel{\Psi^{mot, un}_t }{\longrightarrow}
& DM^{eff}_{gm}(k; \mathbb{Q}) \cr
\mapdown{ } & &\mapdown{} \cr
D^b( Rep_{\mathbb{Q}_p}(Gal(\overline {k(\eta)}/k)) ) & \stackrel{ \Psi^{et, un}_t }{\longrightarrow}
& D^b( Rep_{\mathbb{Q}_p}(Gal(\overline {k}/k)) )
\end{matrix}$$ $$\def\normalbaselines{\baselineskip20pt
\lineskip3pt \lineskiplimit3pt}
\def\mapright#1{\smash{
\mathop{\to}\limits^{#1}}}
\def\mapdown#1{\Big\downarrow\rlap
{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\begin{matrix}
DM^{eff}_{gm}(\eta ; \mathbb{Q}) & \stackrel{\Psi^{mot, un}_t }{\longrightarrow}
& DM^{eff}_{gm}(\mathbb{C}; \mathbb{Q}) \cr
\mapdown{R^{Hodge} } & &\mapdown{R^{Hodge}} \cr
D^b(VMHS_{eff}^{\mathbb{Q}}(\eta)) & \stackrel{ \Psi^{Hodge, un}_t }{\longrightarrow}
& D^b(MHS^{\mathbb Q}_{eff}).
\end{matrix}$$
Assuming this fact we construct $M_{t,m}(X_k)$ as follows. It is proven in \[BK\] the fully faithful functor $D^b({\cal M}_1(k;
\mathbb{Q})) \to DM^{eff}_{gm}(k; \mathbb{Q})$ has a left adjoint: $$LAlb: DM^{eff}_{gm}(k; \mathbb{Q}) \to D^b({\cal M}_1(k; \mathbb{Q})).$$ Set $$M_{t,m}(X_k):= H_m(LAlb \, \Psi_t^{mot, un}(
\mathbb{Q}_{tr}[X_{\eta}])).$$ Let us just explain that $M_{t,m}(X_k)$ has the key property 1. Indeed, by Theorem 2 from \[Vol\] the Albanese functor commutes with the Hodge realization. Thus, we have $$T^{Hodge}(M_{t,m}(X_{\mathbb C}))\simeq H_m(\overline{LAlb} \, R^{Hodge} \Psi_t^{mot, un}(
\mathbb{Q}_{tr}[X_{\eta}]))$$ $$\simeq \overline {LAlb}\,
\Psi_t ^{Hodge, un} H_m(X_{\mathbb{C}}/C_{\mathbb{C}}).$$
[**Example.**]{} [^21] Here we explain an elementary construction of the motive $M_{t,1}(X_k)$ for a family $\pi: X_k \to
C_k$ of curves with a semi-stable reduction. Choose a semi-stable model $\overline \pi: \overline X_k \to \overline C_k$, such that all the irreducible components $Y_{\overline k, \gamma }$ of the special fiber $Y_{\overline k}:= \overline X_k \times _{\overline
C_k} \overline k $ are smooth. Let $\Gamma$ be the free abelian group whose generators $[\gamma]$ correspond to irreducible components of $Y_{\overline k}$. For each singular point $y_{\mu}$ of $Y_{\overline k}$ we denote by $R_{\mu}$ the subgroup of $\wedge
^2 \Gamma$ generated by $[\gamma _1] \wedge [\gamma_2]$, where $Y_{\overline k, \gamma_1 }$ and $Y_{\overline k, \gamma_2 }$ are the two components meeting at $y_{\mu}$ (i.e. $R_{\mu}$ is isomorphic to $\mathbb{Z}$ but the isomorphism depends on the ordering of the components meeting at $y_{\mu}$). Define a homomorphism $$u: R_{\mu} \to Pic(Y_{\overline k})$$ as follows. Consider the invertible sheaf ${\cal O}(y_{\mu,
\gamma_1} - y_{\mu, \gamma_2})$ on the normalization $\tilde
Y_{\overline k} \to Y_{\overline k}$, where $y_{\mu, \gamma_1}$, $y_{\mu, \gamma_2}$ are the preimages of $y_{\mu}$ in $\tilde
Y_{\overline k}$. We claim that ${\cal O}(y_{\mu, \gamma_1} -
y_{\mu, \gamma_2})$ canonically descends to a line bundle $u([\gamma _1] \wedge [\gamma_2])$ over $Y_{\overline k}$: the descend data are trivial outside of points $y_{\mu, \gamma_1}$, $y_{\mu, \gamma_2}$, and the identification $${\cal O}(y_{\mu, \gamma_1} -
y_{\mu, \gamma_2})_{y_{\mu, \gamma_1}} \simeq {\cal O}(y_{\mu,
\gamma_1} - y_{\mu, \gamma_2})_{y_{\mu, \gamma_2}}$$ is given by a canonical isomorphism $$T_{Y_{\overline k, \gamma_1 }, y_{\mu, \gamma_1}} \otimes T_{Y_{\overline k, \gamma_2 }, y_{\mu,
\gamma_2}} \simeq T_{\overline C_{\overline k}, a} \simeq \overline
k. \footnote{The ismorphism $T_{\overline C_{\overline k}, a} \simeq
\overline k$ is determined by the coordinate $t$.}$$ Finally, let $\Lambda \subset \bigoplus _\mu R_\mu $ be the kernel of the degree map: $$\bigoplus _\mu R_\mu \stackrel{u}{\longrightarrow} Pic(Y_{\overline
k}) \stackrel{deg}{\longrightarrow} \bigoplus _\gamma \mathbb{Z}.$$ Then $M_{t,m}(X_k)=(\Lambda, {\bf Pic}^0(Y_{\overline
k}), \Lambda \stackrel{u}{\to} {\bf Pic}^0(Y_{k})(\overline k)).$
[**4.5. Proof of Proposition \[dif\].**]{} a) By (\[deq\]) and (\[pdeq\]) we have $$\label{eqcor}
d\, log \, q_{\mathbb{C}}(t) = d\, log \, q_{\mathbb{Z}_p}(t)\in
(1+ t \mathbb{Q}[[t]] ) \frac{dt}{t} .$$ Thus, it suffices to prove that $$q'_{\mathbb{C} }(0)= q'_{\mathbb{Z}_p}(0) \in \mathbb{Q}^* \otimes \mathbb{Q} .$$
Consider the 1-motive $M_{t,n}(X_{\mathbb{Q}})= (\Lambda, G, \Lambda
\stackrel{u}{\longrightarrow} G(\overline{k})) $. By (\[c1\]) we have $$L^{Hodge}\subset T_{Hodge}^*(M_{t,n}(X_{\mathbb{Q}}))=: (W_{\cdot}\subset V_{\mathbb{Q}}, F^{\cdot}\subset V_{\mathbb{C}})
\subset \Psi_t ^{Hodge}( H^n(X_{\mathbb{C}}/C_{\mathbb{C}}))\footnote{Here
$T_{Hodge}^*(M_{t,n}(X_{\mathbb{Q}}))$ denotes the Hodge structure dual to
$T^{Hodge}(M_{t,n}(X_{\mathbb{Q}}))$.} .$$ Hence, $W_0 L^{Hodge} \otimes \mathbb{Q} \simeq \Lambda ^* \otimes
\mathbb{Q}$ and $W_{-1}M_{t,n}(X_{\mathbb{Q}}) =
W_{-2}M_{t,n}(X_{\mathbb{Q}})$. We claim that the image of the embedding $W_2 L^{Hodge}/W_0 L^{Hodge} \otimes
\mathbb{Q}\hookrightarrow (T_* \otimes \mathbb{Q})^*$ is $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$-invariant. Indeed, this is clear from the commutative diagram
$$\def\normalbaselines{\baselineskip20pt
\lineskip3pt \lineskiplimit3pt}
\def\mapright#1{\smash{
\mathop{\to}\limits^{#1}}}
\def\mapdown#1{\Big\downarrow\rlap
{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\begin{matrix}
L^{Hodge}_{\mathbb{Q}} \otimes \mathbb{Q}_p & \stackrel{ }{\longrightarrow} & V_{\mathbb{Q}}\otimes \mathbb{Q}_p \cr
\mapdown{\simeq } & &\mapdown{\simeq} \cr
L^{et}\otimes _{\mathbb{Z}_p} \mathbb{Q}_p:= Im \, N^{n-1}_{et} & \stackrel{ }{\longrightarrow} &
T^{*et}_{\mathbb{Q}_p}(M_{t,n}(X_{\mathbb{Q}}))
\end{matrix}$$ since all the arrows in the bottom row are morphisms of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$-modules. It follows that there exists a unique quotient $L^{mot} \in {\cal M}_1(\mathbb{Q};
\mathbb{Q})$, $\gamma: M_{t,n}(X_{\mathbb{Q}})\twoheadrightarrow
L^{mot} $ which fits into the following diagram $$\def\normalbaselines{\baselineskip20pt
\lineskip3pt \lineskiplimit3pt}
\def\mapright#1{\smash{
\mathop{\to}\limits^{#1}}}
\def\mapdown#1{\Big\downarrow\rlap
{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\begin{matrix}
T_{Hodge}^*(L^{mot})& \stackrel{ }{\longrightarrow} &
T_{Hodge}^*(M_{t,n}(X_{\mathbb{Q}})) \cr
\mapdown{\simeq } & &\mapdown{Id} \cr
L^{Hodge} & \stackrel{ }{\longrightarrow} &
T_{Hodge}^*(M_{t,n}(X_{\mathbb{Q}} ))
\end{matrix}$$ Observe that the operator $N$ descends to $L^{mot}$ and $$\label{mon}
N: (W_0 L^{mot} /W_{-2} L^{mot})\otimes \mathbb{Q}(1)\stackrel{\simeq }{\longrightarrow} W_{-2} L^{mot}\otimes \mathbb{Q} .$$ Finally, we have from (\[c2\]) a canonical isomorphism $T^{*et}_{\mathbb{Q}_p}(L^{mot}) \simeq L^{et}\otimes
_{\mathbb{Z}_p} \mathbb{Q}_p$ of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$-modules.
Let $[L^{mot}, N^{-1}]\in Ext^1_{{\cal M}_1(\mathbb{Q};
\mathbb{Q})}(\mathbb{Q}(0), \mathbb{Q}(1))$ be the class of the extension $$\label{motext}
0 \to W_{-2} L^{mot} \to L^{mot} \to W_0 L^{mot} /W_{-2} L^{mot} \to 0$$ composed with $N^{-1}$ from (\[mon\]) and let $\kappa$ be the corresponding (by (\[tateext\])) element in $\mathbb{Q}^* \otimes \mathbb{Q}$. The functor $T_{Hodge}^*$ takes this extension to the class $[L^{Hodge}, N^{-1}_B] \in Ext^1_{MHS}(\mathbb{Q}(0),
\mathbb{Q}(1)) \simeq \mathbb{C}^* \otimes \mathbb{Q}$. The latter class is equal to $q'_{\mathbb{C}}(0)$. It follows that $
q'_{\mathbb{C}}(0)= \kappa ^{-1} $.
If we pull back the extension (\[motext\]) on $spec \,
\mathbb{Q}_p$ and then apply the etale realization functor $T_{\mathbb{Q}_p}^*: {\cal M}_1(\mathbb{Q}_p; \mathbb{Q}) \to Rep_{\mathbb{Q}_p}(\Gamma)$ we get the extension $[L^{et}\otimes \mathbb{Q}_p , N^{-1}_{et}]$ equivalent (by [**4.2**]{} , 3)) to the one obtained from $[L^{FL}\otimes \mathbb{Q}_p,
N^{-1}_{DR}]$ by applying the Fontaine-Laffaille functor $U$. Hence $ q'_{\mathbb{Z}_p}(0)= \kappa ^{-1} $, and we are done.
The above argument shows that for any family $X_{\mathbb{Q}} \to C_{\mathbb{Q}} $ over $\mathbb{Q}$ with a maximal degeneracy point at $a\in \overline
C_{\mathbb{Q}}(\mathbb{Q})$ $$q_{\mathbb{C}}\in (\mathbb{Q}((t)))^*\otimes \mathbb{Q}.$$
b\) Assume that the Betti monodromy is small i.e. $$\label{monbetti}
N_B: W_2 L^{Hodge}_{\mathbb{Z}}/ W_0 L^{Hodge}_{\mathbb{Z}} \simeq
W_0 L^{Hodge}_{\mathbb{Z}}(-1)$$ We have to show that, for any prime $p$ in $S$, $$\label{monrham}
N_{DR}: W_2 L^{FL}/ W_0 L^{FL} \to W_0 L^{FL}(-1)$$ is an isomorphism as well. Indeed, by (by [**4.2**]{} , 3)) the functor $U$ takes the morphism (\[monrham\]) to $$N_B\otimes Id:
(W_2 L^{Hodge}_{\mathbb{Z}}/ W_0 L^{Hodge}_{\mathbb{Z}})\otimes
\mathbb{Z}_p \simeq W_0 L^{Hodge}_{\mathbb{Z}} \otimes
\mathbb{Z}_p(-1) .$$ The claim follows.
c\) Let ${\cal E}$ be the quotient of $H^n_{log}(\overline
X_S/\overline C_S)$ modulo torsion, and let $\overline{\cal F}^n
\subset {\cal E}$, $ {\cal W}_0 \subset {\cal E}$ be the Hodge and monodromy filtrations ([**3.3**]{}). As we explained in [*loc. cit.*]{} the Poincare duality identifies the line bundle $\overline{\cal F}^n$ with the dual to $ {\cal W}_0$. It is also shown there that $ {\cal W}_0$ is generated by a parallel section $e^0 \in {\cal W}_0^{\nabla}$. It suffices to prove the claim for a single nonvanishing section $\omega \in \overline {\cal F}^n$. Let us choose $\omega$ such that $(e^0, \omega)=0$. Then the integral $$\frac{1}{(2\pi i)^n} \int
_{\delta _1} \omega$$ is a constant function on $D_{\mathbb{C}}$. We have to show that the square of this constant is in $\mathbb{Z}[N^{-1}]^*$. The following lemma does the job.
Let $E$ (resp. $H$ ) be the torsion free part of $a^*(H^n_{log}(\overline X_S/ \overline C_S))$ (resp. $\Psi_t^{an}(R^n\pi_{\mathbb{C} *}^{an}\mathbb{Z}[N^{-1}])$ ) , and let $$E \hookrightarrow E\otimes \mathbb{C}\simeq H \otimes \mathbb{C} \hookleftarrow H$$ be the isomorphism from (\[derhamv\]). Then the two $\mathbb{Z}[N^{-1}]$-lattices $$(W_{0} E)^{\otimes ^2} \hookrightarrow (W_{0} E)^{\otimes ^2} \otimes \mathbb{C} \simeq (W_{0} H)^{\otimes ^2}
\otimes \mathbb{C}\hookleftarrow (W_{0} H)^{\otimes ^2}$$ coincide.
Indeed, consider the monodromy paring $$\Xi: <\cdot, \cdot >_{mon}: (W_{0} E)^{\otimes ^2} \otimes \mathbb{C}\to \mathbb{C},$$ $$<x,y>_{mon}= <x, N^{-n}_{DR} y>_{DR} =\pm <x, N^{-n}_{B}y >_B,$$ where $N^{-n}_B= (-2\pi i)^{-n}N^{-n}_{DR}: W_{0} E \otimes
\mathbb{C} \to W_{2n} E \otimes \mathbb{C} $. The monodromy paring takes $(W_{0} E)^{\otimes ^2} \otimes \mathbb{Q}$ and $ (W_{0}
H)^{\otimes ^2} \otimes \mathbb{Q}$ into $\mathbb{Q}\subset
\mathbb{C}$. Therefore, since $rk \, W_0 E =1$, $(W_{0}
E)^{\otimes ^2} \otimes \mathbb{Q} = (W_{0} H)^{\otimes ^2} \otimes
\mathbb{Q}$. Moreover, to prove that $(W_{0} E )^{\otimes ^2}=(W_{0}
H )^{\otimes ^2}$, it is enough to show that $\Xi((W_{0} E
)^{\otimes ^2})=\Xi((W_{0} H )^{\otimes ^2})$. Since the pairings $$<\cdot, \cdot>_{DR} : W_{0} E \otimes W_{2n} E \to \mathbb{Z}[N^{-1}], \, <\cdot, \cdot>_B : W_{0} H \otimes
W_{2n} H \to \mathbb{Z}[N^{-1}]$$ are perfect, the claim would follow if we prove that, for any prime $p$ in $S$, the cokernels of the maps $N_{DR}^{n}: W_{0}E \otimes \mathbb{Z}_p \to W_{2n} E
\otimes \mathbb{Z}_p, \, N_{et}^{n}: W_{0}H \otimes \mathbb{Z}_p \to
W_{2n} H \otimes \mathbb{Z}_p$ have the same order. This follows from parts 1) and 3) in [**4.2**]{}.
[**REFERENCES**]{}
\[A1\] J. Ayoub, [*Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique*]{}, 2006. Available electronically at http://www.institut.math.jussieu.fr/ ayoub/These/THESE.pdf
\[A2\] J. Ayoub, [*Private communications.*]{}
\[BK\] L. Barbieri-Viale, B. Kahn, [*On the derived category of 1-motives, I*]{}, arXiv:0706.1498v1 \[math.AG\].
\[BK2\] L. Barbieri-Viale, B. Kahn, [*On the derived category of 1-motives, II*]{}. In preparation.
\[B\] A. Beilinson, [*How to glue perverse sheaves*]{}, K-theory, Arithmetic and Geometry, LNM 1289.
\[BOV\] A. Beilinson, A. Otwinowska, V. Vologodsky, [ *Motivic sheaves over a curve*]{}, work in progress.
\[COGP\] P. Candelas, X. de la Ossa, P. Green, L. Parkes, [*A pair of Calabi-Yau manifolds as an exactly soluble superconformal field theory*]{}, Nuclear Physics, B359(1991) 21
\[D1\] P.Deligne, [*Local behavior of Hodge structures at infinity*]{}, AMS/IP Studies in Advanced Mathematics, Volume 1, 1997.
\[D2\] P.Deligne, [*Théorie de Hodge 2*]{}, Publ. Math. IHES 40 (1971).
\[D3\] P.Deligne, [*Théorie de Hodge 3*]{}, Publ. Math. IHES 44 (1974).
\[Fa\] G. Faltings, [*Crystalline cohomology and p-adic Galois representations*]{}, Algebraic Analysis, Geometry and Number Theory (J.Igusa, ed.) (1989).
\[Hu1\] A. Huber, [*Realization of Voevodsky’s motives*]{}, J. Algebraic Geom. 9 (2000), no. 4.
\[Hu2\] A. Huber, [*Corrigendum to: “Realization of Voevodsky’s motives”* ]{} J. Algebraic Geom. 13 (2004), no. 1.
\[Il\] L.Illusie, [*Logarithmic spaces (according to K.Kato)*]{}, Perspect. Math., 15.
\[KSV\] M. Kontsevich, A. Schwarz, V. Vologodsky, [*Integrality of the instanton numbers and p-adic B-model*]{}, Physics Letters B 637 (2006).
\[LY\] B.Lian, S.T.-Yau, [*Mirror Maps, Modular Relations and Hypergeometric Series,1*]{}, arXiv:hep-th/9507151 v1 27 July 1995.
\[M\] D.Morrison, [*Mirror Symmetry and Rational Curves on Quintic Threefolds: A Guide for Mathematicians*]{}, J. Amer. Math. Soc. 6 (1993), no. 1.
\[O\] F. Orgogozo, [*Isomotifs de dimension inférieure ou égale à un*]{}, Manuscripta Math. 115 (2004), no. 3.
\[Ste\] J. Steenbrink, [*Limits of Hodge structures*]{}, Inv. Math. 31 (1976).
\[Sti\] J. Stienstra, [*Ordinary Calabi-Yau-3 Crystals*]{}, Calabi-Yau varieties and mirror symmetry (Toronto, ON, 2001), Fields Inst. Commun., 38, Amer. Math. Soc., Providence, RI, (2003).
\[V\] V. Voevodsky, [*Triangulated category of motives over a field*]{}, in “Cycles, Transfers, and Motivic Homology Theories”, Annals of Mathematics Studies 143 (2000).
\[Vol\] V. Vologodsky, [*The Albanese functor commutes with the Hodge realization.*]{}
[^1]: i.e. $n_d =
\frac{m}{N^k}$, where $m$ and $k$ are integers.
[^2]: For example, for the quintic family \[COGP\], we can take $N=2\times 3\times 5$.
[^3]: The idea to use the Frobenius action is due, in a slightly different setting, to Jan Stienstra \[Sti\].
[^4]: Logarithmic functions in these formulas have different meanings: in the first formula $log$ is the the usual complex-valued function while in the second formula $log$ takes p-adic values.
[^5]: This amounts to proving relation (\[cperiod\]).
[^6]: The proof is reproduced in 2.2.
[^7]: The nontrivial part is to show that ${ \int _{\delta
_1} \omega}$ does not vanish on a sufficiently small $D^*$ and that that $q$ has regular singularity at $a$. This is another corollary of the existence of the limit Hodge structure. See 2.2.
[^8]: The cycle $\delta_1$ is defined up to sign. But the trivialization of ${\cal
F}^n\otimes {\cal F}^n$ is independent of this choice.
[^9]: This is a corollary of a result of Schmid, which says that the Hodge filtration extends to Deligne’s canonical extension of the underlying vector bundle. (See also 2.1).
[^10]: i.e. locally for the etale topology $\overline{\pi}: \overline X_S \to \overline C_S$ is isomorphic to $spec\, \mathbb{Z}[t, x_1, \cdots x_n]/(x_1\cdots x_r - t) \to
spec \, \mathbb{Z}[t]$, where $r\leq n $ .
[^11]: Observe that dimension of the space of first order deformations of a Calabi-Yau n-fold Y is equal to $dim \, H^1(Y,
T_Y)= dim \, H^{n-1}(Y, \Omega ^1)$. Thus the condition (\[one\]) implies that $\pi: X_{\mathbb{C}}\to C_{\mathbb{C}}$ induces a dominant map from $C_{\mathbb{C}}$ to an irreducible component of the moduli space of Calabi-Yau threefolds. The case of a higher dimensional component in the moduli space of Calabi-Yau threefolds will be considered elsewhere (also see \[KSV\], Section 3).
[^12]: One readily sees that any power series $Y(q)\in \mathbb{C}[[q]]$ can be written in the form (\[instanton\]), with $n_d\in \mathbb{C}$. Thus the content of the theorem is the integrality property of the numbers. These are the instanton numbers the title of our paper refers to.
[^13]: By definition, this is a unique filtration such that the quotients $\Psi_t^{an, un}( {\cal H}_{\mathbb{Z}})/ W_{\cdot}\Psi_t^{an, un}(
{\cal H}_{\mathbb{Z}})$ are torsion free, $N_B(W_i\Psi_t^{an, un}(
{\cal H}_{\mathbb{Z}}))\subset W_{i-2}\Psi_t^{an, un}( {\cal
H}_{\mathbb{Z}})\otimes \mathbb{Q}$ and $N_B^i: Gr^{n+i}_W \Psi_t^{an, un}( {\cal H}_{\mathbb{Z}}) \otimes \mathbb{Q}\simeq Gr^{n-i}_W \Psi_t^{an, un}( {\cal H}_{\mathbb{Z}}) \otimes \mathbb{Q} $.
[^14]: Recall that a variation of mixed Hodge structure $( {\cal W}_{\cdot} {\cal L}_{\mathbb{Z}} \subset {\cal L}_{ \mathbb{Z}}\, , \,
{\cal F}^{\cdot}{\cal L} \subset {\cal L})$ over $D^*$ is called admissible if the Hodge filtration ${\cal
F}^{\cdot}$ extends to the Deligne extension $\overline{\cal L}$ of ${\cal L}$.
[^15]: Except for the last section where we need the category of all Fontaine-Laffaille modules over a point.
[^16]: The group structure on the set of pairs $(\omega, h)$ is defined by the formula $(\omega, h)+ (\omega ', h') = (\omega +
\omega ', h+ h') $.
[^17]: The Dwork’s lemma is the following statement:
Let $\omega \in \Omega ^1 _{log }= \mathbb{Z}_p[[t]] \frac{dt}{t}$ with $Res_{0} \omega \in \mathbb{Z}$. The following two conditions are equivalent:
i)$1/p\tilde F^*\omega - \omega = dh $, for some $h\in
\mathbb{Z}_p[[t]]$
ii\) $\omega = d\,log q$, for some $q\in {\cal O}^*(D^*) $ .
[^18]: The paring ${\cal W}_0\otimes \overline{\cal F}^n \to
{\cal O}_D$ is perfect. For the projection ${\cal W}_0
\hookrightarrow \overline{\cal E} \to \overline{\cal
F}^0/\overline{\cal F}^1$ is an isomorphism.
[^19]: The Galois module $ \Lambda $ can be viewed as a discrete group scheme over $spec \, k$. Giving a homomorphism $\Lambda \longrightarrow G(\overline{k})$ of Galois modules is equivalent to giving a morphism $ \underline \Lambda
\longrightarrow \underline G$ of the étale sheaves represented by $\Lambda$ and $G$. This remark provides a construction of the category ${\cal M}_1(k)$ that is independent of the choice of an algebraic closure $\overline k$ (\[BK\]).
[^20]: We say that a triangulated functor $T: D({\cal A})\to D({\cal B})$ is t-exact if $T({\cal A})$ belongs to the essential image of ${\cal B}$ in $D({\cal B})$.
[^21]: The reader can skip this example: it will not be used in the main text below.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
In [@Kar], a number of critical claims were made concerning our paper [@DS].\
Here we reply to these claims.
---
ł Ł
[**Reply to Comments on Our Paper\
“On the Relation Between Two Approaches to Necessary\
Optimality Conditions in Problems with State Constraints”** ]{}\
[Andrei Dmitruk, Ivan Samylovskiy]{}\
\
\
Reply to Comments by Karamzin {#reply-to-comments-by-karamzin .unnumbered}
=============================
First of all, a highly biased style of these comments leaps to the eye. However, let us set it aside and concentrate on mathematics. One can see that the author’s claims can be essentially reduced to the following: a) lack of novelty (that our results “can simply be derived from the already known results in the literature”), and b) incompleteness of the result (stationarity conditions instead of Maximum Principle). Let us make sense of this.
As was clearly said in our paper [@DS Remark 9.1], its novelty is a complete realization of the idea of obtaining optimality conditions (in this case, conditions of stationarity) in problems with a state constraint by differentiating the state constraint on the contact interval and thus by passing to a problem with a mixed constraint of the equality type. This idea was first proposed by Gamkrelidze [@Pont], but in our paper it was realized, for a specific class of problems, by another approach. It needs to be emphasized that, rather than relying on Gamkrelidze’s [*result*]{}, in order to obtain optimality conditions in the form of Dubovitskii and Milyutin [@DM65] we realized [*his idea*]{} for obtaining these conditions. (Perhaps, this should have been said in [@DS] more clearly.) Our method is based on the replication of the state and control variables in accordance with the number of qualitatively different subarcs of the reference trajectory (interior or boundary w.r.t. the state constraint). This replication trick is quite natural and has been known for a long time (we give several references in [@DS]), but to our knowledge it has never been used in the given situation with complete details and clarity (and with a new feature consisting of a “two-stage varying” of the reference trajectory). Despite the author’s assertion, it is not the “reduction to $v-$problem” in the sense of the $v-$problem proposed by Dubovitskii and Milyutin [@DM65] and used in [@IT Sec. 2.5]. The book [@IT] does not contain the replication method.
Note that this replication trick turned out to be effective in some other problems as well, such as problems with the control system of hybrid type, with intermediate constraints, and with a variable structure, where it allows to reduce these problems to a standard optimal control problem and to apply the already known optimality conditions, but rather surprisingly, it was missed even by highly prominent mathematicians, who obtained optimality conditions in these problems by performing all the heavy procedure of variational analysis (see comments and references in [@DK-SCL; @DK-NCS]). Therefore, in our opinion, this method is quite worthy of attention.
Now, let us dwell on the paper by Neustadt [@Nst]. The author of the note claims that its results allow one to make a passage from the conditions of Gamkrelidze (GamC) to those of Dubovitskii and Milyutin (DMC) by a change of the costate variable, proposed in [@Nst]. However, this is not true, because the proposed passage contains a vicious circle. The matter is that Neustadt’s paper is not based either on the result of Gamkrelidze, or on his idea of differentiating the state constraint on the contact interval. His approach is in fact close to that of Dubovitskii and Milyutin; namely, just like they, he considers the state constraint as an inequality constraint in the space of continuous function, and exactly from here, making the linearization of all the constraints and the cost, and applying then the Farkas-Minkowski theorem, he directly obtains a sign-definite measure possessing all the required properties. He clearly says (p. 134) that his conditions [*coincide*]{} with DMC and [*imply*]{} GamC. In fact, he makes a passage from DMC to GamC. (The same passage is later performed by Arutyunov, Karamzin, and Pereira in [@AKP], with no reference to [@Nst].) But the reverse passage GamC $\to$ DMC (by the reverse transformation of the costate) is valid only if the measure in GamC is completely sign-definite (and in this case one does not need to rely on paper [@Nst], since the transformation, as well as the calculations in [@Kar], is quite simple), whereas GamC do not provide the information about the sign of the measure at its atoms.
Thus, in the claim concerning novelty Mr. Karamzin misleads the reader by incorrect interpretations of methods and results in [@IT; @Nst; @AKP].
Now let’s address the alleged incompleteness of our result. Note again that we confined the study to the [*stationarity conditions*]{} (i.e. necessary first order conditions) for the extended weak minimality. Such conditions constitute an important stage in the study of any optimization problem. (Recall here the Fermat condition $f'(x_0) = 0$ in the problem $f(x)\to \min,$ the Euler and Euler-Lagrange equations in the calculus of variations, the Lagrange multipliers rule in the nonlinear programming, etc.) Like any classical necessary first order conditions, they are not complete in the sense that they can be strengthened (e.g. by second order conditions). But [*as the stationarity conditions*]{} they are complete, since they are equivalent, loosely speaking, to the non-negativity of the cost derivative in all admissible directions, and because of this their importance remains undisturbed even if they are strengthened by other conditions. (The Weierstrass necessary condition does not invalidate the Euler equation!) The issue of obtaining Maximum Principle [*by the given approach*]{} was not considered in our paper, it requires additional study. So, this claim of the author is also inadequate.
Obviously, the presence of extended weak minimality implies the strong minimality in an $\e$-tube around the reference control, and hence, the fulfillment of Maximum Principle in this $\e$-tube, which in turn implies the absence of atoms of the measure at the junction points with the state boundary. In view of this, our sentence in [@DS p. 407], given below in bold italics, is, indeed, not correctly formulated:
[*"Studies show ... that in case of strong (or at least Pontryagin type*]{} \[16,17\]) [*minimality, the adjoint variable and measure do not have jumps under condition*]{} (2). [***However, this result is not, in general, valid in the case of extended weak minimality***]{} [*(the reason is that one cannot rely upon the maximality of Pontryagin function w.r.t. u, having in disposal only the stationarity of the extended Pontryagin function)."*]{}
Here we should have said more precisely: [***“However, this result is not, in general, valid in the case of stationarity”***]{}. In fact, this is then said in brackets. Note also that in p. 410 we give a correct resume:
[*"Thus, the stationarity conditions do not guarantee the absence of atoms, while, according to*]{} \[5,15,19–23\], [*the maximum principle does."*]{} We hope that the attentive reader will be able to understand this point properly.
Having said that [*“the technique of reducing to a mixed constrained problem is obviously too restrictive as important information on the admissible trajectories subject to state constraints is lost by this transition”*]{}, Mr. Karamzin missed the fact that this is just the first stage of our variation procedure. At the second stage we use variations that go inside the state constraints.
As concerns an omission in our assumptions, of course the data functions $f'_u$ and $g'_u$ should be assumed Lipschitz continuous, not just continuous.
The author’s claims concerning “wrong” citations and “confusing” title present just his personal opinion.
[99]{}
\
[The Mathematical Theory of Optimal Processes.]{} [Wiley,]{} [New York/London]{} [(1962)]{}
: [An abstract variational theory with applications to a broad class of optimization problems. II: Applications.]{} [SIAM J. on Control.]{} [5(1),]{} [90–137]{} [(1967)]{}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The embedding diagrams of representations of the [$N=2$]{} superconformal algebra with central charge $c=3$ are given. Some non-unitary representations possess subsingular vectors that are systematically described. The structure of the embedding diagrams is largely defined by the spectral flow symmetry. As an additional consistency check the action of the spectral flow on the characters is calculated.'
---
Introduction {#sec:intro}
============
The representation theory of the $N=2$ superconformal algebra was long thought to be an obvious generalisation of the representation theory of the Virasoro and $N=1$ superconformal algebra. It was first shown in [@doerrzapf; @Dorrzapf:1995wv; @Dorrzapf:1994es] that this is not quite the case (see [@Dobrev:1987hq] for some earlier results). In particular the structure of singular vectors turned out to be more involved than in the $N=1$ supersymmetric generalisation of the Virasoro algebra.
In [@Gato-Rivera:1996ma] it was noticed that some representations of the $N=2$ superconformal algebra contain subsingular vectors. However, only some explicit examples are known which are all constructed using the “topological twisted” algebra and then translated back via the topological “untwisting”. The examples found in this way showed that subsingular vectors exist but did not admit a systematic exploration of these.
In [@Semikhatov:1998gv; @Semikhatov:1997pf; @Feigin:1998sw] the representation theory of the topologically twisted $N=2$ superconformal algebra was treated via the representation theory of $\widehat{sl}(2)$. By this approach some of the embedding diagrams for $c=3$ were found in [@Semikhatov:1997pf] however, as the relation of the $N=2$ superconformal algebra with $\widehat{sl}(2)$ representations breaks down at $c=3$ not all embedding diagrams could be approached in this way. In [@Semikhatov:1998gv] a classification of all subsingular vectors for $c<3$ was tried, however, as was pointed out in [@Gato-Rivera:1999ey], some cases seem to have been overlooked.
Our main aim in this paper is to clarify the structure of the embedding diagrams for central charge $c=3$. To this end we pursue another route. The $N=2$ superconformal algebra possesses a family of outer automorphisms $\alpha_\eta$, called the spectral flow, which map the algebra to itself. For integer values of the flow parameter $\eta$, the Neveu-Schwarz sector is mapped to itself as is the Ramond sector. We study the induced action of the spectral flow on the representations of the algebra. It is shown that the representations are mapped onto each other under spectral flow in a systematic manner. In particular the embedding diagrams can be grouped into a few categories which transform among themselves under spectral flow. Furthermore we obtain an algorithm to derive subsingular vectors from singular vectors via spectral flow transformations. We show that only non-unitary representations posses subsingular vectors. Although we only cover the representations with $c=3$ in detail, we briefly demonstrate that our techniques are applicable to representations with other values of the central charge in the case of the $N=2$ unitary minimal models.
The paper is organised as follows. In chapter \[sec:n2algebra\] we review some well known facts about the $N=2$ algebra and introduce the spectral flow to fix our conventions. In chapter \[sec:rep-flow\] we explore how the spectral flow acts on a given representation and how representations are transformed. In Chapter \[sec:singular-flow\] we describe how singular vectors behave under spectral flow and how subsingular vectors arise under spectral flow transformations. In chapter \[sec:ramond-algebra\] we comment briefly on how the Ramond algebra can be analysed by the same method. In chapter \[sec:characters-c=3\] we derive the characters of a large class of representations and determine the action of the spectral flow on them. In order to check the consistency of the embedding diagrams derived above we also construct the characters for representations with subsingular vectors. In chapter \[sec:minimal\] we briefly describe how the same methods can be used for the unitary minimal models. Chapter \[sec:conclusions\] contains further comments and outlooks. Some technical proofs can be found in the appendix.
The $N=2$ Algebra and Spectral flow {#sec:n2algebra}
===================================
Basic facts {#sec:basic}
-----------
We want to review some basic facts of the $N=2$ superconformal algebra before we proceed.
The $N=2$ superconformal Algebra ${\cal A}$ consists of the Virasoro algebra $\{L_n\}$, a weight one $U(1)$-current $\{J_n\}$ and the modes of two supersymmetric partners $\{G_r^\pm\}$ of conformal dimension $h=\tfrac{3}{2}$, obeying the (anti-)commutation relations [@Ademollo:1976an] $$\begin{aligned}
\left[L_m, L_n \right] &=(m-n)L_{m+n} + \tfrac{{\tilde{c}}}{4}(m^3 -
m)\delta_{m,-n} \\
\left[L_m, J_n \right] &=-nJ_{m+n}\\
\left[L_m, G_r^\pm \right] &=\left( \tfrac{1}{2}m-r \right)
G_{m+r}^\pm \\
\left[J_m, J_n \right] &= {\tilde{c}}m\delta_{m,-n}\\
\left[J_m, G_r^\pm \right] &=\pm G_{r+m}^\pm \\
\left\{ {G^+_{r}}, {G^-_{s}}\right\} &= 2 L_{r + s} + (r -s) J_{r + s} +
{\tilde{c}}(r^2 - \tfrac{1}{4})\delta_{r, -s} \\
\left\{{G^+_{r}}, {G^+_{s}} \right\} &=\left\{{G^-_{r}}, {G^-_{s}}\right\}=0\end{aligned}$$ where ${\tilde{c}}=\tfrac{c}{3}$ and braces denote anticommutators. In the NS sector the modes $L_n$ and $J_n$ are integral and the modes $G_r^\pm$ are half-integral whereas in the R sector all modes are integral.
The determinant formula of the NS sector of this algebra at level $n$ with relative charge $m$ was first written down in [@kent] and proved in [@Kato:1987td]. It is given by $$\label{eq:determinant}
\det M_{n,m}^A({\tilde{c}}, h, q)
= \prod_{1\leq rs\leq 2n \atop s \text{ even}}
(f_{r,s}^A)^{P_A(n-rs/2,m)}
\prod_{k \in {\mathbb{Z}}+ 1/2}(g_k^A)^{\tilde{P}_A(n - |k|, m - \operatorname{sgn}(k);k)},$$ where $P_A$ is defined by $$\sum_{n,m} P_A(n,m)x^ny^m = \prod_{k=1}^\infty
\frac{( 1 + x^{k-1/2} y)(1 + x^{k-1/2}y^{-1})}{(1-x^k)^2}.$$ $P_A(n,m)$ counts the states at level $n$ with relative charge $m$. $\tilde{P}_A(n,m,k)$ is given by $$\sum_{n,m} \tilde{P}_A(n,m,k)x^n y^m = (1+ x^{|k|}y^{\operatorname{sgn}(k)})^{-1}
\sum_{n,m} P_A(n,m)x^ny^m,$$ with $\operatorname{sgn}(k)=1$ for $k>0$ and $\operatorname{sgn}(k)=-1$ for $k<0$. $\tilde{P}_A(n,m,k)$ counts the states build on vectors with relative charge $\operatorname{sgn}(k)$ at level $k$. The functions $f^A$ and $g^A$ are given by $$\begin{aligned}
f_{r,s}^A({\tilde{c}}, h, q) &= 2({\tilde{c}}- 1)h - q^2 - \tfrac{1}{4}({\tilde{c}}-1)^2 +
\tfrac{1}{4}(({\tilde{c}}-1)r + s)^2, & s \text{ even} \\
g_k^A({\tilde{c}}, h, q) &= 2h - 2kq + ({\tilde{c}}-1)(k^2 - \tfrac{1}{4}), & k \in
{\mathbb{Z}}+\tfrac{1}{2}. \end{aligned}$$ If we set ${\tilde{c}}= 1$ these formulas reduce quite drastically to $$\begin{aligned}
f_{r,s}^A({\tilde{c}}=1, h, q) &= \tilde{s}^2 - q^2, & \tfrac{s}{2} =
\tilde{s} \in {\mathbb{Z}}\\
g_k^A({\tilde{c}}=1, h, q) &= 2(h - kq),& k \in {\mathbb{Z}}+\tfrac{1}{2},\end{aligned}$$ and the determinant formula then reads $$\label{eq:determinant2}
\det M_{n,m}^A({\tilde{c}}=1, h, q)
= \prod_{1\leq rs\leq 2n \atop s \text{ even}}
(\tfrac{1}{4}s^2 - q^2)^{P_A(n-rs/2,m)}
\prod_{k \in {\mathbb{Z}}+ 1/2}2(h - kq)^{\tilde{P}_A(n - |k|, m - \operatorname{sgn}(k);k)}.$$
Representations of $N=2$ at $c=3$ {#sec:reps}
---------------------------------
We will denote *Verma modules* by ${\mathcal{V}}_{h,q}$ where $h$ and $q$ denote weight and charge of the highest weight vector (hwv). They are in general not irreducible representations but contain *null vectors* i.e. vectors whose inner product with any other vector vanishes. These vectors have to be quotiened out in order to obtain an irreducible representation. Null vectors which by themselves are highest weight states, i.e. which are annihilated by the action of any lowering operator, are called *singular vectors*. They span submodules inside a Verma module. The ${N=2}$ superconformal algebra furthermore possesses *subsingular vectors* which are null vectors that are neither singular vectors nor descendants of singular vectors. Once the singular vectors are quotiened out subsingular vectors become (new) singular vectors.
Any highest weight representation is determined by the position of its singular (and subsingular) vectors. Vanishing of the determinant formula (\[eq:determinant2\]) signals null vectors at the given level and relative charge. At the level where vanishings first occur we then find a singular vector. As was already pointed out in [@kent], singular vectors can only exist with relative charge $m = 0, \pm 1$.
The position and the relations among the singular vectors can be summarised in the *embedding diagram* of a given representation. We want to study the relations among the embedding diagrams under spectral flow transformations. The embedding diagrams of various highest weight representations were calculated in [@doerrzapf]. Unfortunately some of the results are incomplete as the assumption made that the representations of the $N=2$ superconformal algebra do not contain subsingular vectors does not hold. The first paper where subsingular vectors of the $N=2$ superconformal algebra were discovered was [@Gato-Rivera:1996ma]. The structure of the embedding diagrams depends mainly on the
1. value of $q$: Only for $q\in {\mathbb{Z}}$ (and $q \neq 0$ ) uncharged singular vectors exist, as can be seen from the determinant formula , in particular from the form of $f^A$.
2. value of $\tfrac{h}{q}$: Only if this value is half integer $g^A$ has zeros and thus only then charged singular vectors exist.
There are a few different classes of representations we have to consider in the case of central charge $c=3$.
All representations with charged singular vectors appear in pairs: to any representation with charged singular vectors there exists a representation with the same embedding structure in which the relative charge of the singular vectors is reversed. This is a consequence of the mirror automorphism [@Lerche:1989uy; @Greene:1990ud; @Greene:1996cy] which connects representations that differ only by the sign of the $U(1)$ charge. These representations are mapped onto each other by the mirror automorphism $m$ acting on the modes as follows $$\begin{aligned}
\label{eq:19}
m(L_n) &= L_n \\
m(J_n) &= -J_n, \\
m(G^\pm_r) &= G^\mp_r.\end{aligned}$$ Therefore any statement about a representation with a given charge $q$ holds for the representation with charge $-q$, as well.
In figure \[fig:others\] we have summarised the embedding diagrams for which either $q \not\in {\mathbb{Z}}$ or $\tfrac{h}{q} \not\in
{\mathbb{Z}}+ \tfrac{1}{2}$. These diagrams were already given in [@doerrzapf]. For the remaining cases, i.e for the cases where $\tfrac{h}{q} \in {\mathbb{Z}}+ \tfrac{1}{2}$ and $q \in {\mathbb{Z}}$ we conjecture that the form of the embedding diagrams is as shown in figures \[fig:embeddingpositiv\] and \[fig:embeddingsub\]. In an embedding diagram a black dot denotes a singular vector and a line marks an operator connecting two singular vectors. The unfilled circles denote highest weight vectors. In figure \[fig:embeddingsub\] the boxes mark subsingular vectors and the dashed lines indicate to which singular vector the subsingular vectors are linked[^1]. In the presence of subsingular vectors we have marked some lines with arrows to denote which vectors are connected by operators. In all cases of singular vectors a raising operator connects a singular vector of lower level to a singular vector of higher level. In the case of subsingular vectors a lowering operator maps the subsingular vector to some (descendant of a) singular vector.
In figure \[fig:embeddingpositiv\] and \[fig:embeddingsub\] the relative charge of the charged singular vectors is given by the sign of $h/q$. Uncharged singular vectors arise at levels $n|q|$, charged singular vectors are given at levels $\left|\tfrac{h}{q}\right| + n|q
+ \operatorname{sgn}(q)|$. The diagrams III$^{\pm}_{+}$ (figure \[fig:embeddingpositiv\]) contain infinitely many uncharged and charged singular vectors. Diagram IV in figure \[fig:embeddingpositiv\] is the embedding diagram for the vacuum representation. It contains two charged singular vectors at each level of the form $\ell = \tfrac{1}{2} + k, \, k \in {\mathbb{N}}$ with relative charge $\pm1$. The embedding diagrams given in figure \[fig:embeddingsub\] are the embedding diagrams of the representations with $h<0, \tfrac{h}{q}\in
{\mathbb{Z}}+\tfrac{1}{2}, q\in{\mathbb{Z}}$.
Embedding diagrams of this kind contain only finitely many charged singular vectors. The representations with $h<0, q\in {\mathbb{Z}}$ and $|\tfrac{h}{q}|=\tfrac{1}{2}$ contain subsingular vectors. We have denoted these embedding diagrams by III$_-^{s\pm}$. Representations with $h=-\tfrac{2l + 1}{2}, l \in {\mathbb{N}}$ and $|q|=1$ constitute another special case. They have only one charged singular vector and possess subsingular vectors of higher charge. The associated embedding diagrams are denoted by III$_-^{*\pm}$. We have adopted the notation that the superscript on the label of a diagram denotes the relative charge of the charged singular vectors and a subscript denotes whether the value of $h$ is greater or smaller than zero in cases where the structure of the embedding diagrams is affected by the sign of $h$.
Note that subsingular vectors arise only for representations with $h<0$ and that there exist subsingular vectors with relative charge greater than one. We were not able to prove the actual form of the embedding diagrams but we will show that they are compatible with the spectral flow. Explicit calculations further support our conjectures. For the various values of $h$ and $q$ we obtain the following diagrams:
------------------------------------------------------------------------------------- --------------------------- ---------------------------------------------------------------- -------------------------------
*levels of uncharged* *levels of charged*
*$h$ and $q$* *singular vectors* *singular vectors* *diagram*
$q \not\in{\mathbb{Z}}$, $\tfrac{h}{q} \not\in {\mathbb{Z}}+ \tfrac{1}{2}$ none none 0
$q \not\in{\mathbb{Z}}$, $\tfrac{h}{q} \in {\mathbb{Z}}+ \tfrac{1}{2}$ none $\left|\tfrac{h}{q}\right|$ $\text{I}^\pm$
$q \in {\mathbb{Z}}$, $q\neq 0$ , $\tfrac{h}{q} \not\in {\mathbb{Z}}+ \tfrac{1}{2}$ $n|q|, n\in {\mathbb{N}}$ none II
$q=0, \quad h\neq 0$ none none 0
$q\in {\mathbb{Z}}^+$, $h>0$, $\tfrac{h}{q} \in {\mathbb{Z}}^+ + \tfrac{1}{2}$ $nq, n\in {\mathbb{N}}$ $\tfrac{h}{q} + k(q+1), \,k \in {\mathbb{N}}^0$ III$^+_+$
$q\in {\mathbb{Z}}^-$, $\tfrac{h}{q} \in {\mathbb{Z}}+ \tfrac{1}{2}$, $h>0$ $n|q|, n\in {\mathbb{N}}$ $\left|\tfrac{h}{q}\right| + k(|q|+1), \,k \in {\mathbb{N}}^0$ III$^-_+$
$q=h=0$ none $k + \tfrac{1}{2} \,k \in {\mathbb{N}}^0$ IV
$q\in {\mathbb{Z}}^+$, $\tfrac{h}{q} \in {\mathbb{Z}}+ \tfrac{1}{2}$, $h<0$ $nq, n\in {\mathbb{N}}$ $\ell:=\left|\tfrac{h}{q}\right| + k(q-1), \ell\leq |h|$ III$^{-}_{-}$, III$^{s-}_{-}$
$q\in {\mathbb{Z}}^-$, $\tfrac{h}{q} \in {\mathbb{Z}}+ \tfrac{1}{2}$, $h<0$ $n|q|, n\in {\mathbb{N}}$ $\ell:=\tfrac{h}{q} + k(|q|-1), \ell\leq |h|$ III$^{+}_{-}$, III$^{s+}_{-}$
$q\in {\mathbb{Z}}^+$, $\tfrac{h}{q} =-\tfrac{1}{2}$, $h<0$ $nq, n\in {\mathbb{N}}$ $\ell:=\left|\tfrac{h}{q}\right| + k(q-1), \ell\leq |h|$ III$^{s-}_{-}$
$q\in {\mathbb{Z}}^-$, $\tfrac{h}{q} = \tfrac{1}{2}$, $h<0$ $n|q|, n\in {\mathbb{N}}$ $\ell:=\tfrac{h}{q} + k(|q|-1), \ell\leq |h|$ III$^{s+}_{-}$
$q=1$, $\tfrac{h}{q} \in {\mathbb{Z}}+ \tfrac{1}{2}$, $h<0$ $nq, n\in {\mathbb{N}}$ $\left|\tfrac{h}{q}\right|$ III$^{*-}_{-}$
$q=-1$, $\tfrac{h}{q} \in {\mathbb{Z}}+ \tfrac{1}{2}$, $h<0$ $n|q|, n\in {\mathbb{N}}$ $\tfrac{h}{q}$ III$^{*+}_{-}$
------------------------------------------------------------------------------------- --------------------------- ---------------------------------------------------------------- -------------------------------
![The embedding diagrams for the $c=3$ representations considered here: $\text{0}$: $q \not\in{\mathbb{Z}}$, $\tfrac{h}{q} \not\in {\mathbb{Z}}+
\tfrac{1}{2}$, or $q=0$, $h\neq 0$; I$^+$: $q \not\in{\mathbb{Z}}$, $\tfrac{h}{q} \in {\mathbb{Z}}^+ +
\tfrac{1}{2}$; $\text{I}^-$: $q \not\in{\mathbb{Z}}$, $\tfrac{h}{q} \in {\mathbb{Z}}^- +
\tfrac{1}{2}$; $\text{II}$: $q \in {\mathbb{Z}}$ , $\tfrac{h}{q} \not\in {\mathbb{Z}}+ \tfrac{1}{2}$[]{data-label="fig:others"}](othersmall.eps){width="6cm"}
![The embedding diagrams for $c=3$ representations with $\tfrac{h}{q} \in {\mathbb{Z}}+ \tfrac{1}{2}$ and III$_{+}^{+}:{h,\,q>0}$; III$_{+}^{-}:h>0,\,q<0$; IV: $h=q=0$.[]{data-label="fig:embeddingpositiv"}](embnorsmall.eps){height="8cm"}
![The embedding diagrams for $c=3$ representations with $\tfrac{h}{q} \in {\mathbb{Z}}+ \tfrac{1}{2}$ and $\text{III}_{-}^{-}$: $h<0,\, q \geq 2$; $\text{III}_{-}^{+}$: $h<0,\,q \leq -2$; $\text{III}_{-}^{s-}$: $h=-q/2,\, q \geq 1$; $\text{III}_{-}^{s+}$: $h=q/2,\,q \leq -1$; $\text{III}_{-}^{*-}$: $h<0,\, q=1$; $\text{III}_{-}^{*+}$: $h<0,\, q=-1$.[]{data-label="fig:embeddingsub"}](suballsmall.eps){width="13cm"}
Spectral Flow {#sec:flow}
-------------
There exists a family of outer automorphisms $\alpha_\eta: {\mathcal{A}}\rightarrow {\mathcal{A}}$ which map the $N=2$ superconformal algebra to itself. This family of automorphisms is called [*spectral flow*]{} [@schwimmer; @Lerche:1989uy]. Its action on the modes is given by $$\begin{aligned}
\alpha_\eta({G^+_{r}}) & = {\widetilde{G}^+_{r}} = {G^+_{r-\eta}} \\
\alpha_\eta({G^-_{r}}) & = {\widetilde{G}^-_{r}} = {G^-_{r+\eta}} \\
\alpha_\eta(L_n) & = {\widetilde{L}}_n = L_n - \eta J_n +
{\tilde{c}}\tfrac{\eta^2}{2} \delta_{n,0}\\
\alpha_\eta(J_n) & = {\widetilde{J}}_n = J_n - {\tilde{c}}\eta \delta_{n,0},\end{aligned}$$ where $\eta \in {\mathbb{R}}$ is called *flow parameter*. $\eta \in
{\mathbb{Z}}$ maps the NS and R sectors of the algebra to themselves whereas for $\eta \in {\mathbb{Z}}+
\tfrac{1}{2}$ the spectral flow maps the NS sector to the R sector and vice-versa. We will almost exclusively study the case in which $\eta
\in {\mathbb{Z}}$ acts on the NS sector. Therefore from now on $\eta$ is assumed to be an integer unless otherwise stated.
The action of the spectral flow on the operators of the algebra induces an action on the representations. We will take the point of view that under a spectral flow transformation the highest weight state we build representations upon is unchanged. However the action of the transformed modes on the same state gives rise to a new representation. In particular with respect to the new modes the previous highest weight vector might become a descendant state. This point of view has the advantage that the inner product of the vector space the modes act upon is manifestly unchanged. This implies that null vectors must be mapped to null vectors under spectral flow. As an example let us consider the spectral flow with flow parameter $\eta = 1$ acting on some representation $(h,q)$. By fixing ${\tilde{c}}=1$ we obtain $$\begin{aligned}
\label{example}
\alpha_1({G^+_{1/2}}){|h,q\rangle}&={\widetilde{G}^+_{1/2}}{|h,q\rangle}=
{G^+_{-1/2}}{|h,q\rangle} \\
\alpha_1({G^-_{1/2}}){|h,q\rangle}&={\widetilde{G}^-_{1/2}}{|h,q\rangle}=
{G^-_{3/2}}{|h,q\rangle} \label{example2} \\
\alpha_1(L_0){|h,q\rangle} &={\widetilde{L}}_0{|h,q\rangle}=(L_0 - J_0 +
\tfrac{1}{2} ){|h,q\rangle} = (h - q + \tfrac{1}{2}){|h,q\rangle} \\
\alpha_1(J_0){|h,q\rangle}& ={\widetilde{J}}_0{|h,q\rangle}=(J_0 - 1){|h,q\rangle} =
(q-1){|h,q\rangle}.\end{aligned}$$ This example already reveals a crucial observation, namely the spectral flow does not respect the highest weight condition because in general $\alpha_1({G^+_{1/2}}){|h,q\rangle}\neq 0$.
Let now be $\eta \in {\mathbb{N}}$ arbitrary. The highest weight condition for the fermionic operators expressed in the transformed modes is given by $$\label{eq:hwc}
\forall r>0: {\widetilde{G}^-_{r}}{|h,q\rangle} = {\widetilde{G}^+_{r}}{|h,q\rangle} =0.$$ If we express the transformed modes in terms of the original modes condition (\[eq:hwc\]) becomes $$\begin{aligned}
\forall r > 0:\, {G^-_{r+\eta}} {|h,q\rangle} & = 0 \\
\forall r > 0:\, {G^+_{r-\eta}} {|h,q\rangle} & = 0 \label{eq:8}\end{aligned}$$ from which the second condition is not satisfied for $\eta \in
{\mathbb{N}}$. Therefore the hwv changes under spectral flow and a descendant in the original module serves as the highest weight vector of the transformed modes. The highest weight state for the transformed modes is generically given by $$\label{eq:newhwv}
{G^+_{-\eta + 1/2}} \dots {G^+_{-1/2}}{|h,q\rangle} =: {{|h^\eta,q^\eta\rangle_\eta}}, \qquad \eta\in
{\mathbb{N}}.$$ For $\eta \in {\mathbb{Z}}, \eta < 0$ the ${G^+_{s}}$ in formula (\[eq:newhwv\]) have to be interchanged with ${G^-_{s}}$. With respect to the original modes the weight and charge of ${{|h^\eta,q^\eta\rangle_\eta}}$ are given by $$\begin{aligned}
L_0{{|h^\eta,q^\eta\rangle_\eta}}& = (h + \tfrac{\eta^2}{2} ){{|h^\eta,q^\eta\rangle_\eta}}\\
J_0{{|h^\eta,q^\eta\rangle_\eta}}& = (q + \eta){{|h^\eta,q^\eta\rangle_\eta}}. \end{aligned}$$ The weight and charge of ${{|h^\eta,q^\eta\rangle_\eta}}$ with respect to the transformed modes are given by $$\begin{aligned}
{\widetilde{L}}_0{{|h^\eta,q^\eta\rangle_\eta}}& = (L_0 - \eta J_0 + \tfrac{\eta^2}{2}){{|h^\eta,q^\eta\rangle_\eta}}= (h - \eta q)
{{|h^\eta,q^\eta\rangle_\eta}}\label{eq:newweight} \\
{\widetilde{J}}_0 {{|h^\eta,q^\eta\rangle_\eta}}& = q {{|h^\eta,q^\eta\rangle_\eta}}. \label{eq:newcharge}\end{aligned}$$ Let us for later convenience define $$\begin{aligned}
{\widetilde{L}}_0{{|h^\eta,q^\eta\rangle_\eta}}&= h^\eta {{|h^\eta,q^\eta\rangle_\eta}}, & h^\eta &= h - \eta q \quad
\text{and} \label{eq:23} \\
{\widetilde{J}}_0{{|h^\eta,q^\eta\rangle_\eta}}&= q^\eta {{|h^\eta,q^\eta\rangle_\eta}}, &q^\eta &=
q \label{eq:24}. \end{aligned}$$ The charge of the highest weight state generically does not change under spectral flow. Note that we defined $h^\eta$ and $q^\eta$ to denote the eigenvalues of ${\widetilde{L}}_0$ and ${\widetilde{J}}_0$ respectively.
There exists one exception to this construction namely if equation is satisfied in the sense that the state defined by equation is a null state. This was discussed in a slightly different context for $c<3$ in [@Gaberdiel:1997kf]. In this case the “correction” of the highest weight state is not necessary and for the case $\eta=1$ the new highest weight state has weight and charge given by $$\begin{aligned}
\label{eq:9}
{\widetilde{L}}_0 {{|h^\eta,q^\eta\rangle_\eta}}& = (L_0 - J_0 + \tfrac{1}{2}){|h,q\rangle}
=(h-q+\tfrac{1}{2}){|h,q\rangle}, \\
{\widetilde{J}}_0 {{|h^\eta,q^\eta\rangle_\eta}}& = (J_0 - 1){|h,q\rangle} = (q-1){|h,q\rangle}. \label{eq:10}\end{aligned}$$ The formula for general $\eta$ is slightly more complicated.
Unless $G^\pm_{-1/2}{|h,q\rangle}$ is a singular vector, ${|h^\eta, q^\eta\rangle}_\eta$ can never be a descendant of a singular vector in the original representation. This can be seen as follows. Let $n$ denote the level and $m$ the charge of a vector. For any given level $n$ there is a maximal value for the ratio $\tfrac{m}{n}$. Vectors of the form are exactly the vectors for which the ratio $\tfrac{m}{n}$ is maximal at their level. On the other hand singular vectors have at most charge one at some level $n_s$. Therefore any descendant null vector has a lower ratio of $\tfrac{m}{n}$. The only singular vectors whose descendants could have a maximal ratio of $\tfrac{m}{n}$ are exactly $G^\pm_{-1/2}{|h,q\rangle}$. Therefore if $G^\pm_{-1/2}{|h,q\rangle}$ is not singular, ${{|h^\eta,q^\eta\rangle_\eta}}$ will not be a singular vector with respect to the original representation.
The construction of new singular vectors under spectral flow will prove to be essentially the same as for the highest weight state. As we shall see, the situations where the highest weight state is not shifted under spectral flow sometimes gives rise to subsingular vectors. This happens in particular in the non-unitarizable representations of type III.
Representations under spectral flow {#sec:rep-flow}
===================================
We now discuss how the spectral flow acts on representations. As an easy exercise we will first address the action on the embedding diagrams shown in figure \[fig:others\] before we turn to the representations shown in the other diagrams. Our aim is to identify which representations are connected under spectral flow and therefore to group the various representations into orbits under spectral flow transformations. These orbits will depend on the charge $q$ of the highest weight states and the relative sign between the highest weight and the charge.
The easy case {#sec:easy-case}
-------------
Let us first turn to diagrams without any singular vectors, that is to diagrams of type $0$ in figure \[fig:others\]. The spectral flow only shifts the highest weight vector to another vector in the Verma module. The construction of the highest weight vector is as shown in equation and the connection between the representations is the following $$\label{eq:18}
{|h^\eta,q^\eta\rangle}_\eta = {|h - \eta q,q\rangle}$$ for all values of $\eta$.
The situation is analogous for diagrams of type II. The charge $q$ of the highest weight state does not change and therefore the position of the singular vector remains unchanged and equation describes the orbits of these representations, as well.
For diagrams of type I the situation is slightly more involved. As long as $\left|\tfrac{h}{q}\right|\neq\tfrac{1}{2}$ the situation is unchanged. However for $\left|\tfrac{h}{q}\right|=\tfrac{1}{2}$ one of the states $G^\pm_{-1/2}{|h,q\rangle}$ is a singular vector. For definiteness let us choose ${G^+_{-1/2}}{|h,q\rangle}$ to be the singular vector. If we now perform a spectral flow transformation with $\eta=1$ the highest weight condition is satisfied up to the singular vector. Therefore the highest weight state after spectral flow is given by the original highest weight state and has highest weight and charge according to equations and . Each pair of values for $h$ and $q$ such that $\left|\tfrac{h}{q}\right|=\tfrac{1}{2}$ gives rise to an orbit of the spectral flow. Under spectral flow these representations are then mapped to representations whose value of $h$ differs by integer values $$\label{eq:22}
{|\tfrac{q}{2}^\eta,\, q^\eta\rangle}_{\eta} = \begin{cases}
{|-\tfrac{q-1}{2} - (\eta -1)(q-1),\, q-1\rangle}, & \text{for }
\eta > 0 \\
{|\tfrac{q}{2} - \eta q,\, q\rangle}, & \text{for } \eta < 0 .
\end{cases}$$
The interesting case {#sec:interesting-case}
--------------------
Let us now specify to values of $h$ and $q$ such that $q \in {\mathbb{Z}}$ and $\tfrac{h}{q}\in {\mathbb{Z}}+ \frac{1}{2}$. These representations correspond to the embedding diagrams of type III of figures \[fig:embeddingpositiv\] and \[fig:embeddingsub\]. The representations corresponding to diagrams of this type have uncharged singular vectors at level $n|q|$ with $n\in {\mathbb{N}}$ and charged singular vectors of relative charge $\text{sgn}(\tfrac{h}{q})$ of which the first is at level $\left|\tfrac{h}{q}\right|$. Spectral flow of these representations gives as new value for $\tfrac{h^\eta}{q^\eta}$ $$\frac{h - \eta q}{q} = \frac{h}{q} - \eta \in {\mathbb{Z}}+ \frac{1}{2}.$$ Therefore representations of this class are mapped onto each other under spectral flow. In particular as long as the value of $h^\eta$ does not change sign with respect to $h$, the value of $q$ remains unchanged.
However, the existence of singular vectors might spoil the construction of the highest weight vector in some cases as described in section \[sec:flow\]. For $\eta=1$ the weight and charge with respect to the transformed modes are then given by equations and such that $$\frac{\alpha_1(L_0){|h,q\rangle}}{\alpha_1(J_0){|h,q\rangle}}=\frac{h^\eta}{q^\eta}
= -\frac{1}{2}.$$ This indicates that the transformed representation has a negatively charged vector at level $\tfrac{1}{2}$. This was to be expected as this singular vector is needed to reach the original representation by the inverse spectral flow transformation.
We can now address the question of the orbit of a given representation under spectral flow. Let us first assume that $h >0, q>1 $ and $\tfrac{h}{q} = k + \tfrac{1}{2}\in {\mathbb{N}}+
\tfrac{1}{2}$. Gathering together the various pieces of information we see that for $-n := \eta < 0$ the highest weight vector with respect to the transformed modes has weight and charge $(h + n q, q)$ and for $n=\eta, k \geq n > 0$ it has weight and charge $(h - n q, q)$. For $\eta= k +1$ the values of $(h^\eta,q^\eta)$ are $( h - \eta q + \tfrac{1}{2}, q
- 1) = ( - \tfrac{q -1}{2}, q-1)$. For $\eta = k +1 + n$ the new highest weights and charges are given by $( - \tfrac{q -1}{2} -n(q-1), q-1)$. Therefore what characterises the orbits is the charge $q$ of the original highest weight vector. Spectral flow transformations of representations with the highest weight vectors of the form ${|q/2,q\rangle}$ create distinct orbits. Excluding the case $|q|=1$ at the moment we can summarise the above by saying (remember $q>1$) $$\label{eq:orbit1}
{|(\tfrac{q}{2})^\eta,\, q^\eta\rangle}_{\eta} = \begin{cases}
{|-\tfrac{q-1}{2} - (\eta -1)(q-1),\, q-1\rangle}, & \text{for }
\eta > 0 \\
{|\tfrac{q}{2} - \eta q,\, q\rangle}, & \text{for } \eta < 0 .
\end{cases}$$ For the case $h>0, q<-1$, we obtain $${|(-\tfrac{q}{2})^\eta,\, q^\eta\rangle}_{\eta} = \begin{cases}
{|\tfrac{q+1}{2} - (\eta + 1)(q + 1),\, q +1\rangle}, & \text{for }
\eta < 0 \\
{|-\tfrac{q}{2} - \eta q,\, q\rangle}, & \text{for } \eta > 0.
\end{cases}$$ Thus the roles of $\eta>0$ and $\eta<0$ are interchanged. In figure \[fig:typeIII\] we have sketched which representations of this class are connected via spectral flow. Actually there exists a special case when we start off with a representation with $|q|=2$. These embedding diagrams are mapped to the diagrams of type III$_-^{*\pm}$ under spectral flow if we flow to negative values of $h$. This is depicted in figure \[fig:star\].
The case $h=\tfrac{1}{2}$ and $q=1$ is special. For $\eta = 1$ it is mapped to the vacuum representation: $$\begin{aligned}
\alpha_1(L_0) & =(L_0-J_0 + \tfrac{1}{2}){|\tfrac{1}{2},1\rangle} = 0 \\
\alpha_1(J_0) &= (J_0 -1){|\tfrac{1}{2},1\rangle} = 0.\end{aligned}$$ If we apply $\eta = 1$ on the vacuum representation, we obtain the representation $h=\tfrac{1}{2}, q=-1$. This special case therefore can be summarised as $$\label{eq:null}
{|0^\eta, 0^\eta\rangle}_{\eta}=
\begin{cases}
{|\tfrac{1}{2} + (\eta-1), -1\rangle} \text{ for } \eta >0 \\
{|\tfrac{1}{2} + (1-\eta), 1\rangle} \text{ for } \eta <0. \\
\end{cases}$$ This transition under spectral flow is shown in figure \[fig:type0\].
![Spectral flow of type III diagrams, for $h>0, |q|>2$ to $h<0$[]{data-label="fig:typeIII"}](embeddingflow3.eps){width="12cm"}
![Spectral flow of the vacuum representation to representations with $h>0, |q|=1$[]{data-label="fig:type0"}](vacuumflow.eps){width="14cm"}
![Spectral flow of type III diagrams from $h>0, |q|=2$ to $h<0, |q|=1$[]{data-label="fig:star"}](starflow2.eps){width="12cm"}
Unitarity {#sec:unitarity}
---------
The previous constructions show that a large class of representations for $c=3$ are mapped to representations with $h<0$ which are manifestly not unitary. However the unitary representations are mapped onto each other as the condition that $h>0$ is not sufficient to guarantee unitarity.
In [@kent] the conditions for the existence of unitary representations of the $N=2$ algebra were given. For $c=3$ the relevant condition is given by $$({\tilde{c}},h,q) \text{ such that } g_n^A = 0, g_{n+\operatorname{sgn}(n)}^A < 0,
f^A_{1,2}\geq0 \text{ for some }
n\in {\mathbb{Z}}+ \tfrac{1}{2}.$$ which translates into the following conditions for $h$ and $q$ $$\begin{aligned}
\label{eq:2}
(h - nq) &= 0 \\
(h - (n + \operatorname{sgn}(n))q) & < 0 \\
(1 - q^2) & \geq 0.\end{aligned}$$ This implies that only diagrams of type I (figure \[fig:others\]) for $|q|<1$ and diagrams of type III$_+^{\pm}$ with $|q|=1$ and the vacuum representation (type IV) are unitary.
Under spectral flow unitary representations are mapped onto each other. For the type IV and type III diagrams we can see this behaviour in the way the spectral flow only maps the diagrams with charge 0, 1, and $-1$ onto each other. In diagrams of type I we can see that unitarity is preserved by the spectral flow from the following argument.
Let us consider some representation $(h,q)$ such that $\tfrac{h}{q} \in {\mathbb{N}}+ \tfrac{1}{2}$, $h>0$ and $0<q<1$. After spectral flow towards lower values of $h$ the ratio of $h^\eta$ and $q$ will eventually be given by $\tfrac{h^\eta}{q}=\tfrac{1}{2}$. At this point the existing singular vector at level $\tfrac{1}{2}$ alters the prescription of how the weight of the highest weight vector changes and we obtain (c.f. equation ) $$\label{eq:52}
{|h^\eta, q^\eta\rangle} = {|\tfrac{1-q}{2}, q-1\rangle}$$ which defines again a unitary representation (note that $0<q<1$). For $-1<q<0$ we obtain $${|h^\eta, q^\eta\rangle} = {|\tfrac{1+q}{2}, 1+q\rangle}.$$
Singular vectors under spectral flow {#sec:singular-flow}
====================================
After writing down the orbits of representations it is important to ensure that the singular vectors of a given representation transform in a way consistent with the transformation properties of the highest weight vector. The basic idea behind this section is the observation that the spectral flow does not change the scalar product. In particular this implies that null vectors remain null vectors under spectral flow. We will first discuss the situation for generic singular vectors and then turn our attention to the representations where subsingular vectors arise from spectral flow.
Generic singular vectors {#sec:sing-vect}
------------------------
Let us consider a singular vector ${\mathcal{N}}$ of some representation. If we perform a spectral flow transformation on ${\mathcal{N}}$, it will in general not be mapped to a singular vector. However ${\mathcal{N}}$ must be mapped to a null vector as the inner product is invariant under spectral flow. As only the mode numbers of fermionic operators are changed under spectral flow only some fermionic operator can fail to annihilate ${\mathcal{N}}$ after a spectral flow transformation. If we look at the case for $\eta=1$ we observe, comparing with equations and , that ${\widetilde{G}^+_{1/2}}$ fails to annihilate ${\mathcal{N}}$ and the new singular vector is given by $$\label{eq:3}
{\widetilde{G}^+_{1/2}}{\mathcal{N}}.$$ Therefore singular vectors transform generically in the same way as the highest weight state and the structure of the embedding diagrams is (generically) unchanged.
There are a few situations where the above considerations fail to give meaningful results. The first situation where this method fails is the one described preceding equations and . That is, if the highest weight vector does not change under spectral flow. The second situation can arise in representations with $h<0$ where the number of charged singular vectors changes.
Subsingular Vectors {#sec:subsingular}
-------------------
In this section we want to show how spectral flow transformations dictate the existence and form of subsingular vectors in the representations under consideration. Generally there are two classes of subsingular vectors: those with relative charge one and those with higher relative charge. As the situations where they arise are quite different we study both cases in turn. We will first consider the subsingular vectors of charge one. They can arise if we flow from a representation with $h>0$ to a representation with $h<0$.
### Subsingular vectors of charge one {#sec:subs-vect-charge}
In this subsection we want to give evidence that the spectral flow transformations which are shown in figure \[fig:typeIII\] are correct. In order to do so we have to explain how the subsingular vector arises under spectral flow. We will first explain the spectral flow from diagrams with $h>0$ to diagrams with $h<0$ and then turn our attention to the spectral flow from the diagram with $h<0$ without a subsingular vector to the diagram with $h<0$ with a subsingular vector.
Consider the spectral flow with flow parameter $\eta = 1$ of a representation of type III$_+$ with $h=q/2$ and $q\geq2$ to a representation of type III$_-$, see figure \[fig:flow\]. In the original representation the state ${G^+_{-1/2}}{|q/2,q\rangle}$ is a singular vector and therefore, as explained in section \[sec:flow\], the highest weight vector remains the highest weight vector after spectral flow. With respect to the transformed modes weight and charge are given by $$\begin{aligned}
\label{test}
{\widetilde{L}}_{0}{|q/2,q\rangle} & = - \frac{q-1}{2}{|q/2,q\rangle}, \\
{\widetilde{J}}_0{|q/2,q\rangle} & = (q - 1){|q/2,q\rangle}.\end{aligned}$$ Therefore the embedding diagrams change from an infinite number of positively charged singular vectors to a finite number of negatively charged singular vectors as is shown in diagram \[fig:typeIII\].
Consider now the action of the spectral flow on the singular vectors. The charged singular vectors are all descendants of the vector ${G^+_{-1/2}}{|q/2,q\rangle}$. This vector is mapped to ${\widetilde{G}^+_{1/2}}{|q/2,q\rangle}$ and therefore is set to zero by the highest weight condition. This implies that all its descendants are mapped to zero as well. This is not true for the uncharged singular vector at level $q$. If we call this vector ${\mathcal{S}}$, by general arguments of chapter \[sec:sing-vect\] we know that ${\mathcal{S}}$ must be mapped to a null vector and furthermore ${\widetilde{G}^+_{1/2}}{\mathcal{S}}$ must span some submodule. If we look at the embedding diagram of the representation after spectral flow we observe that all singular vectors are descendants of ${\mathcal{C}}={\widetilde{G}^-_{-1/2}}{|q/2,q\rangle}$. The inverse spectral flow transformation maps ${\mathcal{C}}$ to ${G^-_{1/2}}{|q/2, q\rangle}$ which is set identically to zero. As ${\mathcal{C}}$ is identically zero after the inverse spectral flow transformation it follows that ${\widetilde{G}^+_{1/2}}{\mathcal{S}}$ can not be a descendant of ${\mathcal{C}}$. Nevertheless ${\widetilde{G}^+_{1/2}}{\mathcal{S}}$ is a null vector. The resolution is that ${\widetilde{G}^+_{1/2}}{\mathcal{S}}$ is a subsingular vector. More specifically ${\widetilde{G}^-_{1/2}}{\widetilde{G}^+_{1/2}}{\mathcal{S}}$ is given by the uncharged singular vector ${\mathcal{N}}$, as can be seen as follows: Suppose first of all ${\widetilde{G}^-_{1/2}}{\widetilde{G}^+_{1/2}}{\mathcal{S}}=0$. Then ${\widetilde{G}^+_{1/2}}{\mathcal{S}}$ is itself a singular vector of charge $+1$ which is not allowed for a module with a negative ratio of $\tfrac{h}{q}$. Therefore ${\widetilde{G}^-_{1/2}}{\widetilde{G}^+_{1/2}}{\mathcal{S}}$ must not vanish. However, all lowering operators applied to ${\widetilde{G}^-_{1/2}}{\widetilde{G}^+_{1/2}}{\mathcal{S}}$ vanish and therefore ${\widetilde{G}^-_{1/2}}{\widetilde{G}^+_{1/2}}{\mathcal{S}}$ is proportional to the uncharged singular vector ${\mathcal{N}}$.
On the other hand ${\widetilde{G}^+_{1/2}}{\mathcal{S}}$ can not be a descendant of ${\mathcal{N}}$ because ${\mathcal{N}}$ is a descendant of ${\widetilde{G}^+_{-1/2}}\psi$ which is identically zero after the inverse spectral flow transformation. ${\mathcal{S}}$ is a descendant of ${\widetilde{G}^+_{1/2}}{\mathcal{S}}$ and therefore if ${\widetilde{G}^+_{1/2}}{\mathcal{S}}$ were a descendant of ${\mathcal{N}}$ so were ${\mathcal{S}}$. This would imply that ${\mathcal{S}}$ had to vanish under the inverse spectral flow transformation. That ${\mathcal{S}}$ is indeed a descendant of ${\widetilde{G}^+_{1/2}}{\mathcal{S}}$ can be seen as follows. We compute ${\widetilde{G}^-_{-1/2}}{\widetilde{G}^+_{1/2}}{\mathcal{S}}= (2{\widetilde{L}}_0 + {\widetilde{J}}_0 -
{\widetilde{G}^+_{1/2}}{\widetilde{G}^-_{-1/2}}){\mathcal{S}}$. The commutator in the untilded modes is given by $(2L_0 - J_0){\mathcal{S}}= 2q{\mathcal{S}}$ and ${\widetilde{G}^+_{1/2}}{\widetilde{G}^-_{-1/2}}{\mathcal{S}}=
{G^+_{-1/2}}{G^-_{1/2}}{\mathcal{S}}=0$. Therefore we conclude that ${\widetilde{G}^+_{1/2}}{\mathcal{S}}$ is a subsingular vector.
In order to see how this matches with the spectral flow from representations with $h<0$ let us start with the representation $h=-\tfrac{3}{2}q$, $q \in {\mathbb{N}}$. As has been explained in section \[sec:reps\] the number of charged singular vectors varies for representations of type III$_-$ with $h<0$. The representations with $h$ and $q$ as given above have two charged singular vectors. The first one has level $|\tfrac{h}{q}| = 3/2$ and the second one has level $|\tfrac{h}{q}| + (q-1) = \tfrac{1}{2} + q$. Let us call this charged singular vector ${\mathcal{N}}$. ${\mathcal{N}}$ is a descendant of the first uncharged singular vector ${\mathcal{S}}$ as can be read off from figure \[fig:flowm\]. Comparing the levels of the singular vectors ${\mathcal{N}}$ and ${\mathcal{S}}$ we see that ${\mathcal{N}}$ must be proportional to ${G^-_{-1/2}}{\mathcal{S}}$. Now we consider a spectral flow transformation with flow parameter $\eta=-1$. Under this spectral flow transformation ${\mathcal{N}}$ is mapped to ${\widetilde{G}^-_{1/2}}{\mathcal{S}}$. Comparing this with the general result of formula we observe that after spectral flow ${\widetilde{G}^-_{1/2}}{\mathcal{S}}$ itself already defines an uncharged singular vector. Therefore, under spectral flow ${\mathcal{N}}$ is mapped to an uncharged singular vector. The previous uncharged singular vector ${\mathcal{S}}$ becomes subsingular because $$\begin{aligned}
{\widetilde{G}^+_{1/2}}{\mathcal{S}}&= {G^+_{3/2}}{\mathcal{S}}= 0\\
{\widetilde{G}^+_{-1/2}}{\mathcal{S}}&= {G^+_{1/2}}{\mathcal{S}}= 0 \label{eq:13}\\
{\widetilde{G}^-_{1/2}}{\mathcal{S}}&= {G^-_{-1/2}}{\mathcal{S}}= {\mathcal{N}}.\end{aligned}$$ ${\mathcal{S}}$ is not a descendant of ${\mathcal{N}}$ because ${\widetilde{G}^+_{-1/2}}{\mathcal{N}}=
{G^+_{1/2}}{\mathcal{N}}=0$ although a lowering operator maps ${\mathcal{S}}$ to ${\mathcal{N}}$. By general arguments ${\mathcal{S}}$ is a null vector. Therefore it has to be a subsingular vector since there exist no singular vectors of which ${\mathcal{S}}$ could be a descendant.
Observe that the construction of the subsingular vector above depends crucially on the fact that there exists an uncharged singular vector whose $G^\pm_{-1/2}$ descendant is a charged singular vector. In any representation with $h= -\frac{2l+1}{2}|q|$, $q\in {\mathbb{Z}}$, $l\in{\mathbb{N}}$ there exists an uncharged singular vector which satisfies this condition. This singular vector is mapped to an analogue of ${\mathcal{S}}$ under spectral flow and enjoys the same properties, namely that the application of ${\widetilde{G}}^\mp_{1/2}$ maps it to a singular vector, however it is not the descendant of this singular vector. Unlike in the case described above the analogue of ${\mathcal{S}}$ is for higher values of $|h|$ a descendant of some other singular vector of lower level. It is only for the case $l=1$ that there is no other singular vector of which ${\mathcal{S}}$ could be a descendant after spectral flow. The only singular vector with lower level than ${\mathcal{S}}$ is the highest weight vector. Therefore only in this case the subsingular vector appears in the embedding diagram because in all other cases it is a descendant state.
### Subsingular vectors of higher charge {#sec:more-sub}
In diagrams of type III$_-^{*\pm}$ there exist subsingular vectors of higher relative charge[^2]. The embedding diagrams of the modules of type III$_-^{*\pm}$ are shown in figure \[fig:embeddingsub\]. The easiest example of this type of subsingular vectors is given by $$\label{eq:40}
{\mathcal{S}}= {G^-_{-3/2}}{G^-_{-1/2}}{|-3/2,1\rangle}.$$ As one can check, ${\mathcal{S}}$ is not a descendant of any singular vector, in particular not a descendant of the first uncharged singular vector and application of the operator ${G^+_{1/2}}$ maps ${\mathcal{S}}$ to the charged singular vector of the Verma module.
Subsingular vectors of this type arise in all Verma modules with $|q|=1$, $h=-\tfrac{2l + 1}{2}, l\geq1$. Before we comment on the behaviour under spectral flow let us construct these subsingular vectors for all representations in which they arise.
Let us first define for convenience $$\label{eq:28}
\Phi(-\tfrac{2l+1}{2},1):=
{G^-_{-\frac{2l + 1}{2}}}{G^-_{-\frac{2l - 1}{2}}} \dots
{G^-_{-\frac{1}{2}}}{|-\tfrac{2l+1}{2},1\rangle}.$$ We then observe $$\begin{aligned}
\label{eq:29}
{G^+_{\frac{2l+1}{2}}}\Phi(-\tfrac{2l+1}{2},1) = 0 \\
\{{G^+_{\frac{2l-1}{2}}},{G^-_{-\frac{2l-1}{2}}}\}\Phi(-\tfrac{2l+1}{2},1)
= 0 \label{eq:30}.\end{aligned}$$ Furthermore, inspection shows that $$\label{eq:32}
{G^+_{\frac{2l -1}{2}}}\Phi(-\tfrac{2l+1}{2},1) = {G^-_{-\frac{2l -1}{2}}} \dots
{G^-_{-\frac{1}{2}}}{\mathcal{N}}_1,$$ where ${\mathcal{N}}_1$ denotes the uncharged singular vector at level one, see figure \[fig:example\]. These equations are proved in the appendix. Therefore $\Phi(-\tfrac{2l+1}{2},1)$ defines a subsingular vector as it is a null vector and as it is not possible to reach $\Phi(-\tfrac{2l+1}{2},1)$ from ${\mathcal{N}}_1$, whereas applying an appropriate lowering operator maps $\Phi(-\tfrac{2l+1}{2},1)$ to a descendant of ${\mathcal{N}}_1$. However it is not possible to map $\Phi(-\tfrac{2l+1}{2},1)$ to ${\mathcal{N}}_1$ itself. In the example above the descendant in question ${G^-_{-1/2}}{\mathcal{N}}_1$ is the charged singular vector of the Verma module. Observe that $\Phi(-\tfrac{2l+1}{2},1)$ is not annihilated by the operators ${G^+_{\frac{2l-3}{2}}}, \dots,
{G^+_{\frac{1}{2}}}$. Therefore, in general, it is not clear from the start which vector actually defines the subsingular vector of lowest level. The character formulae we calculate in chapter \[sec:characters-c=3\] seem to indicate that the lowest lying subsingular vector is given by $$\label{eq:50}
{\mathcal{S}}_0:={G^+_{\frac{2l-3}{2}}}\dots{G^+_{\frac{1}{2}}}\Phi(-\tfrac{2l+1}{2},1)$$ and thus has always a relative charge of two. In the examples we checked this vector was not a descendant of the singular vector ${\mathcal{N}}_1$, and therefore it was indeed the subsingular vector of lowest charge and level.
Vectors of the form as given by equation are mapped onto each other under spectral flow. They are the images of the singular vector ${G^-_{-1/2}}{|-1/2, 1\rangle}$ under spectral flow. As mentioned in section \[sec:reps\] there exists only one charged singular vector in these embedding diagrams. By the general construction described in section \[sec:subsingular\] this charged singular vector is mapped to an uncharged singular vector under spectral flow with flow parameter $\eta=-1$. However a preimage of the “new” charged singular vector after spectral flow must exist. This preimage is given by the subsingular vectors. Therefore subsingular vectors of higher charge are needed for consistency under spectral flow transformations.
An ambiguity in the embedding diagrams arises when we consider subsingular vectors of higher charge. All subsingular vectors of this kind are mapped to descendants of the first uncharged singular vector ${\mathcal{N}}_1$ however we can not construct an operator which maps the subsingular vector to the singular vector ${\mathcal{N}}_1$ itself whereas the application of an appropriate lowering operator maps the subsingular vector to the charged singular vector. We have adopted the convention to connect the subsingular vector with the singular vector of lowest level whose descendant can be reached by any lowering operator.
The Ramond algebra {#sec:ramond-algebra}
==================
In this section we will briefly comment on the spectral flow between the Neveu-Schwarz and the Ramond (R) algebra. As has been remarked in e.g. [@schwimmer] the spectral flow connects the R and the NS algebra and they are essentially equivalent. The link between the highest weight representations is conveniently easy as spectral flow with $\eta = \pm 1/2$ maps NS ground states to R ground states.
Consider some highest weight state ${|h,q\rangle}$. Under, say $\eta =
1/2$, the relevant modes are mapped to $$\begin{aligned}
\label{eq:6}
\alpha_{1/2}(L_0){|h,q\rangle} = (L_0 - \tfrac{q}{2} +
\tfrac{1}{8}){|h,q\rangle} \\
\label{eq:25}
\alpha_{1/2}(J_0){|h,q\rangle} = (J_0 - \tfrac{1}{2}){|h,q\rangle} \\
\label{eq:26}
\alpha_{1/2}({G^+_{1/2}}){|h,q\rangle} = {G^+_{0}}{|h,q\rangle} \\
\label{eq:27}
\alpha_{1/2}({G^-_{1/2}}){|h,q\rangle} = {G^-_{1}}{|h,q\rangle}\end{aligned}$$ From this we see that the NS highest weight state remains a R highest weight state, if we employ the condition that ${G^+_{0}}{|h,q\rangle}=0$. This is one of the two possible choices of which fermionic zero mode should annihilate the ground state. The other choice has to be made for $\eta = -1/2$. With this requirement the NS highest weight state is mapped to a R highest weight state with eigenvalue $h - \tfrac{q}{2} + \tfrac{1}{8}$ and charge $q-\tfrac{1}{2}$. Therefore the embedding diagrams of the R algebra have the same form as the embedding diagrams of the NS algebra.
Characters for $c=3$ {#sec:characters-c=3}
====================
As an additional consistency check we calculated the characters of the NS algebra for some of the representations discussed previously and determined their behaviour under spectral flow.
The Characters {#sec:characters}
--------------
The Characters for the $N=2$, $c=3$ algebra can be read off from the embedding diagrams. We discuss here only the characters of the type III diagrams and of the vacuum representation. To obtain the remaining characters is a straightforward exercise.
A general character over a representation ${\mathcal{V}}$ of the superconformal algebra is defined as $$\label{eq:15}
\chi^{{\vphantom{+}}}_{{\mathcal{V}}}(q,z):={\operatorname{Tr}^{{\vphantom{+}}}_{{\mathcal{V}}}} (q^{L_0-c/24}z^{J_0}),$$ where $q:=e^{2\pi i \tau}$ and $z:=e^{2\pi i \nu}$. The generic character of the Verma module ${\mathcal{V}}_{h,Q}$ is given by[^3] (see for example [@Dobrev:1987hq; @Kiritsis:1988rv; @Dorrzapf:1997jh; @Eholzer] where the characters for the minimal models are discussed.) $$\label{eq:5}
\chi^{{\vphantom{+}}}_{h,Q}(q,z) = q^{h-c/24} z^Q \prod_{n=1}^\infty \frac{
(1+q^{n-\frac{1}{2}}z) (1+q^{n-\frac{1}{2}}z^{-1})}{(1-q^n)^2}.$$ The character of the vacuum representation for $c=3$ is given by $$\label{eq:4}
\chi^{{\vphantom{+}}}_{0,0}(q,z) = q^{-\frac{1}{8}}
\prod_{n=1}^\infty \frac{
(1+q^{n-\frac{1}{2}}z) (1+q^{n-\frac{1}{2}}z^{-1})}{(1-q^n)^2}
\left(
1- \frac{q^{\frac{1}{2}}z}{1+q^{\frac{1}{2}}z} -
\frac{q^{\frac{1}{2}}z^{-1}}{1+q^{\frac{1}{2}}z^{-1}} \right)$$ as we have to subtract the subrepresentations spanned by the singular vectors from the generic character. As discussed in [@Dorrzapf:1997jh; @Eholzer] the character of a submodule spanned by a charged singular vector of level $n$ is given by $q^n/(1+q^{n-n'})$ where $n'<n$ is the level of the uncharged singular vector which is connected to the charged singular vector by some operator. The characters for type III embedding diagrams without subsingular vectors are given by $$\begin{gathered}
\label{eq:7}
\chi_{h,Q}^{III}(q,z) = q^{h-\frac{1}{8}}z^Q \prod_{n=1}^\infty \frac{
(1+q^{n-\frac{1}{2}}z) (1+q^{n-\frac{1}{2}}z^{-1})}{(1-q^n)^2} \\
\times\left( 1 - q^{|Q|} -
\frac{q^{\left|\frac{h}{Q}\right|}z^{\operatorname{sgn}(\frac{h}{Q})}}{ 1 +
q^{\left|\frac{h}{Q}\right|}z^{\operatorname{sgn}(\frac{h}{Q})}} +
\frac{q^{\left|\frac{h}{Q}\right|+|Q+\operatorname{sgn}(h)|}z^{\operatorname{sgn}(\frac{h}{Q})}}{
1 +
q^{\left|\frac{h}{Q}\right|+|Q+\operatorname{sgn}(h)|-|Q|}
z^{\operatorname{sgn}(\frac{h}{Q})}} \right). \end{gathered}$$ We have to divide out the first charged and uncharged singular vectors and add the second charged singular vector in again. Otherwise we would subtract the charged singular vectors of higher level twice as they are descendants of both the first uncharged and charged singular vectors, see figure \[fig:embeddingpositiv\] and \[fig:embeddingsub\].
The characters of the embedding diagrams with subsingular vectors are special. For the type III$^s$ diagrams the characters are given by $$\begin{gathered}
\label{eq:12}
\chi_{h,Q}^{s}(q,z) = q^{h-\frac{1}{8}}z^Q \prod_{n=1}^\infty \frac{
(1+q^{n-\frac{1}{2}}z) (1+q^{n-\frac{1}{2}}z^{-1})}{(1-q^n)^2} \\
\times\left( 1 -
\frac{q^{\frac{1}{2}}z^{\operatorname{sgn}(\frac{h}{Q})}}{ 1 +
q^{\frac{1}{2}}z^{\operatorname{sgn}(\frac{h}{Q})}} - \frac{q^{|Q| +
\frac{1}{2}}z^{-\operatorname{sgn}(\frac{h}{Q})}}{1 +
q^{\frac{1}{2}}z^{-\operatorname{sgn}(\frac{h}{Q})}} \right).\end{gathered}$$ The first fraction has the usual form and accounts for the singular vectors which have to be divided out. The second fraction accounts for the subsingular vector which is annihilated by the operator $G^{\pm}_{-1/2}$, as can be seen from equation . Therefore we have to divide out the states produced by the application of this particular operator. This character can be seen as a generalisation of the character for the vacuum representation as it has (sub)singular vectors of opposite charge which have to be subtracted from the generic character.
The characters for the embedding diagrams of type III$^*$ are more involved. For $h=-\tfrac{2l+1}{2}$ and $Q=1$ they are given by $$\begin{gathered}
\label{eq:59}
\chi_{-\frac{2l+1}{2},1}^{*}(q,z) =
zq^{-\frac{2l+1}{2}-\frac{1}{8}} \prod_{n=1}^\infty \frac{
(1+q^{n-\frac{1}{2}}z) (1+q^{n-\frac{1}{2}}z^{-1})}{(1-q^n)^2} \\
\times\left( 1 - q - \frac{q^{2l} z^{-2}}{(1 + q^{\frac{2l+1}{2}}z^{-1}) (1
+ q^{\frac{2l-1}{2}}z^{-1})} + \frac{q^{2l+1} z^{-2}}{(1 +
q^{\frac{2l+1}{2}}z^{-1}) (1 + q^{\frac{2l-1}{2}}z^{-1})}\right).\end{gathered}$$ All singular vectors are descendants of the first uncharged singular vector and are thus subtracted by the first $q$ in . The subsingular vector is annihilated by two charged operators as described in section \[sec:more-sub\]. Therefore analogously to the procedure for ordinary singular vectors both operators have to be divided out. In this case the modules created by the singular and the subsingular vector have an overlap and in order to prevent subtracting some vectors twice the correction given by the last fraction has to be added in again. In the easiest case for $h=-3/2$ and $Q=1$ the character formula reads $$\begin{gathered}
\label{eq:14}
\chi_{-3/2,1}^{*}(q,z) = zq^{-\frac{3}{2}-\frac{1}{8}}
\prod_{n=1}^\infty \frac{
(1+q^{n-\frac{1}{2}}z) (1+q^{n-\frac{1}{2}}z^{-1})}{(1-q^n)^2} \\
\times\left( 1 - q - \frac{q^2 z^{-2}}{(1 + q^{\frac{1}{2}}z^{-1}) (1
+ q^{\frac{3}{2}}z^{-1})} + \frac{q^3 z^{-2}}{(1 + q^{\frac{1}{2}}z^{-1}) (1
+ q^{\frac{3}{2}}z^{-1})}\right)\end{gathered}$$ and the overlap is given by the state $$\label{eq:61}
{G^-_{-3/2}}{G^-_{-1/2}}{\mathcal{N}}_1 = 2(L_{-1} + J_{-1}){\mathcal{S}}.$$ For representations with higher values of $|h|$ this formula gets rather cumbersome but can be calculated explicitly, most conveniently via the application of spectral flow to equation .
Spectral flow of characters {#sec:char-flow}
---------------------------
As an additional consistency check we calculated the behaviour of the characters under spectral flow transformations. If we consider equation the obvious way how characters should transform under spectral flow is given by $$\label{eq:20}
{\operatorname{Tr}^{{\vphantom{+}}}_{{\mathcal{V}}_{h,Q}}}(q^{{\widetilde{L}}_0-c/24}z^{{\widetilde{J}}_0}) =
{\operatorname{Tr}^{{\vphantom{+}}}_{{\mathcal{V}}_{h^\eta,Q^\eta}}}(q^{L_0-c/24}z^{J_0}).$$ That is, the trace of the transformed operators over the original representation should equal the character of the representation defined by the eigenvalues $h^\eta$ and $Q^\eta$ of ${\widetilde{L}}_0$ and ${\widetilde{J}}_0$, respectively as defined in equation and . The term on the left hand side of equation can be rewritten to give (for $\eta = 1$) $$\label{eq:39}
{\operatorname{Tr}^{{\vphantom{+}}}_{{\mathcal{V}}_{h,Q}}}(e^{2 \pi i (L_0 - J_0 + \frac{1}{2} - \frac{1}{8})\tau} e^{2 \pi i (J_0 - 1)\nu})
= {\operatorname{Tr}^{{\vphantom{+}}}_{{\mathcal{V}}_{h,Q}}}( e^{2 \pi i(\tau/2 -\nu)} e^{2 \pi i (L_0 -\frac{1}{8})\tau}
e^{2 \pi i J_0(\nu - \tau)}).$$ If we define $\tilde{z}:=e^{2 \pi i(\nu - \tau)}=zq^{-1}$ the trace over the shifted operators can be written as $$\label{eq:44}
{\operatorname{Tr}^{{\vphantom{+}}}_{{\mathcal{V}}_{h,Q}}}(q^{{\widetilde{L}}_0-c/24}z^{{\widetilde{J}}_0})=
{\operatorname{Tr}^{{\vphantom{+}}}_{{\mathcal{V}}_{h,Q}}}(q^{L_0}{\tilde{z}}^{J_0}q^{-\frac{1}{2}}{\tilde{z}}^{-1}).$$ The factor $q^{-\frac{1}{2}}{\tilde{z}}^{-1}$ gives just a constant contribution to the trace. If we substitute this expression into the generic character of equation and then rewrite it again in terms of $q$ and $z$ we obtain $$\begin{aligned}
\label{eq:62}
q^{-\frac{1}{2}}{\tilde{z}}^{-1} \chi^{{\vphantom{+}}}_{h,Q}(q,{\tilde{z}}) &=
q^{\frac{1}{2}}z^{-1} q^{h-Q-\frac{1}{8}} z^Q \frac{1 +
q^{-\frac{1}{2}}z}{1+q^{\frac{1}{2}}z^{-1}} \prod_{n=1}^\infty \frac{
(1+q^{n-\frac{1}{2}}z) (1+q^{n-\frac{1}{2}}z^{-1})}{(1-q^n)^2} \\ \nonumber
& = q^{h-Q-\frac{1}{8}} z^Q \prod_{n=1}^\infty \frac{
(1+q^{n-\frac{1}{2}}z) (1+q^{n-\frac{1}{2}}z^{-1})}{(1-q^n)^2} \\ \nonumber
& = \chi^{{\vphantom{+}}}_{h^\eta, Q}(q,z)\end{aligned}$$ where we have used the fact that $\tfrac{1 + q^{-\frac{1}{2}}z}
{1+q^{\frac{1}{2}}z^{-1}}=q^{-\frac{1}{2}}z$.
Now let us consider the spectral flow of the vacuum representation. To be more explicit we choose $\eta = 1$ and therefore we expect to obtain the character of the representation $h=1/2$, $Q=-1$. Indeed, we get $$\begin{gathered}
\label{eq:54}
q^{-\frac{1}{2}}{\tilde{z}}^{-1}\chi^{{\vphantom{+}}}_{0,0}(q,{\tilde{z}}) =
q^{\frac{1}{2}} z^{-1}
q^{-\frac{1}{8}} \frac{1 + q^{-\frac{1}{2}} z}{1+q^{\frac{1}{2}}z^{-1}}
\prod_{n=1}^\infty
\frac{(1+q^{n-\frac{1}{2}}z)(1+q^{n-\frac{1}{2}}z^{-1})}{(1-q^n)^2} \\
\times \left( 1- \frac{q^{\frac{1}{2}-1}z}{1+q^{\frac{1}{2}-1}z} -
\frac{q^{\frac{1}{2}+1}z^{-1}}{1+q^{\frac{1}{2}+1}z^{-1}} \right)\end{gathered}$$ which can be rewritten as $$\begin{gathered}
\label{eq:53}
q^{\frac{-1}{2}}{\tilde{z}}^{-1}\chi^{{\vphantom{+}}}_{0,0}(q,{\tilde{z}}) =
q^{\frac{1}{2}-\frac{1}{8}}z^{-1}
\prod_{n=1}^\infty \frac{
(1+q^{n-\frac{1}{2}}z)(1+q^{n-\frac{1}{2}}z^{-1})}{(1-q^n)^2} \\
\shoveright{ \left( \frac{1}{1 + q^{\frac{1}{2}}z^{-1}} - \frac{q(1 +
q^{\frac{1}{2}}z^{-1})}{(1+ q^{\frac{1}{2}}z^{-1})(1+
q^{\frac{3}{2}}z^{-1})} \right)} \\ = q^{\frac{1}{2}-\frac{1}{8}}z^{-1}
\prod_{n=1}^\infty
\frac{(1+q^{n-\frac{1}{2}}z)(1+q^{n-\frac{1}{2}}z^{-1})}{(1-q^n)^2} \\
\shoveright{ \left(1 - q -
\frac{q^{\frac{1}{2}}z^{-1}}{1+q^{\frac{1}{2}}z^{-1}} +
\frac{q^{\frac{5}{2}}z^{-1}}{1+q^{\frac{3}{2}}z^{-1}} \right) }\\
= \chi^{{\vphantom{+}}}_{1/2, -1}(q,z).\end{gathered}$$
In the same fashion one can show that the other characters transform appropriately under spectral flow. In particular we can show that $$\label{eq:11}
q^{-\frac{1}{2}}{\tilde{z}}^{-1}\chi^{III}_{Q/2,Q}(q,{\tilde{z}})
=\chi^{s}_{-(Q-1)/2,Q-1}(q,z)$$ and $$\label{eq:21}
q^{-\frac{1}{2}}{\tilde{z}}^{-1}\chi^{s}_{-1/2,1}(q,{\tilde{z}})
=\chi^{*}_{-3/2,1}(q,z).$$ As the calculations are essentially the same we will not demonstrate them here. We therefore conclude that the spectral flow transforms the characters among each other as expected.
Unitary minimal models {#sec:minimal}
======================
Definitions and embedding diagrams {#sec:unitary-def}
----------------------------------
As an example how this technique can be applied to other values of the central charge, let us briefly demonstrate how spectral flow transformations act on the embedding diagrams of the unitary minimal models. The representation theory of the minimal models was extensively discussed, see for example [@Feigin:1998sw; @Semikhatov:1998gv; @Semikhatov:1997pf; @Dorrzapf:1997jh] or [@Kiritsis:1988rv; @Dobrev:1987hq] for earlier results.
Our definitions will closely follow [@Dorrzapf:1997jh]. The unitary minimal models of the $N=2$ algebra can be parametrised by three parameters $m,j,k$ such that $$\begin{aligned}
\label{eq:u1}
c &=3(1-\tfrac{2}{m}), \qquad m\in {\mathbb{N}}, m\geq 2 \\
h &=\frac{jk-\frac{1}{4}}{m}, \qquad j,k \in {\mathbb{N}}+ \tfrac{1}{2},
0<j,k,j+k\leq m-1\\
q &=\frac{j-k}{m}.
\end{aligned}$$ The embedding diagrams for the minimal models were given in [@Dorrzapf:1997jh]. They have the form shown in figure \[fig:unitary\]. In this diagram we denote the highest weight vector by a square, singular vectors corresponding to Kac-determinant vanishings are denoted by a filled circle and descendant singular vectors which do not correspond to Kac-determinant vanishings are denoted by an unfilled circle.
![Embedding Diagram for the Unitary series with weights for the singular vectors.[]{data-label="fig:unitary"}](unitaryweights.eps)
The unitary models have the feature that they possess ‘uncharged fermionic singular vectors’ [@Dorrzapf:1997jh]. These (uncharged singular) vectors possess charged singular vector descendants of only positive or negative charge. In the embedding diagram the uncharged fermionic singular vectors are denoted by arrows pointing to the right or to the left, respectively.
Spectral flow {#sec:minimal-flow}
--------------
The action of the spectral flow on the highest weight states of minimal models has been discussed in [@Gaberdiel:1997kf]. Analogously to the $c=3$ case one has to distinguish between a generic case where the highest weight vector after spectral flow is given by a descendant of the highest weight vector before spectral flow, and a special case where the descendant which would be the new highest weight vector is itself a singular vector. More precisely, a representation $(j,k)$ for fixed $m$ is generically transformed under spectral flow with flow parameter $\eta=1$ to the representation $(j+1, k - 1)$ where the highest weight state is then given by ${\widetilde{G}^+_{1/2}}{|(j,k)\rangle}={G^+_{-1/2}}{|(j,k)\rangle}$. If $k=\tfrac{1}{2}$, ${G^+_{-1/2}}{|(j,\tfrac{1}{2})\rangle}$ is a singular vector and the state ${|(j,\tfrac{1}{2})\rangle}$ is mapped to the highest weight state after spectral flow. In this case we find that the the representation $(j,\tfrac{1}{2})$ is mapped to the representation $\alpha_{1}(j,\tfrac{1}{2})=(\tfrac{1}{2}, m-j-1)$. The same phenomenon occurs for $\eta = -1$ and $j=\tfrac{1}{2}$. This situation is analogous to the case for $c=3$, if ${G^-_{-1/2}}{|h,q\rangle}$ is a singular vector (c.f. section \[sec:rep-flow\]).
Let us first analyse the generic case $j,k \neq \tfrac{1}{2}$. For a generic spectral flow transformation with $\eta=1$, the highest weight state after spectral flow is given by ${\widetilde{G}^+_{1/2}}{|(j,k)\rangle}$. In this case a singular vector ${\mathcal{N}}$ of the representation $(j,k)$ is mapped to the singular vector ${\widetilde{G}^+_{1/2}}{\mathcal{N}}$. The levels of uncharged singular vectors are derived in [@Dorrzapf:1997jh]. They are given by $$\Delta^0 = n(nm + j + k), \qquad n\in {\mathbb{Z}}\backslash\{0\}.$$ The combination $(j + k)$ is invariant under the generic spectral flow, therefore the levels of uncharged singular vectors remain unchanged. This should be the case as the level of an uncharged singular vector operator is not changed by spectral flow. The levels of charged singular vectors are given by $$\begin{aligned}
\label{eq:u2}
\Delta^{+} &= k + n((n+1)m + j + k) \qquad n\in
{\mathbb{Z}}\backslash\{-1,0\}\\
\Delta^{-} &= j + n((n+1)m + j + k) \qquad n\in
{\mathbb{Z}}\backslash\{-1,0\} \nonumber\end{aligned}$$ As has been discussed in section \[sec:rep-flow\], the levels of charged singular vectors are altered under spectral flow according to $$\alpha_{\eta}(L_0){\mathcal{N}}_\eta^{\pm} = (h_{{\mathcal{N}}} \mp \eta){\mathcal{N}}_\eta^{\pm},$$ where ${\mathcal{N}}_\eta^{\pm}$ is the singular vector analogue of . As we can read off from the levels of the charged singular vectors in the embedding diagrams transform accordingly. It can be shown that in the generic case, the spectral flow respects the splitting of the uncharged singular vectors into left and right fermionic uncharged singular vectors. The existence of uncharged fermionic singular vectors is due to the fact, that some products of singular vector operators vanish identically [@Dorrzapf:1997jh]. Under (generic) spectral flow transformations these vanishings are preserved. Therefore the splitting of the uncharged singular vectors in left and right uncharged fermionic vectors is preserved under generic spectral flow transformations.
Let us now analyse the spectral flow of $(j, \tfrac{1}{2})$ to $(\tfrac{1}{2}, m-j-1)$. In the representation $(j, \tfrac{1}{2})$ the vector ${G^+_{-1/2}}{|(j, \tfrac{1}{2})\rangle}$ is singular and therefore the highest weight vector remains a highest weight state after spectral flow.
In order to obtain a candidate for a singular vector from a singular vector after spectral flow, we have to correct the shifted modes, and the new singular vector would be given by ${\widetilde{G}^+_{1/2}}{\mathcal{N}}$. If we compare the levels of singular vectors before and after spectral flow and keep in mind that ${G^+_{-1/2}}{|(j, \tfrac{1}{2})\rangle}$ is mapped to ${\widetilde{G}^+_{1/2}}{|(j, \tfrac{1}{2})\rangle}$ and is therefore the new highest weight condition, the images of the uncharged singular vectors are at the levels of the positively charged singular vectors after spectral flow and the images of the negatively charged singular vectors are at the levels of the uncharged singular vectors. The descendants of ${G^+_{-1/2}}{|(j,
\tfrac{1}{2})\rangle}$ vanish after spectral flow. However the highest weight condition ${G^-_{1/2}}{|(j,
\tfrac{1}{2})\rangle}$ is mapped to the singular vector ${\widetilde{G}^-_{-1/2}}{|(j,
\tfrac{1}{2})\rangle}$. Due to this fact the vectors ${\widetilde{G}^+_{1/2}}{\mathcal{N}}$ which used to be singular vectors in the generic case are now mapped to null vectors, which are [*at least subsingular vectors*]{}. I.e. in general they are mapped to vectors of the appropriate level and charge to be singular vectors, however their image under spectral flow in general can be written as ${\mathcal{N}}_{k}^{\rm{new}} = +\Theta_{-k}{|(1/2,
m-j-1)\rangle} + \Theta_{-k+1/2}{G^-_{-1/2}}{|(1/2, m-j-1)\rangle}$, where $\Theta_{-k}$ is a (sub)singular vector operator. Therefore these vectors are at least singular after dividing out ${G^-_{-1/2}}{|(1/2,
m-j-1)\rangle}$. However the appearance of additional subsingular vectors would result in additional submodules of all representations. Therefore we can conclude that singular vector are flown to linear combinations of (descendants of) singular vectors.
As a final remark let us mention that we checked the transformation properties of the characters of the minimal models (for their form see e.g. [@Dorrzapf:1997jh]) under spectral flow. Similarly to the $c=3$ case it can be shown that they transform accordingly under spectral flow giving another argument that the spectral flow respects the embedding structure.
Conclusions {#sec:conclusions}
===========
We have analysed in detail the behaviour of representations and singular vectors of the $N=2$ superconformal algebra with central charge $c=3$ under spectral flow. The representations are mapped to representations of the same kind in orbits determined by the charge of the highest weight state. A careful analysis of the spectral flow of singular vectors predicts subsingular vectors in representations with $h<0$. It would be interesting to understand if non-unitarity is a necessary condition for the existence of subsingular vectors in the $N=2$ algebra.
Furthermore it was possible to construct subsingular vectors with relative charge greater than two which were previously unknown. As an example how our technique can be used to analyse representations with other values of $c$ we applied it to the unitary minimal models and found the results consistent with the embedding diagrams.
As an additional check we determined the action of the spectral flow on the characters and found complete agreement with the behaviour of the embedding diagrams. To this end we had to construct the characters for representations with subsingular vectors which, to our knowledge, have not been written down in the literature before.
As mentioned in the beginning we were not able to prove the actual form of the embedding diagrams. However the behaviour under spectral flow at least suggests that they are of the suggested form. As an additional check we performed some numerical tests. We calculated the inner product matrices up to level four and compared the dimension of their null-space with the dimension given by the null vectors we assume to exist. The dimensions of the null-spaces matched nicely with our expectations based on the embedding diagrams. We performed this test on various representations up to level four and found complete agreement.
Acknowledgements {#sec:acknoledgements .unnumbered}
================
It is a pleasure to thank my PhD supervisor Matthias Gaberdiel for support and helpful discussions, furthermore I would like to thank Andreas Recknagel, Gerard Watts and in particular Matthias D[ö]{}rrzapf and Kevin Graham for interesting discussions and Matthew Hartley for a very helpful Perl script. The computation of the inner product matrices were done using Maple and <span style="font-variant:small-caps;">Reduce</span>. This work was supported by a scholarship of the Marianne und Dr. Fritz Walter Fischer-Stiftung and a Promotionsstipendium of the DAAD.
Proofs for section \[sec:more-sub\] {#sec:proofs}
===================================
We want to prove equations – . Let $\Phi$ be defined as $$\label{eq:33}
\Phi(-\tfrac{2l+1}{2},1):=
{G^-_{-\frac{2l + 1}{2}}}{G^-_{-\frac{2l - 1}{2}}} \dots
{G^-_{-\frac{1}{2}}}{|-\tfrac{2l+1}{2},1\rangle}$$ as in equation . For ease of notation let us furthermore define the abbreviation $$\label{eq:37}
\Gamma(2l-1) {|-\tfrac{2l+1}{2},1\rangle}:= {G^-_{-\frac{2l-1}{2}}}
{G^-_{-\frac{2l - 3}{2}}} \dots
{G^-_{-\frac{1}{2}}}{|-\tfrac{2l+1}{2},1\rangle}$$ for the product of fermionic operators. This product of operators satisfies: $$\label{eq:1}
{G^+_{\frac{2l+1}{2}}} \Gamma(2l-1) {|h,q\rangle} = 0$$ for any $h, q$, because the commutator of ${G^+_{a}}$ with any of the operators ${G^-_{-b}}$ is proportional to some bosonic lowering operator $\Theta_{a-b}$. The commutator of this bosonic operator with any of the fermionic raising operators ${G^-_{-c}}$ is either proportional to some fermionic lowering operator if $a>b+c$ which anticommutes with the other lowering operators and annihilates the ground state or to some raising operator ${G^-_{a-b-c}}$. As $a>0$ and because of the particular form of $\Gamma(2l-1)$, ${G^-_{a-b-c}}$ will occur twice and therefore this expression will vanish.
The first equation we want to show is
**Equation** $$\label{eq:41}
\{{G^+_{\frac{2l-1}{2}}},{G^-_{-\frac{2l-1}{2}}}\}\Phi(-\tfrac{2l+1}{2},1)
= 0.$$
To show this we have to compute the anticommutator which is given by $$\label{eq:17}
\left(2L_0 + (2l-1)J_0 + \tfrac{(2l-1)^2 -1}{4} \right)
\Phi(-\tfrac{2l+1}{2},1)$$ The operators in the anticommutator evaluated on the given expressions are $$\begin{aligned}
\label{eq:36}
L_0 \Phi(-\tfrac{2l+1}{2},1)
&= \left(\frac{(l+1)^2}{2} - \frac{2l + 1}{2}\right)
\Phi(-\tfrac{2l+1}{2},1),\\
J_0 \Phi(-\tfrac{2l+1}{2},1) &= -l
\Phi(-\tfrac{2l+1}{2},1). \label{eq:42}
\end{aligned}$$ Adding everything up proves equation .
Now we turn our attention to
**Equation** $$\label{eq:34}
{G^+_{\frac{2l + 1}{2}}}\Phi(-\tfrac{2l+1}{2},1) = 0.$$
We compute $$\begin{gathered}
\label{eq:35}
{G^+_{\frac{2l + 1}{2}}}\Phi(-\tfrac{2l+1}{2},1) =
{G^+_{\frac{2l + 1}{2}}} {G^-_{-\frac{2l + 1}{2}}}
\Gamma(2l-1){|-\tfrac{2l+1}{2},1\rangle} = \\
\left( \{{G^+_{\frac{2l + 1}{2}}}, {G^-_{-\frac{2l + 1}{2}}} \} -
{G^-_{-\frac{2l + 1}{2}}} {G^+_{\frac{2l + 1}{2}}} \right)
\Gamma(2l-1){|-\tfrac{2l+1}{2},1\rangle} = \\
\left(2L_0 + (2l+1)J_0 + \tfrac{(2l+1)^2 -1}{4}
- {G^-_{-\frac{2l + 1}{2}}} {G^+_{\frac{2l + 1}{2}}} \right)
\Gamma(2l-1){|-\tfrac{2l+1}{2},1\rangle}.\end{gathered}$$ The last term of equation vanishes by equation . The terms stemming from the anticommutator vanish, as can be shown using equations and and adjusting some factors to take into account that the mode numbers differ slightly and we can conclude $$\label{eq:38}
\left(2L_0 + (2l+1)J_0 + \tfrac{(2l+1)^2 -1}{4} \right)
\Gamma(2l-1){|-\tfrac{2l+1}{2},1\rangle} = 0,$$ which proves our statement.
Furthermore we have to prove equation , i.e.
**Equation** $$\label{eq:31}
{G^+_{\frac{2l -1}{2}}}\Phi(-\tfrac{2l+1}{2},1) = {G^-_{-\frac{2l -1}{2}}} \dots
{G^-_{-\frac{1}{2}}}{\mathcal{N}}_1.$$
The first step is to give the general form of the singular vector at level 1 of the Verma modules with $h=-\tfrac{2l + 1}{2}$ and $q=1$. Explicit calculations show that the first uncharged singular vector at level one is given by $$\begin{aligned}
\label{eq:43}
{\mathcal{N}}_1 & = \left( 2L_{-1} - (2h + 1)J_{-1} -
{G^-_{-1/2}}{G^+_{-1/2}}\right){|h,1\rangle} \\
& = \left(2L_{-1} +2l J_{-1} -
{G^-_{-1/2}}{G^+_{-1/2}}\right){|-\frac{2l+1}{2},1\rangle}.
\end{aligned}$$ The next step is to evaluate $$\label{eq:46}
{G^+_{\frac{2l -1}{2}}}\Phi(-\tfrac{2l+1}{2},1)$$ which gives $$\begin{aligned}
\label{eq:47}
{G^+_{\frac{2l -1}{2}}}\Phi(-\tfrac{2l+1}{2},1) &= (2L_{-1} + 2lJ_{-1})
{G^-_{-\frac{2l-1}{2}}}\dots{G^-_{-\frac{1}{2}}}{|-\frac{2l+1}{2},1\rangle}
\nonumber \\
&- {G^-_{-\frac{2l + 1}{2}}} \left(2L_0 + (l-1)J_0 +
\frac{(2l-1)^2-1}{4}\right)
{G^-_{-\frac{2l-3}{2}}}\dots{G^-_{-\frac{1}{2}}}{|-\frac{2l+1}{2},1\rangle}
\nonumber \\
&+ {G^-_{-\frac{2l+1}{2}}}{G^-_{-\frac{2l-1}{2}}}{G^+_{\frac{2l -1}{2}}}
{G^-_{-\frac{2l-3}{2}}}\dots{G^-_{-\frac{1}{2}}}{|-\frac{2l+1}{2},1\rangle}
\end{aligned}$$
The first line gives $$\label{eq:49}
-2 {G^-_{-\frac{2l + 1}{2}}} {G^-_{-\frac{2l-3}{2}}} \dots
{G^-_{-\frac{1}{2}}}{|-\frac{2l+1}{2},1\rangle} +{G^-_{-\frac{2l-1}{2}}} \dots
{G^-_{-\frac{1}{2}}}{\mathcal{N}}_1$$ because all other commutators of $L_{-1}$ and $J_{-1}$ with the fermionic operators are proportional to some other fermionic operator ${G^-_{a-1}}$. Apart from the first term this operator already exists in the row of operators and therefore the commutator vanishes. Only the first term and the last term are exceptional and remain. The second line of equation gives $$\label{eq:48}
2 {G^-_{-\frac{2l + 1}{2}}} {G^-_{-\frac{2l-3}{2}}} \dots
{G^-_{-\frac{1}{2}}}{|-\frac{2l+1}{2},1\rangle}$$ The last line of equation is of the type ${G^+_{\frac{2l+1}{2}}}\Gamma(2l-1){|h,q\rangle}$ and therefore vanishes. So we are left with $$\label{eq:16}
(2 - 2) {G^-_{-\frac{2l + 1}{2}}} {G^-_{-\frac{2l-3}{2}}} \dots
{G^-_{-\frac{1}{2}}}{|-\frac{2l+1}{2},1\rangle} +
{G^-_{-\frac{2l-1}{2}}} \dots {G^-_{-\frac{1}{2}}}{\mathcal{N}}_1.$$ The first two terms cancel out and thus we have proved our statement.
[10]{}
M. D[ö]{}rrzapf, *Superconformal [F]{}ield [T]{}heories and [T]{}heir [R]{}epresentations*, Ph.D. thesis, University of Cambridge, 1995.
M. D[ö]{}rrzapf, *[Analytic Expressions for Singular Vectors of the $N=2$ Superconformal Algebra]{}*, Commun. Math. Phys. **180** (1996) 195, hep-th/9601056.
M. D[ö]{}rrzapf, *[Singular Vectors of the N=2 Superconformal Algebra]{}*, Int. J. Mod. Phys. **A10** (1995) 2143, hep-th/9403124.
V. K. Dobrev, *[Characters of the Unitarizable Highest Weight Modules over the N=2 Superconformal Algebras]{}*, Phys. Lett. **B186** (1987) 43.
B. Gato-Rivera and J. I. Rosado, *[New Interpretation for the Determinant Formulae of the N=2 Superconformal Algebras]{}*, (1996) hep-th/9602166.
A. M. Semikhatov and I. Y. Tipunin, *[The structure of Verma modules over the N = 2 superconformal algebra]{}*, Commun. Math. Phys. **195** (1998) 129, hep-th/9704111.
A. M. Semikhatov and V. A. Sirota, *[Embedding diagrams of N = 2 Verma modules and relaxed]{} $\widehat{sl(2)}$ [Verma modules]{}*, (1997) hep-th/9712102.
B. L. Feigin, A. M. Semikhatov, V. A. Sirota, and I. Yu. Tipunin, *[Resolutions and Characters of Irreducible Representations of the N=2 Superconformal Algebra]{}*, Nucl. Phys. **B536** (1998) 617, hep-th/9805179.
B. Gato-Rivera, *A note concerning subsingular vectors and embedding diagrams of the n=2 superconformal algebras*, (1999) hep-th/9910121.
M. Ademollo, L. Brink, A. D’Adda, R. D’Auria, E. Napolitano, S. Sciuto, E. Del Giudice, P. Di Vecchia, S. Ferrara, F. Gliozzi, R. Musto, and R. Pettorino, *[Supersymmetric Strings and Color Confinement]{}*, Phys. Lett. **B62** (1976) 105.
W. Boucher, D. Friedan, and A. Kent, *[Determinant Formulae and Unitarity for the N=2 Superconformal Algebras in Two Dimensions or Exact Results on String Compactification]{}*, Phys. Lett. **B172** (1986) 316.
M. Kato and S. Matsuda, *[N]{}ull [F]{}ield [C]{}onstruction and [K]{}ac [F]{}ormulae of [N=2]{} [S]{}uperconformal [A]{}lgebras in [T]{}wo-[D]{}imensions*, Phys. Lett. **B184** (1987) 184.
W. Lerche, C. Vafa, and N. P. Warner, *[Chiral Rings in N=2 Superconformal Theories]{}*, Nucl. Phys. **B324** (1989) 427.
B. R. Greene and M. R. Plesser, *[Duality in Calabi-Yau Moduli Space]{}*, Nucl. Phys. **B338** (1990) 15.
B. R. Greene, *[String Theory on Calabi-Yau manifolds]{}*, (1996) hep-th/9702155.
A. Schwimmer and N. Seiberg, *[Comments on the N=2, N=3, N=4 Superconformal Algebras in Two Dimensions]{}*, Phys. Lett. **B184** (1987) 191.
M. R. Gaberdiel, *[Fusion of Twisted Representations]{}*, Int. J. Mod. Phys. **A12** (1997) 5183, hep-th/9607036.
M. D[ö]{}rrzapf and B. Gato-Rivera, *[Transmutations between Singular and Subsingular Vectors of the [N = 2]{} Superconformal Algebras]{}*, Nucl. Phys. **B557** (1999) 517, hep-th/9712085.
E. Kiritsis, *[Character Formulae and the Structure of the Representations of the N=1, N=2 Superconformal Algebras]{}*, Int. J. Mod. Phys. **A3** (1988) 1871.
M. D[ö]{}rrzapf, *[The Embedding Structure of Unitary N = 2 Minimal Models]{}*, Nucl. Phys. **B529** (1998) 639, hep-th/9712165.
W. Eholzer and M. R. Gaberdiel, *[Unitarity of Rational N=2 Superconformal Theories]{}*, Commun. Math. Phys. **186** (1997) 61, hep-th/9601163.
[^1]: This can be ambiguous, see section \[sec:more-sub\].
[^2]: For $c<3$ subsingular vectors of higher relative charge were first discussed in [@rivera]. However the example given there is not subsingular for $c=3$. The subsingular vector constructed in [@rivera] is still a null vector for $c=3$ but it becomes a descendant state of a singular vector.
[^3]: In this section we will use a capital $Q$ to denote the charge.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A 2003 counterexample to a conjecture of Auslander brought attention to a family of rings—colloquially called AC rings—that satisfy a natural condition on vanishing of cohomology. Several results attest to the remarkable homological properties of AC rings, but their definition is barely operational, and it remains unknown if they form a class that is closed under typical constructions in ring theory. In this paper, we study transfer of the AC property along local homomorphisms of Cohen–Macaulay rings. In particular, we show that the AC property is preserved by standard procedures in local algebra. Our results also yield new examples of Cohen–Macaulay AC rings.'
address:
- 'Department of Math. and Stat., Texas Tech University, Lubbock, TX 79409, U.S.A.'
- 'Department of Basic Sciences and Environment, University of Copenhagen, Thorvaldsensvej 40, DK-1871 Frederiksberg C, Denmark'
author:
- Lars Winther Christensen
- Henrik Holm
date: 23 February 2012
title: |
Vanishing of cohomology\
over [C]{}ohen–[M]{}acaulay rings
---
Introduction {#introduction .unnumbered}
============
Vanishing of Ext groups and functors play a crucial role in the study of rings and their modules. For a fixed module $M$, vanishing of ${\operatorname{Ext}_{}^{n}(M,N)}$ for all modules $N$ and integers $n \gg 0$ says that $M$ has finite projective dimension, in which case the functors ${\operatorname{Ext}_{}^{n}(M,-)}$ vanish for all $n > {\operatorname{proj.\!dim}_{}(M)}$. Finiteness of ${\operatorname{proj.\!dim}_{R}(M)}$ for all finitely generated $R$-modules is a lot to ask from a ring—in the noetherian case it means that $R$ is regular. Auslander conjectured that for every finitely generated module $M$, also those of infinite projective dimension, there would be a sort of an upper bound for non-vanishing of the groups ${\operatorname{Ext}_{}^{n}(M,N)}$; see [@mas1 ch. V]. The exact conjecture is that every Artin algebra $R$ satisfies the following:
[<span style="font-variant:small-caps;">(ac)</span>]{} For every finitely generated $R$-module $M$ there exists an integer ${b_{M}} {\geqslant}0$ such that for every finitely generated $R$-module $N$ one has: ${\operatorname{Ext}_{R}^{n}(M,N)}=0$ for $n \gg 0$ implies ${\operatorname{Ext}_{R}^{n}(M,N)}=0$ for $n > {b_{M}}$.
An integer ${b_{M}}$ with this property is called an *Auslander bound* for $M$; the least such bound may be perceived as a “latent projective dimension” of $M$.
An example, discovered by Jorgensen and Şega [@DAJLMS04], disproved Auslander’s conjecture. However, several classes of rings do satisfy Auslander’s condition [<span style="font-variant:small-caps;">(ac)</span>]{}, and remarkable homological properties that flow from [<span style="font-variant:small-caps;">(ac)</span>]{} have been uncovered, for example, in work of Huneke and Jorgensen [@CHnDAJ03]; see also [@LWCHHlc].
The counterexample in [@DAJLMS04] is a commutative Gorenstein local ring, which is a finite dimensional algebra, and even Koszul. Nevertheless, the condition [<span style="font-variant:small-caps;">(ac)</span>]{} is satisfied by all commutative local rings that are complete intersection or Golod, and that is striking: with regard to homological characteristics these two classes of rings usually belong on opposite sides of the spectrum.
Among commutative noetherian local rings, those that satisfy [<span style="font-variant:small-caps;">(ac)</span>]{} are emerging as a family with intriguing homological properties, albeit one whose position relative to the traditional classes of rings is not easily described. In this paper, we give new examples—explicit and abstract—of Cohen–Macaulay local rings that satisfy [<span style="font-variant:small-caps;">(ac)</span>]{}.
$*\:*\:*$
From this point on, all rings considered are commutative, noetherian, and local. In particular, $(R,{\mathfrak{m}})$ is such a ring, and ${\widehat{R}}$ denotes its ${\mathfrak{m}}$-adic completion.
Most Cohen–Macaulay rings of finite CM type are Golod and, therefore, known from [[@DAJLMS04 prop. 1.4]]{} to satisfy even a uniform version of Auslander’s condition:
[<span style="font-variant:small-caps;">(uac)</span>]{} There exists an integer $b {\geqslant}0$ such that for all finitely generated $R$-modules $M$ and $N$ one has: ${\operatorname{Ext}_{R}^{n}(M,N)}=0$ for $n\gg 0$ implies ${\operatorname{Ext}_{R}^{n}(M,N)}=0$ for $n >
b$.
In [Section \[sec:cm\]]{} we show by a direct argument that every Cohen–Macaulay ring of finite CM type satisfies [<span style="font-variant:small-caps;">(uac)</span>]{}.
An explicit collection of new examples of rings that satisfy [<span style="font-variant:small-caps;">(uac)</span>]{} is constructed via results about preservation of the [<span style="font-variant:small-caps;">(ac)</span>]{} and [<span style="font-variant:small-caps;">(uac)</span>]{}properties under standard operations in local algebra—such as completion and reduction modulo regular elements. These results were advertised in [@LWCHHlc rmk. 5.7]; they are proved in [Section \[sec:stab\]]{}, and the examples follow in [Section \[sec:example\]]{}.
The canonical maps and , where $x$ is a regular element, are archetypes of local homomorphisms of finite flat dimension; they are even c.i. homomorphisms in the sense of Avramov [@LLA99]. A classical chapter of local algebra studies transfer of ring theoretic properties along local homomorphisms. In [Section \[sec:homos\]]{} we add to it as we prove, essentially, that the [<span style="font-variant:small-caps;">(ac)</span>]{} and [<span style="font-variant:small-caps;">(uac)</span>]{} properties descend along homomorphisms of finite flat dimension and ascend along c.i. homomorphisms.
Maximal Cohen–Macaulay modules {#sec:cm}
==============================
For a finitely generated $R$-module $M$, the $n$th syzygy in its minimal free resolution is denoted ${\operatorname{syz}^R_{n}(M)}$. For $i > n {\geqslant}0$ there are isomorphisms: $$\label{eq:is1}
{\operatorname{Ext}_{R}^{i-n}({\operatorname{syz}^R_{n}(M)},N)} {\cong}{\operatorname{Ext}_{R}^{i}(M,N)}.$$
Let $R$ be Cohen–Macaulay of dimension $d$. A finitely generated $R$-module $M$ is called *maximal Cohen–Macaulay*, abbreviated MCM, if the equality ${\operatorname{depth}_{R}M}=d$ holds. Every $d$th syzygy of a finitely generated $R$-module is either $0$ or an MCM module, and every syzygy of an MCM module is an MCM module; see [[@yos prop. (1.16) and (1.3)]]{}. These facts together with the theory of MCM approximations, which is due to Auslander and Buchweitz [@MAsROB89], are central to this section.
\[rmk:mcmab\] Let $R$ be Cohen–Macaulay of dimension $d$. Dimension shifting [([\[eq:is1\]]{})]{} shows that $R$ satisfies [<span style="font-variant:small-caps;">(ac)</span>]{} if and only if every MCM $R$-module has an Auslander bound. Every MCM module is a (finite) direct sum of indecomposable MCM modules, so $R$ satisfies [<span style="font-variant:small-caps;">(ac)</span>]{} if and only if every indecomposable MCM $R$-module has an Auslander bound. In particular, $R$ satisfies [<span style="font-variant:small-caps;">(uac)</span>]{} if and only if the indecomposable MCM $R$-modules have a common Auslander bound.
A Cohen–Macaulay ring $R$ is said to be of *finite CM type* if there are only finitely many indecomposable MCM $R$-modules, up to isomorphism.
\[thm:cm\] A Cohen–Macaulay local ring of finite CM type satisfies [<span style="font-variant:small-caps;">(uac)</span>]{}.
Let $R$ be Cohen–Macaulay of finite CM type. By [Remark [\[rmk:mcmab\]]{}]{}, it suffices to prove that each of the finitely many indecomposable MCM $R$-modules has an Auslander bound. For an MCM $R$-module $X$, let $n(X)$ be the number of distinct, up to isomorphism, indecomposable MCM $R$-modules that occur as direct summands of any one of the syzygies ${\operatorname{syz}^R_{i}(X)}$, for $i {\geqslant}0$.
Let $M$ be an indecomposable MCM $R$-module. If $n(M)$ is $1$, then $M$ is either free of rank $1$ or a direct summand of every syzygy module ${\operatorname{syz}^R_{i}(M)}$ for $i {\geqslant}0$, whence it has Auslander bound $0$; cf. [([\[eq:is1\]]{})]{}. Let $t {\geqslant}1$ and assume that every MCM $R$-module $X$ with $n(X) {\leqslant}t$ has an Auslander bound. Assume $n(M) = t+1$. If $M$ occurs as a direct summand of one of its syzygies, say, ${\operatorname{syz}^R_{l}(M)}$ for some $l {\geqslant}1$, then—by uniqueness of minimal free resolutions—$M$ occurs as a summand of infinitely many syzygies, namely ${\operatorname{syz}^R_{lm}(M)}$ for all $m\in{\mathbb{N}}$; whence $M$ has Auslander bound $0$. If $M$ does not occur as a direct summand of ${\operatorname{syz}^R_{i}(M)}$ for any $i {\geqslant}1$, then one has $n({\operatorname{syz}^R_{1}(M)}) {\leqslant}t$. By the induction hypothesis, ${\operatorname{syz}^R_{1}(M)}$ has an Auslander bound, whence $M$ has one, again by dimension shifting [([\[eq:is1\]]{})]{}.
The *embedding dimension* of $R$, ${\operatorname{edim}R}$, is the minimal number of generators of its maximal ideal, and the *codimension* is the difference ${\operatorname{codim}R} = {\operatorname{edim}R} - {\operatorname{dim}R}$. The multiplicity ${\operatorname{e}(R)}$ of a Cohen–Macaulay ring is at least ${\operatorname{codim}R} +1 $; if equality holds, then $R$ is said to have *minimal multiplicity.*
Eisenbud and Herzog [@DEsJHr88] raise the question whether every complete Cohen–Macaulay ring of finite CM type and dimension at least $2$ has minimal multiplicity. Cohen–Macaulay rings of minimal multiplicity are Golod, see Avramov [[@ifr prop. 5.2.4]]{}, and satisfy [<span style="font-variant:small-caps;">(uac)</span>]{} for a strong reason, see [[@DAJLMS04 prop. 1.4]]{}.
\[obs:Syz\] Let $R$ be Cohen–Macaulay and assume that it has a dualizing module. By [[@MAsROB89 thm. A]]{}, every finitely generated $R$-module $N$ has a maximal Cohen–Macaulay approximation: an exact sequence of finitely generated $R$-modules, $$0 {\longrightarrow}I {\longrightarrow}Y {\longrightarrow}N {\longrightarrow}0,$$ where $Y$ is MCM, and $I$ is of finite injective dimension. If $X$ is an MCM module, then one has ${\operatorname{Ext}_{R}^{n}(X,I)} =0$ for $n{\geqslant}1$; see Yoshino [[@yos cor. (1.13)]]{} and [[@MAsROB89 cor. 6.4]]{}. Thus, for all $n {\geqslant}1$ there are isomorphisms $${\operatorname{Ext}_{R}^{n}(X,Y)} {\cong}{\operatorname{Ext}_{R}^{n}(X,N)}.$$ From [([\[eq:is1\]]{})]{} and the displays above, it is not hard to see that it suffices to verify the conditions [<span style="font-variant:small-caps;">(ac)</span>]{} and [<span style="font-variant:small-caps;">(uac)</span>]{} on MCM modules. That is, $R$ satisfies [<span style="font-variant:small-caps;">(ac)</span>]{} if and only if it satisfies:
For every MCM $R$-module $X$ there exists ${b_{X}} {\geqslant}0$ such that for every MCM $R$-module $Y$ one has: ${\operatorname{Ext}_{R}^{n}(X,Y)}=0$ for $n\gg
0$ implies ${\operatorname{Ext}_{R}^{n}(X,Y)}=0$ for $n> {b_{X}}$.
And $R$ satisfies [<span style="font-variant:small-caps;">(uac)</span>]{} if and only if it satisfies:
There exists an integer $b {\geqslant}0$ such that for all MCM $R$-modules $X$ and $Y$ one has: ${\operatorname{Ext}_{R}^{n}(X,Y)}=0$ for $n\gg 0$ implies ${\operatorname{Ext}_{R}^{n}(X,Y)}=0$ for $n> b$.
Completion and reduction modulo regular sequences {#sec:stab}
=================================================
Let $R$ be Cohen–Macaulay; we can now show that $R$ and ${\widehat{R}}$ satisfy [<span style="font-variant:small-caps;">(ac)</span>]{} simultaneously and—in a generalization of [[@CHnDAJ03 prop. 3.3.(1)]]{}—that $R$ and $R/(x)$ satisfy [<span style="font-variant:small-caps;">(ac)</span>]{}simultaneously, if $x$ is a regular element.
\[lem:regelt\] Let $x$ be a regular element in $R$. If $R$ satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}, then $R/(x)$ satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}.
Let $M$ and $N$ be finitely generated $R/(x)$-modules. The change of rings spectral sequence [@careil XVI.§$5.(2)_3$] $$\mathrm{E}_2^{p,q} = {\operatorname{Ext}_{R/(x)}^{q}({\operatorname{Tor}^{R}_{p}(R/(x),M)},N)} \Rightarrow
{\operatorname{Ext}_{R}^{p+q}(M,N)}$$ has zero differentials on the second and all subsequent pages. For each $n{\geqslant}0$ it follows from [[@careil prop. 5.5]]{} that there is an exact sequence $$0 \to {\operatorname{Ext}_{R/(x)}^{n-1}(M,N)} \to {\operatorname{Ext}_{R}^{n}(M,N)} \to
{\operatorname{Ext}_{R/(x)}^{n}(M,N)} \to 0.$$ If $b$ is an Auslander bound for $M$ as an $R$-module, then it follows that $\max\{0,b-1\}$ is an Auslander bound for $M$ as an $R/(x)$-module.
A slightly different argument for the result above appeared in the proof of [[@CHnDAJ03 prop. 3.2.(1)]]{}; the converse for Cohen–Macaulay local rings is established in [[\[cor:cmreg\]]{}]{}.
The remaining proofs in this section use computations in the derived category over $R$; first we recall some notation. A complex of $R$-modules is graded cohomologically, $$M {\:=\:}\cdots {\longrightarrow}M^{n-1} {{\xrightarrow[]{\;{{\partial}_{M}^{n-1}}\;}}} M^{n} {{\xrightarrow[]{\;{{\partial}_{M}^{n}}\;}}}
M^{n+1} {\longrightarrow}\cdots.$$ The suspension of $M$ is the complex ${\mathsf{\Sigma}^{}{M}}$ with ${({\mathsf{\Sigma}^{}{M}})}^n =
M^{n+1}$ and ${{\partial}_{{\mathsf{\Sigma}^{}{M}}}^{}} = - {{\partial}_{M}^{}}$. The $n$th cohomology module of $M$ is denoted . Isomorphisms in the derived category over $R$ are marked by the symbol ${\simeq}$; they induce isomorphisms at the level of homology. We use the standard notation, ${\operatorname{\mathbf{R}Hom}_{R}(-,-)}$ and ${\nobreak{-\otimes_{R}^{\mathbf{L}}-}}$, for the right derived Hom functor and the left derived tensor product functor. For all $R$-modules $M$ and $N$ and all integers $n$ there are isomorphisms $${\operatorname{Ext}_{R}^{n}(M,N)} {\cong}{\operatorname{H}^{n\!}{\operatorname{\mathbf{R}Hom}_{A}(M,N)}} {{\quad\text{and}\quad}}{\operatorname{Tor}^{R}_{n}(M,N)} {\cong}{\operatorname{H}^{-n}({\nobreak{M\otimes_{R}^{\mathbf{L}}N}})}.$$
In the next work-hose lemma, the ring $S$ need not be local.
\[lem:descent\] Let ${(R,{\mathfrak{m}})}$ be Cohen–Macaulay and assume that it has a dualizing module. Let $S$ be an $R$-algebra of finite flat dimension and with $S/{\mathfrak{m}}S \ne 0$. If $S$ satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}, then $R$ satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}.
By [Observation [\[obs:Syz\]]{}]{} it suffices to consider cohomology of MCM $R$-modules. For such modules $X$ and $Y$ there are isomorphisms in the derived category over $R$, $$\begin{aligned}
{\nobreak{{\operatorname{\mathbf{R}Hom}_{R}(X,Y)}\otimes_{R}^{\mathbf{L}}S}} & {\simeq}{\operatorname{\mathbf{R}Hom}_{S}({\nobreak{X\otimes_{R}^{\mathbf{L}}S}},{\nobreak{Y\otimes_{R}^{\mathbf{L}}S}})} \\
& {\simeq}{\operatorname{\mathbf{R}Hom}_{S}({\nobreak{X\otimes_{R}S}},{\nobreak{Y\otimes_{R}S}})}.
\end{aligned}$$ The first isomorphism is tensor evaluation, see Avramov and Foxby [[@LLAHBF91 lem. 4.4.(F)]]{}, and adjointness of Hom and tensor. The second isomorphism follows as one has ${\operatorname{Tor}^{R}_{n}(X,S)} =0=
{\operatorname{Tor}^{R}_{n}(Y,S)}$ for $n{\geqslant}1$; see [[@CFF-02 thm. (3.4)]]{}. These isomorphisms and the assumption $S/{\mathfrak{m}}S \ne 0$ yield: $$\begin{aligned}
\sup{\{\,n \mid {\operatorname{Ext}_{R}^{n}(X,Y)} \ne 0\,\}} =
\sup{\{\,n \mid {\operatorname{Ext}_{S}^{n}({\nobreak{X\otimes_{R}S}},{\nobreak{Y\otimes_{R}S}})}\ne 0\,\}};
\end{aligned}$$ see Foxby [[@HBF77b lem. 2.1.(2)]]{}. Thus, if $b$ is an Auslander bound for the $S$-module ${\nobreak{X\otimes_{R}S}}$, then $b$ is also an Auslander bound for the $R$-module $X$.
\[thm:completion\] Let ${(R,{\mathfrak{m}})}$ be Cohen–Macaulay. Any one of the local rings $$R, \ \, {\widehat{R}}, \ \, {R[\mspace{-2.3mu}[X]\mspace{-2.3mu}]}, \ \,\text{and} \ \, {R[X]}_{({\mathfrak{m}},X)}$$ satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{} if and only if they all satisfy [<span style="font-variant:small-caps;">(ac/uac)</span>]{}.
The rings $R$, ${\widehat{R}}$, ${R[\mspace{-2.3mu}[X]\mspace{-2.3mu}]}$, and ${R[X]}_{({\mathfrak{m}},X)}$ are all local and Cohen–Macaulay.
We first argue that $R$ and ${\widehat{R}}$ satisfy [<span style="font-variant:small-caps;">(ac/uac)</span>]{} simultaneously. If ${\widehat{R}}$ satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}, then so does $R$ by [[@LWCHHlc prop. 5.5]]{}, as ${\widehat{R}}$ is a faithfully flat $R$-algebra. For the converse, assume that $R$ satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}, and let ${\pmb{x}}=x_1,\dots,x_d$ be a maximal $R$-regular sequence; its image in ${\widehat{R}}$ is a maximal ${\widehat{R}}$-regular sequence. There is an isomorphism of rings $R/({\pmb{x}}) {\cong}{\widehat{R}}/({\pmb{x}})$, and this ring is [<span style="font-variant:small-caps;">(ac/uac)</span>]{} by [Lemma [\[lem:regelt\]]{}]{}. As ${\widehat{R}}$ has a dualizing module, it satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{} by [Lemma [\[lem:descent\]]{}]{}.
If ${R[\mspace{-2.3mu}[X]\mspace{-2.3mu}]}$ or ${R[X]}_{({\mathfrak{m}},X)}$ satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}, then so does $R$ by [Lemma [\[lem:regelt\]]{}]{}. For the converse, assume that $R$ and, therefore, ${\widehat{R}}$ satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}. The local ring ${{\widehat{R}}[\mspace{-2.3mu}[X]\mspace{-2.3mu}]}$, which is Cohen–Macaulay and has a dualizing module, satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{} by [Lemma [\[lem:descent\]]{}]{} applied to the surjection ${{\widehat{R}}[\mspace{-2.3mu}[X]\mspace{-2.3mu}]} {\twoheadrightarrow}{\widehat{R}}$. The isomorphisms $$\widehat{{R[X]}_{({\mathfrak{m}},X)}} {\cong}{{\widehat{R}}[\mspace{-2.3mu}[X]\mspace{-2.3mu}]} {\cong}\widehat{{R[\mspace{-2.3mu}[X]\mspace{-2.3mu}]}}$$ and the assertion established in the first part of the proof now show that ${R[\mspace{-2.3mu}[X]\mspace{-2.3mu}]}$ and ${R[X]}_{({\mathfrak{m}},X)}$ satisfy [<span style="font-variant:small-caps;">(ac/uac)</span>]{}.
\[cor:cmreg\] Let $R$ be Cohen–Macaulay and ${\pmb{x}}=x_1,\dots,x_n$ be an $R$-regular sequence. The ring $R/({\pmb{x}})$ satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{} if and only if $R$ satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}.
The “if” part is immediate from [Lemma [\[lem:regelt\]]{}]{}. For the converse, assume that $R/({\pmb{x}})$ satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}. It is a Cohen–Macaulay ring, so by [Theorem [\[thm:completion\]]{}]{} the completion satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}. As the image of ${\pmb{x}}$ in ${\widehat{R}}$ is a regular sequence, the ring ${\widehat{R}}$ satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{} by [Lemma [\[lem:descent\]]{}]{}, and then another application of [[\[thm:completion\]]{}]{} shows that $R$ satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}.
A Cohen–Macaulay ring reduces modulo a regular sequence to an artinian ring, and by [Corollary [\[cor:cmreg\]]{}]{} the two rings satisfy [<span style="font-variant:small-caps;">(ac/uac)</span>]{}simultaneously. In a clear analogy, the conditions [<span style="font-variant:small-caps;">(ac)</span>]{} and [<span style="font-variant:small-caps;">(uac)</span>]{} can be verified on modules of finite length:
Let $R$ be Cohen–Macaulay. It satisfies [<span style="font-variant:small-caps;">(ac)</span>]{} if and only if it satisfies the condition:
For every $R$-module $K$ of finite length there exists an integer ${b_{K}} {\geqslant}0$ such that for every $R$-module $L$ of finite length one has: ${\operatorname{Ext}_{R}^{n}(K,L)}=0$ for $n\gg 0$ implies ${\operatorname{Ext}_{R}^{n}(K,L)}=0$ for $n> {b_{K}}$.
Furthermore, $R$ satisfies [<span style="font-variant:small-caps;">(uac)</span>]{} if and only if it satisfies the condition:
There exists an integer $b {\geqslant}0$ such that for all $R$-modules $K$ and $L$ of finite length one has: ${\operatorname{Ext}_{R}^{n}(K,L)}=0$ for $n\gg
0$ implies ${\operatorname{Ext}_{R}^{n}(K,L)}=0$ for $n> b$.
Assume that $R$ satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{} for modules of finite length. For ${\widehat{R}}$-modules $K$ and $L$ of finite length and integers $n {\geqslant}0$ there are isomorphisms $${\operatorname{Ext}_{{\widehat{R}}}^{n}(K,L)} {\cong}{\operatorname{Ext}_{{\widehat{R}}}^{n}({\nobreak{{\widehat{R}}\otimes_{R}K}},L)} {\cong}{\operatorname{Ext}_{R}^{n}(K,L)}.$$ As $R$-modules, $K$ and $L$ also have finite length; hence it follows that ${\widehat{R}}$ satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{} for modules of finite length. In view of [Theorem [\[thm:completion\]]{}]{} we can now assume that $R$ is complete and, in particular, that it has a dualizing module.
By [Observation [\[obs:Syz\]]{}]{} it is sufficient to prove that $R$ satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{} for MCM modules. Let $X$ and $Y$ be MCM $R$-modules, and let ${\pmb{x}}=x_1,\dots,x_d$ be a maximal $R$-regular sequence. As ${\pmb{x}}$ is regular on $X$ and $Y$, there are isomorphisms $$\label{eq:aa}
\tag{1}
X/{\pmb{x}}X {\simeq}{\nobreak{X\otimes_{R}^{\mathbf{L}}R/({\pmb{x}})}} {{\quad\text{and}\quad}}Y/{\pmb{x}}Y {\simeq}{\nobreak{Y\otimes_{R}^{\mathbf{L}}R/({\pmb{x}})}}$$ in the derived category over $R$. Nakayama’s lemma, see [[@HBF77b lem. 2.1.(2)]]{}, yields the first equality below, while the second follows from [[@LLAHBF91 lem. 4.4.(F)]]{} and [([\[eq:aa\]]{})]{}. $$\label{eq:ab}
\tag{2}
\begin{split}
\sup{\{\,n \mid {\operatorname{H}^{n\!}&{\operatorname{\mathbf{R}Hom}_{R}(X,Y)} \ne 0}\,\}}\\ &= \sup{\{\,n \mid {\operatorname{H}^{n}({\nobreak{{\operatorname{\mathbf{R}Hom}_{R}(X,Y)}\otimes_{R}^{\mathbf{L}}R/({\pmb{x}})}})}\ne 0\,\}}\\
&= \sup{\{\,n \mid {\operatorname{H}^{n\!}{\operatorname{\mathbf{R}Hom}_{R}(X,Y/{\pmb{x}}Y)} \ne 0}\,\}}.
\end{split}$$ From [([\[eq:aa\]]{})]{} and adjointness of Hom and tensor one gets $$\label{eq:ac}
\tag{3}
{\operatorname{\mathbf{R}Hom}_{R}(X/{\pmb{x}}X,Y/{\pmb{x}}Y)} {\simeq}{\operatorname{\mathbf{R}Hom}_{R}(X,{\operatorname{\mathbf{R}Hom}_{R}(R/({\pmb{x}}),Y/{\pmb{x}}Y)})}.$$ The Koszul complex on ${\pmb{x}}$ is a free resolution of $R/({\pmb{x}})$, so there is an isomorphism $$\label{eq:ad}
\tag{4}
{\mathsf{\Sigma}^{d}{{\operatorname{\mathbf{R}Hom}_{R}(R/({\pmb{x}}),-)}}} {\simeq}{\nobreak{-\otimes_{R}^{\mathbf{L}}R/({\pmb{x}})}}.$$ The next equalities follow from [([\[eq:ab\]]{})]{} and Nakayama’s lemma, from [[@LLAHBF91 lem. 4.4.(F)]]{} and [([\[eq:ad\]]{})]{}, and from [([\[eq:ac\]]{})]{}, respectively. $$\begin{aligned}
\sup{\{\,n \mid {\operatorname{H}^{n\!}&{\operatorname{\mathbf{R}Hom}_{R}(X,Y)}}\ne 0\,\}}\\
&= \sup{\{\,n \mid {\operatorname{H}^{n}({\nobreak{{\operatorname{\mathbf{R}Hom}_{R}(X,Y/{\pmb{x}}Y)}\otimes_{R}^{\mathbf{L}}R/({\pmb{x}})}})}\ne 0\,\}}\\
&= \sup{\{\,n \mid {\operatorname{H}^{n\!}{\operatorname{\mathbf{R}Hom}_{R}(X,{\mathsf{\Sigma}^{d}{{\operatorname{\mathbf{R}Hom}_{R}(R/({\pmb{x}}),Y/{\pmb{x}}Y)}}})}}\ne 0 \,\}}\\
&= \sup{\{\,n \mid {\operatorname{H}^{n\!}{\operatorname{\mathbf{R}Hom}_{R}(X/{\pmb{x}}X,Y/{\pmb{x}}Y)}}\ne 0 \,\}} -d.
\end{aligned}$$ That is, one has $$\begin{aligned}
\sup{\{\,n \mid {\operatorname{Ext}_{R}^{n}(X,Y)}\ne 0\,\}}= \sup{\{\,n \mid {\operatorname{Ext}_{R}^{n}(X/{\pmb{x}}X,Y/{\pmb{x}}Y)}\ne 0 \,\}} -d.
\end{aligned}$$ As the $R$-modules $X/{\pmb{x}}X$ and $Y/{\pmb{x}}Y$ have finite length, a straightforward argument finishes the proof, cf. proof of [Lemma [\[lem:descent\]]{}]{}.
An example {#sec:example}
==========
One can construct new examples of Cohen–Macaulay local rings that satisfy [<span style="font-variant:small-caps;">(ac/uac)</span>]{} from existing ones through the process of adjoining power series variables and killing regular elements. To get examples not covered in the literature, see the list [@LWCHHlc A.1], the new ring $R$ should not be Golod or complete intersection (c.i.); and if $R$ is Gorenstein, then its codimension, ${\operatorname{codim}R}$, should be at least $5$, and its multiplicity, ${\operatorname{e}(R)}$, should be at least ${\operatorname{codim}R}+3$. Furthermore, $R$ should not be of finite CM type, cf. [Theorem [\[thm:cm\]]{}]{}.
\[prp:heitmann\] Let $Q$ be a Cohen–Macaulay local ring and assume that it satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}. For integers $d,n {\geqslant}1$ and $s_i {\geqslant}2$, the local ring $$R = {Q[\mspace{-2.3mu}[X_1,\dots,X_d,Y_1,\dots,Y_n]\mspace{-2.3mu}]}/(Y_1^{s_1},\dots,Y_n^{s_n})$$ is Cohen–Macaulay with ${\operatorname{codim}R} = {\operatorname{codim}Q} + n$, and it satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}. Moreover, the following hold:
$R$ is not of finite CM type.
If $Q$ is not c.i., then $R$ is not c.i. and $R$ is not Golod.
If $Q$ has infinite residue field and one has $s_i {\geqslant}4$ for some $i\in{\{\, 1,\ldots,n \,\}}$, then the inequality ${\operatorname{e}(R)} {\geqslant}{\operatorname{codim}R} + 3$ holds.
Clearly, $R$ is Cohen–Macaulay with ${\operatorname{codim}R} = {\operatorname{codim}Q} + n$, and it is immediate from [Theorem [\[thm:completion\]]{}]{} and [Corollary [\[cor:cmreg\]]{}]{} that $R$ satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}.
(a): Let ${\mathfrak{q}}$ be the maximal ideal in $Q$. The image of $Y_1$ in $R_{({\mathfrak{q}},Y_1,\dots,Y_n)}$ is a zero-divisor, so $R$ is not an isolated singularity; in particular, $R$ is not of finite CM type; see Huneke and Leuschke [[@CHnGJL02 cor. 2]]{}.
(b): Assume that $Q$ is not c.i. By [@LLA99 (5.10)] the ring $R$ is not c.i. As $Q$ is singular, one has ${\operatorname{codim}R} ={\operatorname{codim}Q} + n
{\geqslant}2$. The $R$-module $R/(Y_1)$ has constant Betti numbers equal to $1$. It follows from [[@ifr thm. 5.3.3.(5)]]{} that $R$ is not Golod.
(c): Let $\mathfrak{a}$ be a minimal reduction of the maximal ideal ${\mathfrak{q}}$ in $Q$. It is elementary to verify that $\mathfrak{b} =
(\mathfrak{a},X_1,\ldots,X_d)$ is a reduction of ${\mathfrak{m}}=
({\mathfrak{q}},X_1,\ldots,X_d,Y_1,\ldots,Y_n)$, the maximal ideal in $R$. Assume that $Q/{\mathfrak{q}}$ is infinite. It follows that $\mathfrak{a}$ is minimally generated by ${\operatorname{dim}_{}Q}$ elements and, therefore, $\mathfrak{b}$ is generated by ${\operatorname{dim}_{}Q}+d={\operatorname{dim}R}$ elements. Hence, $\mathfrak{b}$ is a minimal reduction of ${\mathfrak{m}}$; see Swanson and Huneke [[@icirm prop. 8.3.7 and cor. 8.3.5.(1)]]{}. As $R$ is Cohen–Macaulay, [[@icirm prop. 11.2.2]]{} yields ${\operatorname{e}(R)} =
{\operatorname{length}_{}R/\mathfrak{b}}$. If one has $s_i {\geqslant}4$ for some $i$, then $({\mathfrak{m}}/\mathfrak{b})^3$ is non-zero, whence $${\operatorname{length}_{}R/\mathfrak{b}} {\geqslant}{\operatorname{edim}R/\mathfrak{b}} + 3 {\geqslant}{\operatorname{edim}R} - {\operatorname{dim}R}+ 3 = {\operatorname{codim}R} + 3. \qedhere$$
We thank Louiza Fouli for clarifying for us the basics of multiplicities that are used in the proof of part (c) above.
All assumptions on $Q$ in [Proposition [\[prp:heitmann\]]{}]{} are satisfied by the ring in [Example [\[exa:Q\]]{}]{}; thus for all choices of $d,n {\geqslant}1$ and $s_i {\geqslant}2$ the resulting ring $R$ in [[\[prp:heitmann\]]{}]{} is not among the rings on the list [@LWCHHlc A.1] nor is it of finite CM type.
\[exa:Q\] Let $k$ be a field of characteristic $0$ and consider the local $k$-algebra $$Q = {k[U,V,W]}/(U^2-V^2,V^2 - W^2,UV,VW).$$ It has radical cube zero and ${\operatorname{edim}Q}=3$, so it satisfies [<span style="font-variant:small-caps;">(uac)</span>]{} by [[@DAJLMS04 prop. 1.1.(3)]]{}. Moreover, $Q$ is not Gorenstein; in particular, it is not c.i.
Through a result of Heitmann [@RCH93], the construction in [Proposition [\[prp:heitmann\]]{}]{} also provides examples of unique factorization domains that satisfy [<span style="font-variant:small-caps;">(ac/uac)</span>]{}.
Let $Q$ be a complete Cohen–Macaulay local ring that satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}. Assume that $Q$ is not c.i. and that no integer is a zero divisor in $Q$; the ring in [[\[exa:Q\]]{}]{} serves as an example.
Let $R$ be as in [Proposition [\[prp:heitmann\]]{}]{}. If $Q$ is not artinian or $d$ is at least $2$, then $R$ has depth at least $2$. By [[@RCH93 thm. 8]]{}, there is a local unique factorization domain, $D$, with $\widehat{D} {\cong}R$. It follows from [Theorem [\[thm:completion\]]{}]{} that $D$ satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}. Of course, $R$ is also the completion of the ring ${Q[X_1,\dots,X_d,Y_1,\dots,Y_n]}/(Y_1^{s_1},\dots,Y_n^{s_n})$ localized at the irrelevant maximal ideal, but that ring is not a domain.
Local homomorphisms {#sec:homos}
===================
Let ${(R,{\mathfrak{m}})}$ and ${(S,{\mathfrak{n}})}$ be local rings, and let ${\nobreak{{\varphi}\colon R \rightarrow S}}$ be a *local* homomorphism, that is, a ring homomorphism with ${\varphi}({\mathfrak{m}})
\subseteq {\mathfrak{n}}$. The *semicompletion* of ${\varphi}$ is its composite with the embedding $S {\hookrightarrow}{\widehat{S}}$; it is written ${\nobreak{\grave{{\varphi}}\colon R \rightarrow {\widehat{S}}}}$. The semicompletion of ${\varphi}$ admits a Cohen factorization; that is, there exist local homomorphisms $$R {\xrightarrow[]{\;\dot{{\varphi}}\;}} R' {\xrightarrow[]{\;{\varphi}'\;}} {\widehat{S}},$$ such that $\grave{{\varphi}}={\varphi}'\dot{{\varphi}}$, where $\dot{{\varphi}}$ is flat with regular closed fiber $R/{\mathfrak{m}}R'$, the ring $R'$ is complete, and ${\varphi}'$ is surjective; see Avramov, Foxby, and Herzog [[@AFH-94 thm. (1.1)]]{}. If ${\varphi}$ is of finite flat dimension—that is, $S$ has finite flat dimension as an $R$-module via ${\varphi}$—then the surjection ${\varphi}'$ is of finite projective dimension; see [@AFH-94 (3.3)].
A Gorenstein local ring satisfies [<span style="font-variant:small-caps;">(ac)</span>]{} if and only if it satisfies [<span style="font-variant:small-caps;">(uac)</span>]{}; following [@CHnDAJ03] such a ring is called an *AB ring*. The [<span style="font-variant:small-caps;">(ac)</span>]{} and [<span style="font-variant:small-caps;">(uac)</span>]{} properties descend along local homomorphisms of finite flat dimension:
Let ${\nobreak{{\varphi}\colon R \rightarrow S}}$ be a local homomorphism of finite flat dimension.
If $S$ is Cohen–Macaulay and satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}, then $R$ is Cohen–Macaulay and satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}.
If $S$ is an AB ring, then $R$ is an AB ring.
Assume that $S$ is Cohen–Macaulay and satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}, then ${\widehat{S}}$ has the same properties; see [Theorem [\[thm:completion\]]{}]{}. Consider a Cohen factorization $R \to R' \to {\widehat{S}}$ of the semicompletion $\grave{{\varphi}}$. By [@AFH-94 (3.10)] the rings $R'$ and $R$ are Cohen–Macaulay. The ring $R'$ is complete, hence it has a dualizing module, and by [Lemma [\[lem:descent\]]{}]{} it satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}. Finally, $R$ satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{} by [[@LWCHHlc prop. 5.5]]{}; this proves part (a). For part (b) it suffices to note that if $S$ is Gorenstein, then $R$ is Gorenstein; see Avramov and Foxby [[@LLAHBF90 thm. (2.4)]]{}.
The notation ${(R,{\mathfrak{m}},k)}$ specifies that $k$ is the residue field of the local ring ${(R,{\mathfrak{m}})}$.
\[lem:ascent\] Let ${\nobreak{{\varphi}\colon {(R,{\mathfrak{m}},k)}\rightarrow (S,{\mathfrak{n}},k)}}$ be a local homomorphism of complete $k$-algebras. If $R$ is Cohen–Macaulay and satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}, then ${\varphi}$ has a Cohen factorization $R \to R' \to S$, where $R'$ is Cohen–Macaulay and satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}.
Let $y_1,\ldots,y_n$ be a generating set for ${\mathfrak{n}}$. As the field $k$ is a Cohen ring for $S$, in the sense of [@AFH-94 (1.0.2)], the inclusion $k {\hookrightarrow}S$ extends by the assignment $Y_i \mapsto y_i$ to a surjective homomorphism $\pi\colon {k[\mspace{-2.3mu}[Y_1,\ldots,Y_n]\mspace{-2.3mu}]} {\twoheadrightarrow}S$. Set $R' = {R[\mspace{-2.3mu}[Y_1,\ldots,Y_n]\mspace{-2.3mu}]}$, then $R'$ is complete and a flat local extension of $R$ with regular closed fiber $R'/{\mathfrak{m}}R' {\cong}{k[\mspace{-2.3mu}[Y_1,\ldots,Y_n]\mspace{-2.3mu}]}$. The homomorphism ${\varphi}$ extends by the assignment $Y_i \mapsto y_i$ to a homomorphism $R' \to S$; it is surjective because $\pi$ is surjective. Thus, $R {\hookrightarrow}R' {\twoheadrightarrow}S$ is a Cohen factorization of ${\varphi}$. If $R$ is Cohen–Macaulay and satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}, then $R'$ has the same properties; see [Theorem [\[thm:completion\]]{}]{}.
Following [@LLA99 3.1], a local homomorphism ${\nobreak{{\varphi}\colon R \rightarrow S}}$ is called *c.i.*, for complete intersection, if there is a Cohen factorization $R \to R' \to {\widehat{S}}$ of the semicompletion $\grave{{\varphi}}$ in which the kernel of the surjection $R' {\twoheadrightarrow}S$ is generated by an $R'$-regular sequence.
Here is an ascent result for the [<span style="font-variant:small-caps;">(ac)</span>]{} and [<span style="font-variant:small-caps;">(uac)</span>]{} properties:
Let ${\nobreak{{\varphi}\colon {(R,{\mathfrak{m}},k)}\rightarrow (S,{\mathfrak{n}},k)}}$ be a c.i. local homomorphism of $k$-algebras.
If $R$ is Cohen–Macaulay and satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}, then $S$ is Cohen–Macaulay and satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}.
If $R$ is an AB ring, then $S$ is an AB ring.
Assume that $R$ is Cohen–Macaulay and satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}, then ${\widehat{R}}$ has the same properties; see [Theorem [\[thm:completion\]]{}]{}. By [Lemma [\[lem:ascent\]]{}]{}, the semicompletion $\grave{{\varphi}}$ has a Cohen factorization $R \to R' \to {\widehat{S}}$, in which $R'$ is Cohen–Macaulay and satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}. As ${\varphi}$ is c.i., the kernel of the surjection $R' {\twoheadrightarrow}{\widehat{S}}$ is generated by an $R'$-regular sequence; see [@LLA99 (3.3)]. It follows from [Corollary [\[cor:cmreg\]]{}]{} that ${\widehat{S}}$ is Cohen–Macaulay and satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{} and, therefore, $S$ is Cohen–Macaulay and satisfies [<span style="font-variant:small-caps;">(ac/uac)</span>]{}; see [Theorem [\[thm:completion\]]{}]{}. This proves part (a).
For part (b), recall that c.i. homomorphisms are Gorenstein by [[@LLAHBF90 thm. (2.4)]]{}. Thus, if $R$ is Gorenstein, it then follows from [@AFH-94 (3.11)] that $S$ is Gorenstein.
\[1\]
[10]{}
Maurice Auslander, *Selected works of [M]{}aurice [A]{}uslander. [P]{}art 1*, American Mathematical Society, Providence, RI, 1999, Edited and with a foreword by Idun Reiten, Sverre O. Smalø, and Øyvind Solberg. [MR ]{}[MR1674397]{}
Maurice Auslander and Ragnar-Olaf Buchweitz, *The homological theory of maximal [C]{}ohen-[M]{}acaulay approximations*, Mém. Soc. Math. France (N.S.) (1989), no. 38, 5–37, Colloque en l’honneur de Pierre Samuel (Orsay, 1987). [MR ]{}[MR1044344]{}
Luchezar L. Avramov, *Infinite free resolutions*, Six lectures on commutative algebra (Bellaterra, 1996), Progr. Math., vol. 166, Birkhäuser, Basel, 1998, pp. 1–118. [MR ]{}[MR1648664]{}
[to3em]{}, *Locally complete intersection homomorphisms and a conjecture of [Q]{}uillen on the vanishing of cotangent homology*, Ann. of Math. (2) **150** (1999), no. 2, 455–487. [MR ]{}[MR1726700]{}
Luchezar L. Avramov and Hans-Bj[ø]{}rn Foxby, *Gorenstein local homomorphisms*, Bull. Amer. Math. Soc. (N.S.) **23** (1990), no. 1, 145–150. [MR ]{}[MR1020605]{}
[to3em]{}, *Homological dimensions of unbounded complexes*, J. Pure Appl. Algebra **71** (1991), no. 2-3, 129–155. [MR ]{}[MR1117631]{}
Luchezar L. Avramov, Hans-Bj[ø]{}rn Foxby, and Bernd Herzog, *Structure of local homomorphisms*, J. Algebra **164** (1994), no. 1, 124–145. [MR ]{}[MR1268330]{}
Henri Cartan and Samuel Eilenberg, *Homological algebra*, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1999, With an appendix by David A. Buchsbaum, Reprint of the 1956 original. [MR ]{}[MR1731415]{}
Lars Winther Christensen, Hans-Bj[ø]{}rn Foxby, and Anders Frankild, *Restricted homological dimensions and [C]{}ohen-[M]{}acaulayness*, J. Algebra **251** (2002), no. 1, 479–502. [MR ]{}[MR1900297]{}
Lars Winther Christensen and Henrik Holm, *Algebras that satisfy [A]{}uslander’s condition on vanishing of cohomology*, Math. Z. **265** (2010), no. 1, 21–40. [MR ]{}[MR2606948]{}
David Eisenbud and J[ü]{}rgen Herzog, *The classification of homogeneous [C]{}ohen-[M]{}acaulay rings of finite representation type*, Math. Ann. **280** (1988), no. 2, 347–352. [MR ]{}[MR929541]{}
Hans-Bj[ø]{}rn Foxby, *Isomorphisms between complexes with applications to the homological theory of modules*, Math. Scand. **40** (1977), no. 1, 5–19. [MR ]{}[MR0447269]{}
Raymond C. Heitmann, *Characterization of completions of unique factorization domains*, Trans. Amer. Math. Soc. **337** (1993), no. 1, 379–387. [MR ]{}[MR1102888]{}
Craig Huneke and David A. Jorgensen, *Symmetry in the vanishing of [E]{}xt over [G]{}orenstein rings*, Math. Scand. **93** (2003), no. 2, 161–184. [MR ]{}[MR2009580]{}
Craig Huneke and Graham J. Leuschke, *Two theorems about maximal [C]{}ohen-[M]{}acaulay modules*, Math. Ann. **324** (2002), no. 2, 391–404. [MR ]{}[MR1933863]{}
Craig Huneke and Irena Swanson, *Integral closure of ideals, rings, and modules*, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006. [MR ]{}[MR2266432]{}
David A. Jorgensen and Liana M. [Ş]{}ega, *Nonvanishing cohomology and classes of [G]{}orenstein rings*, Adv. Math. **188** (2004), no. 2, 470–490. [MR ]{}[MR2087235]{}
Yuji Yoshino, *Cohen-[M]{}acaulay modules over [C]{}ohen-[M]{}acaulay rings*, London Mathematical Society Lecture Note Series, vol. 146, Cambridge University Press, Cambridge, 1990. [MR ]{}[MR1079937]{}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We have studied the distribution of $B_J<20.5$ galaxies from the ROE/NRL COSMOS/UKST catalogue around two samples of $z>0.3$ QSOs with similar redshift distributions. The first sample is formed by 144 radio-loud QSOs from the Parkes Catalogue, and the other contains 167 optically selected QSOs extracted from the Large Bright Quasar Survey. It is found that there is a $\approx
99.0\%$ significance level excess of COSMOS/UKST galaxies around the PKS QSOs, whereas there is a marginal defect of galaxies around the LBQS QSOs. When the distribution of galaxies around both samples is compared, we found that there is an overdensity of galaxies around the PKS sample with respect to the LBQS sample anticorrelated with the distance from the QSOs at a $99.7\%$ significance level. Although this result apparently agrees with the predictions of the multiple magnification bias, it is difficult to explain by gravitational lensing effects alone; dust in the foreground galaxies and selection effects in the detection of LBQS QSOs should be taken into account. It has been established that the lines of sight to PKS flat-spectrum QSOs go through significatively higher foreground galaxy densities than the directions to LBQS quasars, what may be partially related with the reported reddening of PKS QSOs.
author:
- 'N. Ben'' itez & E. Mart'' inez-González'
title: ' Large Scale QSO-galaxy correlations for radio loud and optically selected QSO samples'
---
Introduction {#intr}
============
In the last few years, several studies have established the existence of a statistical excess of line-of-sight companion galaxies around high redshift quasars. Although it has been suggested that these objects belong to clusters or groups which are physically associated to the quasars ([@hin91]; [@tys86]), in order to be detected at such high redshifts they should be undergoing strong luminosity evolution. This seems unlikely on the light of the recent data on galaxy evolution obtained through the study of absorption-selected galaxy samples ([@ste95]), which shows that the most plausible (and often the unique) interpretation for many of these observations is the existence of a magnification bias caused by gravitational lensing (see the reviews [@sch92]; [@sch95]; [@wu95]).
The density of a population of flux-limited background sources (e.g. QSOs) behind a gravitational lens is affected by the lens magnification $\mu$ in two opposite ways. One of the effects is purely geometrical: as the angular dimensions of a lensed patch of the sky are expanded by a factor $\mu$, the physical size of a region observed through a fixed angular aperture will be smaller than in the absence of the lens. Because of this, the QSO surface density will decrease by a factor $\mu$ with respect to the unlensed background density [@Na89]). On the other hand, the lens will magnify faint quasars (which would not have been detected otherwise ) into the sample and increase the number of detected QSOs ([@can81]; [@sch86], etc.). If the slope of the quasar number-counts cumulative distribution is steep enough, this effect would dominate over the angular area expansion and there would be a net excess of QSOs behind the lens. Foreground galaxies trace the matter overdensities acting as lenses and thus there will be a correlation between the position in the sky of these galaxies (or other tracers of dark matter as clusters) and the background quasars. This QSO-galaxy correlation is characterized by the overdensity factor $q$ ([@sch89]), which is defined as the ratio of the QSO density behind a lens with magnification $\mu$ to the unperturbed density on the sky. Its dependence on the effective slope $\alpha$ of the QSO number counts distribution (which has the form $n(>S)\propto S^{-\alpha}$, or $n(<m)\propto 10^{0.4\alpha m}$) and the magnification $\mu$ can be expressed as ([@Na89]) $$q \propto \mu^{\alpha-1}$$ We see that the value of q critically depends on the number counts slope of the background sources. For instance, if the number counts are shallow enough, ($\alpha < 1$), there would be negative QSO-galaxy associations. It is clear that in order to detect strong, positive QSO-galaxy correlations due to the magnification bias, we have to use QSO samples with very steep number counts slopes. [@bo91] have shown that for a QSO sample which is flux-limited in two bands (with uncorrelated fluxes), $\alpha$ is substituted by $\alpha_{eff}$, the sum of the number counts-flux slopes in those bands. This effect is called ’double magnification bias’. Since $\alpha_{eff}$ is usually $>1$ for samples which are flux-limited in both the optical and radio bands (e.g. radio-loud QSOs), a strong positive QSO-galaxy correlation should be expected for them.
It is important to understand when QSO samples may be affected by the double magnification bias. The usual identification procedure for a X-ray or radio selected QSO sample involves setting up a flux threshold in the corresponding band and obtaining follow-up optical images and spectra of the QSO candidates. The observer is limited in this step by several circumstances (e.g. the sensitivity of the detectors or the telescope diameter), and even if the QSO sample was not intended to be optically selected, in the practice there will be an optical flux threshold for the QSOs to enter the sample. Therefore the existence of an explicit and homogeneus flux-limit in the optical band is not as essential for the presence of the magnification bias as the vaue of the effective slope of the unperturbed number counts. If this slope is steep enough, the effect should be detectable even in incomplete samples, and often more strongly than in complete catalogues: within such samples, the optically brightest QSOs (i.e., those more likely to be lensed) are usually the first to be identified, as they are easier to study spectroscopically or through direct imaging.
At small angular scales, ($\theta\lesssim$ few $\arcsec$) the existence of QSO-galaxy correlations is well documented for several QSO samples obtained with different selection criteria ([@web88]; see also [@ha90] and [@wu95] for reviews). As expected due to the double magnification bias effect, the correlations are stronger for radio-loud quasars ([@tho94]). In the cases where no correlation is found, e.g. for optically-selected and relatively faint quasars, the results are still consistent with the magnification bias effect and seem to be due to the shallowness of the QSO number counts distribution at its faint end ([@wu94]).
[@ha90] reviewed the studies on QSO-galaxy correlations (on both small and large scales). After assuming that the galaxies forming the excess are physical companions to the QSO, they showed that while the amplitude of the radio-quiet QSO-galaxy correlation quickly declines at $z\gtrsim 0.6$, the inferred radio-loud QSO-galaxy correlation function steadily increases with redshift, independently of the limiting magnitude of the study. It should be noted that such an effect will be expected, if a considerable part of the galaxy excess around radio-loud QSOs is at lower redshifts. If a foreground group is erroneously considered to be physically associated with a QSO, the higher the redshift of the QSO, the stronger the 3-D clustering amplitude that will be inferred. This source of contamination should be taken into account carefully when carrying out these studies, as there is evidence that part of the detected excess around high redshift radio-loud QSOs is foreground and related with the magnification bias.
The association of quasars with foreground galaxies on scales of several arcmin may arise as a consequence of lensing by the large scale structure as proposed by [@ba91] (see also [@sch95] and references therein). Several authors have also investigated QSO-cluster correlations: [@roho94], [@ses94] and [@wuha95]. There are not many studies of large scale QSO-galaxy correlation, mainly because of the difficulty in obtaining apropriate, not biased, galaxy samples. Besides, the results of these studies may seem contradictory, as they radically differ depending on the QSO type and limiting magnitude.
[@boy88] found a slight anticorrelation between the positions of optically selected QSOs and COSMOS galaxies. When they cross-correlated the QSOs with the galaxies belonging to clusters, the anticorrelation become highly significant ($4\sigma$) on $4\arcmin$ scales. Although this defect was interpreted as due to the presence of dust in the foreground clusters which obscured the quasars, the recent results of [@mao95] and [@lrw95] have imposed limits on the amounts of dust in clusters which seem to contradict this explanation. It seems more natural, taking into account the rather faint flux-limit of the QSOs of [@boy88] to explain this underdensity as a result of the magnification bias effect.
[@smi95] do not find any excess of foreground, $B_J< 20.5$ APM galaxies around a sample of $z>0.3$ X-ray selected QSOs. It should be noted that although the objects in this sample are detected in two bands (X-ray and optical), the expected excess should not be increased by the double magnification bias effect, because the X-ray and optical fluxes are strongly correlated ([@boy96]). For these QSOs, the overdensity should be roughly the same as for the optically selected ones.
On the other hand, there have been strong evidences of positive large scale associations of foreground galaxies with high redshift radio-loud QSOs. As e.g., between the radio-loud quasars from the 1Jy catalogue ([@sti94]) and the galaxies from the Lick ([@fug90], [@bar93]), IRAS ([@bbs96], [@bar94]) and APM ([@ben95a]) catalogues. In the latter paper, we found that APM galaxies are correlated with the 1Jy QSO positions, but did not have enough statistics to reach the significance levels reported in the present paper. Unlike the COSMOS catalogue, the APM catalogue provides magnitudes in two filters, $O$(blue) and $E$(red). When we considered only the reddest, $O-E>2$ galaxies, the overdensity reached a significance level of $99.1\%$. Afterwards, [@ode95] showed (using a similar, but much more processed catalog, the APS scans of POSS-I) that the galaxies which trace the high-density cluster and filamentary regions have redder $O-E$ colors than the galaxies drawn from low-density interfilamentary regions. If the fields containing Abell clusters were removed from the sample of [@ben95a], the results did not change significantly, so it seems that the detected effect was caused by the large scale structure as a whole. [@fort] confirmed the existence of gravitational lensing by the large scale structure by directly detecting the large foreground invisible matter condensations responsible for the magnification of the QSOs through the polarization produced by weak lensing in several 1Jy fields.
These differences in the nature of the QSO-galaxy associations for the several QSO types seem to arise quite naturally when we take into account the effects of the double magnification bias. There is not any strong correlation between the radio and optical fluxes for radio-loud quasars, so for these objects the overdensity will be $\propto \mu^{\alpha_{opt}+\alpha_{rad}-1}$ ([@bo91]), where $\alpha_{opt}$ and $\alpha_{rad}$ are the number-counts slope in the optical and in radio respectively. If we assume that $\alpha_{opt}$ is roughly the same for radio-loud and optically selected QSOs (although this is far from being clear), the overdensity of foreground galaxies should be higher for the radio-loud QSOs. For optically and X-ray selected samples (because of the X-ray-optical flux correlation ), $\alpha_{eff}$ and therefore the overdensity, will be smaller.
In any case, it is difficult to compare conclusively the published analyses of QSO-galaxy correlations. They have been performed using galaxy samples obtained with different filters and magnitude limits, and which therefore may not trace equally the matter distribution because of color and/or morphological segregation or varying depths. Besides, the QSO samples differ widely in their redshift distributions. As the magnification of a background source depends on its redshift, QSO samples with different redshift distributions will not be magnified equally by the same population of lenses. It would thus be desirable, in order to disentangle these factors from the effects due to the magnification bias caused by gravitational lensing, to reproduce the mentioned studies using galaxy samples obtained with the same selection criteria and QSO samples which are similarly distributed in redshift. This is the purpose of the present paper: we shall study and compare the distribution of COSMOS/UKST galaxies around two QSO samples, one radio-loud and the other radio-quiet with practically identical redshift distributions.
It is also interesting to mention in this context the results of [@web95]. These authors observed that a sample of flat-spectrum radio-loud quasars from the Parkes catalogue ([@pks]) displays a wide range of $B_J-K$ colours, apparently arising from the presence of varying amounts of dust along the line of sight. Optically selected quasars do not present such a scatter in $B_J - K$ and lie at the lower envelope of the $B_J - K$ colours, so Webster and collaborators suggested that the selection criteria used in optical surveys miss the reddest quasars. Although there seems to be several other, more plausible, reasons for this phenomenon ( see for instance [@boy96]), it is not adventurous to suppose that it may be partially related to the differences between the foreground projected environments of radio-loud and optically selected quasars. If a considerable part of the absorbing dust belongs to foreground galaxies, a greater density of these galaxies would imply more reddening.
The structure of the paper is the following: Section 2 describes the selection procedures of both QSO samples and galaxy fields and discuss the possible bias which could have been introduced in the data. Section 3 is devoted to the discussion of the statistical methods used in the literature for the study of this problem and applies them to our data. Section 4 discusses the results obtained in Sec 3. Section 5 lists the main conclusions of our work.
The Data {#data}
========
As we have explained above, the aim of our paper is the study of the distribution of foreground galaxies around two QSO samples, one radio-loud and the other radio quiet, which have similar redshift distributions. It would also be interesting to find out if these differences in the foreground galaxy density occur for the QSO samples of [@web95]. This, as we have mentioned, could be related to the differential reddening of the radio-loud quasars. Therefore, in order to form a radio-loud sample suitable for our purposes but as similar as possible to the one used by [@web95], we collect all the quasars from the PKS catalogue which are listed in the [@ver96] QSO catalogue and obey the following constraints: a) flux $>$ 0.5Jy at 11cm; b) $-45 < \delta
< 10$ and c) galactic latitude $|b| > 20$. So far, we do not constrain the radio spectral index of the quasars. This yields a preliminary sample of 276 quasars.
The galaxy sample is taken from the ROE/NRL COSMOS/UKST Southern Sky object catalogue, (see [@yen92] and references therein). It contains the objects detected in the COSMOS scans of the glass copies of the ESO/SERC blue survey plates. The UKST survey is organized in $6\times6$ square degree fields on a 5 degree grid and covers the zone $-90 < \delta <0$ excluding a $\pm 10$ deg zone in the galactic plane. COSMOS scans cover only $5.4\times 5.4$ square degrees of a UKST field. The scan pixel size is about 1 arcsec. Several parameters are supplied for each detected object, including the centroid in both sky and plate coordinates, $B_J$ magnitude and the star-galaxy image classification down to a limiting magnitude of $B_J\approx21$.
We are going to study the galaxy distribution in $15 \arcmin $ radius fields centered on the quasars of our sample. Due to several factors as vignetting and optical distorsions, the quality of the object classification and photometry in Schmidt plate based surveys degrades with increasing plate radius. Therefore, we constrain our fields to be at a distance from the plate center of $r=\sqrt{\Delta x^2 +\Delta y^2} < 2.5$ degrees. Besides, to avoid the presence of fields which extend over two plates we further restrict our sample to have $|\Delta x_k|, |\Delta y_k| < 2.25$ degrees, where $\Delta x_k$ and $\Delta y_k$ are the distances, in the $\alpha$ and $\delta$ directions respectively, from the center of the plate ( because of the UKST survey declination limits, this also constrains our quasar sample to have $\delta < 2.25$). After visually inspecting all the fields, several of them (6) are excluded from the final sample because they present meteorite traces. We end up with 147 circular fields with a $15\arcmin$ radius centered on an equal number of Parkes Quasars. This subsample of radio-loud quasars is, as far as we know, not biased towards the presence of an excess of foreground galaxies, which is the essential point for our investigation. In order to avoid contamination from galaxies physically associated with the quasars, we also exclude three $z<0.3$ quasars from our radio-loud sample ([@smi95] point out that only $5\%$ of $B_J < 20.5$ galaxies have $z>0.3$), which is finally formed by 144 fields.
We have excluded a region of $12\arcsec$ around the positions of the quasars (which may have an uncertainty up to $\pm 5$ arcsec). This is done to avoid the possibility of counting the quasar as a galaxy (there are six cases in which the quasar is classified as an extended object) and because of the great number of ’blended’ objects at the quasar position. Most of these pairs of objects are classified as ’stars’ when deblended, but taking into account the pixel resolution, it would be desirable to examine the original plates or, better yet, perform high resolution CCD imaging in order to properly deblend and classify these objects as many of them could be galaxies.
The optically selected sample is taken from the Large Bright Quasar Survey as in [@web95]. This prism-selected catalogue contains 1055 quasars brighter than $B_J \approx 18.6$ on several equatorial and southern fields (for details see [@hew90]). In order to form an optically selected subsample of quasars we have begun by choosing the 288 quasars from the LBQS catalogue which were closest in redshift to our final sample of 144 PKS quasars. We impose to them exactly the same constraints in sky and plate position as to the radio-loud quasar fields. Finally we visually examined the fields and excluded six of them because of meteorite traces. The resulting number of fields is 167. As the LBQS extends over relatively small areas of the sky, several of these fields overlap. We have checked that their exclusion from the statistical tests performed below does not affect significantly the result, so we leave them in the sample.
The resulting redshift distribution for both QSO samples is plotted in Fig 1. A Kolmogorov-Smirnov test cannot distinguish between them at a 94.5% significance level. We merge all the fields in each sample into two ’superfields’ which contain all the objects classified as galaxies with $B_J<20.5$. This is a reasonable limiting magnitude, and has been already used by other investigators ([@smi95]). The PKS merged field contains 15235 galaxies whereas the LBQS field only contains 14266. This is a difference of $24\%$ in the average object background density, well over a possible Poissonian fluctuation, and seems to be caused by the presence of misclassified stars in our galaxy sample at low galactic latitudes. The Parkes fields extend over a much wider range of galactic latitudes $(|b| > 20^o)$ than the LBQS ones, which are limited to high galactic latitudes $(|b|>45^o)$ and thus much less affected. In fact, we measure the existence of a significant anticorrelation between the absolute value of the galactic latitude $|b_k|$ of the fields and the total number of objects in each field $N_{gal}$. The correlation is still stronger between $N_{gal}$ and sec$(90-|b|$), as shown in Fig. 2, with a correlation coefficient $\rho=0.4, (p>99.99\% )$. This contamination should be randomly distributed over the field and would lower the significance of any possible correlation and make it more difficult to detect. In order to check this, we have correlated the overdensity $n_{in}/n_{out}$ of objects within the inner half of every individual field, ($n_{in}$ is the number of objects within the inner half of the field surface and $n_{out}$ is the number of objects in the outer half) with sec$(90^o-|b|)$, as can be seen in Fig. 3. If anything, there might be a slight anticorrelation (the Spearman’s rank correlation test only gives a significance of $80\%$) in the sense that the fields with more star contamination are the ones which show less excess of galaxies in the inner half of the field. This is what could be expected if there were a genuine galaxy excess close to the QSO positions; this excess should be diluted by the presence of stars randomly distributed with respect to the QSOs. Excluding the fields with $|b|\leq 30^o$, as in [@smi95], does not change significantly the main results, as we show in Fig 4. Because of this contamination by stars, there is a slight bias in the data which makes harder to detect QSO-galaxy correlations for the PKS QSOs than for the LBQS ones. We have also checked that there are no other correlations between $N_k$ and $q_k$ and other possible relevant parameters as the plate or sky position of the fields.
Statistical Analysis {#stats}
====================
The study of QSO-galaxy correlations due to the magnification bias effect is complicated by several circumstances. The amplitude of the correlation function $w_{qg}$ is expected to be rather weak, and strongly dependent on the limiting magnitude of the galaxies and the QSOs. Besides, the shape of $w_{qg}$ as a function of $\theta$ is unknown ( it seems that the interesting theoretical estimation of [@ba94] has not been confirmed empirically by [@bbs96]).
In the past, several methods have been used to detect and statistically establish the existence of these correlations. One of the most simple and widespread approaches has consisted in counting the number of galaxies $N_{in}$ in a somehow arbitrarily defined region centered on the quasars and comparing the value found with its statistical expectation, which is measured either from the outer regions of the fields or from some other comparison fields which are assumed to have the same density of galaxies on average. The significance can be inferred empirically ([@ben95a]) or just by considering that N has a poissonian distribution with $\sqrt{N}$ rms. This last assumption seems to hold well in some cases, when the number of individual fields is very large, but for other samples, usually smaller, the rms is found to be $\alpha\sqrt{N}$, where $\alpha\approx 1.1-1.5$ ([@ben95b]). A shortcoming of this method is that it does not extract all the information contained in the fields as it only explores the distribution of galaxies around the quasar in a small number of scales, and often uses the rest of the field just to determine the average density. Besides, if the correlation scale is comparable with the dimensions of the fields, the average density measured on the fields would be increased with respect to the unperturbed one and thus an artificially lowered signification will be obtained.
Another method, the rank-correlation test was used in [@bar93], [@bar94]. All the individual galaxy fields are merged into a single ’superfield’, which is subsequently divided into $N_{bins}$ annular intervals of equal surface. A Spearman’s rank order test is applied to determine if the number of galaxies in each bin $n_i,( i=1,N_{bins})$ is anticorrelated with the bin radius $r_i$. This test does not require any assumption about the underlying probability distribution of the galaxies and takes into account all the information contained in the fields. However it has several drawbacks. The rings have all equal surface, so we end up with more bins in the outer parts of the fields, which are less sensitive from the point of view of detecting the correlation. Besides, the method only ’senses’ the relative ordering of $n_i$ and $r_i$ over the whole field. For instance if $w_{qg}(\theta)$ is very steep and goes quickly to zero, there will be only a few bins clearly over the average in the central region, and the correlation coefficient could then be dominated by the more numerous outer bins with nearly no excess galaxies. The value of the correlation coefficient and its significance, depend thus critically on the number of chosen bins and the dimension of the fields. However, this test can still be useful if the correlation scale is similar to the scale of the field.
Recently, Bartsch et al. (1996) have introduced the weighted-average test. They define the estimator $r_g$ as $$r_g={1\over N}\sum^N_{j=1}g(\theta_j),$$ where N is the total number of galaxies in the merged field, and $\theta_j$ are the radius from the QSO of each galaxy. They show, under certain general assumptions, that if the galaxies in the merged field are distributed following a known QSO-galaxy correlation function $w_{gq}(\theta)$, for $g(\theta) \propto w_{gq}(\theta)$ the quantity $r_g$ is optimized to distinguish such a distribution of galaxies from a random one. They take $w_{gq}(\theta)=(0.24+h\theta/deg)^{-2.4}$ from the theoretical results of [@ba94] ($h=H_o/100$ Mpc km s$^{-1}$), and show with simulated fields that this method is slightly more sensitive than Spearman’s rank order test. However, when they study the correlation between IRAS galaxies and the QSOs from the 1Jy catalogue with the weighted-average test they obtain a much higher significance for their result than using the rank order test. They conclude that although the IRAS galaxies do not seem to be clustered around the QSOs following Bartelmann’s correlation function, the weighted average method seems to be a much more efficient estimator than the rank order test. This is not surprising if we consider that, when calculating the estimator $r_g$ (as long as we use a steep enough form for $g(\theta)$) this test gives a much higher weight to the galaxies closer to the QSO, that is, to the regions where the excess signal-to-noise is higher. An extreme case would be to use a top hat function with a given width $\theta_o$ as $g(\theta)$ (which would be equivalent to counting galaxies in a central circle of dimension $\theta_o$). This arbitrariness in the choice of $g(\theta)$ when we do not know the real shape of the QSO-galaxy correlation is a drawback of this method. Another problem is that the probability distribution of $r_g$ is unknown a priori. Because of that, the significance has to be determined using simulations, and as we have seen before, the real galaxy distribution is not always easy to know and model. Nevertheless, when we know theoretically the correlation, this test should be optimal, and it may also be useful in many other cases.
We have applied a variant of the rank order test to study the distribution of galaxies around our PKS and LBQS samples. We also use the Spearman’s rank order test as statistical estimator (in the implementation of [@nrec]), but instead of fixing a number of bins and dividing the field in rings of equal surface as in [@bar94], the variables to be correlated will be $w(\theta_i)$ and $\theta_i$, where $w(\theta_i)$ is the value of the empirically determined angular correlation function in rings of equal width and $\theta_i$ is the distance of the bins from the QSO. Now, in general, each ring will have a different number of galaxies, but the values of $\theta_i$ are going to be uniformly distributed in radius, and thus we will not give more weight to the outer regions of the field. As a statistical estimator we shall use $Z_d$, the number of times by which $D$, the so-called sum squared difference of ranks, deviates from its null-hypothesis expected value. $D$ has an approximately normal distribution and is defined as $$D=\sum^N_{i=1}(R_i-S_i)^2$$ where $R_i$ is the rank of the radius of the i-th ring and $S_i$ is the rank of the density in that same ring. Trying to avoid the dependence of the result on the concrete number of bins, we have followed this procedure: we have chosen a minimal ring width ($0.4\arcmin$) in order to have at least $\approx 20$ galaxies in the first ring, and a maximal width ($0.75\arcmin$) so that there are at least 20 rings within the field. Then we perform 8 different binnings changing the ring width in steps of $0.05\arcmin$, estimate $Z_d$ for each of them and calculate its average $<Z_d>$. This estimator should be very robust as it does not depend so strongly on the concrete value obtained for a binning, and the significance can be estimated directly from the value of $Z_d$ without the need of simulations. The value for the PKS field is $<Z_d>=2.33\sigma$, $p=99.0\%$ and for the LBQS field $<Z_d>=-0.68\sigma$, $p=75.2\%$. We have also confirmed this by estimating $<Z_d>$ for $10^5$ simulations with randomly distributed galaxies for each of both fields: the empirical significance for the PKS field is $p=99.01\%$ whereas the LBQS field gives $p=72.46\%$. This similarity in the values of the probabilities also confirms that the distribution of the galaxies in the fields is practically Poissonian. The QSO-galaxy correlation function for the PKS and LBQS sample is shown in Fig. 4 and 5 respectively. Error bars are poissonian and the bin width is $0.75\arcmin$. In Fig. 4 we also show, without error bars, the correlation function obtained for the PKS fields with $|b|>30^o$
In order to further test our results, we have also applied Bartsch et al. (1996) test to our data using Bartelmann’s $g(\theta)$ and have estimated the significance with $10^5$ simulated fields for each sample with the same number of galaxies as the real fields randomly distributed. Strictly speaking this is an approximation, as the galaxies are clustered among themselves, but we have studied the number of galaxies on rings of equal surface (excluding a few central rings) and their distribution is marginally consistent with being Gaussian with a rms $\approx \sqrt{N}$, what is not surprising if we take into account the great number of fields contributing to each bin. The existence of a positive QSO-galaxy correlation for the PKS sample is detected at a significance level of $98.85 \%$. On the other hand, when we apply this test to the LBQS merged field, a slight anti-correlation is found at a level of $88.85\%$. These results are comparable to the ones obtained with the previous test. We have also tried other arbitrarily chosen variants of the function $g(\theta)$ to see the dependence of significance of the PKS excess on the concrete shape of $g(\theta)$: a Gaussian with a $2\arcmin$ width (analogous to a box of this size) yields $p=99.66\%$ and a $\propto \theta^{-0.8}$ law ( the slope of the galaxy-galaxy correlation function) gives $p=99.5\%$. We see that for our galaxies, the shape of $g(\theta)$ proposed by Bartelmann is not optimal, and the significance depends sensitively on the shape of the function. However, tinkering with the form of $g(\theta)$ may be dangerous, as it could lead to creating an artificially high significance if we overadjust the shape of the function to the data.
Thus, it seems that there is a positive QSO-galaxy correlation in the PKS fields, and what appears to be a slight anticorrelation in the LBQS ones. In order to measure how much these two radial distributions differ, we have performed a series of binnings as the one described above for our test and defined $q_{i}$ in each ring as $q_i\propto n^{PKS}_i/n^{LBQS}_i$, where $n^{PKS}_i$ is the number of objects in each ring of the PKS field and $n^{LBQS}_i$ is the number of objects in the same bin of the LBQS field, and normalize by the mean density of each field. We apply the rank order test to all the resulting sequences of $q_i$ and bin radii $r_i$ as described above and find that $<Z_d>=2.77$, $p=99.7\%$. $10^5$ simulations of field pairs give a significance of $p=99.74$. This means that the density of foreground galaxies around the radio-loud quasars is higher than in front of the optically selected sample, and is anticorrelated with the distance from the QSOs at a $99.7\%$ significance level.
Discussion {#disc}
==========
As shown above, we have confirmed the existence of large scale positive correlations between high-redshift radio-loud QSOs and foreground galaxies, whereas for optically selected QSOs with the same redshift distribution the correlation is null or even negative. Can these results be explained by the double magnification bias mentioned in the introduction? In order to answer this question the value of the number-counts distribution slopes in expression (1) must be determined. These slopes can be estimated from the empirical distribution functions of our QSO samples. The cumulative number-radio flux distribution for the PKS QSOs is plotted in Fig. 6. A linear squares fit gives an approximate slope $\alpha^{PKS}_{rad} \approx 1.6$. The histogram of the distribution of $B_J$ magnitudes for the LBQS and the PKS QSOs is plotted in Fig 7a. For the PKS QSOs we do not use the magnitudes quoted in [@ver96] as they have been obtained with different filters and photometric systems. Instead, we have obtained $B_J$ magnitudes from the ROE/NRL COSMOS/UKST catalog, which should be reasonably homogeneous and accurate for $16<B_J<20.5$, apart from the intrinsic variability of the QSOs. Fig. 7a shows that PKS QSOs extend over a wider range of magnitudes than the LBQS ones, which have $B_J \lesssim 18.6$. In Fig 7b we show the cumulative distributions of both QSO samples, $N(<B_J)$ as a function of $B_J$. The LBQS distribution (crosses) can be well aproximated by a power law $\propto 10^{0.4\alpha^{LBQS}_{opt}}$ with $\alpha^{LBQS}_{opt}\approx 2.5$. The PKS distribution (filled squares) is more problematic and cannot be approximated reasonably by a single power law. Although at brighter magnitudes it seems to have a slope similar to the LBQS ones, it quickly flattens and has a long tail towards faint magnitudes. Due to the incompleteness of the PKS sample, this distribution can be interpreted in two ways: either the flattening is caused by the growing incompleteness at fainter optical magnitudes and the slope of the underlying unperturbed distribution for the radio loud QSOs is the same as for LBQS ones, or the distribution function is intrisically shallow, and we are approximately observing its true form. Fortunately this is not a critical question; as it will be shown below, the difference between the slope values obtained in both cases is not enough to change significantly our main conclusions about the causes of the overdensity. Then, we roughly estimate the optical slope for the PKS distribution function with a linear squares fit between $16 < B_J < 17.75$ which yields $\alpha^{PKS}_{opt}\approx 1.9$. These slopes imply an overdensity of galaxies around the LBQS and PKS QSOs $$\begin{aligned}
&q^{LBQS} =\mu^{\alpha_{opt}^{LBQS}-1}\approx \mu^{1.5}\nonumber\\
&\\
&q^{PKS} =\mu^{\alpha_{opt}^{PKS}+\alpha_{rad}^{PKS}-1}\approx \mu^{2.5}\nonumber\end{aligned}$$ That is, $q^{PKS}/q^{LBQS}\approx\mu$. At e.g. $\theta=2\arcmin$, for the LBQS we found $q^{LBQS}=0.968\pm0.063$. This yields a value for the magnification $\mu = 0.98 \pm 0.04$. Then for the PKS QSOs, the overdensity should be $\approx 0.95 \pm 0.1$. However at $\theta=2\arcmin$, we measure $q_{PKS}=1.164\pm 0.061$. If we assume that the intrinsic PKS $B_J$ number-counts slope is the same as for the LBQS QSOs, $\alpha_{opt}^{PKS}=2.5$, we still cannot make both overdensities agree with a same magnification. In order to obtain these results with ’pure’ gravitational lensing, a slope $\alpha_{PKS}^{opt}> 4$ would be needed. For smaller scales, the situation does not change, since $q^{PKS}/q^{LBQS}$ is still higher. Therefore, we must conclude that it is unlikely that the multiple magnification bias alone explains the results found.
As mentioned above, some authors have explained the optically selected QSO-cluster anticorrelations as due to the existence of dust in clusters ([@mao95] and references therein). What would be the expected overdensity when we consider the combined effects of magnification and dust absorption? Let’s consider a patch of the sky which has an average magnification of $\mu$ for background sources and an average flux extinction of $\tau$ for a given optical band, i.e. the observed flux $S$ from the background sources in that band would be $S\approx(1-\tau)S_o$, where $S_o$ is the flux that we would measure in the absence of absorption. If we consider that the radio emission suffers no attenuation by the dust, the overdensity estimations for our samples would be $$\begin{aligned}
&q^{PKS}=\mu^{\alpha_{opt}^{PKS}+\alpha_{rad}^{PKS}-1}(1-\tau)^
{\alpha_{opt}^{PKS}}
\approx \mu^{2.5}(1-\tau)^{1.9}\nonumber\\
&\\
&q^{LBQS}=\mu^{\alpha_{opt}^{LBQS}-1}(1-\tau)^{\alpha_{opt}^{LBQS}}
\approx \mu^{1.5}(1-\tau)^{2.5}\nonumber\end{aligned}$$ Therefore, from our results on scales of $2\arcmin$ we find $\mu\approx 1.139$, and $\tau\approx 0.089$. This extinction is consistent with the results of [@mao95]. Although these values should be considered only as rough estimations, they show that considering dust absorption together with the multiple magnification bias produces new qualitative effects in the behavior of the overdensities of the different QSO types. The strength of the overdensity is attenuated in both samples of QSOs, but the effect is stronger for the LBQS QSOs, which have a steeper optical counts slope. If we consider that the dust approximately traces the matter potential wells acting as lenses, i.e. that there is a strong correlation between magnification and extinction, the QSOs which are more magnified are also the ones which are more absorbed. However, if the product $\mu(1-\tau)$ is greater than unity, the net effect for each QSO will be a flux increment.
An alternative explanation is the existence of the bias suggested by [@rm92] and [@mao95]. They interpret that the avoidance of foreground clusters by optically selected QSOs is probably a selection effect due to the difficulty in identyfing quasars in crowded fields. In that case, apart from the slight QSO-galaxy anticorrelation generated by this effect, the LBQS samples would avoid the zones where the lensing by the large scale structure is stronger and thus their average magnification $\mu$ would be smaller than that of the PKS, which would not be affected by this selection bias. Besides, if dust and magnification are correlated, radio-loud QSOs would also be more reddened on average than optically selected QSOs.
Regarding flat-spectrum QSOs, if we set an additional constraint for our QSOs, $\gamma > -0.5$, where $\gamma$ is the slope of the spectral energy distribution, the resulting sample of 107 $z>0.3$ QSOs should be a fair representation of the radio-loud sample used by [@web95] for the study of reddening in QSOs. We apply again our rank correlation test to the field obtained by merging these 107 fields and conclude that the COSMOS/UKST galaxies are correlated with flat-spectrum QSOs at a $98.5\%$ level. The QSO-galaxy correlation function is plotted in Fig. 8 with $0.75 \arcmin$ bins. The value of the overdensity is similar, but as we have now fewer fields, the significance level is lower. Nevertheless, if we take into account the small amounts of dust allowed by the observations of [@mao95], it seems very unlikely that all the reddening measured by [@web95] for the PKS QSOs is due to dust absorption by foreground galaxies, although in some cases this effect could contribute considerably, as it has been shown by [@sti96]. This question could be clarified by cross-correlating the reddening of the QSOs with the density of foreground galaxies on small scales.
Conclusions
===========
We have studied the clustering of galaxies from the ROE/NRL COSMOS/UKST catalogue up to 15 $\arcmin$ scales around two QSO samples with $z>0.3$. One of them contains 144 radio-loud QSOs from the Parkes Catalogue, and the other is formed by 167 optically selected QSOs obtained from the Large Bright Quasar Survey.
There is a $\approx 99.0\%$ significance level excess of COSMOS $B_J<20.5$ galaxies around the PKS QSOs, whereas there is a slight defect of galaxies around the LBQS QSOs. We have compared the distribution of galaxies around both samples, and found that there is an overdensity around the PKS sample with respect to the LBQS sample anticorrelated with the distance from the QSO at a $99.7\%$ significance level. Whilst this result could be thought to agree qualitatively with the theoretical predictions of the multiple magnification bias effect, we show that it is difficult to explain it through gravitational lensing effects alone, and dust in the foreground galaxies and/or selection effects in the detection of LBQS QSOs must be considered.
Finally, we have established that the lines of sight to PKS flat-spectrum QSOs go through significantly higher foreground galaxy densities than the directions to LBQS quasars. This may be related, at least partially, with the reddening of the PKS QSOs observed by [@web95].
The authors acknowledge financial support from the Spanish DGICYT, project PB92-0741. NB acknowledges a Spanish M.E.C. Ph.D. fellowship. The authors are grateful to Tom Broadhurst, José Luis Sanz and Ignacio Ferreras for carefully reading the manuscript and making valuable suggestions, and Sebastian Sanchez for useful comments. They also thank D.J. Yentis for his help.
Bartelmann, M. 1995, , 298, 661
Bartelmann, M., & Schneider, P. 1994, , 248, 349
Bartelmann, M., & Schneider, P. 1994, , 271, 421
Bartelmann, M., & Schneider, P. 1994, , 284, 1
Bartsch, A., Schneider, P. & Bartelmann, M., 1996, in press, preprint astro-ph/9601125
Benítez, N., & Martínez-González,1995, , 339, 53
Benítez, N., Martínez-González, E., González-Serrano, J.I., & Cayón L. 1995, , 109, 935
Canizares, C.R., 1981, Nature, 291, 620
Borgeest, U., von Linde, J., Refsdal, S. 1991, , 251, L35
Boyle, B.J., Fong, R. & Shanks, T. 1988, , 231, 897
Boyle, B.J. & di Matteo, T. 1996, , 277, L63
Fort, B., Mellier, Y., Dantel-Fort, M., Bonnet, H. & Kneib, J.P. 1995, , submitted; preprint astro-ph/9507076
Fugmann, W. 1990, , 240, 11
Hartwick, F.D.A & Schade, D. 1990, , 28, 437
Hewett, P.C., Foltz, C.B.& Chafee, F.H., 1995, , 109, 1498
Hintzen, P., Romanishin, W., & Valdés, F., 1991, , 371, 49
Maoz, D., 1995, , 455, 115
Narayan, R. 1989, , 341, L1
Odewahn, S.C. & Aldering, G., 1995, , 2009
Press, W.H., Teukolsky, S.A., Vetterling, W.T. & Flannery, B.P., 1992, Numerical Recipes in Fortran, Cambridge University Press, 634
Rodrigues-Williams, L.L. & Hawkins, M.R.S, 1995, preprint astro-ph/9412044
Rodrigues-Williams, L.L. & Hogan, C.J., 1994, , 107, 451
Romani, R.W., & Maoz, D. 1992, , 386, 36
Schneider, P., 1986,
Schneider, P., 1989, , 221, 221
Schneider, P., Ehlers, J., & Falco, E.E. 1992, Gravitational Lenses (Heidelberg: Springer)
Schneider, P. 1995, in ’The Universe at high redshift, Large Scale structure and the Cosmic Microwave Background’, Ed. E. Martínez-González & J.L. Sanz, Springer-Verlag, p.148, in press.
Seitz, S., & Schneider, P. 1995, , in press
Smith, R.J., Boyle, B.J.& Maddox, S.J., preprint astro-ph/9506093
Steidel, C.C. 1995, in QSO Absorption Lines, proc. of the ESO Workshop held at Garching, Germany, ed. G. Meylan, (Springer-Verlag), p. 139
Stickel, M., Meisenheimer, K., & Kühr, H. 1994, , 105, 211
Stickel, M., Rieke, M.J., Rieke, G.H. & Kühr, H. 1996, , 306, 49
Thomas, P.A., Webster, R.L., & Drinkwater, M.J. 1994, , 273, 1069
Tyson, J.A., 1986, , 92, 691
Webster, R.L., Hewett, P.C., Harding, M.E., & Wegner, G.A. 1988, Nature, 336, 358
Webster, R.L., Francis, P.J., Peterson, B., Drinkwater, M.J. & Masci, F.J., 1995, Nature, 375, 469
Wright, A. & Otrupcek, R.E. PKSCAT90: Radio Source Catalogue and Sky Atlas ( Australia Telescope National Facility, Epping, NSW, 1990)
Wu, X.P. 1994, , 286, 748
Wu, X.P., 1995, , 272, 705
Wu, X.P. 1995, Fundamentals of Cosmic Physics, in press
Veron-Cetty, M.P. & Veron, P., 1996, to be published as a ESO Scientific Report.
Yentis, D.J, Cruddace, R.G., Gursky, H., Stuart, B.V., Wallin, J.F., Mac-Gillivray, H.T. & Thompson, E.B., 1992, Digitised Optical Sky Surveys, Ed. H.T. Mac-Gillivray and E.B. Thompson, Kluwer,Dordrecht, 67
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present a re-analysis of the cosmic confusion hypothesis, elucidating the degree to which “confusion” can be expected to hold in a class of flat, adiabatic models. This allows us to devise a simple and accurate fitting function for the height of the first peak in the radiation power spectrum in a wide range of inflationary inspired models. The range preferred by current data is given, together with a discussion of the impact of measurements of the peak height on constraining models of structure formation.'
address: |
Enrico Fermi Institute, University of Chicago\
5640 S. Ellis Ave, Chicago IL 60637
author:
- Martin White
title: Cosmic Confusion and Structure Formation
---
-.2in -.6in 8.5in
epsf
\#1[to 0pt[\#1]{}]{}
Introduction
============
The picture of structure formation through gravitational instability now appears to be well established. Initial fluctuations grow through gravitational instability to form the structures which we observe today, and leave their imprint at redshift $z\sim1000$ in the cosmic microwave background (CMB) radiation anisotropy. Within this framework any theory which purports to explain the large-scale structures we observe today, must simultaneously fit the increasing amount of data on CMB anisotropies. The combination of these two constraints is especially powerful, since they probe a large lever arm both in scale ($k\simlt10^{-4}\hMpc$ to $k\simgt1\hMpc$) and time ($z\simgt1000$ to $z\simeq0$).
Calculations of both CMB anisotropy spectra and matter power spectra are now well developed for inflationary models, with the former approaching 1% precision [@Sug; @HSSW]. (In Fig. \[fig:cl\_ref\] we show the anisotropy power spectrum for the “standard” CDM model, see [@Sel; @echoes; @SSW; @HuSug] for a discussion of the physics behind these anisotropies). This kind of precision in theoretical predictions is necessary to interpret the results envisioned from a future satellite CMB mission, which could accurately measure the power spectrum over a broad range of scales, allowing one to measure most of the standard cosmological parameters with unprecedented accuracy [@echoes; @JunKKS]. However such a measurement is still (optimistically) several years away, and in the meantime the data on CMB anisotropies continues to be amassed at a steady rate. For the interpretation of current data, such a high level of precision is not necessary, and semi-analytic techniques exist which can give spectra accurate to $\sim10\%$ [@HuSug].
An important constraint on structure formation models can be obtained without even this amount of effort however. It has been pointed out [@Bondetal; @HuSug; @Sel] that for measurements of the spectrum constrained to scales of $\sim0\degrees5$ or above, the anisotropy power spectra exhibit a degeneracy in parameter space, which can be both a hindrance and a boon. This degeneracy has been dubbed “cosmic confusion” [@Bondetal] as it provides a limit to how well cosmological parameters can be extracted from measures of the CMB power spectra on scales larger than $0\degrees5$.
Notice the promise this degeneracy has in simplifying the interpretation of degree scale CMB data. As noted by Dodelson & Kosowsky [@DodKos], if this degeneracy holds it is possible to characterize the constraints on a wide range of models in terms of two parameters: an overall amplitude and the value of the degenerate combination of parameters $\nu$ (see §\[sec:confusion\] for details. A part of this approximate degeneracy was implicitly used in [@SSW] to constrain the spectral slope and ionization history from the peak height). The amplitude of the [*potential*]{} fluctuations, measured on the largest scales by [*COBE*]{}, has become the preferred method for normalizing theories of large-scale structure [@cobenorm]. (We show an example of this in Fig. \[fig:deltah\], where the dimensionless amplitude of the matter power spectrum at horizon crossing, $\delta_H$, is shown vs. $\Omega_0=1-\Omega_\Lambda$ and spectral tilt $n$.) The value of $\nu$ will constrain other parameters on which the predictions of the theory depend. Furthermore, to the extent that cosmic confusion holds, generating spectra for a wide range of parameters can be reduced to simple modifications of a standard spectrum (however with the advent of semi-analytic techniques for quickly generating spectra over the entire angular range, this may be of limited utility). An obvious application of this is a fitting formula for the height of the peak in the power spectra at $\ell\simeq220$. The current data provide limits on the height of the peak (see e.g. [@SSW]), which are becoming stronger. A quick method of estimating the height of the peak in the CMB spectrum can allow one to estimate the constraints from degree-scale measurements quickly and efficiently over a broad range of parameter space.
The outline of this paper is as follows. In §\[sec:confusion\] we discuss how well cosmic confusion works, and slightly modify the original formulation [@Bondetal]. §\[sec:reion\] discusses the special case of late reionization in processing the spectra. In §\[sec:fit\] we use a compilation of CMB data to provide a limit on the peak height and discuss implications for models of structure formation. §\[sec:conclusions\] presents the conclusions.
Cosmic Confusion {#sec:confusion}
================
In this section we would like to make some comments about, and minor improvements upon, the degeneracy of CMB spectra with respect to certain parameters which has been dubbed “cosmic confusion” [@Bondetal]. Specifically, it is claimed that for a range of parameters the CMB anisotropy spectra are degenerate in $$\widetilde{n} = n - 0.28\ln(1+0.8r)
- 0.52\left( \Omega_0^{1/2}h - 0.5 \right)
- 0.00036 z_R^{3/2}$$ where $n$ is the (scalar) spectral index, $r$ is the ratio of tensors to scalars (see below), $h$ is the Hubble constant in units of $100{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$ and $z_R$ is the reionization redshift (see §\[sec:reion\]). The degeneracy is supposed to hold up to the first peak in the power spectrum, $\ell\simeq 220$, though some spectra can be degenerate over an even larger range of $\ell$ [@Bondetal; @Crossroads].
To start let us take as our guiding principle the following conjecture:
- For reasonable input parameters all CMB anisotropy spectra with the same peak-to-plateau ratio will be approximately degenerate.
The interpretation of this statement clearly depends upon what range of cosmological models are adopted, over what range of scales the degeneracy is required to hold (not beyond the first peak), and the precise meaning of “approximately”. We will discuss all of these issues below.
Inflation in its most generic form predicts a flat universe, so in this paper we will restrict ourselves to models with vanishing spatial curvature. Several questions relating to open inflation are still unresolved, for example the predictions for super-curvature modes, spectral tilt and tensor modes[^1]. While for any given model the calculations of the power spectrum are straightforward [@open; @SSW], we feel that these uncertainties make fitting to spectra premature for these models.
The simplest implementation of cosmic confusion is in terms of an apparent spectral index, which we shall call $\nu$ (this is related to the $\widetilde{n}$ of [@Bondetal] by $\nu\simeq\widetilde{n}-1$). As our base model we shall adopt ‘standard’ CDM ($h=0.5$, $\Omega_0=1$, $\Omega_\Lambda=0$, $n=1$, $C_2^{(T)}=0$) for which we define $\nu=0$. This spectrum is shown in Fig. \[fig:cl\_ref\] and tabulated in Table \[tab:scdm\]. The spectrum for a model whose parameters combine to give $\nu$ is then related to that of standard CDM by the simple formula $$D_\ell(\nu) = \left({\ell\over 10}\right)^\nu\ D_\ell(\nu=0)
\label{eqn:tiltapp}$$ where $D_\ell\equiv\ell(\ell+1)C_\ell$ for notational convenience. We have chosen to pivot around $\ell=10$ as this is approximately the pivot point of the [*COBE*]{} data [@Goretal]. Alternatively one could use $\sigma(10^{\circ})$ [@Bondetal], which turns out to be almost the same for the range of spectra under consideration.
To understand the expression for $\nu$ in terms of the parameters of the model, we need only consider the effect of changing each parameter on the peak height of the model. We will now consider each of these effects in turn, concentrating first on those parameters for which the conjecture works well and then turning to those for which the deviations are larger.
As discussed in [@HSSW] changing the spectral index, $n$, of the primordial spectrum ($\propto k^n$) is equivalent to multiplying the anisotropy spectrum by $\ell^{n-1}$ as in Eq. \[eqn:tiltapp\]. For $n\ge0.8$ the worst deviations induced by this approximation occur around $\ell=30$–50, and are $\simlt10\%$. Hence tilting the model away from scale invariance changes $\nu$ by $n-1$.
Associated with tilt is the possibility that some of the CMB anisotropy comes from long-wavelength, inflation-produced gravitational waves (tensor fluctuations). The fraction, $r$, of tensors is usually quoted in terms of the contribution to the quadrupole: $r\equiv C_2^{(T)}/C_2^{(S)}$. Unfortunately due to the decaying potentials in a $\Lambda$ model the scalar quadrupole is very sensitive to $\Omega_\Lambda$ [@SacWol; @KofSta] for a fixed “initial” power spectrum. The tensor quadrupole is much less sensitive [@TWL] and hence the ratio has a strong (artificial) $\Lambda$ dependence[^2] [@Knox]. We choose instead to write $r=1.4 C_{10}^{(T)}/C_{10}^{(S)}$, where the prefactor has been chosen to make the definitions agree for $\Omega_0=n=1$. The $\ell=10$ mode is not as strongly affected by the decaying potentials [@HuWhi] making this a more robust measure of the underlying tensor/scalar ratio. Since the tensor spectrum damps rapidly on scales smaller than the horizon at last scattering [@TWL; @CBDES] the effect of introducing a tensor component is to increase the plateau by $(1+0.76r)$ while leaving the peak unchanged. \[The factor 0.76 is roughly $D_{10}^{(T)}/D_2^{(T)}$ for a scale-invariant spectrum with $\Omega_0=1$.\] An equivalent reduction in the peak height relative to the plateau can be accomplished by reducing the spectral index (from $\ell=10$ to $\ell\simeq220$) by, $$\nu \ni -\ln(1+0.76r) / \ln(220/10) \simeq -0.32\ln(1+0.76r)$$
In the semi-analytic, two-fluid models of CMB anisotropies [@DZS; @JKNN; @WSS; @Sel; @HuSug], the height of the first peak depends primarily on the combinations $\Omega_Bh^2$ (which sets the speed of sound in the coupled baryon-photon fluid) and the redshift of matter-radiation equality. The latter scales with $\Omega_0h^2f_{\gamma}$, where $f_\gamma\equiv1.68\,\rho_\gamma/\rho_{\rm rad}=3.36/g_{*}$ is the fraction of the radiation made up of photons, scaled to be 1 in the standard model with 3 massless neutrino species. In models in which more relativistic species are included, such as decaying neutrino models [@decneut], $f_\gamma$ can be reduced leading to a larger peak height, just as if $\Omega_0$ were lowered. As a warning we point out that this ‘degeneracy’ is not exact, due to the effect of $\Omega_\Lambda$ at low redshift (which is included implicitly in the formulation of the semi-analytic models [@HuSug]). In Fig. \[fig:cl\_compare\] we show spectra for two models with the same (reasonable) values of $\Omega_0h^2f_\gamma$ and $\Omega_Bh^2$, which differ even up to the first peak by 5–10%. Specifically the models have $\Omega_0h^2=0.1$ and $\Omega_Bh^2=0.0125$, the first with ($\Omega_0$,$h^2$)=(0.4,0.25) and the second with (1,0.1). As one can see from the figure, the exact height and position of the first peak are complicated functions of the cosmological parameters! The reason for the breaking of the degeneracy in this particular case is that the large-angle temperature fluctuations (from decaying potentials at $\Lambda$ domination) and the projection of physical scales to $\ell$ both depend on $\Omega_\Lambda$, not just $(1-\Omega_\Lambda)h^2$.
The original statement of cosmic confusion was meant to hold for models with $\Omega_Bh^2=0.0125$, meaning that models with equal $\Omega_0h^2f_\gamma$ would be degenerate, and furthermore that they would coincide with models with some $n\ne1$. For models with scale invariant initial perturbations, $\Omega_0=1$, $\Omega_Bh^2=0.0125$ and $h=0.3$ to 0.75, we find that the height of the peak (relative to $D_{10}$) is very well fit by $$\left( {D_{\rm peak}\over D_{10}} \right) \propto h^{-1.19} \quad .$$ We note that $\ell_{\rm peak}$ ranges from 260 to 200 over this range of $h$, since a changing sound speed causes the sound horizon at last scattering to subtend a varying angular scale [@Sel; @HuSug]. This small movement of the peak should not matter if we redefine “cosmic confusion” to hold when averaged over a reasonably broad window in $\ell$, as is the case in most experiments to date. Under the “confusion” assumption then we predict that $$\nu \ni -0.37 \ln(2h)$$ which is roughly equivalent to the original statement given in [@Bondetal]. Both this approximation, and the original statement [@Bondetal], give $\simlt10\%$ deviations in $D_\ell$ for $10\le\ell\le250$ and $0.35\le h\le0.75$. \[The percentage deviations for $\ell<10$ can be larger than 10%, but here cosmic variance is also larger.\]
Similarly we find that the variation with $\Omega_0=1-\Omega_\Lambda$ at fixed $\Omega_Bh^2$ and $h$ works at the $\sim10\%$ level in power for $\ell\ge10$ and $\Omega_0\ge0.4$, with $$\nu \ni -0.16 \ln(\Omega_0) - 0.15\ln(f_\gamma)$$ This is not exactly what we would have expected based on a degeneracy in $\sqrt{\Omega_0f_\gamma}h$: the scaling with $\Omega_0$ is slightly weaker, and that with $f_\gamma$ weaker still. \[This can be traced to the same non-degeneracy mentioned above and illustrated in Fig. \[fig:cl\_compare\].\]
Note that at present $\Omega_Bh^2$ as determined by Big Bang Nucleosynthesis (BBN) is uncertain to a factor of $\sim2$ (c.f. the Hubble constant!) [@CopST]. Thus we are compelled to study the variation of the peak height with the sound speed, or $\Omega_Bh^2$, which was not included in the original statement of cosmic confusion [@Bondetal]. For $\Omega_0=1$ and $0.01\le\Omega_B\le0.10$ the height of the peak is relatively well fit by an exponential in $\Omega_B$ for $h=0.5$ and less well fit by an exponential for $h=0.8$. Unfortunately the slopes of the fits for these two cases (which have different $\Omega_0h^2$) disagree: $$\begin{aligned}
\ln\left( {D_{\rm peak}\over D_{10}} \right)
&\simeq 19\Omega_Bh^2 + 1.388 & \quad\mbox{for}\ h=0.5\\
&\simeq 23\Omega_Bh^2 + 0.824 & \quad\mbox{for}\ h=0.8\end{aligned}$$ As a compromise then we take the average of the two coefficients to arrive at our dependence of $\nu$ on $\Omega_Bh^2$ $$\nu \ni 6.8 (\Omega_Bh^2 - 0.0125)$$ which holds relatively well over the range preferred by BBN: $0.01\le\Omega_Bh^2\le0.02$ [@CopST].
In summary then we can define a “spectral tilt” $\nu$ through $$\begin{array}{lcc}
\nu &\equiv& n-1 - 0.32\ln(1+0.76r) + 6.8( \Omega_Bh^2-0.0125 ) \\
&& -0.37\ln(2h) - 0.16\ln(\Omega_0) - 0.15\ln(f_\gamma)
\end{array}
\label{eqn:nudef}$$ For a large range of $\Omega_{\rm tot}=1$ CDM models, we find with this $\nu$ that “cosmic confusion” holds at the $\sim10\%$ level in power (5% in temperature) up to $\ell\sim200$ for those models with $\Omega_0\ge0.4$, $h\ge0.4$ and $0.01\le\Omega_Bh^2\le0.02$ (we discuss $\tau>0$ in §\[sec:reion\]). This deviation usually occurs at $\ell\sim30$–50 and some of it can be attributed to the approximation for tilt of the primordial spectrum that we are using (Eq. \[eqn:tiltapp\]). For low-$\Omega_0$ or $h$ models the shape of the rise into the peak and the position of the peak are sufficiently different that a tilted model is not a good fit (though our fit is a good approximation to the height of the peak even for $\Omega_0\sim0.3$). On the other hand for some models in the range mentioned the degeneracy is good to 1% in power for $10\le\ell\le200$. The logarithmic dependence of $\nu$ on $\Omega_0h^2f_\gamma$ differs from the linear dependence of $\widetilde{n}$ on the same quantity found by [@Bondetal]. Both approximations give similar “worst fit” $D_\ell$, but the logarithm is a better fit to the height of the peak ($D_{220}$ relative to $D_{10}$) over the range of models considered; it works to better than 5%.
With the rapid progress being made in measurements of the CMB anisotropy, it is likely that we will need to do better than 10% in the very near future. A more complicated fit or a better treatment of spectral tilt could alleviate matters slightly. However, bearing in mind that the position of the peak also changes with the input parameters and that the degeneracies (e.g. $\Omega_0^{1/2}h$) built into “confusion” themselves only hold at the 5–10% level, one is lead to conclude that “one number” summaries of the CMB data are becoming a thing of the past. Of course, Eq. \[eqn:nudef\] is still useful as a quick-and-dirty method of estimating the height of the peak (to 5%) in the power spectrum for quite a wide range of parameters. This can be useful for narrowing the large parameter space down to a smaller region which can be searched more carefully (as one often uses linear theory estimates in large-scale structure work). If the experiments are chosen to probe scales for which “confusion” works well, then constraints on $\nu$ will still encode much of the information that the CMB has for large-scale structure.
As more experiments probe scales “beyond the first peak” ($\ell>200$), the utility of cosmic confusion as a summary of [*all*]{} CMB constraints will decrease. In these cases one must search a multi-dimensional parameter space and compute the full CMB spectrum either by numerical evolution of the coupled Boltzmann equations or fast semi-analytic methods which can be accurate to $\simlt10\%$ (e.g. [@HuSug]). Moving beyond this degeneracy will allow us to obtain more information about the cosmological parameters from study of the detailed structure of the power spectrum.
Reionization {#sec:reion}
============
In the original statement of cosmic confusion [@Bondetal], there was a term for the reionization redshift. Reionization in the adiabatic models of interest is likely to occur very rapidly [@reion] and at relatively low redshift, $z_R$. Since $z_R$ the universe has probably been fully ionized. We show in Table \[tab:zr\] a “reasonable” estimate of the reionization redshift for some models by way of example. These determinations rely on several assumptions and so should only be taken as illustrative. The results are also very sensitive to the value of $\Omega_Bh^2$ and $\sigma_8$ chosen.
In such a scenario, the amplitude of the fluctuations on small scales is reduced by $e^{-2\tau}$ [@WSS; @HuSS; @HuWhi], where $$\tau=0.035\,{\Omega_B\over\Omega_0}h\,x_e
\left[ \sqrt{\Omega_0(1+z_R)^3+1-\Omega_0}-1 \right]$$ (for $\Omega_0+\Omega_\Lambda=1$) is the optical depth to Thomson scattering from $z=0$ to $z_R$ (also shown in Table \[tab:zr\]). Note that $\tau$ depends not only on $z_R$ but also on $\Omega_B$ and $h$, which was neglected in the original treatment of cosmic confusion [@Bondetal]. (For the accuracy to which the spectrum depends on and scales with $\tau$ see [@HSSW].) In addition new fluctuations are generated on larger angular scales for $\tau\simgt0.1$ [@WSS; @SugVitSil]. Reionized spectra are [*not*]{} well described by a simple tilting of the sCDM spectrum. An approximation which has the correct asymptotic forms [@HSSW] is to multiply the spectrum by $\exp[-2\tau(z_\ell)]$, where $\tau(z')$ is the optical depth from $z=0$ to redshift $z=z'$ and $z_\ell$ defines a mapping from angles to redshift (e.g. the redshift at which the horizon subtends an angle $\ell^{-1}$). While this approximation has the right asymptotic form, it fails to be a good approximation to a reionized spectrum for $\tau\simgt0.1$ since it does not take into account the fluctuations generated on the new last scattering surface. These fluctuations typically do not extend to $\ell>100$, so if one is interested only in scales near the peak of the spectrum a reduction of $\exp[-2\tau]$ is appropriate.
Because reionization is such an important part of interpreting CMB anisotropy measurements in the context of cosmological models, we will defer a more detailed discussion of reionization to a future paper [@WhiHu].
Current limits on $\nu$ {#sec:fit}
=======================
In this section we discuss what we can infer about $\nu$ from current observations of CMB anisotropy. We shall use the data tabulated in [@SSW] since the newer data is (mostly) not yet available. As discussed in [@SSW; @WSSD] and below, the constraints even without including the new data are very interesting for large-scale structure modelling.
Several issues regarding foreground contamination and possible systematic errors intrude in the analysis of current CMB data. It was shown in [@DodSte] that for the experiments dominating the fit near the peak, removing the foregrounds does not increase the error bars by more than $\sim10\%$. To account for foreground removal we have (conservatively) multiplied the errors in [@SSW] by 10% (except for [*COBE*]{}). Note that since [@SSW], several experiments have reproduced earlier observations, indicating that systematic errors are not as severe as might have been thought. In one case however, that of the MSAM experiment [@MSAM], one of the two channels (the “single-difference”) showed a discrepancy. To be very conservative we have dropped this point from our analysis (shown in Fig. \[fig:the\_data\] as the solid square at $\ell\simeq150$) and updated the “double-difference” point to include the new data (which is slightly lower than the older data). Also we do not include the new data from the Python experiment (though we show it in Fig. \[fig:the\_data\] as the solid triangles with the small error bars [@Python]) since the points are correlated in an unknown way.
As it is not the purpose of this paper to revisit the data analysis, especially without access to the latest data, we shall use the data set as tabulated in [@SSW], with the simple modifications discussed above. One point deserves special mention. In [@SSW] we used symmetric error bars on all of the points. This is a conservative method if the inference is the presence of a peak, since most of the observations have skew positive error bars. However this leads to a slightly stronger than warranted [*upper*]{} limit on the height of the peak, and a slightly weaker lower limit. As most of the skewed likelihood functions have not been published in tabular or graphical form we have not tried to correct for this bias. Since for most of the models under consideration (especially the models with high $\Omega_\Lambda$) the large-scale structure data is best fit with a tilt ($n<1$), this treatment remains the most conservative.
A fit to the data as in [@SSW], but with spectra generated from Eq. \[eqn:tiltapp\], gives a likelihood function for $\nu$ shown in Fig. \[fig:nulik\]. Here we have integrated, or marginalized, over the normalization $D_{10}^{1/2}$, which is well fixed by [*COBE*]{} [@cobenorm]. The mean and standard deviation are $\nu=-0.05\pm0.07$. The absolute goodness of fit to both the [*COBE*]{} data alone [@WhiBun] and the other data [@SSW] show no indications that the fit should be suspect for purely [*statistical*]{} reasons.
Notice that this result has several immediate implications for structure formation models. That the preferred peak height is near that of a CDM model with scale invariant initial conditions puts a [*lower*]{} limit on the amount of tilt which can be accommodated [@SSW; @WSSD]. The lower limit clearly depends on the values of $\Omega_0$, $h$ and $\Omega_Bh^2$ assumed, with lower $\Omega_0$ and $h$ acting to loosen the constraint and lower $\Omega_Bh^2$ acting to strengthen it. The probability of early reionization [@reion] in these models will only tighten the lower limit on $n$.
The increased height of the peak in low-$\Omega_0$ models or models with $f_\gamma<1$ can cause conflict with the [*upper*]{} limit on $\nu$, unless the models are tilted, the baryon fraction is lowered or there is some reionization. We note in passing that for $\Lambda$CDM, some tilt is probably necessary to provide a good fit with the large scale structure data in any case [@SSW; @WhiBun; @KlyPriHol; @WhiSco]. In general, while the degree scale data provide a strong constraint on models with high $\Omega_0$, the possibility of early reionization (the degree scale power is exponentially sensitive to the very uncertain $z_R$) and the large uncertainty in $\Omega_Bh^2$ from BBN significantly weaken the constraints from this data for low $\Omega_0$ or $f_\gamma$ models.
Conclusions {#sec:conclusions}
===========
In this paper we have compared the accuracy of spectra generated using the “cosmic confusion” [@Bondetal] assumption to those calculated using numerical evolution of the Boltzmann hierarchy. For a range of parameters of current interest the spectra are found to agree to $\sim10\%$ for all multipoles up to the first peak ($\ell\simeq220$). Over the same range the height of the peak is reproduced to $\simlt5\%$. As pointed out by Dodelson & Kosowsky [@DodKos], the presence of this degeneracy for scales greater than $0\degrees5$ allows a simple statement of the parameter constraints arising from CMB data. Given current data we find that $\nu$ (see Eq. \[eqn:nudef\]) is constrained to be $\nu=-0.05\pm0.07$. By limiting the amount of spectral tilt, this result in combination with large-scale structure measurements (on somewhat smaller scales) strongly constrains models of structure formation (see e.g. [@SSW; @WSSD; @WhiSco] and §\[sec:fit\]).
I would like to thank Wayne Hu for conversations.
N. Sugiyama, Astrophys. J. Supp. [**100**]{} 281 (1995). W. Hu, D. Scott, N. Sugiyama, M. White, Phys. Rev. [**D52**]{} 5498 (1995). U. Seljak, Astrophys. J. [**435**]{}, L87 (1994). D. Scott, M. White, General Relativity & Gravitation [**27**]{}, 1023 (1995). D. Scott, J. Silk, M. White, Science [**268**]{}, 829 (1995); D. Scott, M. White, in [*CMB Anisotropies Two Years After COBE*]{}, ed. L.M. Krauss, World Scientific, New York, p.214 W. Hu, N. Sugiyama, Astrophys. J. [**444**]{}, 489 (1995). G. Jungman, M. Kamionkowski, A. Kosowsky, D.N. Spergel, preprint, astro-ph/9507080 J.R. Bond, R. Crittenden, R.L. Davis, G. Efstathiou, P.J. Steinhardt, Phys. Rev. Lett. [**72**]{}, 13 (1994). S. Dodelson, A. Kosowsky, Phys. Rev. Lett. [**75**]{}, 604 (1995). E. Bunn, D. Scott, M. White, Astrophys. J. [**441**]{}, L9 (1995); E. Bunn, N. Sugiyama, Astrophys. J. [**446**]{}, 49 (1995); K. Gorski, G. Hinshaw, A.J. Banday, C.L. Bennett, E.L. Wright, A. Kogut, G.F. Smoot, P. Lubin, Astrophys. J. [**430**]{}, L89 (1994); M. White, E. Bunn, Astrophys. J. [**450**]{}, 477 (1995); K.M. Gorksi, B. Ratra, N. Sugiyama, A. Banday, Astrophys. J. [**446**]{}, L67 (1995); R. Stompor, K. Gorski, A.J. Banday, to appear in MNRAS; M. White, D. Scott, to appear in Comments on Astrophysics. L. Cayon, E. Martinez-Gonzalez, J.L. Sanz, N. Sugiyama, S. Torres, to appear in Astrophys. J.; K. Yamamoto, E. Bunn, to appear in Astrophys. J. P. Steinhardt, “Cosmology at the Crossroads”, in [*Proceedings of the 1994 Snowmass Workshop*]{}. M.L. Wilson, Astrophys. J. [**273**]{}, 2 (1983); N. Gouda, N. Sugiayama, M. Sasaki, Prog. Theor. Phys. [**85**]{}, 1023 (1991); M. Kamionkowski, D. Spergel, N. Sugiyama, Astrophys. J. [**426**]{}, L57 (1994); M. Kamionkowski, B. Ratra, D.N. Spergel, N. Sugiyama, Astrophys. J. [**434**]{}, L1 (1994); K.M. Gorksi, B. Ratra, N. Sugiyama, A. Banday, Astrophys. J. [**446**]{}, L67 (1995); M. White, D. Scott, to appear in Astrophys. J.; K. Gorski, G. Hinshaw, A.J. Banday, C.L. Bennett, E.L. Wright, A. Kogut, G.F. Smoot, P. Lubin, Astrophys. J. [**430**]{}, L89 (1994). R.K. Sachs, A.M. Wolfe, Astrophys. J. [**147**]{}, 73 (1967). L. Kofman, A. Starobinsky, Sov. Astron. Lett. [**11**]{}, 271 (1985). M.S. Turner, M. White, J.E. Lidsey, Phys. Rev. [**D48**]{}, 4613 (1993). L. Knox, Phys. Rev. [**D52**]{}, 4307 (1995). M. White, E. Bunn, Astrophys. J. [**450**]{}, 477 (1995); (erratum in press). R. Crittenden, J.R. Bond, R.L. Davis, G. Efstathiou, P.J. Steinhardt, Phys. Rev. Lett. [**71**]{}, 324 (1993). A.G. Doroshkevich, Ya.B. Zel’dovich, R.A. Sunyaev, Sov. Astron. [**22**]{}, 523 (1978). H.E. Jorgensen, E. Kotok, P.D. Nasel’skii, I.D. Novikov, Astr. Astrophys. [**294**]{}, 639 (1995) M. White, D. Scott, J. Silk, Ann. Rev. Astron. & Astrophys. [**32**]{}, 319 (1994). J. Bardeen, J.R. Bond, G. Efstathiou, Astrophys. J. [**321**]{}, 28 (1987); S. Dodelson, G. Gyuk, M. Turner, Phys. Rev. Lett. [**72**]{}, 3754 (1994); Phys. Rev. [**D49**]{}, 5068 (1994); M. White, G. Gelmini, J. Silk, Phys. Rev. [**D51**]{}, 2669 (1995); S.J. McNally, J.A. Peacock, MNRAS 277, 143 (1995). C. Copi, D.N. Schramm, M.S. Turner, Science [**267**]{}, 192 (1995). H.M.P. Couchman, MNRAS [**214**]{}, 137 (1985). H.M.P. Couchman, M.J. Rees, MNRAS [**221**]{}, 53 (1986); M. Fukugita, M. Kawasaki, MNRAS [**269**]{}, 563 (1994); S. Sasaki, F. Takahara, Y. Suto, Prog. Theor. Phys. [**90**]{}, 85 (1993); M. Tegmark, J. Silk, Blanchard, Astrophys. J. [**420**]{}, 484 (1993) (erratum [**434**]{}, 395); A. Liddle, D.H. Lyth, MNRAS [**273**]{}, 1177 (1995). W. Hu, D. Scott, J. Silk, Phys. Rev. [**D49**]{}, 648 (1994). W. Hu, M. White, to appear in Astron. & Astrophys. N. Sugiyama, N. Vittorio, J. Silk, Astrophys. J. [**419**]{} L1 (1993). W. Hu, M. White, to appear in Astrophys. J. M. White, D. Scott, J. Silk, M. Davis, MNRAS, [**276**]{}, L69 (1995). S. Dodelson, A. Stebbins, Astrophys. J. [**433**]{}, 440 (1994). E.S. Cheng, et al., to appear in Astrophys. J. M. Dragovan, private communication A. Klypin, J. Primack, J. Holtzman, preprint, astro-ph/9510042. M. White, D. Scott, to appear in Comments on Astrophysics. E. Bunn, Ph.D. Thesis, University of California, Berkeley (1995).
$\ell$ $D_\ell$ $\ell$ $D_\ell$ $\ell$ $D_\ell$
-------- ---------- -------- ---------- -------- ----------
2 0.91 54 1.63 178 4.63
4 0.92 64 1.78 195 4.93
6 0.95 74 1.94 212 5.07
10 1.00 84 2.12 230 5.03
14 1.06 96 2.38 248 4.80
19 1.13 108 2.67 267 4.39
25 1.22 120 3.00 287 3.87
31 1.31 134 3.41 308 3.29
38 1.41 148 3.83 329 2.79
46 1.52 163 4.26 351 2.42
: The multipole moments, $D_\ell\equiv\ell(\ell+1)C_\ell$, for the “standard” Cold Dark Matter model with $\Omega_0=1$, $\Omega_B=0.05$, $n=1$ and $C_2^{(T)}=0$. The moments have been normalized to have $D_{10}=1$. Enough moments have been given to interpolate the behaviour of the curve.[]{data-label="tab:scdm"}
$\Omega_0$ $h$ $n$ $\Omega_B$ $z_R$ $\tau$
------------ ------ ------ ------------ ------- --------
0.3 0.90 0.92 0.02 28 0.165
0.4 0.70 0.93 0.03 23 0.141
0.5 0.57 0.94 0.05 20 0.126
0.6 0.49 0.95 0.06 18 0.115
1.0 0.45 0.80 0.10 11 0.065
1.0 0.50 1.00 0.05 45 0.271
: A “reasonable” estimate of the ionization redshift, $z_R$, for some models. The optical depth $\tau$ from $z=0$ to $z_R$ assuming full ionization of the hydrogen is also shown. The models with $\Omega_0<1$ have been chosen to be “best fits” to LSS data. Specifically they have $\Omega_Bh^2=0.015$, a “shape parameter” $\Gamma=0.25$ and $\sigma_8=0.7\Omega_0^{-0.6}$ (which fixes $n$). Since $n<1$ the effective shape measured by the galaxy correlation function will be $\Gamma_{\rm eff}\simeq0.21$–$0.24$. The last two models are a “best-fitting” and a “standard” CDM model.[]{data-label="tab:zr"}
[^1]: The contribution to the large-angle CMB anisotropy from tensor perturations will probably be small in open models, since a combination of the [*COBE*]{} and cluster abundance normalizations predicts the spectral index $n$ to be very close to (or greater than) 1.
[^2]: This dependence was originally neglected in [@WhiBun] thus overestimating the damping effect of tensors for the tilted models.
| {
"pile_set_name": "ArXiv"
} |
The electron tunneling experiments are powerful tools to study the spectroscopy of superconductors. These experiments measure the dynamical conductance $dI/dV $ through a junction as a function of applied voltage $V$ and temperature [@tunn; @tunn2]. For superconductor-insulator-normal metal (SIN) junctions, the measured dynamical conductance is proportional to the electron density of states (DOS) in a superconductor $N(\omega) = -(1/\pi) \int d k ~Im G(k, \omega)$ at $\omega =eV~\cite{mahan}$. For superconductor-insulator-superconductor (SIS) junctions, the conductance $dI/dV \propto G(\omega = eV)$, where $G(\omega) = \int^{\omega}_0 d \Omega N(\omega -\Omega)~ \partial_\Omega
N(\Omega)$ is proportional to the derivative over voltage of the convolution of the two DOS [@mahan].
For conventional superconductors, the tunneling experiments have long been considered as one of the most relevant ones for the verification of the phononic mechanism of superconductivity [@carbotte]. In this communication we discuss to which extent the tunneling experiments on cuprates may provide the information about the pairing mechanism in high-$T_c$ superconductors. More specifically, we discuss the implications of the spin-fluctuation mechanism of high-temperature superconductivity on the forms of SIN and SIS dynamical conductances.
The spin-fluctuation mechanism implies that the pairing between electrons is mediated by the exchange of their collective spin excitations peaked at or near the antiferromagnetic momentum $Q$. This mechanism yields a $d-$wave superconductivity [@dwave], and explains [@expl; @ac] a number of measured features in superconducting cuprates, including the peak/dip/hump features in the ARPES data near $(0,\pi)$ [@arpes], and the resonance peak below $2\Delta $ in the inelastic neutron scattering data [@neutrons]. Moreover, in the spin-fluctuation scenario, the ARPES and neutron features are related: the peak-dip distance in ARPES equals the resonance frequency in the dynamical spin susceptibility [@ac]. This relation has been experimentally verified in optimally doped and underdoped $YBCO$ and optimally doped $Bi2212$ materials [@norman]. Here we argue that the resonance spin frequency can also be inferred from the tunneling data by analyzing the derivatives of SIN and SIS conductances.
The SIN and SIS tunneling experiments have been performed on $YBCO$ and $Bi2212$ materials [@tunn; @tunn2]. At low/moderate frequencies, both SIN and SIS conductances display a behavior which is generally expected in a $d-$wave superconductor: SIN conductance is linear in voltage for small voltages, and has a peak at $eV = \Delta$ where $\Delta$ is the maximum value of the $d-$wave gap [@tunn], while SIS conductance is quadratic in voltage for small voltages, and has a near discontinuity at $eV = 2 \Delta$ [@tunn2]. These features have been explained by a weak-coupling theory, without specifying the nature of the pairing interaction [@maki]. However, above the peaks, both SIN and SIS conductances have extra dip/hump features which become visible at around optimal doping, and grow with underdoping [@tunn; @tunn2]. We argue that these features are sensitive to the type of the pairing interaction and can be explained in the spin-fluctuation theory. As a warm-up for the strong coupling analysis, consider first SIN and SIS tunneling in a $d-$wave superconductor in the weak coupling limit. In this limit, the fermionic self-energy is neglected, and the superconducting gap does not depend on frequency. For simplicity, we consider a circular Fermi surface for which $\Delta_k = \Delta \cos 2\phi$.
We begin with the SIN tunneling. Integrating $G(k, \omega) =
(\omega + \epsilon_k)/(\omega^2 - \epsilon_k^2 - \Delta^2_k)$ over $\epsilon_k = v_F (k-k_F)$ we obtain $$\begin{aligned}
N (\omega ) &=& Re~ \frac{\omega }{2\pi }\int _{0}^{2\pi }
\frac{d\phi }{\sqrt{\omega ^{2}-\Delta ^{2}\cos ^{2}(2\phi )}} \nonumber\\
&=&\frac{2}{\pi }\left\{
\begin{array}{ll}
K(\Delta /\omega ) & \mbox{for $\omega >\Delta $}\\
(\omega /\Delta )K(\omega /\Delta ) & \mbox{for $\omega <\Delta $}
\end{array}\right. , \label{sin-gas}\end{aligned}$$ where $K(x)$ is the elliptic integral. We see that $N(\omega)
\sim \omega $ for $\omega \ll \Delta $ and diverges logarithmically as $(1/\pi)\ln (8\Delta/|\Delta -\omega|)$ for $\omega \approx \Delta$. At larger frequencies, $N(\omega)$ gradually decreases to a frequency independent, normal state value of the DOS, which we normalized to $1$. The plot of $N(\omega)$ is presented in Fig \[fig1\]a.
\[t\]
=3.3in =1.8in
. \[fig1\]
We now turn to the SIS tunneling. Substituting the results for the DOS into $G(\omega)$ and integrating over $\Omega$, we obtain the result presented in Fig \[fig1\]b. At small $\omega$, $G(\omega)$ is quadratic in frequency, which is an obvious consequence of the fact that the DOS is linear in $\omega$. At $\omega = 2\Delta$, $G(\omega)$ undergoes a finite jump. This discontinuity is related to the fact that near $2\Delta$, the integral over the two DOS includes the region $\Omega \approx \Delta$ where both $N(\Omega)$ and $N(\omega -\Omega)$ are logarithmically singular, and $\partial_\Omega N(\Omega)$ diverges as $1/(\Omega-\Delta)$. The singular contribution to $G(\omega)$ from this region can be evaluated analytically and yields $$G(\omega) = -\frac{1}{\pi ^{2}}P\int _{-\infty }^{\infty }
\frac{d x~\ln |x|}{x+\omega -2\Delta }
= -\frac{1}{2}\mbox{sign}(\omega -2\Delta )
\label{sis-gas}
$$ We see that the amount of jump in the SIS conductance is a universal number which does not depend on $\Delta$.
The results for the SIN and SIS conductances in a $d-$wave gas agree with earlier studies [@maki]. In previous studies, however, SIS conductance was computed numerically, and the universality of the amount of the jump at $2\Delta$ was not discussed, although it is clearly seen in the numerical data.
We now turn to the main subject of the paper and discuss the forms of SIN and SIS conductances for strong spin-fermion interaction.
We first show that the features observed in a gas are in fact quite general and are present in an arbitrary Fermi liquid as long as the impurity scattering is weak. Indeed, in an arbitrary $d-$wave superconductor, $$N(\omega) \propto Im~\int d \phi~
\frac{\Sigma (\phi ,\omega )}{(F^2 (\phi,\omega) -
\Sigma^{2} (\phi ,\omega ))^{1/2}},
\label{eq}$$ where $\phi $ is the angle along the Fermi surface, and $F (\phi, \omega)$ and $\Sigma (\phi, \omega)$ are the retarded anomalous pairing vertex and retarded fermionic self-energy at the Fermi surface (the latter includes a bare $\omega$ term in the fermionic propagator). The measured superconducting gap $\Delta (\phi)$ is a solution of $F (\phi,\Delta (\phi)) = \Sigma (\phi, \Delta (\phi))$.
In the absence of impurity scattering, $Im \Sigma$ and $Im F$ in a superconductor both vanish at $T=0$ up to a frequency which for arbitrary strong interaction exceeds $\Delta$. The Kramers-Kronig relation then yields at low frequencies $Re \Sigma (\phi, \omega) \propto \omega$, $Re F(\phi,
\omega) \propto (\phi - \phi_{node})$ where $\phi_{node}$ is a position of the node of the $d-$wave gap. Substituting these forms into (\[eq\]) and integrating over $\phi$ we obtain $N(\omega) \propto \omega$ although the prefactor is different from that in a gas. The linear behavior of the DOS in turn gives rise to the quadratic behavior of the SIS conductance.
Similarly, expanding $\Sigma^2 - F^2$ near each of the maxima of the gap we obtain $\Sigma^2 (\phi, \omega) -F^2 (\phi, \omega) \propto (\omega - \Delta)
+ B (\phi-\phi_{max})^2$, where $B>0$. Then $$N (\omega ) \propto
Re\int \frac{d{\tilde\phi} }{\sqrt{B {\tilde\phi}^{2} +
(\Omega -\Delta)}} \approx -\frac{\ln |\Omega -\Delta |}{\sqrt{B}}
\label{sin-strong}
$$ This result implies that the SIN conductance in an arbitrary Fermi liquid still has a logarithmic singularity at $eV = \Delta$, although its residue depends on the strength of the interaction. The logarithmical divergence of the DOS causes the discontinuity in the SIS conductance by the same reasons as in a Fermi gas.
In the presence of impurities, the logarithmical singularity is smeared out, and the DOS acquires a nonzero value at zero frequency (at least, in the self-consistent $T-$matrix approximation [@t-mat]). However, for small concentration of impurities, this affects the conductances only in narrow frequency regions near singularities while away from these regions the behavior is the same as in the absence of impurities.
We now show that a strong spin-fermion interaction gives rise extra features in the SIS and SIN conductances, not present in a gas. The qualitative explanation of these features is the following. At strong spin-fermion coupling, a $d$-wave superconductor possesses [*propagating*]{}, spin-wave type collective spin excitations near antiferromagnetic momentum $Q$ and at frequencies below $2\Delta$. These excitations give rise to a sharp peak in the dynamical spin susceptibility at a frequency $\Omega _{res} <2\Delta$ [@neutrons], and also contribute to the damping of fermions near hot spots (points at the Fermi surface separated by $Q$), where the spin-mediated $d-$wave superconducting gap is at maximum. If the voltage for SIN tunneling is such that $eV=\Omega _{res}+\Delta$, then an electron which tunnels from the normal metal, can emit a spin excitation and fall to the bottom of the band (see Fig. \[fig2\]a) loosing its group velocity. This obviously leads to a sharp reduction of the current and produce a drop in $dI/dV$.
Similar effect holds for SIS tunneling. Here however one has to first break an electron pair, which costs the energy $2\Delta$. After a pair is broken, one of the electrons becomes a quasiparticle in a superconductor and takes an energy $\Delta$, while the other tunnels. If $eV = 2\Delta + \Omega_{res}$, the electron which tunnels through a barrier has energy $\Delta + \Omega_{res}$, and can emit a spin excitation and fall to the bottom of the band. This again produces a sharp drop in $dI/dV$ (see Fig. \[fig2\]b).
\[t\]
=3.3in =1.8in
In the rest of the paper we consider this effect in more detail and make quantitative predictions for the experiments. Our goal is to compute $dI/dV$ for SIN and SIS tunneling for strong spin-fermion interaction.
The point of departure for our analysis is the set of two Eliashberg-type equations for the fermionic self-energy $\Sigma_\omega$, and the spin polarization operator $\Pi_\Omega$. The later is related to the dynamical spin susceptibility at the antiferromagnetic momentum by $\chi^{-1} (Q, \Omega) \propto 1- \Pi_\Omega$. The same set was used in our earlier analysis of the relation between ARPES and neutron data [@ac]. In Matsubara frequencies these equations read (${\tilde \Sigma}_{\omega_m} = i \Sigma ({\omega_m})$) $$\begin{aligned}
{\tilde\Sigma}_{\omega_m} &=& {\omega_m} + \frac{3R}{8 \pi^{2}}~\int
\frac{{\tilde\Sigma}_{{\omega_m} +{{\Omega_m}}}}
{q_{x}^{2}+{\tilde\Sigma}^{2}_{{\omega_m} +{{\Omega_m}}}+F^2}
\frac{d {{\Omega_m}}}{\sqrt{q_{x}^{2}+1 -\Pi_\Omega }}
\nonumber \\
\Pi_\Omega&=& \frac{1}{2}
\int \frac{d {\omega_m}}{\omega_{sf}}~\left(
\frac{{\tilde\Sigma}_{{{\Omega_m}} +{\omega_m}}~{\tilde\Sigma}_{{\omega_m}} + F^2}
{\sqrt{{\tilde\Sigma}^{2}_{{{\Omega_m}} +{\omega_m}}+F^2}~
\sqrt{\tilde\Sigma^{2}_{\omega_m}+F^2}}-1 \right).
\label{set2b}\end{aligned}$$ This set is a simplification of the full set of Eliashberg equations that includes also the equation for the anomalous vertex $F(\omega)$ [@acf]. As in [@ac] we assume that near optimal doping, the frequency dependence of $F(\omega)$ is weak at $\omega
\sim \Delta$ relevant to our analysis, and replace $F(\omega)$ by a frequency independent input parameter $F$. Other input parameters in (\[set2b\]) are the dimensionless coupling constant $R = {\bar g}/(v_F \xi^{-1})$ and a typical spin fluctuation frequency $\omega_{sf} = (\pi/4) (v_F \xi^{-1})^2/{\bar g}$. They are expressed in terms of the effective spin-fermion coupling constant ${\bar g}$, the Fermi velocity at a hot spot $v_F$, and the magnetic correlation length $\xi$. By all accounts, at and below optimal doping, $R \geq 1$ [@chubukov], i.e., the system behavior falls into the strong coupling regime.
Strictly speaking, the set (\[set2b\]) is valid near hot spots where $\phi \approx
\phi_{max}$. Away from hot spots $F (\phi)$ is reduced compared to $F$. We, however, will demonstrate that the new features due to spin-fermion interaction are produced solely by fermions from hot regions. As in [@ac], we consider the solution of (\[set2b\]) for the experimentally relevant case $F \gg R \omega_{sf}$ when the measured superconducting gap $\Delta \sim F^2/(R^2 \omega_{sf}) \gg \omega_{sf}$. In this situation, at frequencies $\sim \Delta$, fermionic excitations in the normal state are overdamped due to strong spin-fermion interaction. In a superconducting state, the form of the spin propagator is modified at low frequencies because of the gap opening, and this gives rise to a strong feedback from superconductivity on the electron DOS.
More specifically, we argued in [@ac] that in a superconductor, $\Pi_\Omega$ at low frequencies $\Omega \ll 2\Delta$ behaves as $\Omega^2/\Delta$, i.e., collective spin excitations are undamped, propagating spin waves. This behavior is peculiar to a superconductor – in the normal state, the spin excitations are completely overdamped. The propagating excitations give rise to the resonance in $\chi (Q,\Omega)$ at $\Omega _{res}\sim (\Delta \omega _{sf})^{1/2} \ll \omega_{sf}$ where $Re\Pi (\Omega _{res})=1$ [@ac]. This resonance accounts for the peak in neutron scattering [@neutrons].
The presence of a new magnetic propagating mode changes the electronic self-energy for electrons near hot spots. In the absence of a propagating mode, an electron can decay only if its energy exceeds $3\Delta$. Due to resonance, an electron at a hot spot can emit a spin wave already when its energy exceeds $\Delta + \Omega_{res}$. It is essential that contrary to a conventional electron-electron scattering, this process gives rise to a discontinuity in $Im \Sigma (\omega)$ at the threshold. Indeed, using the spectral representation to transform from Matsubara to real frequencies in the first equation in (\[set2b\]), integrating over momentum and neglecting for simplicity unessential $q^2_x$ in the spin susceptibility, we obtain for $\omega \geq \omega_{th} = \Delta + \Omega_{res}$ $$\begin{aligned}
Im \Sigma (\omega) &\propto& \int^{\omega-\Delta}_{\Omega_{res}}
d \Omega \frac{1}{\sqrt{\omega - \Omega -
\Delta}}~\frac{1}{\sqrt{\Omega - \Omega_{res}}} \nonumber \\
&&\propto \int^{(\omega - \omega_{th})^{1/2}}_0 dx
\frac{1}{\sqrt{\omega - \omega_{th} -x^2}} = \frac{\pi}{2},\end{aligned}$$ We see that $Im \Sigma (\omega)$ jumps to a finite value at the threshold. This discontinuity is peculiar to two dimensions. By Kramers-Kronig relation, the discontinuity in $Im \Sigma$ gives rise to a logarithmical divergence of $Re
\Sigma$ at $\omega = \omega_{th}$. This in turn gives rise to a vanishing spectral function near hot spots, and accounts for a sharp dip in the ARPES data [@arpes].
We now show that the singularity in $Re \Sigma (\omega)$ causes the singularity in the derivatives over voltages of both SIN and SIS conductances $d^2I/dV^2$. Indeed, near a hot spot, $F(\phi) = F (1- \lambda {\tilde \phi}^2)$ where ${\tilde \phi} = \phi - \phi_{max}$, and $\lambda >0$. Then, quite generally, $Re \Sigma (\phi,
\omega) \propto \ln |\omega - \omega_{th} (\phi)|$ where $\omega_{th} (\phi) = \omega_{th} + C {\tilde \phi}^2$, and $C>0$. Substituting this expression into the DOS and differentiating over frequency, we obtain after a simple algebra $$\begin{aligned}
\frac{\partial N (\omega )}{\partial \omega }
&\sim& -\int \frac{F^{2}(\phi )}{\Sigma^{3}(\phi ,\omega ) }
\partial_{\omega } \Sigma (\phi ,\omega )d\phi
\nonumber \\
&&\sim \frac{1}{\ln ^{3}|\omega - \omega_{th}|}
\frac{\Theta (\omega_{th}-\omega)}
{\sqrt{\omega_{th}-\omega}}, \label{sin-strong-spin}\end{aligned}$$ where $\Theta(x)$ is a step function. We see that $\partial N(\omega)/\partial\omega$ has a one-sided, square-root singularity at $\omega = \omega_{th}$. Physically, this implies that the conductance drops when propagating electrons start emitting spin excitations. Note that the typical $\phi$ which contribute to the singularity are small (of order $|\omega_{th} - \omega|^{1/2}$), which justifies our assertion that the singularity is confined to hot spots.
The singularity in $\partial N(\omega)/\partial \omega$ is likely to give rise to a dip in $N(\omega)$ at $\omega \geq \omega_{th}$. The argument here is based on the fact that if the angular dependence of $\omega_{th} (\phi)$ is weak (i.e., $C$ is small), then $\Sigma (\omega_{th}) \gg F(\omega_{th})$, and $N(\omega_{th})$ reaches its normal state value with infinite negative derivative. Obviously then, at $\omega > \omega_{th}$, $N(\omega)$ goes below its value in the normal state and should therefore have a minimum at some $\omega \geq \omega_{th}$. Furthermore, at larger frequencies, we solved (\[set2b\]) perturbatively in $F(\omega)$ and found that $N(\omega)$ approaches a normal state value [*from above*]{}. This implies that besides a dip, $N(\omega)$ should also display a hump somewhere above $\omega_{th}$. The behavior of the SIN conductance is schematically shown in Fig. \[fig3\]a.
\[t\]
=3.3in =1.8in
Similar results hold for SIS tunneling. The derivative of the SIS current, $d^{2}I/dV^{2} \sim
\partial G(\omega)/\partial \omega$, is given by $$\frac{\partial G(\omega)}{\partial \omega} =
\int _{0}^{\omega}
\partial _{\omega}N (\omega -\Omega )
\partial _{\Omega }N (\Omega )d\Omega
\label{sis2}$$ Evaluating the integral in the same way as for SIN tunneling, we find a square-root singularity at $\omega = \omega^*_{th} = 2\Delta +\Omega _{res}$. $$\begin{aligned}
\frac{d^{2}I}{dV^{2}} &\sim& -P\int_{0}^{\omega }
\frac{d\Omega }{\omega -\Omega-\Delta}~
\frac{1}{\ln ^{3}|\omega_{th}-\Omega |}
\frac{\Theta (\omega_{th}-\omega)}
{\sqrt{\omega_{th}-\omega}} \nonumber \\
&\sim &-\frac{1}{\ln^{3}|\omega^*_{th}-\omega |}
\frac{\Theta (\omega^*_{th} - \omega)}
{\sqrt{\omega^*_{th} - \omega}}
\label{sis-strong-spin}\end{aligned}$$ The singularity comes from the region where $\Omega \approx \omega_{th}$ and $\omega-\Omega \approx \Delta$, and both $\partial_{\omega} N (\omega -\Omega )$ and $\partial _{\omega }N (\omega )$ are singular.
Again, it is very plausible that the singularity of the derivative causes a dip at a frequency $\omega \geq
\omega^*_{th}$, and a hump at even larger frequency. We stress, however, that at exactly $\omega^*_{th}$, the SIS conductance has an infinite derivative, while the dip occurs at a frequency which is somewhat larger than $\omega^*_{th}$. The behavior of the SIS conductance is presented in Fig \[fig3\].
Qualitatively, the forms of conductances presented in Fig \[fig3\] agree with the SIN and SIS data for YBCO and Bi2212 materials [@tunn; @tunn2]. Moreover, recent SIS tunneling data for $Bi2212$ [@tunn2] indicate that the relative distance between the peak and the dip ($\Omega_{res}/(2\Delta)$ in our theory) decreases with underdoping. More data analysis is however necessary to quantitatively compare tunneling and neutron data.
To summarize, in this paper we considered the forms of SIN and SIS conductances both for noninteracting fermions, and for fermions which strongly interact with their own collective spin degrees of freedom. We argue that for strong spin-fermion interaction, the resonance spin frequency $\Omega_{res}$ measured in neutron scattering can be inferred from the tunneling data by analyzing the derivatives of SIN and SIS conductances. We found that the derivative of the SIN conductance diverges at $eV = \Delta + \Omega_{res}$ while the derivative of the SIS conductance diverges at $eV = 2\Delta + \Omega_{res}$, where $\Delta$ is the maximum value of the $d-$wave gap.
It is our pleasure to thank G. Blumberg, A. Finkel’stein and particularly J. Zasadzinski for useful conversations. The research was supported by NSF DMR-9979749.
Ch. Renner et al, Phys. Rev. Lett. [**80**]{}, 149 (1998); Y. DeWilde et al, ibid [**80**]{}, 153 (1998).
N. Miyakawa et al, Phys. Rev. Lett. [**83**]{}, 1018 (1999); L. Ozyurer, J.F. Zasadzinski, and N. Miyakawa, cond-mat/9906205; J.F. Zasadzinski, private communication.
G.D. Mahan, Many-Particle Physics, Plenum Press, 1990.
J.P. Carbotte, Rev. Mod. Phys. [**62**]{}, 1027 (1990) and references therein.
D.Z. Liu, Y Zha, and K. Levin, Phys. Rev. Lett. [**75**]{}, 4130 (1995); I. Mazin and V. Yakovenko, ibid [**75**]{}, 4134 (1995); A. Millis and H. Monien, Phys. Rev. B [**54**]{}, 16172 (1996); N. Bulut and D.J. Scalapino, ibid [**53**]{}, 5149 (1996); Z-X Shen and J.R. Schrieffer, Phys. Rev. Lett. [**78**]{}, 1771 (1997), M.R. Norman and R. Ding, Phys. Rev. B [**57**]{}, R11089 (1998).
Ph. Monthoux and D. Pines, Phys. Rev. B [**47**]{}, 6069 (1993); D.J. Scalapino, Phys. Rep. [**250**]{}, 329 (1995).
Ar. Abanov and A. Chubukov, Phys. Rev. Lett., [**83**]{}, 1652
J.C. Campuzano et al., Nature [**392**]{}, 157 (1988); Z-X. Shen et al, Science, [**280**]{}, 259 (1998).
H.F. Fong et al, Phys. Rev. B [**54**]{}, 6708 (1996); P. Dai et al, Science [**284**]{}, 1344 (1999).
M.R. Norman et al, Phys. Rev. Lett. [**79**]{}, 3506 (1997)
H. Won and K. Maki, Phys. Rev. B [**49**]{}, 1397 (1994).
P.J. Hirshfeld, P. Wolfle, and D. Einzel, Phys. Rev. B [**37**]{}, 83 (1988); T.P. Devereaux and A.P. Kampf, Int. J. Mod. Phys. [**11**]{}, 2093 (1997) and references therein. Ar. Abanov, A. Chubukov, and A. M. Finkel’stein, cond-mat/9911445
A. Chubukov, Europhys. Lett. [**44**]{}, 655 (1997).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Torus-fibered Calabi-Yau threefolds $Z$, with base $d{\mathbb P}_9$ and fundamental group $\pi_1(Z)={\mathbb Z}_2 \times {\mathbb Z}_2 $, are reviewed. It is shown that $Z=X/({\mathbb Z}_2 \times {\mathbb Z}_2)$, where $X=B \times_{\cp{1}} B^{''}$ are elliptically fibered Calabi-Yau threefolds that admit a freely acting ${\mathbb Z}_2 \times {\mathbb Z}_2$ automorphism group. $B$ and $B^{''}$ are rational elliptic surfaces, each with a ${\mathbb Z}_2 \times {\mathbb Z}_2$ group of automorphisms. It is shown that the ${\mathbb Z}_2 \times {\mathbb Z}_2$ invariant classes of curves of each surface have four generators which produce, via the fiber product, seven ${\mathbb Z}_2 \times {\mathbb Z}_2$ invariant generators in $H_4(X,{\mathbb Z})$. All invariant homology classes are computed explicitly. These descend to produce a rank seven homology group $H_4(Z,{\mathbb Z})$ on $Z$. The existence of these homology classes on $Z$ is essential to the construction of anomaly free, three family standard-like models with suppressed nucleon decay in both weakly and strongly coupled heterotic superstring theory.'
author:
- |
Burt A. Ovrut$^1$, Tony Pantev$^2$ and René Reinbacher$^1$\
[$^1$Department of Physics, University of Pennsylvania]{}\
[Philadelphia, PA 19104–6396]{}\
[$^2$Department of Mathematics, University of Pennsylvania]{}\
[Philadelphia, PA 19104–6395, USA]{}\
title: |
\
\
[Invariant Homology on Standard Model Manifolds]{}\
\[1em\]
---
Introduction
============
In [@hw1; @hw2; @w1] Hořava and Witten showed that the simplest vacuum of strongly coupled heterotic superstring theory consists of an eleven dimensional bulk space bounded on either end of the eleventh dimension by a ten dimensional orbifold fixed plane. Each fixed plane supports an $E_8$ supergauge theory on its worldvolume. This theory was compactified on Calabi-Yau threefolds to produce, at low energy, an effective five dimensional bulk space bounded in the fifth dimension by two three-branes. When the Calabi-Yau space supports a non-trivial $E_8$ gauge instanton with structure group $G \subset E_8$, the gauge group on that three-brane will no longer be $E_8$. Rather, it is reduced to a group $H$ which is the commutant of $G$ in $E_8$. In addition to these “end-of-the-world” three-branes, there are, generically, five-branes in the bulk space which are wrapped on a holomorphic curve in the Calabi-Yau threefold. This five-dimensional theory with three-branes is called heterotic M-theory. It is a fundamental paradigm for the so called “brane-world” theories of particle physics. The basic construction of heterotic M-theory was presented in [@losw1; @losw2; @low1].
To produce phenomenologically relevant models of particle physics on one of the three-branes, called the “observable brane”, it is essential to construct the appropriate gauge instantons on the Calabi-Yau threefold. This is similar to constructing instantons on a flat ${\mathbb R}^4$ manifold, using for example, the ADHM method [@adhm]. Now, however, instantons satisfying the hermitian Yang-Mills equations must be found on a curved Calabi-Yau threefold, a much more challenging problem. The basic ideas for such a construction were presented in [@fmw1; @fmw2] for elliptically fibered Calabi-Yau threefolds using the equivalent concept of stable, holomorphic vector bundles. Extending this work, it was shown in [@low2; @dlow] that phenomenologically relevant grand unified theories (GUTs) can, indeed, be constructed on the observable brane. This is accomplished by compactifying Hořava-Witten theory on elliptically fibered Calabi-Yau threefolds with trivial fundamental group that support holomorphic vector bundles with structure group $G=SU(n)$. Choosing $n$ to be $5$ and $4$, for example, leads to GUT groups $H$ of $SU(5)$ and $SO(10)$ respectively.
Constructing holomorphic vector bundles that break the $E_8$ gauge theory to standard-like models is more difficult. However, it was shown in [@dopw-i; @dopw-ii; @dopw-iii; @dopw-iv] that three-family models with gauge group $H=SU(3)_C\times SU(2)_L \times U(1)_Y$ do exist in heterotic M-theory. This was accomplished as follows. First, elliptically fibered Calabi-Yau threefolds with non-trivial fundamental group ${\mathbb Z}_2$ were constructed. It was found to be convenient to choose the base surface of these threefolds to be $B=d{\mathbb P}_9$. See also [@schoenCY]. Second, new methods were introduced for producing stable, holomorphic vector bundles with structure group $G=SU(5)$ over these Calabi-Yau spaces. Such instantons break the $E_8$ group to $SU(5)$, as mentioned above. Now, however, the non-trivial ${\mathbb Z}_2$ fundamental group allows topologically stable flat bundles, that is, ${\mathbb Z}_2$ Wilson lines, which break $SU(5)$ down to $SU(3)_C\times SU(2)_L \times U(1)_Y$. Importantly, by sufficiently restricting the Calabi-Yau threefolds, it was shown that new homology classes can appear that allow the theory to be anomaly free and to admit three families of quarks and leptons.
Although very compelling in many ways, standard-like models constructed from $G=SU(5)$ instantons and ${\mathbb Z}_2$ Wilson lines have a potential problem. It is well-known that superstring vacua associated with GUT theories with gauge group $SU(5)$ may allow nucleon decay more rapid than the experimental bounds. Although the nucleon decay rate can be reduced by imposing certain discrete symmetries and by superstring effects, such vacua do not exhibit a natural mechanism for suppressing nucleon decay. This problem was emphasized by Witten [@w2; @w3] and discussed in [@rt]. The question of nucleon decay has motivated the authors of this paper to attempt to construct alternative standard-like models of particle physics in heterotic M-theory based on $G=SU(4)$ instantons and ${\mathbb Z}_2 \times {\mathbb Z}_2$ Wilson lines. As mentioned above, $SU(4)$ instantons break $E_8$ to an $SO(10)$ GUT group, which is further broken to a standard-like model by a ${\mathbb Z}_2 \times {\mathbb Z}_2$ Wilson line. The key point is that the gauge group of this standard-like model now contains an unbroken $U(1)_{B-L}$ factor, which naturally suppresses nucleon decay.
This program was begun in [@opr], where the requisite torus-fibered Calabi-Yau threefolds with fundamental group ${\mathbb Z}_2 \times {\mathbb Z}_2$ were constructed. Before proceeding with the construction of $SU(4)$ instantons on these manifolds, however, it is essential that one compute the transformation laws of certain elements of the Calabi-Yau homology ring under the ${\mathbb Z}_2 \times {\mathbb Z}_2$ automorphism group. This is necessary in order to establish the existence and properties of homology classes that are invariant under ${\mathbb Z}_2 \times {\mathbb Z}_2$. As discussed in [@dopw-i; @dopw-ii; @dopw-iii; @dopw-iv], such invariant classes are essential to produce three-family, anomaly free standard-like models. This rather technical, but very important, step in our program is carried out in this paper in detail.
To be specific, in Section \[Z\] we briefly review the construction of torus-fibered Calabi-Yau threefolds with non-trivial fundamental group ${\mathbb Z}_2 \times {\mathbb Z}_2$. It is shown that such threefolds have the form $Z=X/({\mathbb Z}_2 \times {\mathbb Z}_2)$, where $X$ is an elliptically fibered Calabi-Yau threefold that is the fiber product of two rational elliptic surfaces $B$ and $B^{'}$. Section \[rat\] is devoted to a detailed review of the properties of restricted surfaces $B$ that admit at least a ${\mathbb Z}_2 \times {\mathbb Z}_2$ automorphism group. In addition, the relevant involutions $(-1)_B$, $\alpha_B$, $t_{e_6}$ and $t_{e_4}$ on $B$, as well as the ${\mathbb Z}_2 \times {\mathbb Z}_2$ generating automorphisms $\tau_{B1}$ and $\tau_{B2}$, are defined and discussed. In [@opr] and Section \[rat\], it is shown that generic $B=d{\mathbb P}_9$ surfaces allowing an involution $\alpha_B$ are elliptically fibered with projection map $\beta: B \to \cp{1}$ and are four-fold covers of a surface $Q=\cp{1}\times \cp{1}$. However, the strong restrictions on $B$ necessary to allow a ${\mathbb Z}_2 \times {\mathbb Z}_2$ automorphism group, also produce extra structure on these surfaces. In Section \[double\], it is shown that $B$ can, alternatively, be expressed as a two-fold cover of another surface $\bar{Q}=\cp{1}\times \cp{1}$. Using this result, we demonstrate in Section \[ruled\] that the restricted surfaces are “ruled”. That is, these surfaces have a second fibration with projection $\delta: B \to \cp{1}$ whose generic fiber is not elliptic but, rather, the projective space $\cp{1}$. These, admittingly technical, results play an important role. They allow one to identify, and give the properties of, a canonical set of generators of $H_2(B,{\mathbb Z})$, which we denote by $e_i,\;i=1,\dots,9$ and $l$. This is done in Section \[basis\]. Sections \[-1\], \[alpha\] and \[trans\] are devoted to computing the explicit transformation laws of these generators under the involutions $(-1)_B$, $\alpha_B$, and $t_{e_6}$, $t_{e_4}$ respectively. These results are then composed to produce the action of the ${\mathbb Z}_2\times {\mathbb Z}_2$ automorphism group on the canonical set of generators. This is carried out in Section \[comp\], where an explicit table of the transformations of the generators $e_1,\dots,e_9$ and $l$ of $H_2(B,{\mathbb Z})$ under all relevant involutions is presented. Having found the ${\mathbb Z}_2\times {\mathbb Z}_2$ action on $H_2(B,{\mathbb Z})$, we can search for the curve classes that are invariant under this action. In Section \[comp\], we show that these are generated by four invariant classes, which are presented explicitly. Finally, in Section \[XZ\], using the construction outlined in Section \[Z\], we “lift” the four invariant classes on each of $B$ and $B^{'}$ to the fiber product manifold $X$. By construction, these classes are in $H_4(X,{\mathbb Z})$. It is shown that $H_4(X,{\mathbb Z})$ has seven independent homology generators which are invariant under the ${\mathbb Z}_2\times {\mathbb Z}_2$ automorphism group on $X$. These classes are, of course, constructed from the lift of invariant classes on $B$ and $B^{'}$. We explicitly exhibit these seven generators. Clearly, they survive the modding out of $X$ by the ${\mathbb Z}_2\times {\mathbb Z}_2$ action. Therefore, the torus-fibered Calabi-Yau threefolds $Z=X/({\mathbb Z}_2\times {\mathbb Z}_2)$ with fundamental group $\pi_1(Z)={\mathbb Z}_2\times {\mathbb Z}_2$ admit a rank seven homology group $H_4(Z,{\mathbb Z})$, as required to produce anomaly free, three family standard-like models.
Having established these requisite results, we can turn to the problem of constructing stable, holomorphic vector bundles with structure group $G=SU(4)$ on the Calabi-Yau threefolds $Z$. This will be accomplished in [@dopr-i]. A detailed discussion of the associated particle physics will appear in [@dopr-iii; @dlor]. A more precise mathematical discussion of the entire program is also in preparation [@dopr-ii]. We emphasize to the reader that, although much of our discussion was within the context of the strongly coupled heterotic superstring, that is, M-theory, our results are equally applicable to weakly coupled heterotic vacua.
The Calabi-Yau Threefolds $Z$ {#Z}
=============================
To make the paper self contained and to fix notation, we recall in this section the construction of torus fibered Calabi-Yau threefolds $Z$ with fundamental group \_1(Z)=[Z]{}\_2 \_2. For a more complete exposition, the reader is referred to [@opr]. To construct torus fibered Calabi-Yau spaces $Z$ with non-trivial first homotopy group, one starts with simply connected elliptically fibered Calabi-Yau threefolds $X$. Denote the projection map by : X B\^[’]{}. The generic fiber of $\pi$ is isomorphic to a torus $T^2$. Furthermore, there exists a zero section $\sigma: B^{'}\to X$ which fixes a unique point on each fiber. This point acts as the identity of an Abelian group, turning $T^2$ into an elliptic curve. In our construction, the base $B^{'}$ is chosen to be isomorphic to a rational elliptic surface $d{\mathbb P}_9$. Being a rational elliptic surface, $B^{'}$ is itself elliptically fibered over $\cp{1}$. We denote its projection map by \^[’]{}: B\^[’]{}\^[1’]{}. It can be shown, see for example [@opr], that every elliptically fibered Calabi-Yau threefold $X$ over a base $B^{'}$ which is isomorphic to $d{\mathbb P}_9$ is actually a fiber product of two rational elliptic surfaces $B$ and $B^{'}$. That is, X=B\_B\^[’]{}, where B\_B\^[’]{}={(b,b\^[’]{})B B\^[’]{} | (b)=\^[’]{}(b\^[’]{})} and $\beta: B \to \cp{1}$ and $\beta^{'}:B^{'} \to \mathbb{P}^{1'}$ are the projection maps of the two rational elliptic surfaces. Note that hidden in the definition of $B\times_{\cp{1}}B^{'}$ is an isomorphism $i: \cp{1}\to {\mathbb P}^{1'}$ which identifies these spaces. The fiber product structure allows us to construct involutions $\tau_X$ on $X$ in terms of involutions $\tau_{B}$ and $\tau_{B^{'}}$ on $B$ and $B^{'}$ respectively.
As proven in [@opr], when $B^{}$ and $B^{'}$ are each restricted to a two-parameter subset of $d{\mathbb P}_9$ surfaces, there exists a freely acting automorphism group ${\mathbb Z}_2 \times {\mathbb Z}_2$ on $X$. This group is generated by two commuting involutions $\tau_{X1}$ and $\tau_{X1}$ defined as \[invX\] \_[X1]{}=\_[B1]{}\_\_[B\^[’]{}1]{},\_[X2]{}=\_[B2]{}\_\_[B\^[’]{}2]{}, where $\tau_{Bi}$ and $\tau_{B^{'}i}$ for $i=1,2$ are involutions on $B$ and $B^{'}$. It follows that one can define a smooth quotient manifold Z=X/([Z]{}\_2 \_2). It was proven in [@opr; @dopr-ii] that $Z$ is indeed a Calabi-Yau threefold, albeit one with no global sections. Hence, it is only a torus fibration and is not elliptically fibered. Clearly, to understand these Calabi-Yau threefolds $Z$, it is essential to understand the properties of the Calabi-Yau manifolds $X$ and the action of the automorphism group ${\mathbb Z}_2 \times {\mathbb Z}_2$. However, it follows from the above discussion that the properties of $X$ and its automorphism group are determined by the choice of the rational elliptic surfaces $B $ and $B^{'}$, the involutions $\tau_{Bi}$ and $\tau_{B^{'}i}$ for $i=1,2$ on these surfaces and the specification of an identification map $i: \cp{1}\to {\mathbb P}^{1'}$. We turn, therefore, to the properties of rational elliptic surfaces and the classification of their involutions.
Rational Elliptic Surfaces {#rat}
==========================
We begin this section by considering generic rational elliptic surfaces. As already mentioned, a rational elliptic surface $B$ is an elliptic fibration over $\cp{1}$. We denote the projection map by \[beta\] : B . The generic fiber of $\beta$ is isomorphic to a torus $T^{2}$. Furthermore, there is a zero section e: B which fixes a unique zero point on each fiber, turning it into an elliptic curve. In addition, there exists a second projection map, denoted by $\beta_2$ in [@opr], such that \_2: B . The pre-image of $\beta_2$ for a generic point $p \in \cp{2}$, that is $\beta^{-1}_2(p)$, consists of one point. However, at nine points, which we denote by $\omega_a$ for $a=1,\dots,9$, \_2\^[-1]{}(\_a)e\_a. Hence, $B$ is a blow-up of $\cp{2}$ at nine points. This description of $B$ is very useful in determining several geometric properties of the surface. We find that the generators for the homology classes of $H_2(B,{\mathbb Z})$ are the nine blow-up $\cp{1}$ lines $e_1,\dots,e_9$, also called exceptional divisors, and the pull-back $\beta_2^{-1}(l)$ of a line $l$ in $\cp{2}$. We simply denote $\beta_2^{-1}(l)$ by $l$ as well. That is, the lattice $H_2(B,{\mathbb Z})$ is spanned by these ten elements. Since $B$ is connected, it follows that $H_0(B,{\mathbb Z})\cong {\mathbb Z}$ and, hence, $H_0(B,{\mathbb Z})$ is generated by a point. Additionally, since $B$ is compact and orientable, $H_4(B,{\mathbb Z})\cong {\mathbb Z}$. Using, for example, the Van Kampen theorem, one can show that $H_1(B,{\mathbb Z})$ and $H_3(B,{\mathbb Z})$ vanish. Therefore, the topological Euler characteristic of $B$ is given by (B)=\_[i=0]{}\^[4]{} (-1)\^[i]{} H\_[i]{}(B,[Z]{})=1+10+1=12. It is easy to calculate the canonical bundle $K_B=\wedge^2(T^{*}B)$, where we have denoted the holomorphic cotangent bundle of $B$ by $T^{*}B$. The result is \[can\] K\_B=\_2\^[\*]{}K\_\_B(\_[i=1]{}\^[9]{}e\_i)=[O]{}\_B(-3l+\_[i=1]{}\^[9]{}e\_i).
As shown in [@opr], these geometric properties, derived using the projection $\beta_2 : B \to \cp{2}$, have consequences for the fibration $\beta: B \to \cp{1}$. The first consequence is that one can consider the exceptional divisors $e_1,\dots,e_9$ as sections of this fibration. For concreteness, we choose the zero section to be e=e\_9. It was shown in [@opr] that the class of a generic fiber of (\[beta\]) is given by \[df\] f=3l-\_[i=1]{}\^9 e\_i. Therefore, (\[can\]) can be written as K\_B=[O]{}\_B(-f). Since the Euler characteristic of a smooth torus vanishes, the fact that $\chi(B)=12$ requires the occurrence of singular fibers of (\[beta\]). For generic surfaces $B$, these singular fibers are of Kodaira type [@kodaira-casIII] $I_1$ with $\chi(I_1)=1$. Hence, there are twelve such fibers.
Having discussed the basic properties of generic rational elliptic surfaces $B$, we can proceed to determine all possible involutions on them. As shown in [@opr], all involutions \_B: B B induce an involution on the base $\cp{1}$, namely, $\tau_{\cp{1}}:\cp{1}\to \cp{1}$, obeying \_B = \_. Hence, $\tau_B$ maps each fiber $f$ to some possibly different fiber $\tau_B(f)$. Since involutions $\tau_{\cp{1}}:\cp{1}\to \cp{1}$ are either the identity map $\tau_{\cp{1}}=id$ or a non-trivial map with two fixed points, where the fixed points determine $\tau_{\cp{1}}$ uniquely, the induced action on the base $\cp{1}$ gives a crude classification of involutions on $B$. Let us first consider those involutions which act as the identity on the base.
One such involution is canonically given for any elliptic fibration, namely (-1)\_B : B B. It is defined as follows. Since the zero section $e$ fixes a unique zero on each fiber, which we denote by $e$ as well, each fiber forms an Abelian group. We will denote the operation of addition on each fiber by ${\stackrel{.}{+}}$ and the operation of inverse on any element $a$ by ${\stackrel{.}{-}}a$. The restriction of $(-1)_B$ to a fiber maps each point $a$ of that fiber to its Abelian group inverse. Therefore, we obtain (-1)\_B|\_f (a)= a. Since the choice of the identity element $e$ is determined by the global section $e$, $(-1)_B$ is defined globally. Furthermore, $(-1)_B$ induces an automorphism on the global sections of $B$. That is (-1)\_B : (B)(B), where $\Gamma(B)$ denotes the set of global sections of the fibration $\beta: B\to \cp{1}$. More specifically, for any section $\xi \in \Gamma(B)$ \[3.15\] (-1)\_B()=, where ${\stackrel{.}{-}}\xi$ is defined as follows. Consider any point $p \in \cp{1}$ and let $\xi(p)=a$. Then the section ${\stackrel{.}{-}}\xi$ is defined by $({\stackrel{.}{-}}\xi)(p)={\stackrel{.}{-}}a$.
Another automorphism of $B$ which acts trivially on the base $\cp{1}$ is given, for any section $\xi$, by the translation $t_{\xi}$. If we denote the point of intersection of $\xi$ with a fiber $f$ by $\xi$ as well, then the action of $t_\xi$ is given by t\_|\_f: a afor each point $a$ on $f$. Again, the action $t_\xi$ is globally defined and induces an automorphism on the global sections of $B$, namely \[3.17\] t\_(s) = s for any section $s$ in $\Gamma(B)$. For $p \in \cp{1}$ and $s(p)=a$, $s{\stackrel{.}{+}}\xi$ is defined by $(s{\stackrel{.}{+}}\xi)(p)=a{\stackrel{.}{+}}\xi$. Note that (\[3.15\]) and (\[3.17\]) induce an Abelian group structure on $\Gamma(B)$. Since this structure is defined in terms of the group law on each fiber, we continue to denote addition and inverse on $\Gamma(B)$ by ${\stackrel{.}{+}}$ and ${\stackrel{.}{-}}$.[^1] It is clear, that $t_\xi$ is generically not an involution. A necessary condition for it to be an involution is that the section $\xi$ intersect each fiber at a point of order two, that is, $\xi{\stackrel{.}{+}}\xi=e$. There are four such points on each fiber, which we denote by $e,e^{'},e^{''}$ and $e^{'''}$. Of course, the identity element $e$ is one such point. Observe that points of order two are also fixed under the action of $(-1)_B$. For example (-1)\_B (e\^[’]{})=e\^[’]{}=(e\^[’]{}e\^[’]{})e\^[’]{}=e\^[’]{}. However, the above condition is not sufficient to guarantee that global sections $\xi$ with $t_{\xi}^{2}=id$ exist. In fact, for generic elliptic surfaces $B$ they do not. They can occur, however, in a restricted subset of surfaces $B$. We will use such restricted surfaces to construct the Calabi-Yau threefolds $X$.
A further type of involution $\tau_B$ on $B$ which induces a trivial action on the base $\cp{1}$ is given by \_B=t\_(-1)\_B. However, since it is impossible to lift these involutions to a fixed point free involution on $X$, we will not consider them further.
We now discuss involutions on $B$ which act non-trivially on the base $\cp{1}$. By a theorem of [@dopw-i], each such involution can be written as \_B=t\_\_B, where $\alpha_B : B \to B$ is an involution of $B$ leaving the zero section $e$ fixed and inducing a non-trivial action on $\cp{1}$. That is, $\tau_{\cp{1}}\circ \beta=\beta\circ \alpha_B$ where $\tau_{\cp{1}}$ is a non-trivial involution on $\cp{1}$ with two fixed points, which we call $0$ and $\infty$. The section $\xi$ must have the property that \[xi\] \_B()=(-1)\_B(). As shown in [@opr], a generic rational elliptic surface $B$ does not admit such involutions $\alpha_B$. However, there is five dimensional sub-family in the eight dimensional family of rational elliptic surfaces which does allow such involutions. Furthermore, it was shown in [@dopw-i] that each member of this five dimensional sub-family has a rank four lattice of sections $\xi$ fulfilling condition (\[xi\]).
As described in [@opr], $\alpha_B$ leaves two fibers of $B$ stable, namely $f_0=\beta^{-1}(0)$ and $f_\infty = \beta^{-1}(\infty)$. Furthermore, it was shown that $\alpha_B$ acts as the identity map on $f_0$ and as $(-1)_B$ on $f_\infty$. Hence, the double cover \[3.22\] : B B/\_B has the image of $f_0$ and the image of the four points of order two of the fiber $f_\infty$ under $\kappa$ as its branch locus. Hence, the image of $f_\infty$, that is $f_\infty/\alpha_B \subset B/\alpha_B$, contains four isolated $A_1$ surface singularities of $B/\alpha_B$.
These can be resolved by blowing up and one obtains an $I_0^{*}$ fiber. It was shown in [@opr] that, for generic $B$ in the five parameter family of rational elliptic surfaces, the quotient $B/\alpha_B$ can be described as the double cover of $Q=\cp{1}\times \cp{1}$. That is \[3.23\] : B/\_B Q, where the branch locus $M $ of $\pi$ consists of the union of a $(1,4)$ curve $T$ and a $(1,0)$ curve $r$. These are shown in Figure \[gen-M\], where the two natural projections of $Q$, $p_i : Q \to \cp{1}_i,\;i=1,2$ to both $\cp{1}$ factors of $Q$ are also defined.
Having described all possible involutions on rational elliptic surfaces $B$, we now further restrict ourselves to a two dimensional sub-family of rational elliptic surfaces. These were used in [@opr] to construct Calabi-Yau threefolds $X$ with two freely acting involutions. Since this two dimensional family is a subset of the five dimensional family mentioned above, these surfaces continue to admit involutions $\alpha_B$. However, for these restricted surfaces, the curve $T$ splits into $T=t^{''}\cup s\cup i\cup j$, where $t^{''}$ is a $(1,1)$ curve and $s,i$ and $j$ are $(0,1)$ rulings. See Figure \[2df\] for a pictorial description of them.
Note that $t^{''}$ intersects each of $r,s,i$ and $j$ at one point. Call these points $p_1,p,p^{'}$ and $p^{''}$ respectively. Furthermore, $s,i$ and $j$ each intersect $r$ at a single point, which we denote by as $p_2,p_3$ and $p_4$.
What is the double cover of $Q$ corresponding to this restricted branch locus? For maximal clarity, we will here denote this double cover by $W_M$, relating it to $B/\alpha_B$ only later. We continue to indicate the covering map by $\pi: W_M \to Q$. Let us consider the composition map $\tilde{p}_1=p_1\circ \pi$. Clearly, we have the projection map \[tildep\] \_1: W\_M . The fiber of $\tilde{p}_1$, for a generic point $x \in \cp{1}$, is the double cover of the fiber of projection $p_1$ at this point, branched over the four points where this fiber intersects the branch locus $M$, namely, at its intersection with $t^{''},s,j$ and $i$. It was shown in [@opr] that this generic fiber of $\tilde{p}_1$ is isomorphic to a torus. Hence, $W_M$ is a torus fibration. What is the double cover of the fiber corresponding to the point $p_1(r)=a$? Since $r$ is in the branch locus itself, $\tilde{p}^{-1}_1(a)$ is a double $\cp{1}$ line. Furthermore, since $r$ intersects the other components of the branch curve $M$, namely, $t^{''},s,i$ and $j$ at $p_1,p_2,p_3$ and $p_4$ respectively, $W_M$ has four $A_1$ surface singularities on $\tilde{p}_1^{-1}(a)$ located at $\pi^{-1}(p_i)$ for $i=1,\dots,4$. Recall from [@opr] that resolving these singularities would produce an $I_0^{*}$ fiber. In addition to $\tilde{p}_1^{-1}(a)$, there are three other fibers of $\tilde{p}_1$ which contain singular points of $W_M$. Singular points occur when different components of the branch locus of $W_M $ intersect. Clearly, the remaining intersection points are in the fibers ${p}_1^{-1}(p_1(p)),\;{p}_1^{-1}(p_1(p^{'}))$ and ${p}_1^{-1}(p_1(p^{''}))$. As explained in [@opr], the double covers of these three fibers in $W_M $ are singular, each containing an $A_1$ surface singularity. After resolving the singularities, each of these fibers consists of two components, the proper transform of the original fiber and an exceptional divisor which is isomorphic to $\cp{1}$. This is called an $I_2$ fiber in the Kodaira classification and is sketched in Figure \[I2\].
To conclude, the fibration $(\ref{tildep})$ has four singular fibers, one containing four $A_1$ singularities and the remaining three with one $A_1$ singularity each. Resolving these singularities, we obtain a smooth surface, which we denote by $\widehat{W_M}$, with one $I_0^{*}$ fiber and three $I_2$ fibers. Recall from [@opr] that $\chi(I_0^{*})=6$ and $\chi(I_2)=2$. Therefore $\chi(\widehat{W_M})=12$. Furthermore, the fibration $\tilde{p}_1 : W_M \to \cp{1}$ has four special sections, namely $\pi^{-1}(s),\pi^{-1}(i)$, $ \pi^{-1}(j)$ and $\pi^{-1}(t^{''})$. These sections, each of which is isomorphic to $\cp{1}$, are described graphically in Figure \[four-s\]. Choosing one of them, which we take to be $\pi^{-1}(s)$, as the zero section completes the proof that $\widehat{W_M}$ is a rational elliptically fibered surface with one $I_0^{*}$ fiber and three $I_2$. As discussed in [@opr], $W_M$ is its Weierstrass model. We can summarize the projections $\pi, p_1$ and $\tilde{p}_1$ in the diagram Recall from [@opr] that one can choose a smooth fiber, which we call $f_0$, in $W_M$ and construct the double cover of $W_M$ branched over $f_0$ and the four singular points in $\tilde{p}_1^{-1}(a)$. One then obtains the double cover of $W_M$, which we denote by $\bar{B}$ with the cover map |: |[B]{} W\_M. This double covering map $\bar{\kappa}$ induces an involution $\alpha_{\bar{B}}$ on $\bar{B}$, which acts non-trivially as $\tau_{\cp{1}}$ on the base $\cp{1}$. Defining sq: /\_ and noting that $\cp{1}/\tau_{\cp{1}}\cong \cp{1}$, we obtain \[d2\] which defines the fibration $\bar{\beta}: \bar{B} \to \cp{1}$. It is important to note that $\bar{B}$, the double cover of $W_M$, is not yet the surface $B$. The reason is the following.
Let us recall from [@opr] some of the properties of $\bar{B}$. First, from the diagram (\[d2\]) it is easy to see that a generic fiber of $W_M$ has two disjoint pre-images in $\bar{B}$. These form two fibers of the fibration $\bar{\beta}: \bar{B} \to \cp{1}$ and get exchanged by the involution $\alpha_{\bar B}$. Hence, only the fibers $\bar{\kappa}^{-1}(f_0)$ and $\bar{\kappa}^{-1}(\tilde{p}_1^{-1}(a))$ are stable in $\bar{B}$. We denote these fibers by $f_0$ and $f_\infty$ respectively. Of course, these are the fibers over the fixed points $0$ and $\infty$ of $\tau_{\cp{1}}$, the involution on $\cp{1}$ induced by $\alpha_{\bar{B}}$. Although $\tilde{p}_1^{-1}(a)$ in $W_M$ contains four surface singularities of type $A_1$, its pre-image $f_\infty$ in $\bar{B}$ is smooth. This is explained in [@opr] and follows from the fact that the pre-images of the four singularities of $\tilde{p}_1^{-1}(a) $ in $W_M$ are fixed under the covering involution $\alpha_{\bar B}$. Be that as it may, $\bar{B}$ does have singular fibers. Recall that $W_M$ had three additional singular fibers containing an $A_1$ surface singularity. Since these fibers are generically not $f_0$ and $\tilde{p}_1^{-1}(a)$ in $W_M$, their pre-images each consists of two fibers in $\bar{B}$. Clearly, each such fiber contains an $A_1$ surface singularity. Therefore, there are six fibers in $\bar{B}$ containing a surface singularity of type $A_1$. We conclude that the surface ${\bar B}$ is singular, whereas the rational elliptic surface $B$ is smooth. Therefore, $B$ differs from $\bar{B}$, being obtained from it by blowing up each of the six $A_1$ singularities, as we will discuss below.
What are the pre-images of the four special sections of $W_M$? First consider $\pi^{-1}(s)$. Recall that $\pi^{-1}(s)$ intersects the fiber $f_0$ in $W_M$ and contains the singular point $\pi^{-1}(p_2)$. Hence, it intersects the branch locus of $\bar{B}$ in two points and its pre-image $\bar{\kappa}^{-1}(\pi^{-1}(s))$ is the double cover of $\pi^{-1}(s)$. Since $\pi^{-1}(s)\cong \cp{1}$, it follows that $\bar{\kappa}^{-1}(\pi^{-1}(s))$ is isomorphic to $\cp{1}$ as well. Furthermore, since $\bar{\kappa}^{-1}(\pi^{-1}(s))$ intersects each fiber of $\bar{B}$ once, it is a section of the fibration $\bar{\beta}: \bar{B} \to \cp{1}$. Note that $\bar{\kappa}^{-1}(\pi^{-1}(s))$ also contains two $A_1$ surface singularities, namely the pre-image $\bar{\kappa}^{-1}(\pi^{-1}(p))$.
The pre-images under $\bar{\kappa}$ of $\pi^{-1}(i)$ and $\pi^{-1}(j)$ have similar properties. They form sections of $\bar{B}$ each containing two $A_1$ surface singularities, namely the two points $\bar{\kappa}^{-1}(\pi^{-1}(p^{'}))$ and the two points $\bar{\kappa}^{-1}(\pi^{-1}(p^{''}))$ respectively. The pre-image $\bar{\kappa}^{-1}(\pi^{-1}(t^{''}))$ in $\bar{B}$, the double cover of $\pi^{-1}(t^{''})$ branched over $\pi^{-1}(p_1)$ and $f_0\cap \pi^{-1}(t^{''})$, is isomorphic to $\cp{1}$ as well. Clearly, it is also a section of $\bar{B}$. However, unlike the other three sections $\bar{\kappa}^{-1}(\pi^{-1}(s))$, $\bar{\kappa}^{-1}(\pi^{-1}(i))$ and $\bar{\kappa}^{-1}(\pi^{-1}(j))$, $\bar{\kappa}^{-1}(\pi^{-1}(t^{''}))$ contains all six $A_1$ surface singularities. This follows from the fact that $\pi^{-1}(t^{''})$ contains $\pi^{-1}(p),\;\pi^{-1}(p^{'})$ and $\pi^{-1}(p^{''})$.
Having discussed the singular surfaces $\bar{B}$, we finally can construct their resolution, the rational elliptic surfaces $B$ that we wish to consider. If we denote by $B$ the blow-up of $\bar{B}$ at the six $A_1$ singularities and let $c : B \to \bar{B}$ be the projection map, then clearly \[blow\] which defines the fibration $\beta: B \to \cp{1}$ as $\beta=\bar{\beta}\circ c$. Combining diagrams (\[d2\]) and (\[blow\]), we see that : B W\_M is the covering of $W_M$ by $B$ where $\kappa=\bar{\kappa}\circ c$. We denote by $\alpha_B$ the induced involution on $B$ of $\alpha_{\bar B}$.
What are the fibers of $B$? As described in [@opr], each singular fiber with an $A_1$ surface singularity can be resolved into an $I_2$ fiber. Hence, $B$ has six $I_2$ fibers, namely, the pre-images of $\kappa^{-1}(\tilde{p}_1^{-1}(p_1(p)))$, $\kappa^{-1}(\tilde{p}_1^{-1}(p_1(p^{'})))$ and $\kappa^{-1}(\tilde{p}_1^{-1}(p_1(p^{''})))$. Let us denote the exceptional divisors corresponding to $\kappa^{-1}(\tilde{p}_1^{-1}(p_1(p^{})))$ by $n_1$ and $n_2$, the exceptional divisors corresponding to $\kappa^{-1}(\tilde{p}_1^{-1}(p_1(p^{'})))$ by $n_3$ and $n_4$ and the exceptional divisors corresponding to $\kappa^{-1}(\tilde{p}_1^{-1}(p_1(p^{''})))$ by $n_5$ and $n_6$. The associated proper transforms are denoted $o_1,\dots,o_6$. Hence, the six $I_2$ fibers in $B$ have the structure $n_i\cup o_i,\;i=1,\dots,6$.
Next, consider the proper transform of the four distinguished sections in $\bar{B}$. Clearly, they are still sections of $B$. Following the notation of [@opr], we denote the proper transform of $\kappa^{-1}(\pi^{-1}(s))$ by $e_9$, the proper transform of $\kappa^{-1}(\pi^{-1}(i))$ by $e_6$ and the proper transform of $\kappa^{-1}(\pi^{-1}(j))$ by $e_4$. From the distribution of the singularities at the image of these sections in $\bar{B}$, as described above, it is clear that $e_9$ intersects $n_1$ and $n_2$, $e_6$ intersects $n_3$ and $n_4$ and $e_4$ intersects $n_5$ and $n_6$. We will show in the next section that the proper transform of $\kappa^{-1}(\pi^{-1}(t^{''}))$ is $e_4{\stackrel{.}{+}}e_6$.
At this point, we want to make an important technical comment. As discussed above, $\bar{B}$ is the singular four-fold cover of $Q$. $B$, on the other hand, is a smooth surface obtained by blowing up the singularities in $\bar{B}$. The exact relationship is expressed in (\[blow\]). However, in most of the remaining discussion in this paper, one can readily distinguish whether a surface is $\bar{B}$ or its blow-up $B$ from context. With this in mind, we find that it leads to enormous simplification of our notation if we simply use the symbol $B$ to denote both $B$ and $\bar{B}$. In keeping with this notation, we will also denote both sets of mappings $\bar{\beta}, \bar{\kappa}, \alpha_{\bar B}$ and $\beta, \kappa, \alpha_B$ by $\beta, \kappa, \alpha_B$. We do this henceforth. With this simplified notation, we can identify W\_MB/\_B, as in (\[3.22\]) and (\[3.23\]). In a few places when it is helpful to specify the surface and the associated mappings exactly, we will do so.
As in [@opr], we choose $e_9$ to be the zero section of the rational elliptic surface $B$. Furthermore, as we will show in the next section, $e_4$ and $e_6$ are sections of $B$ which intersect each fiber at a point of order two. Hence (-1)\_B (e\_9)=e\_9,(-1)\_B (e\_6)=e\_6,(-1)\_B (e\_4)=e\_4. It is clear from the construction that $e_9$, $e_6$ and $e_4$ are invariant under $\alpha_B$. Hence, they fulfill condition (\[xi\]). Note, in addition, that $e_4{\stackrel{.}{+}}e_6$ is a section of $B$ which intersects each fiber at a point of order two. It follows that it too satisfies (\[xi\]). Using these sections, we can define the two involutions \[inv\] \_[B1]{}=t\_[e\_6]{}\_B,\_[B2]{}=t\_[e\_4]{}\_B. Using the fact that \_[B1]{}\_[B2]{}=t\_[e\_4 e\_6]{} and t\_[e\_4 e\_6]{}\^2=id, it was shown in [@opr] that $\tau_{B1}$ and $\tau_{B2}$ commute. It was explained in [@opr] that one can lift these commuting involutions to freely acting involutions on the Calabi-Yau threefolds $X$. Henceforth, we will assume (\[inv\]) to be the generators of the $\mathbb{Z}_2 \times \mathbb{Z}_2$ automorphism group on the rational elliptic surface $B$.
We choose the rational elliptic surface $B^{'}$ to be within the same two dimensional family. We denote the components of its $I_2$ fibers by $n_i^{'}$ and $o_i^{'},\;i=1,\dots,6$. The zero section is denoted by $e^{'}\equiv e_9^{'}$ and the other sections of order two by $e_6^{'}$, $e_4^{'}$ and $e_4^{'}{\stackrel{.}{+}}e_6^{'}$. The generators of the $\mathbb{Z}_2 \times \mathbb{Z}_2$ automorphism group on $B^{'}$ are then given by \_[B\^[’]{}1]{}=t\_[e\_6\^[’]{}]{}\_B\^[’]{},\_[B\^[’]{}2]{}=t\_[e\_4\^[’]{}]{}\_B\^[’]{}.
This concludes our review of rational elliptic surfaces. As discussed in Section \[Z\], the fiber product $X=B \times_{\cp{1}} B^{'}$ is a Calabi-Yau threefold and admits the two commuting involutions \_[X1]{}=\_[B1]{}\_\_[B\^[’]{}1]{},\_[X2]{}=\_[B2]{}\_\_[B\^[’]{}2]{}. They can be shown [@opr] to generate a freely acting automorphism group $\mathbb{Z}_2 \times \mathbb{Z}_2$ on $X$. Hence, the quotient manifold Z=X/(\_2 \_2) is smooth and, as shown in [@opr; @dopr-ii], is a Calabi-Yau threefold.
$B$ as a Double Cover of $\bar{Q}=\cp{1}\times \cp{1}$ {#double}
======================================================
In this section, we begin the process of computing the transformation laws of the elements of $H_4(X,{\mathbb Z})$ of the Calabi-Yau threefolds $X$ under the automorphism group ${\mathbb Z}_2 \times {\mathbb Z}_2$. To do this, we need to find the transformation laws of the second homology groups of the rational elliptic surfaces $B$ and $B^{'}$ under the respective ${\mathbb Z}_2 \times {\mathbb Z}_2$ automorphisms.
Let us consider a rational elliptic surface within the two dimensional sub-family described in the previous section. To find the transformation laws of $H_2(B,{\mathbb Z})$ under ${\mathbb Z}_2\times {\mathbb Z}_2$, we must first identify a set of generating elements of $H_2(B,{\mathbb Z})$. For a generic surface, $B$ is a blow-up of $\cp{2}$ at nine separated points. Hence, for a generic $B$ one can choose the nine exceptional divisors $e_1,\dots,e_9$ and $l$, the pre-image of a line in $\cp{2}$, as a set of generators for $H_2(B,{\mathbb Z})$. Note that each of these exceptional curve is isomorphic to $\cp{1}$ and, hence, irreducible. Furthermore, the intersection numbers of $e_1,\dots,e_9$ and $l$ are given by \[intersection\] e\_ie\_j =-\_[ij]{},e\_jl=0,ll=1,i,j=1,…,9. However, the rational elliptic surfaces we are going to consider are not generic. Rather, they are chosen to be in a restricted two dimensional sub-family. It can be shown that one effect of this restriction is to collide together several of the nine points in $\cp{2}$. That is, not all of the nine points are distinct. The blow-ups of these degenerate points continue to contain exceptional curves $e_i$, but their properties and linear dependence are more complicated. It turns out that for surfaces $B$ in the two dimensional sub-family, $l$ and the nine curves $e_1,\dots e_9$ with intersection numbers (\[intersection\]) still form a set of generators of $H_2(B,{\mathbb Z})$. However, these curves are no longer necessarily irreducible, that is, they may consists of several components. This will be discussed in detail in Section \[basis\]. This fact greatly complicates the process of determining their transformation laws under the ${\mathbb Z}_2\times {\mathbb Z}_2$ automorphisms. So much so, that it becomes necessary to use properties of rational elliptic surfaces not discussed in [@opr] or in the previous section.
To find the transformation laws of $e_1,\dots e_9$ under ${\mathbb Z}_2\times {\mathbb Z}_2$, it is important to notice that, in addition to the fibration $\beta: B \to \cp{1}$ discussed in [@opr] and Section \[rat\], $B$ admits a second fibration : B , where the generic fiber of $\delta$ is isomorphic to $\cp{1}$ and not to a torus. This second fibration is very helpful in determining the ${\mathbb Z}_2\times {\mathbb Z}_2$ transformation laws since, first of all, its fibers turn out all to be invariant under the involution $\alpha_B$ and, secondly, all curves $e_1,\dots e_9$ with exception of $e_7$ are contained in some of its singular fibers. This fibration is described in great detail in Section \[ruled\]. However, as a prerequisite, we need to prove the following property of $B$. Recall from Section \[rat\] that $B$ is a four-fold cover, with mapping $\pi\circ \kappa$, of $Q=\cp{1}\times \cp{1}$. In this section, we will show that, viewed differently, $B$ can actually be expressed as a double cover of another $\cp{1}\times \cp{1}$ surface, which we denote by $\bar{Q}$. This result will then allow us to construct the fibration $\delta: B \to \cp{1}$.
We begin by analyzing the covering map $\pi: W_M\to Q$ more closely. Recall from Section \[rat\] that $B$ is a double cover of $W_M$ with covering map $\kappa: B \to W_M$. By construction, the involution on $B$ which is associated with this covering is $\alpha_B$. That is, $W_M\cong B/\alpha_B$. One refers to $\alpha_B$ as the covering involution of the map $\kappa$. Next we need to analyze the covering involution on $W_M$ associated with the map $\pi: W_M \to Q$? Determining this will afford us deeper insight into the structure of $B$. To find this involution, first remember that \[pi\]
where the branch locus of $\pi$ in $Q$ is $M=s\cup i \cup j\cup t^{''}\cup r$. In addition, recall that the branch locus contains the image of the zero section, $\pi^{-1}(s)$, of $W_M$. It is clear from this and the structure of diagram (\[pi\]) that the covering involution on $W_M$ of $\pi$ must act as the identity on the base $\cp{1}$ and leave the zero section point-wise fixed. There is a unique involution on $W_M$ with this property. To see this, consider the involution $(-1)_B$ on $B$ defined in Section \[rat\]. Clearly, this induces a unique involution $(-1)_{W_M}$ on $W_M$ under the covering map $\kappa: B \to W_M$. It is straightforward to see that $(-1)_{W_M}$ has the required properties and, hence, is the covering involution for $\pi: W_M \to Q$. That is QW\_M/(-1)\_[W\_M]{}. Combining this result with the discussion in Section \[rat\], we see that $B$ is a four-fold cover of $Q$ given by \[l\] Note that we have first modded out by $\alpha_B$ and then by $(-1)_{W_M}$. This is shown graphically as the left hand routing in Figure \[tricom\].
Expressing $B$ as the four-fold covering of $Q$ in this manner leads to the following important insight. Namely, note that it should not matter whether we first mod out by $\alpha_B$ and then by $(-1)_{W_M}$, the involution induced on $W_M$ from $(-1)_B$, as in (\[l\]), or first mod out by $(-1)_B$ and then by the induced involution of $\alpha_B$ on $B/(-1)_B$. Either approach represents $B$ as the same four-fold covering of $Q$. Let us specify this second decomposition of the covering more concretely. First, we denote B/(-1)\_B|[Q]{} and the covering map associated with involution $(-1)_B$ by : B |[Q]{}. Now denote the induced involution of $\alpha_B$ on $\bar{Q}$ by $\alpha_{\bar{Q}}$. Modding out $\bar{Q}$ by $\alpha_{\bar{Q}}$ must produce a surface isomorphic to $Q$. That is, |[Q]{}/\_[|[Q]{}]{}Q. We write the associate covering map as : |[Q]{}Q. Combining these results, we see that the four-fold covering of $Q$ by $B$ can be expressed in an alternative manner to (\[l\]) given by \[m\] Here, we have first modded out by $(-1)_B$ and then by $\alpha_{\bar{Q}}$, which is induced from $\alpha_B$ on $B$. This is shown graphically as the middle routing in Figure \[tricom\]. Expressions (\[l\]) and (\[m\]) are not the only ways of writing $B$ as the four-fold cover of $Q$. A third way is as follows. First mod out by the composite involution $(-1)_B\circ \alpha_B$ on $B$ to produce the surface B/((-1)\_B\_B)W\_[M\^[’]{}]{} and the covering map \^[’]{}:B W\_[M\^[’]{}]{}. Now denote the induced involution of $(-1)_B$ on $W_{M^{'}}$ by $(-1)_{W_{M^{'}}}$. Modding out $W_{M^{'}}$ by $(-1)_{W_{M^{'}}}$ must produce a surface isomorphic to $Q$. That is, W\_[M\^[’]{}]{}/(-1)\_[W\_[M\^[’]{}]{}]{}Q. We write the associated covering map as \^[’]{}:W\_[M\^[’]{}]{} Q. Combining these results, we see that \[r\] where we have first modded out by $(-1)_B\circ \alpha_B$ and then by $(-1)_{W_{M^{'}}}$, which is induced from $(-1)_B$ on $B$. This is shown graphically as the right hand routing in Figure \[tricom\].
[$$\xymatrix@C=-0.6in@R=1.5in{
&B \ar[dr]^-{\kappa^{'}} \ar[dl]_-{\kappa} \ar[d]_-{\psi} & \\
B/\alpha_B\cong W_M \ar[dr]_-{\pi} & B/(-1)_B\cong\bar{Q} \ar[d]_-{\Delta} & W_{M^{'}}\cong B/( (-1)_B\circ\alpha_B) \ar[dl]^-{\pi^{'}}\\
& Q \cong W_M/(-1)_{W_M}\cong \bar{Q}/\alpha_{\bar{Q}}\cong W_{M^{'}}/(-1)_{W_{M^{'}}}&
}$$ ]{}
What are the branch loci of $\kappa,\; \pi$ in (\[l\]), $\psi,\;\op{\Delta}$ in (\[m\]) and $\kappa^{'},\; \pi^{'}$ in (\[r\])? The branch loci of $\kappa$ and $\pi$ were discussed in the previous section and are pictured in the left hand routing of Figure \[tri-fg\]. Let us now discuss the branch loci for $\kappa^{'}$ and $\pi^{'}$, beginning with $\kappa^{'}$. The covering involution for $\kappa^{'}$ is $(-1)_B \circ \alpha_B$. Since $\alpha_B$ acts as the identity map on $f_0$ and as $(-1)_B$ on $f_\infty$, the composition $(-1)_B\circ \alpha_B$ acts as $(-1)_B$ on $f_0$ and as the identity map on $f_\infty$. Hence, the branch locus for $\kappa^{'}$ in $W_M^{'}$ consists of the image of the four fixed points under $(-1)_B$ on $f_0$ and the fiber $f_\infty$. Hence, $W_M^{'}$ has similar properties to $W_M$, one fiber $f_0$ which contains four $A_1$ surface singularities and which can be resolved into an $I_0^{*}$ fiber, and three singular fibers which can be resolved into $I_2$ fibers. The branch locus in $Q$ of the double cover $\pi^{'}$ now clearly consists of the three $(0,1)$ rulings $s,i$ and $j$, the $(1,1)$ curve $t^{''}$ and the image $\pi^{'}(f_0)$ of the fiber $f_0$ in $W_M^{'}$, which is a $(1,0)$ ruling that we denote by $r^{'}$. That explains the right hand routing of Figure \[tri-fg\]. Finally, we consider the branch loci of $\psi $ and $\Delta$. Let us start with $\Delta$. To proceed, we need to invoke a mathematical theorem which states that the branch locus of $\Delta$ in ${Q}$ consists of the union of the branch loci of $\pi$ and $\pi^{'}$ in $Q$ minus their intersection. Hence, the branch locus of $\Delta$ in ${Q}$ consists of the two $(1,0)$ rulings $r$ and $r^{'}$. Now consider the covering map $\psi$. To find its branch locus, we first must elucidate the structure of surface $\bar{Q}$ more explicitly. To do this, recall that $Q=\cp{1}\times\cp{1}$ has two natural projections $p_i:Q\to \cp{1}_i$ for $i=1,2$ to the $i$th $\cp{1}$ component. Choose any point $x \in \cp{1}_2$ and consider the fiber $\Delta^{-1}(p_2^{-1}(x))$ in $\bar{Q}$. It is the double cover the fiber $p_2^{-1}(x)$, which is isomorphic to $\cp{1}$, branched over the two points where $p^{-1}_2(x)$ intersects the rulings $r$ and $r^{'}$. Hence $\Delta^{-1}(p_2^{-1}(x))$ is isomorphic to $\cp{1}$. It follows that $\bar{Q}$ is isomorphic to $\cp{1}\times \cp{1}$. That is |[Q]{}=. We denote the two natural projections of $\bar{Q}$ by $\bar{p}_i:\bar{Q}\to \cp{1}_i$ for $i=1,2$. Of course, $\bar{Q}$ is smooth since the branch locus of $\bar{Q}$ in $Q$ is smooth.
[$$\xymatrix@C=2.5in@R=1.5in{ & B
\ar[ld]|-{\boxed{\epsfig{file=wm.ps,height=.75in}}}
\ar[d]|-{\boxed{\epsfig{file=Q.eps,height=.75in}}}
\ar[rd]|-{\boxed{\epsfig{file=wmprime.ps,height=.75in}}} &\\
W_{M} \ar[rd]|-{\boxed{\epsfig{file=psi.eps,height=.75in}}} &
{\bar{Q}} \ar[d]|-{\boxed{\epsfig{file=sq.ps,height=.75in}}} &
W_{M'} \ar[ld]|-{\boxed{\epsfig{file=psiprime.eps,height=.75in}}}\\
& Q &
}$$ ]{}
With this information, we can now determine the branch locus of the double cover $\psi$. It is simply the pre-image under $\Delta$ of the curves $s,i,j$ and $t^{''}$. Since each of the rulings $s,i$ and $j$ pulls back to a unique fiber of $\bar{p}_2:\bar{Q}\to \cp{1}$, they are $(0,1)$ rulings in $\bar{Q}$. The pre-image of $t^{''}$ is isomorphic to $\cp{1}$ as well, since $t{''}$ intersects the branch locus for $\Delta$ in two points, namely $t^{''}\cap r$ and $t^{''}\cap r^{'}$. However, $t^{''}$ intersects the generic fiber of $p_2 : Q\to \cp{1}$ at a point not contained in the branch locus of $Q$. Therefore, its pre-image $\op{\Delta}^{-1}(t^{''})$ intersects the fiber $\bar{p}_2^{-1}(x)$ in two points. It follows that $\Delta^{-1}(t^{''})$ is a bidegree $(2,1)$ curve in $\bar{Q}$. The branch loci for $\Delta$ and $\psi$ are described in the middle routing of Figure \[tri-fg\]. This finishes our explanation of the branch loci of $B$ as a degree four cover of $Q$. We have shown that there are three different ways of describing it, a fact that will prove to be important later when, for a chosen set of generating elements of $H_2(B,{\mathbb Z})$, we try to compute their ${\mathbb Z}_2 \times {\mathbb Z}_2$ transformation laws.
We will finish this section with the proof that the sections $e_4$ and $e_6$ of $\beta : B \to \cp{1}$ intersect each fiber $f$ in a point of order two. This fact turned out to be crucial in [@opr] to construct commuting involutions on $B$. First, it follows from the commutativity of Figure \[tricom\] that \^[-1]{}\^[-1]{}=\^[-1]{}\^[-1]{}. But, we have shown in Section \[rat\] that \[sec\] \^[-1]{}\^[-1]{}(s),\^[-1]{}\^[-1]{}(i),\^[-1]{}\^[-1]{}(j),\^[-1]{}\^[-1]{}(t\^[”]{}) correspond to sections of $\beta : B \to \cp{1}$. Hence, \^[-1]{}\^[-1]{}(s),\^[-1]{}\^[-1]{}(i),\^[-1]{}\^[-1]{}(j),\^[-1]{}\^[-1]{}(t\^[”]{}) correspond to sections of $\beta : B \to \cp{1}$. Secondly, recall from above that $\Delta^{-1}(s),\;\Delta^{-1}(i)$, $\Delta^{-1}(j)$ and $\Delta^{-1}(t^{''})$ are in the branch locus of : B |[Q]{}B/(-1)\_B. Hence, the sections (\[sec\]) are point-wise fixed under $(-1)_B$. Of course, this has to hold for the section $e_9$ corresponding to $\kappa^{-1}\circ \pi^{-1}(s)$, since we had chosen $e_9$ to be the zero section. In addition, this proves the invariance under $(-1)_B$ of the sections $e_6$ and $e_4$ corresponding to $\kappa^{-1}\circ \pi^{-1}(i) $ and $\kappa^{-1}\circ \pi^{-1}(j)$ respectively. Hence, these sections intersect each fiber $f$ at a point of order two. Finally, since $e_4{\stackrel{.}{+}}e_6$ is the last possible section invariant under $(-1)_B$, this proves that $e_4{\stackrel{.}{+}}e_6$ corresponds to $\kappa^{-1}\circ \pi^{-1}(t^{''})$.
Studying the branch locus of $\psi : B \to \bar{Q}$ in $\bar{Q}$, we see that the branch curve has six singularities. Hence, without resolving, $B$ will have six $A_1$ type singularities at the pre-image of these points. Thus, each section $e_9, e_4$ and $e_6$ contains two of these singularities. This is consistent with the explanation in the previous section. Furthermore, the section $e_4{\stackrel{.}{+}}e_6$ contains all six singularities.
$B$ as a Ruled Surface {#ruled}
======================
In this section, we will show that $B$ is a ruled surface. A ruled surface is a two dimensional fibration whose generic fiber is not a torus $T^2$ but, rather, $\cp{1}$. Expressing $B$ as a ruled surface will be very helpful in determining the transformation laws of the elements of $H_2(B,{\mathbb Z})$ under the ${\mathbb Z}_2\times {\mathbb Z}_2 $ automorphism group. Recall that B is elliptically fibered with respect to the mapping $\beta: B \to \cp{1}$. This mapping is defined by $sq\circ\beta= p_1\circ\pi\circ\kappa$ where where $p_1: Q \to \cp{1}$ is the natural projection of $Q=\cp{1}\times \cp{1}$ onto its first $\cp{1}$ factor.
Now recall that there is a second natural projection, namely, $p_2: Q \to \cp{1}$ where $\cp{1}$ is the second $\cp{1}$ factor of $Q$. Clearly, then, we can construct a second fibration of $B$, \[delta1\] : B , where =p\_2and What is the generic fiber of the mapping $\delta$? To answer this, it is very helpful to use an alternative way that $B$ covers $Q$, namely, the covering through the surface $\bar{Q}=\cp{1}\times \cp{1}$ given in (\[m\]). This is expressed as the middle routing in Figure \[tri-fg\]. Using the commutativity of Figure \[tricom\], we can write =p\_2, where now We showed in the previous section that for a generic $x\in \cp{1}_2 \subset Q$, $\Delta^{-1}(p_2^{-1}(x))\cong \cp{1}$. Furthermore, note that if we identify $\cp{1}_2 \subset Q $ with $\cp{1}_2 \subset \bar{Q} $ then $\bar{p}_2^{-1}(x)=\Delta^{-1}(p_2^{-1}(x))$. For a generic point $x\in \cp{1}_2 \subset \bar{Q}$, the fiber $\bar{p}_2^{-1}(x)$ intersects the branch locus of $\psi$ in $\bar{Q}$ only at two points, each contained in the bidegree $(2,1)$ curve $\Delta^{-1}(t^{''})$. This is indicated by the dashed line in Figure \[psibranch\]. Therefore, the double cover $\psi^{-1}(\bar{p}_2^{-1}(x))$ of $\Delta^{-1}(t^{''})$ is two $\cp{1}$ lines identified at two points and, hence, is isomorphic to $\cp{1}$. But \^[-1]{}(|[p]{}\_2\^[-1]{}(x))=(p\_2)\^[-1]{}(x)=\^[-1]{}(x) is the generic fiber of (\[delta1\]). We conclude that $B$ can be written as the fibration $\delta: B \to \cp{1}$ whose generic fiber is $\cp{1}$ and, hence, $B$ is a ruled surface. Note that the fiber of $\delta$ is a degree four cover of the fiber of $p_2$ or, equivalently, a degree two cover of the fiber of $\bar{p}_2$. Adopting the second point of view will simplify many calculations.
What are the singular fibers of $\delta: B \to \cp{1}$? First consider the three bidegree $(0,1)$ curves $\Delta^{-1}(s)$, $\Delta^{-1}(i)$ and $\Delta^{-1}(j)$ in the branch locus of $\psi$ in $\bar{Q}$. These are shown in Figure \[psibranch\], where we have defined $\bar{p}_2(\Delta^{-1}(s))=a$, $\bar{p}_2(\Delta^{-1}(i))=b$ and $\bar{p}_2(\Delta^{-1}(j))=c$. Note that each of these curves is a $\cp{1}$ that intersects $\Delta^{-1}(t^{''})$ at two points. Hence, the pre-image under $\psi$ in $B$ of each of these points must be blown-up to obtain a smooth surface. It follows that the pre-image under $\psi$ of each $(0,1)$ curve is a double $\cp{1}$ line intersecting two exceptional divisors in two distinct points. As discussed previously, we can identify these fibers as e(n\_1 n\_2)=\^[-1]{}(a),e\_6(n\_3 n\_4)=\^[-1]{}(b),e(n\_5 n\_6)=\^[-1]{}(c) respectively. They are shown explicitly in Figure \[delta\].
Are there other singular fibers of $\delta: B \to \cp{1}$? Consider the $(2,1)$ curve $\Delta^{-1}(t^{''})$ in $\bar{Q}$. Clearly, under the projection $\bar{p}_1: \bar{Q}\to \cp{1}_1$ this curve is a degree one cover of $\cp{1}$ and, hence, \[4.9\] (\^[-1]{}(t\^[”]{}))=()=2. However, under the projection $\bar{p}_2: \bar{Q}\to \cp{1}$, $\Delta^{-1}(t^{''})$ is a degree two cover of $\cp{1}$. Then the Riemann-Hurwitz formula states that \[4.10\] (\^[-1]{}(t\^[”]{}))=2({b\_i})+A, where $\{b_i\}$ for $i=1,\dots,A$ are the branch points of the simple ramification points of $\Delta^{-1}(t^{''})$ over $\cp{1}_2$. By definition, above each $b_i \in \cp{1}_2$, $\Delta^{-1}(t^{''})$ intersects $\bar{p}_2^{-1}(b_i)$ in only one point, not in two. At this point, the ramification point $R_i$, the fiber $\bar{p}_2^{-1}(b_i)$ intersects $\Delta^{-1}(t^{''})$ transversally. This is shown in Figure \[psibranch\].
Therefore, the pre-image under $\psi$ of each of $\bar{p}_2^{-1}(b_i)$ consists of two distinct $\cp{1}$ lines in $B$ intersecting in a point. Note, however, that this point is smooth in $B$. How many such fibers are there? Combining equations (\[4.9\]) and (\[4.10\]) we find A=2. We conclude that there are two such singular fibers, $\delta^{-1}(b_i)$ for $i=1,2$. Although not immediately obvious, we will show in the next section that one $\cp{1}$ component of $\delta^{-1}(b_1)$ can be identified with $e_1$, whereas one $\cp{1}$ component of $\delta^{-1}(b_2)$ is $e_2$. These two singular fibers are described in Figure \[delta\].
In summary, we have shown that, in addition to the fibration $\beta: B\to \cp{1}$ discussed in [@opr] and Section \[rat\], there is a second way to fiber $B$, namely, $\delta: B\to \cp{1}$. The $\delta$ fibers are generically $\cp{1}$ and, hence, $B$ is a ruled surface. This second fibration has five singular fibers, all shown explicitly in Figure \[delta\]. This fibration will be very helpful in identifying the set of generators of $H_2(B,{\mathbb Z})$, which will be done in the next section.
Generators of $H_2(B,{\mathbb Z})$ {#basis}
==================================
Recall that a generic rational elliptic surfaces $B$ is a $\cp{2}$ blown-up by a $\cp{1}$ at each of nine separate points. Furthermore, each of these $\cp{1}$ blown-up curves is an irreducible, exceptional divisor of $B$ with self-intersection $-1$. We denote these nine exceptional divisors by $e_1,\dots,e_9$, and have shown that $e_1,\dots,e_9$ and $l$, where $l$ is the pre-image of a line in $\cp{2}$, form a set of generators of $H_2(B,{\mathbb Z})$. However, the rational elliptic surfaces $B$ that we wish to consider are not generic. Rather, they form a restricted two-dimensional sub-family of such surfaces. One can show that, in this case, some of the nine points in $\cp{2}$ coalesce. This greatly obscures the construction of a surface $B$ in this two dimensional sub-family as a blow-up of a $\cp{2}$. In fact, as argued in [@opr] and Section \[rat\], it is more convenient to obtain these surfaces by different means, namely, as the four-fold cover of $Q$ with a specific branch locus. Be that as it may, for surfaces in this two dimensional sub-family it is now essential that we exactly express $B$ as a blow-up of $\cp{2}$. In doing this, we will explicitly identify nine divisors with self-intersection $-1$, which we again denote as $e_1,\dots,e_9$, and give their structure in $B$. These nine divisors, along with the pre-image of a line $l$ in $\cp{2}$, will again form the set of generators of $H_2(B,{\mathbb Z})$. To proceed, we find it essential to use the $\cp{1}$ fibration $\delta: B \to \cp{1}$ presented in the previous section.
To identify the rational elliptic surfaces $B$ as a blow-up of $\cp{2}$, we need to invoke the so-called Castelnuovo-Enriques criterion. This involves a process called blow-down, the inverse of blowing-up. The Castelnuovo-Enriques criterion states that for any surface containing a $-1$ curve, that is, a curve isomorphic to $\cp{1}$ of self-intersection $-1$, one can blow down this curve and obtain a smooth surface. For more details on the Castelnuovo-Enriques criterion see [@gh]. To apply this process to the $d{\mathbb P}_9$ surfaces of our two dimensional sub-family, we must identify nine $-1$ curves and successively blow them down. To do this, recall that the surface $B$ can be considered as a ruled surface, that is : B with the generic fiber of $\delta$ being isomorphic to $\cp{1}$. We will denote the class of such a fiber by $F$. As discussed in Section \[ruled\], this fibration has five singular fibers, each graphically described in Figure \[delta\]. First, we consider the two singular fibers each of which consists of two $\cp{1}$ components which intersect in a single point. Choose one of these two fibers and denote its components by $c_1$ and $c_2$. Then, as homology classes F=c\_1 + c\_2. Note that here and henceforth, $+$ and $-$ denote the group operations in the homology group $H_2(B,{\mathbb Z})$. These must not be confused with the addition and inverse of elements in $\Gamma(B)$, defined in Section \[rat\], which, to distinguish them, were denoted using ${\stackrel{.}{+}}$ and ${\stackrel{.}{-}}$. Recalling that for any fiber class $F^2=0$, and using $c_1\cdot c_2=1$, we find 0=F\^2=(c\_1+ c\_2)\^2=c\_1\^2+2+c\_2\^2. Since the components $c_1$ and $c_2$ are completely symmetric in $B$, it follows that c\_1\^2=c\_2\^2=-1. Therefore, this singular fiber contains two $-1$ curves. By the Castelnuovo-Enriques criterion explained above, we can blow down either of them. Choose one component, for example $c_1$, and shrink it to a point. What is the self-intersection of the remaining component? Since this component now forms a complete fiber of the fibration induced by $\delta$ on the blown down surface, its self-intersection is zero. Hence, there is no longer any $-1$ curve in this fiber. Of course, the second fiber of $\delta : B \to \cp{1}$ consisting of two $\cp{1}$ components intersecting at a point did not get changed in the process of blowing down. Therefore, it still admits two $-1$ curves, one of which can now be blown down. Hence, we have identified two $-1$ curves in $B$ which can be blown down simultaneously.
We now show that there are additional $-1$ curves in the remaining three singular fibers of $\delta : B \to \cp{1}$. Start with the fiber containing the double component $e_9$ and the two exceptional divisors $n_1$ and $n_2$. For a pictorial representation, we refer the reader to Figure \[delta\]. For this fiber we can write F=2e\_9+n\_1+n\_2. Recall that we obtained $n_1$ and $n_2$ by blowing up two $A_1$ surface singularities. Hence, blowing them down would not lead to a smooth surface. Therefore, by the Castelnuovo-Enriques criterion, neither exceptional divisor can be a $-1$ curve. It can be shown, however, that, since they arise by blowing up $A_1$ surface singularities, their self-intersection number is $-2$. Using this, and the fact that $e_9$ intersects each of $n_1$ and $n_2$ in one point, we find that 0=(2e\_9+n\_1+n\_2)\^2=4e\_9\^2-2-2+2(2+2) and, hence e\_9\^2=-1. Therefore, we can blow down $e_9$ and obtain a smooth surface. What are the self-intersection numbers of the images of $n_1$ and $n_2$ in the blown down surface? Since this induced fiber in the blown down surface consists of two $\cp{1}$ components which intersect at one point, we are in the same situation as described above and can conclude that the image of each of $n_1$ and $n_2$ in the blown down surface self-intersects as $-1$. Hence, we can pick one of them and blow it down.
Of course, the reasoning applied to the fiber $2e_9+n_1+n_2$ can be applied to the fibers $2e_6+n_3+n_4$ and $2e_4+n_5+n_6$. Therefore, the three singular fibers each containing a double component and two exceptional divisors allow us to blow down six $-1$ curves. Taking the previous two singular fiber of $\delta : B \to \cp{1}$ into account as well, we have found eight $-1$ curves which can be simultaneously blown down. But since $B\cong d\mathbb{P}_9$, we need to find one additional $-1$ curve to obtain a $\cp{2}$. Here, we will simply state a result proven in [@dopr-i]. Recall the map : B |[Q]{}, which is a double cover branched over the three $(0,1) $ rulings $\Delta^{-1}(s)$, $\Delta^{-1}(i)$, $\Delta^{-1}(j)$ and the $(2,1)$ curve $\Delta^{-1}(t^{''})$. Consider a $(1,1)$ curve in $\bar{Q}$ passing through one of the two intersection points of each of $\Delta^{-1}(s)\cap \Delta^{-1}(t^{''}) $, $\Delta^{-1}(i)\cap \Delta^{-1}(t^{''}) $ and $\Delta^{-1}(j)\cap \Delta^{-1}(t^{''}) $. Specifically, we choose this curve to pass through the points whose pre-image under $\psi$ blows-up to $n_2,\;n_4$ and $n_6$ respectively. It can be shown that such a curve in $\bar{Q} $ exists and that its proper transform in $B$ is a $-1$ curve which is a section of the fibration $\delta : B \to \cp{1}$. We denote this $-1$ curve by $e_7$.
We are finally in a position to construct our restricted surface $B$ as a blow-up of $\cp{2}$. In the process, we will specifically identify, and give the properties of, nine $-1$ curves in $B$. To do this, begin with one such surface $B$. First, consider the two singular fibers of $\delta : B \to \cp{1}$ each consisting of two components. Denote by $e_1$ the component of one of these fibers which does not intersect $e_7$ and by $e_2 $ the component of the other fiber which does not intersect $e_7$. Blow them down. Next, blow the $-1$ curves $e_9,\;e_6$ and $e_4$ down. Continuing, blow the images of the curves $n_1,n_3$ and $n_5$ down. Note that none of the curves $e_9, e_6, e_4, n_1, n_3$ or $n_5$ intersects $e_7$. Finally, upon blowing down $e_7$ we obtain $\cp{2}$. Of the curves $e_1, e_2, e_4, e_6, e_7, e_9, n_1, n_3$ and $ n_5$, all are $-1$ curves with the exception of $n_1, n_3$ and $ n_5$. In addition, some of the $e$-type and $n$-type curves intersect. This motivates us to define \[def1\] e\_3=e\_4+n\_5,e\_5=e\_6+n\_3,e\_8=e\_9+n\_1. It is straightforward to verify that each of $e_3, e_5$ and $e_8$ are $-1$ curves and that $e_1,\dots,e_9$ are all non-intersecting. To reiterate, if we start with a restricted surface $B$ and blow down the curves $e_1, e_2, e_4, e_6, e_9, n_1, n_3, n_5$ and $e_7$ we obtain a $\cp{2}$. Reversing the process, we have constructed $B$ as the blow-up of a $\cp{2}$ surface, as desired.
If we denote by $l$ the pull back of a curve on $\cp{2}$, then $e_1,\dots,e_9$ and $l$ form a set of generators for $H_2(B,{\mathbb Z})$ with the same intersection numbers as given in (\[intersection\]). That is, H\_2(B,[Z]{})=[Z]{}l(\_[i=1]{}\^9 [Z]{}e\_i). Therefore, we have successfully described a set of generators of the curve homology of $B$ in terms of the nine $-1$ curves and the pre-image of a $\cp{1}$ line in $\cp{2}$. Note that this motived our original choice in [@opr] and Section \[rat\] to name the two non-zero sections of order two as $e_4$ and $e_6$.
In this basis, the components of the $I_2$ fibers can be written as \[def\]
n\_[1]{} & = e\_[8]{} - e\_[9]{}\
o\_[1]{} & = f - e\_[8]{} + e\_[9]{}\
n\_[2]{} & = l - e\_[7]{} - e\_[8]{} - e\_[9]{}\
o\_[2]{} & = 2l - e\_[1]{} - e\_[2]{} - e\_[3]{} - e\_[4]{} - e\_[5]{} - e\_[6]{}\
n\_[3]{} & = e\_[5]{} - e\_[6]{}\
o\_[3]{} & = f - e\_[5]{} + e\_[6]{}\
n\_[4]{} & = l - e\_[7]{} - e\_[6]{} - e\_[5]{}\
o\_[4]{} & = 2l - e\_[1]{} - e\_[2]{} - e\_[3]{} - e\_[4]{} - e\_[8]{} - e\_[9]{}\
n\_[5]{} & = e\_[3]{} - e\_[4]{}\
o\_[5]{} & = f - e\_[3]{} + e\_[4]{}\
n\_[6]{} & = l - e\_[7]{} - e\_[4]{} - e\_[3]{}\
o\_[6]{} & = 2l - e\_[1]{} - e\_[2]{} - e\_[5]{} - e\_[6]{} - e\_[8]{} - e\_[9]{}.\
where $f$ is the fiber class of the elliptic fibration $\beta: B \to \cp{1}$. Let us make a remark on these equations. The expressions for $n_1, n_3$ and $n_5$ follow easily from (\[def1\]). To understand the expressions for $n_2, n_4$ and $n_6$, observe the following. Take, for example, $n_2$. It is the component of the singular fiber of $\delta: B \to \cp{1}$ intersecting $e_7$ which is left after blowing down $e_9,\; e_7$ and $n_1$. Hence, the image of $n_2$ in $\cp{2}$ must be $l$. It follows that $n_2$ in $B$ is given by $l$ reduced by the exceptional divisors $2e_9,\;e_7,$ and $n_1$. That is n\_2=l-2e\_9-e\_7-n\_1, which is the expression given in (\[def\]). The equations for $n_4$ and $n_6$ are derived in a similar way. Once we have defined $n_1,\dots,n_6$, the proper transforms $o_1,\dots,o_6$ are given by $o_i=f-n_i,\;i=1,\dots,6$. Recall from Section \[rat\] that the fiber class $f$ is $f=3l-\sum_{i=1}^9 e_i$. Note that the section $e_7$ intersects $n_2, n_4, n_6$ and $o_1, o_3, o_5$. Finally, even through $e_1,\dots,e_9$ and $l$ form a canonical set of generators, the set $e_1, e_2, e_4, e_6$, $ e_7, e_9, n_1, n_3, n_5$ and $l$ also generates $H_2(B,{\mathbb Z})$. We will use these two sets of generators interchangeably depending on the context.
Transformations under $(-1)_B$ {#-1}
==============================
In the following sections, we will determine the transformation laws of the elements of $H_2(B,\mathbb{Z})$ under the involutions $\tau_{B1}$ and $\tau_{B2}$ defined in Section \[rat\]. To determine these transformations, we will use both the elliptic fibration $\beta: B \to \cp{1}$, whose fiber we continue to call $f$, and the ruling $\delta : B \to \cp{1}$, whose fiber we continue to call $F$. We will also need all three descriptions of $B$ shown in Figure \[tri-fg\]. In this section, for completeness and to establish methodology, we will determine the induced action of $(-1)_B$ on $H_2(B,\mathbb{Z})$ .
To begin, observe that \[-f\] (-1)\_B (f)=f. This is clear since $(-1)_B$ acts fiber-wise on the elliptic fibration. In addition, note that \[-F\] (-1)\_B (F)=F. This follows from the fact that $(-1)_B$ is the covering involution for $ \psi : B \to \bar{Q}$ and $F=\delta^{-1}(x)$ is the double cover of the $\cp{1}$ line $\bar{p}_2^{-1}(x)$, where $\bar{p}_2$ is the second natural projection of $\bar{Q}$, namely, $\bar{p}_2: \bar{Q}\to \cp{1}_2$.
Now consider the two singular fibers of $\delta :B \to \cp{1}$ containing the sections $e_1$ and $e_2$ respectively. Choose one of them, say the fiber containing $e_1$. As discussed above, this fiber can be written as F=e\_1+c\_2, where $c_2$ intersects $e_7$. Since this fiber arises as the pre-image under $\psi$ of a single $\cp{1}$ line which intersects the branch locus of $\psi$ in $\bar{Q}$ at a single point, it follows that $e_1$ and $c_2$ must be exchanged under $(-1)_B$. That is, c\_2=(-1)\_B(e\_1). Exactly the same statement is true for the singular fiber containing $e_2$. Therefore, the homology class of the fiber $F$ can be written as \[1\] F=e\_i +(-1)\_B (e\_i),i=1,2. Note that this is consistent with the expression (\[-F\]). Now consider a generic fiber of $\delta: B \to \cp{1}$. On might imagine that $l$ would be the class of this fiber. However, it is clear from the previous discussion that the generic fiber must intersect $e_7$ at a single point. Using the intersections $l\cdot e_7=0$ and $e_7^2=-1$, we see that the fiber class can be written as \[2\] F=l-e\_7. Note that it follows from (\[-F\]) that \[l-e\] (-1)\_B(l-e\_7)=l-e\_7. From (\[1\]) and (\[2\]), we conclude that (-1)\_B (e\_i)=l-e\_7-e\_i,i=1,2.
Next consider the six exceptional divisors $n_i,\; i=1,\dots,6$. Since the image under $\psi$ of the six surface singularities in $\bar{B}$ are contained in the branch locus of $\psi$ in $\bar{Q}$, it follows that the six exceptional divisors $n_i,\; i=1,\dots,6$ are fixed under $(-1)_B$. That is \[-n\] (-1)\_B(n\_i)=n\_i,i=1,…,6. Recall that the proper transforms are defined to be $o_i=f-n_i$ for $ i=1,\dots,6$. Then, it follows from (\[-f\]) and (\[-n\]) that (-1)\_B(o\_i)=o\_i,i=1,…,6. The sections $e_4, e_6$ and $e_9$ are invariant under $(-1)_B$ by definition. Applying these results to the sections $e_3, e_5$ and $e_8$ defined in (\[def1\]) using the linearity of $(-1)_B$, the invariance of $e_4, e_6$ and $e_9$ and (\[-n\]), we find, for example, that (-1)\_B(e\_3)=(-1)\_B(e\_4 +n\_5)=e\_4 +n\_5=e\_3. Similarly, it follows that (-1)\_B(e\_5)=e\_5,(-1)\_B(e\_8)=e\_8. Finally, using $o_2$ given in (\[def\]), the fact that $(-1)_B(o_2)=o_2$ and the invariance of $e_3, e_4, e_5$ and $e_6$ under $(-1)_B$, it follows that \[-l\] (-1)\_B(l)=2l-(e\_1+e\_2+e\_7). Furthermore, from (\[l-e\]) and (\[-l\]) we find (-1)\_B(e\_7)=l-(e\_1+e\_2). This completes the analysis for the action of $(-1)_B$ on the set of generators $e_1,\dots,e_9$ and $l$ of $H_2(B,{\mathbb Z})$. These results are summarized in Table \[tab(-1)\]. The transformation of an arbitrary class in $H_2(B,{\mathbb Z})$ follows immediately from these results.
$e_1$ $e_2$ $e_3$ $e_4$ $e_5$ $e_6$ $e_7$ $e_8$ $e_8$ $l$
------------ ------------- ------------- ------- ------- ------- ------- ------------- ------- ------- --------------------
$(-1)_{B}$ $l-e_7-e_1$ $l-e_7-e_2$ $e_3$ $e_4$ $e_5$ $e_6$ $l-e_1-e_2$ $e_8$ $e_9$ $2l-(e_1+e_2+e_7)$
: The action of $(-1)_{B}$ on the canonical set of generators of $H_{2}(B,{\mathbb Z})$.[]{data-label="tab(-1)"}
The Action of $\alpha_B$ {#alpha}
========================
In this section, we begin our analysis of the transformations of the elements of $H_2(B,{\mathbb Z})$ under $\tau_{B1}$ and $\tau_{B2}$. Recalling from (\[inv\]) that $\tau_{B1}=t_{e_6}\circ \alpha_B$ and $\tau_{B2}=t_{e_4}\circ \alpha_B$, we start by finding the action of $\alpha_B$ on $H_2(B,{\mathbb Z})$. First, observe that \[af\] \_B(f)=f. This follows from the fact that $\alpha_B$ preserves the fiber of the elliptic fibration $\beta: B \to \cp{1}$. The commutativity of Figure \[tri-fg\] implies that $\delta^{-1}(x)=\kappa^{-1}(\pi^{-1}(p_2^{-1}(x)))$. Since $\alpha_B$ is the covering involution for map $\kappa$, this shows that the fiber class of $\delta : B \to \cp{1}$ is also invariant under $\alpha_B$. That is, \[aF\] \_B(F)=F.
Let us first discuss the generic representation of $F$ given in (\[2\]). Then this and (\[aF\]) imply that under $\alpha_B$ \_B(l-e\_7)=l-e\_7. Now, consider the singular fiber $2e_9+n_1+n_2$ of $\delta: B \to \cp{1}$. Note that $\alpha_{\bar B}$ exchanges the two surface singularities in $\bar{B}$ which blow up into $n_1$ and $n_2$. Hence, $\alpha_B$ exchanges $n_1$ and $n_2$. That is, \[n1\] \_B(n\_1)=n\_2,\_B(n\_2)=n\_1. Since, by (\[aF\]), $2e_9+n_1+n_2$ must be invariant under $\alpha_B$, it follows that \_B(e\_9)=e\_9. Similar remarks hold for the singular fibers $2e_6+n_3+n_4$ and $2e_4+n_5+n_6$. We see, therefore, that \[n3\] \_B(n\_3)=n\_4, \_B(n\_4)=n\_3 and \[n4\] \_B(n\_5)=n\_6, \_B(n\_6)=n\_5. Combining this with the invariance of both singular fibers under $\alpha_B$, we conclude that \[a6\] \_B(e\_6)=e\_6,\_B(e\_4)=e\_4. Next consider the proper transform $o_i=f-n_i$ for $ i=1,\dots,6$. Then it follows from (\[af\]), (\[n1\]), (\[n3\]) and (\[n4\]) that \_B(o\_[2j-1]{})=o\_[2j]{},\_B(o\_[2j]{})=o\_[2j-1]{} for $j=1,2,3$. It is easy to read off the transformation laws for $e_3, e_5$ and $e_8$ defined in (\[def1\]). For example, using (\[n4\]), (\[a6\]) and (\[def\]) we see that \[a3\] \_B(e\_3)=\_B(e\_4 +n\_5)=e\_4 +n\_6=l-e\_3-e\_7. Similarly, we have \[a5\] \_B(e\_5)=\_B(e\_6 +n\_3)=e\_6 +n\_4=l-e\_5-e\_7 and \[a8\] \_B(e\_8)=\_B(e\_9 +n\_1)=e\_9 +n\_2=l-e\_7-e\_8.
Let us now discuss the transformation law of the sections $e_1$ and $e_2$. Each of these is contained in one of the two singular fibers of $\delta : B \to \cp{1}$ consisting of two components. Consider either one of these singular fibers. Since it must be stable under $\alpha_B$, $\alpha_B$ can either exchange its two components or leave them invariant. To answer this question, we must analyze these two fibers of $\delta$ more closely. By definition, these fibers are double covers of the $(0,1)$ rulings in $\bar{Q}$ which intersect the $(2,1) $ curve $\Delta^{-1}(t^{''})$ tangentially. It is easy to see that these $(0,1)$ rulings in $\bar{Q}$ are double covers of the two $(0,1)$ rulings in ${Q}$ with pass through either $r^{'}\cap t^{''}$ or $r\cap t^{''}$. Let us assume that $e_1$ is contained in the fiber of $\delta$ corresponding to the degree four cover of the $(0,1)$ ruling in ${Q}$ which passes through the point $r^{'}\cap t^{''}$. It follows that $e_2$ is contained in the fiber of $\delta$ corresponding to the degree four cover of the $(0,1)$ ruling in ${Q}$ which passes through the point $r\cap t^{''}$. From the commutativity of Figure \[tri-fg\], we can discuss these two fibers of $\delta$ using the relation $\delta^{-1}(x)=\kappa^{-1}(\pi^{-1}(p_2^{-1}(x)))$. To determine the transformation law of $e_1$, consider the $(0,1)$ ruling in ${Q}$ which passes through $r^{'}\cap t^{''}$. This ruling intersects the branch divisor of the map $\pi$ at two different points. Hence, its double cover in $W_M$ is a smooth $\cp{1}$. This curve intersects the branch divisor of $\kappa $ at a single point in $f_0$ in $W_M$. Therefore, its pre-image in $B$ consists of two components which are exchanged by the covering involution for $\kappa$ and, hence, by $\alpha_B$. Since one of the two components is $e_1$, F= e\_1+\_B(e\_1) represents the class of a full fiber of $\delta$. It follows from this and (\[2\]) that e\_1+\_B(e\_1)=l-e\_7, or equivalently \[a1\] \_B(e\_1)=l-e\_1-e\_7. What about the fiber of $\delta$ containing $e_2$? Consider the $(0,1)$ ruling in ${Q}$ which passes through $r\cap t^{''}$. This is the only point where this ruling intersects the branch locus of $\pi$ in $Q$. Hence, the pre-image of this ruling in $W_M$ consists of two components, both isomorphic to $\cp{1}$. What is the pre-image of these two $\cp{1}$ lines in $B$? Since each component intersects the branch locus of $\kappa$ in two points, namely its intersection with $f_0$ and the point $\pi^{-1}(p_1)$ in $W_M$, the pre-image of each component is a smooth $\cp{1}$. Clearly, these components are not exchanged under the covering involution of $\kappa$ and, hence, under $\alpha_B$. One of the components we called $e_2$. Therefore, \[a2\] \_B(e\_2)=e\_2.
Finally, solving the equations $\alpha_B(o_1)=o_2$ and $\alpha_B(l-e_7)=l-e_7$ using (\[df\]), (\[def\]), (\[a6\]), (\[a3\]), (\[a5\]), (\[a1\]) and (\[a2\]), we obtain \_B(l)=3l-e\_1-e\_3-e\_5-2e\_7-e\_8 and \_B(e\_7)=2l-e\_1-e\_3-e\_5-e\_7-e\_8. This concludes our calculation of the transformation laws of the canonical set of generators of $H_2(B,\mathbb{Z})$ under $\alpha_B$. The results are summarized in Table \[tab(al)\]. The action of $\alpha_B$ on any class in $H_2(B,\mathbb{Z})$ follows immediately from these results.
$e_1$ $e_2$ $e_3$ $e_4$ $e_5$
-------------- --------------- ----------------------------- --------------- ------- ----------------------------- -- -- -- -- --
$\alpha_{B}$ $l -e_7-e_1 $ $e_2$ $ l-e_3-e_7$ $e_4$ $l-e_ 5-e_7$
$e_6$ $e_7$ $e_8$ $e_9$ $l$
$\alpha_{B}$ $e_6 $ $ 2l-(e_1+e_3+e_5+e_7+e_8)$ $l-e_7-e_8 $ $e_9$ $3l-(e_1+e_3+e_5+2e_7+e_8)$
: The action of $\alpha_{B}$ on the canonical set of generators of $H_{2}(B,{\mathbb Z})$.[]{data-label="tab(al)"}
The Action of $t_{e_6}$ and $t_{e_4}$ {#trans}
=====================================
In this section, we complete the analysis of the transformations of the elements of $H_{2}(B,{\mathbb Z})$ under $\tau_{B1}$ and $\tau_{B2}$ by computing the actions of $t_{e_6}$ and $t_{e_4}$. We will proceed in several steps, first computing the transformations of the $I_2$ fiber components $n_i$ and $o_i$ for $i=1,\dots,6$ and then using these results to find the transformations of the canonical set of generators of $H_{2}(B,{\mathbb Z})$.
The Action of $t_{e_6}$ and $t_{e_4}$ on Fiber Components
---------------------------------------------------------
Recall that $t_{e_6}$ and $t_{e_4}$ preserve the fibers of the elliptic fibration $\beta : B \to \cp{1}$. That is \[t6f\] t\_[e\_6]{}(f)=f,t\_[e\_4]{}(f)=f. As discussed in Section \[rat\], any surface $B$ in the restricted two parameter family has six single $I_2$ fibers. Each such $I_2$ fiber is composed of two $\cp{1}$ components, $n$ and $o$, intersecting at two singular points. This is shown in Figure \[I2\]. That is, the $I_2$ singular fibers of $B$ can be written as $n_i\cup o_i,\,i=1,\dots,6$. The relationship of these fiber components to the canonical set of generators $e_1,\dots,e_9$ and $l$ of $H_{2}(B,{\mathbb Z})$ was given in (\[def\]). It follows from these remarks and (\[t6f\]) that the action of $t_{e_6}$ and $t_{e_4}$ either exchanges the two components of an $I_2$ fiber or leaves them invariant. The remainder of this sub-section is devoted to deciding which of these possibilities occur.
We begin our analysis with the first $I_2$ fiber, defined as $n_1\cup o_1$, and consider the action on it of $t_{e_6}$. Recall that each smooth fiber of $\beta : B \to \cp{1}$ admits an Abelian group structure whose identity element is the point of intersection of the fiber with the zero section $e=e_9$. It can be shown that, after removing the two singular points, each $I_2$ fiber also carries an Abelian group structure. Using (\[def\]) and the intersection numbers of the canonical set of generators of $H_{2}(B,{\mathbb Z})$, we see that n\_1e=1,o\_1e=0. Hence, the identity element is in the $n_1$ component. In addition, note that n\_1e\_6=0,o\_1e\_6=1. These results are represented pictorially in Figure \[tn1\].
Now consider a specific point on $n_1$, which we choose to be the identity element $n_1\cap e$. Translating this point by $t_6$ gives t\_6(n\_1e)=(n\_1o\_1)e\_6=o\_1e\_6, which is a point in $o_1$. If follows the $t_{e_6}$ must exchange $n_1$ and $o_1$. That is t\_[e\_6]{}(n\_1)=o\_1,t\_[e\_6]{}(o\_1)=n\_1. It is straightforward to see that $t_{e_6}$ will exchange the components of any $I_2$ fiber for which $e$ and $e_6$ intersect different components, as in Figure \[tn1\].
What happens for an $I_2$ fiber for which $e$ and $e_6$ intersect the same component? As an example, consider the fifth $I_2$ fiber defined as $n_5 \cup o_5$. Using (\[def\]) and the intersection numbers of the canonical set of generators of $H_{2}(B,{\mathbb Z})$, it is easy to show that n\_5 e= n\_5e\_6=0 and o\_5 e= o\_5e\_6=1. Here, the identity element is in the $o_5$ component. These results are represented in Figure \[tn5\]. Again, consider the zero element $o_5\cap e$. Translating this point by $t_{e_6}$ gives t\_[e\_6]{}(o\_5e)=(n\_5o\_5)e\_6= o\_5e\_6, which remains a point in $o_5$. We conclude that $t_{e_6}$ leaves $n_5$ and $o_5$ stable. That is t\_[e\_6]{}(n\_5)=n\_5,t\_[e\_6]{}(o\_5)=o\_5. Clearly, this will be the case for any $I_2$ fiber for which $e$ and $e_6$ intersect the same component, is in Figure \[tn5\].
It is easy now to derive the remaining transformation laws of $n_i$ and $o_i$ for $i=1,\dots,6$ under $t_{e_6}$ and $t_{e_4}$. From (\[def\]) one can read off the intersection numbers of the various $I_2$ fiber components with $e, e_6$ and $e_4$. By the above remarks, it is then straightforward to determine the transformation laws of the fiber components under $t_{e_6}$ and $t_{e_4}$. We summarize our findings in Table \[tI2\], where we also specify the action of the compositions $\tau_{B1}= t_{e_6}\circ \alpha_{B}$ and $\tau_{B2}= t_{e_4}\circ \alpha_{B}$ on $n_i$ and $o_i$ for $i=1,\dots,6$. To do the latter, we combine the results of this section with those of Section \[alpha\].
$e_9$ $ e_6$ $ e_4$ $t_{e_6}$ $t_{e_4}$ $\alpha_B$ $\tau_{B1}=t_{e_6}\circ \alpha_{B} $ $\tau_{B2}=t_{e_4}\circ \alpha_{B} $
------- ------- -------- -------- ----------- ----------- ------------ -------------------------------------- --------------------------------------
$n_1$ $1$ $0$ $0$ $o_1$ $o_1$ $n_2$ $o_2$ $o_2$
$n_2$ $1$ $0$ $0$ $o_2$ $o_2$ $n_1$ $o_1$ $o_1$
$n_3$ $0$ $1$ $0$ $o_3$ $n_3$ $n_4$ $o_4$ $n_4$
$n_4$ $0$ $1$ $0$ $o_4$ $n_4$ $n_3$ $o_3$ $n_3$
$n_5$ $0$ $0$ $1$ $n_5$ $o_5$ $n_6$ $n_6$ $o_6$
$n_6$ $0$ $0$ $1$ $n_6$ $o_6$ $n_5$ $n_5$ $o_5$
$o_1$ $0$ $1$ $1$ $n_1$ $n_1$ $o_2$ $n_2$ $n_2$
$o_2$ $0$ $1$ $1$ $n_2$ $n_2$ $o_1$ $n_1$ $n_1$
$o_3$ $1$ $0$ $1$ $n_3$ $o_3$ $o_4$ $n_4$ $o_4$
$o_4$ $1$ $0$ $1$ $n_4$ $o_4$ $o_3$ $n_3$ $o_3$
$o_5$ $1$ $1$ $0$ $o_5$ $n_5$ $o_6$ $o_6$ $n_6$
$o_6$ $1$ $1$ $0$ $o_6$ $n_6$ $o_5$ $o_5$ $n_5$
: The intersection of the components of the singular fibers with the sections $e, e_6$ and $e_4$, and their transformation under the automorphisms $t_{e_6}$, $t_{e_4}$, $\alpha_B$, $\tau_{B1}$ and $\tau_{B2}$. []{data-label="tI2"}
Transformations of the Canonical Generators under $t_{e_6}$
-----------------------------------------------------------
Having found the action of $t_{e_6}$ and $t_{e_4}$ on the fiber components, we now use these results to compute the transformation laws of the canonical set of generators of $H_2(B,{\mathbb Z})$ under each translation. In this sub-section, we consider the action of $t_{e_6}$. Some of these laws are easy to read off, others are more complicated. Let us begin with the easy ones. First, it follows from the definition of a translation, and the fact that $e_9=e$ is the zero section, that \[4.1\] t\_[e\_6]{}(e\_9)=e\_9e\_6=e\_6. Furthermore, since $e_6$ is a section of order two, we have \[4.2\] t\_[e\_6]{}(e\_6)=e\_6e\_6=e\_9. With help of (\[4.1\]), (\[4.2\]) and Table \[tI2\], it is now straightforward to determine the transformation laws of $e_5$ and $e_8$. Using, in addition, (\[def1\]), the definition of $o_i=f-n_i$ for $i=1,\dots,6$ and (\[def\]), we find that t\_[e\_6]{}(e\_[5]{}) = t\_[e\_6]{}(e\_[6]{} + n\_[3]{}) = e\_6 + o\_3 = e\_9+f-n\_3 = f-e\_5+e\_6+e\_9 and t\_[e\_6]{}(e\_[8]{}) = t\_[e\_6]{}(e\_[9]{} + n\_[1]{}) = e\_6 + o\_1 = e\_6+f-n\_1 = f+e\_6-e\_8+e\_9.
To find the transformation laws of $e_1, e_2, e_4$ and $e_7$ under $t_{e_6}$, however, we need to exploit the relationship between elements in $H_2(B,{\mathbb Z})$ and line bundles which, until now, was unnecessary. First, by Poincare duality, we have a canonical isomorphism H\_2(B,[Z]{})H\^2(B,[Z]{}). Furthermore, it can be shown that on a simply connected smooth surface H\^2(B,[Z]{})Pic(B), where $Pic(B)$ is the space of all line bundles up to isomorphism on $B$. Hence, we obtain H\_2(B,[Z]{})Pic(B). More explicitly, for any element $\xi \in H_2(B,{\mathbb Z})$, we have \_B(). What is the relationship between an involution $\tau: B \to B$ on $H_2(B,{\mathbb Z})$ and its action on $Pic(B)$? To answer this, consider a class $\xi \in H_2(B,{\mathbb Z})$ and its associated line bundle ${\mathcal O}_B(\xi) \in Pic(B)$. If $\tau$ is an involution on $B$, then we can define the pull-back bundle, $\tau^{*}{\mathcal O}_B(\xi)$, under $\tau$ which satisfies the commutative diagram Map $\pi_2$ takes the fiber $\pi^{'-1}(p) \subset \tau^{*}{\mathcal O}_B(\xi)$ to the fiber $\pi^{-1}(\tau(p)) \subset {\mathcal O}_B(\xi)$ for all points $p\in B$. As is, the bundle $\tau^{*}{\mathcal O}_B(\xi)$ is abstractly defined. However, one can identify $\tau^{*}{\mathcal O}_B(\xi)$ more concretely, as we now show. We will assume that the class in $H_2(B,{\mathbb Z})$ can be represented by a curve, which we also call $\xi$. This implies that ${\mathcal O}_B(\xi)$ has a section s: B \_B() whose vanishing locus is on $\xi$, that is, $s(\xi)=0$. Now consider the map s\^[’]{}: B \^[\*]{}[O]{}\_B() defined by \_2s\^[’]{}=s. Then, it is clear that $s^{'}$ is a section of $\tau^{*}{\mathcal O}_B(\xi)$ satisfying s\^[’]{}(\^[-1]{}())=0. Hence, the curve, or homology cycle, representing $\tau^{*}{\mathcal O}_B(\xi) $ is $\tau^{-1}(\xi)$. That is, \^[\*]{}[O]{}\_B () \_B(\^[-1]{}())\_B(()), where we have used the fact that, since $\tau$ is an involution, $\tau^{-1}=\tau$. We can now answer the above question definitely. The action $\xi \to \tau(\xi)$ of an involution on $H_2(B,{\mathbb Z})$ induces the action \[mappic\] [O]{}\_B()\_B(\^[-1]{}())\_B(()) on $Pic(B)$. We will use this relationship to determine the transformation laws of $e_1, e_2, e_4$ and $e_7$ under translations.
To do this, we first need to review some relevant facts about line bundles. Recall that $B$ is an elliptic fibration $\beta: B \to \cp{1}$. We want to discuss the group structure of both the smooth elliptic fibers, $T^2$, and the space of sections, $\Gamma(B)$, in terms of line bundles on $T^{2}$ and $B$ respectively. First, we consider a smooth elliptic curve $T^{2}$ with a fixed zero point, which we call $e$. Each point $p \in T^2$ determines a line bundle, namely p \_[T\^2]{}(p), which is of degree one since it is defined by a single point. This specifies a map $T^2 \to Pic^1(T^2)$, where $Pic^1(T^2)\subset Pic(T^2)$ is the space of degree one line bundles on $T^2$ up to isomorphism. Note that since $Pic^1(T^2)$ is not closed under tensor product multiplication, it does not form a group. However, exploiting the fact that we have a marked point $e \in T^2$, we can, alternatively, define a map for each point $p$ to a degree zero line bundle, namely p \_[T\^2]{}(p-e). ${\mathcal O}_{T^2}(p-e)$ is of degree zero since it is defined as the difference of two points. This specifies a map $T^2 \to Pic^0(T^2)$, which can be shown to be an isomorphism. $Pic^0(T^2)$ is the space of all degree zero line bundles on $T^2$ up to isomorphism. It is clear that the bundle associated to the zero point $e$ under this map is the trivial bundle. Furthermore, $Pic^0(T^2)$ is closed under tensor product multiplication and forms a group. Note that the map $T^2 \to Pic^0(T^2)$ is a group homomorphism. This means that for any given three points $p, p^{'},p^{''} \in T^2$ which obey pp\^[’]{}=p\^[”]{}, we have \_[T\^2]{}(p-e)\_[T\^2]{}(p\^[’]{}-e)\_[T\^2]{}(p\^[”]{}-e).
Thus far, we have considered points in a smooth fiber $T^2$. Using the zero section $e=e_9$ of $B$, we can try to extend our discussion to a family of elliptic curves, that is, to sections of our elliptic fibration $\beta: B \to \cp{1}$. Consider any section $\xi$ in $\Gamma(B)$. We can, as mentioned above, define a map \[0a\] \_B(). This specifies a map from $ \Gamma(B)\to Pic(B)$. But similarly to the discussion above, we can, alternatively, use the zero section $e$ to define, for any section $\xi$, the mapping \_B(-e). This map has a similar property to the case of one elliptic curve. That is, for any three sections $\xi,\xi^{'},\xi^{''} \in \Gamma(B)$ which obey \[a\] \^[’]{}=\^[”]{} we have \[b\] [O]{}\_B(-e)|\_f\_B(\^[’]{}-e)|\_f= [O]{}\_B(\^[”]{}-e)|\_f, where the equation only holds for the restriction to any smooth fiber $f$. However, care has to be taken at the singular fibers.
Let us now calculate the class $t_{e_6}(\xi)$ for any section $\xi$. To do this, we have to calculate the line bundle ${\mathcal O}_B(t_{e_6}^{-1}(\xi))$, as discussed in (\[mappic\]). Using the map (\[0a\]), this line bundle corresponds to the section (e\_6)=e\_6=t\^[-1]{}\_[e\_6]{}(). Using equations (\[a\]) and (\[b\]), we find that \[great\] [O]{}\_B(-e)|\_f\^[-1]{}\_B(e\_6-e)|\_f=[O]{}\_B(t\_[e\_6]{}\^[-1]{}()-e)|\_f where we have used ${\mathcal O}^{-1}_B(e_6-e)$ as the second factor since, in this case, $\xi^{'}={\stackrel{.}{-}}e_6$. Tensoring (\[great\]) with ${\mathcal O}_B(e)$ gives an equation in terms of divisors, namely \[4.5\] t\_[e\_6]{}()= - e\_[6]{} + e\_9 + a\_[1]{}n\_[1]{} + a\_[2]{}n\_[2]{} + a\_[3]{}n\_[3]{}+ a\_[4]{}n\_[4]{}+ a\_[5]{}n\_[5]{}+ a\_[6]{}n\_[6]{}+ a\_0f, where $a_0,\dots,a_6$ are integers. Note that this equation holds for both smooth and singular fibers. It even holds without restriction to any fiber at all. We have assured this by adding the fiber components. The specific coefficients $a_0,\dots,a_6$ will now be determined.
To begin with, intersect both sides of (\[4.5\]) with the exceptional divisors $n_{i}$ for $i=1,...,6$. Taking into account the results of Table \[tI2\], that is, $(t_{e_6})(n_{k}) = o_{k},\;k=1,...,4$, $(t_{e_6})(n_{k}) = n_{k},\;k=5,6$, the intersection numbers and the fact that $n_i\cdot t_{e_6}(\xi)=t_{e_6}(n_i)\cdot\xi $, we obtain
o\_1&= n\_1+1 -2a\_1\
o\_2&= n\_2+1 -2a\_2\
o\_3&= n\_3-1 -2a\_3\
o\_4&= n\_4-1 -2a\_4\
n\_5&= n\_5-2a\_5\
n\_6&= n\_6-2a\_6.\
Hence, a\_5=a\_6=0 independently of the choice of section $\xi$. Recall that $e_1,e_2,e_4$ and $e_7$ are sections of $B$, that is, they intersect each $I_2$ fiber exactly once. Furthermore, $e_1$ and $e_2$ intersect none of the exceptional divisors, the only exceptional divisors which are intersected by $e_4$ are $n_5$ and $n_6$, and $e_7$ intersects only $n_2,n_4$ and $n_6$. Using these facts, we can determine the coefficients $a_1,...,a_4$ for $e_1,e_2,e_4$ and $e_7$. The results are presented in the first four columns of Table \[te6exept\]. Note that these coefficients are identical for $e_1, e_2$ and $e_4$, but differ for $e_7$.
$a_1 $ $a_2 $ $a_3 $ $a_4 $ $a_0$
--------- -------- -------- -------- -------- -------
$ e_1 $ $0 $ $0$ $-1$ $-1 $ $1$
$ e_2 $ $0 $ $0$ $-1$ $-1 $ $1$
$ e_4 $ $0 $ $0$ $-1$ $-1 $ $1$
$ e_7 $ $0 $ $1$ $-1$ $0 $ $0$
: The coefficients $a_0,...,a_4$ for the expansion of the pull-back of the sections $e_1,e_2,e_4$ and $e_7$ under the action of $t_{e_6}$.[]{data-label="te6exept"}
To determine the remaining coefficients $a_0$, we use the fact that $t_{e_6}({\xi})$ is again a section and, hence, $t_{e_6}({\xi})\cdot t_{e_6}({\xi})=-1 $. Let us begin with $e_1, e_2$ and $e_4$. Using (\[4.5\]) and the results in Table \[te6exept\], we have \[t6i\] t\_[e\_6]{}[(e\_i)]{}=e\_i-e\_6+e\_9-n\_3-n\_4+a\_0fi=1,2,4. Squaring this expression then tells us that $a_0=1$ for $e_1,e_2$ and $e_4$. Hence, we have at long last determined the transformation laws t\_[e\_6]{}(e\_i)=f-l+e\_i+e\_6+e\_7+e\_9i=1,2,4, where we used the definition of $n_3$ and $n_4$ in term of the sections $e_1,\dots,e_9$ and $l$ given in (\[def\]). Similarly, using the expansion of $t_{e_6}(e_7)$ determined by (\[4.5\]) and Table \[te6exept\], we find that $a_0=0$ for $e_7$. Hence, t\_[e\_6]{}(e\_7)=l-e\_5-e\_8. Our results for the coefficients $a_0$ are listed in the last column of Table \[te6exept\]. We can now use $t_{e_6}(n_5)$ determined in the previous sub-section and (\[t6i\]) to show that
t\_[e\_6]{}(e\_[3]{}) &= t\_[e\_6]{}(e\_[4]{} + n\_[5]{}) =f-l+e\_4+e\_6+e\_7+e\_9+n\_5=f-l+e\_3+e\_6+e\_7+e\_9.\
Finally, using the transformation laws of $f$ and $e_1,\dots,e_9$ it is easy to read off the transformation of $l$. We find that t\_[e\_6]{}(l)=2f-e\_5+2e\_6+e\_7-e\_8+2e\_9. A general summary of our findings for $t_{e_6}$ is presented in Table \[table-auts\].
Transformations of the Canonical Generators under $t_{e_4}$
-----------------------------------------------------------
The calculation of the transformation laws of the canonical set of generators of $H_2(B,{\mathbb Z})$ under $t_{e_4}$ closely follows the calculation of the transformations under $t_{e_6}$. First, we determine the transformations of $e_9, e_4, e_8$ and $e_3$. Using the definition of a translation, and the facts that $e=e_9$ is the zero section and that $e_4$ is a section of order two, we find \[g\] t\_[e\_4]{}(e\_[9]{})=e\_9e\_4 =e\_[4]{} and \[f\] t\_[e\_4]{}(e\_4) = e\_[4]{} e\_[4]{} = e\_[9]{}. Using this, (\[def\]) and Table \[tI2\], it follows that t\_[e\_4]{}(e\_[8]{}) = t\_[e\_4]{}(e\_[9]{} + n\_[1]{}) = e\_4 + o\_1 = e\_4+f-n\_1 = f+e\_4-e\_8+e\_9 and t\_[e\_4]{}(e\_[3]{}) = t\_[e\_4]{}(e\_[4]{} + n\_[5]{}) = e\_9 + o\_5 = e\_9+f-n\_3 = f-e\_3+e\_4+e\_9. To finish our calculation of the transformation laws of the canonical set of generators of $H_2(B,{\mathbb Z})$ under $t_{e_4}$, we have to use a formula similar to (\[4.5\]) which, for $t_{e_4}$, is given by \[te4\] t\_[e\_4]{}() = - e\_[4]{} + e\_[9]{} + a\_[1]{}n\_[1]{} + a\_[2]{}n\_[2]{} + a\_[3]{}n\_[3]{}+ a\_[4]{}n\_[4]{}+ a\_[5]{}n\_[5]{}+ a\_[6]{}n\_[6]{}+ a\_0f. To determine the coefficients for specific sections $\xi $, we intersect both sides of (\[te4\]) with $n_{i}$ for $i=1,...,6$. Taking into account the results of Table \[tI2\], that is, $(t_{e_4})(n_{k}) = o_{k}$ for $k=1,2,5,6$ and $(t_{e_4})(n_{k}) = n_{k}$ for $\;k=3,4$, we obtain
o\_1&= n\_1+1 -2a\_1\
o\_2&= n\_2+1 -2a\_2\
n\_3&= n\_3-2a\_3\
n\_4&= n\_4-2a\_4\
o\_5&= n\_5-1 -2a\_5\
o\_6&= n\_6-1 -2a\_6.\
Hence, a\_3=a\_4=0 independently of the section $\xi$. Recall that $e_1,e_2,e_6$ and $e_7$ are sections, that is, they intersect each $I_2$ fiber exactly once. Furthermore, $e_1$ and $e_2$ intersect none of the exceptional divisors and the only exceptional divisors which are intersected by $e_6$ are $n_3$ and $n_4$. Using these facts, we can determine the coefficients $a_1,...,a_4$ for $e_1,e_2,e_6$ and $e_7$. The first four columns of Table \[te4exept\] summarize our findings for the coefficients $a_1,\dots,a_4$.
$a_1 $ $a_2 $ $a_3 $ $a_4 $ $a_0$
--------- -------- -------- -------- -------- -------
$ e_1 $ $0 $ $0$ $-1$ $-1$ $1$
$ e_2 $ $0 $ $0$ $-1$ $-1$ $1$
$ e_6 $ $0 $ $0$ $-1$ $-1$ $1$
$ e_7 $ $0 $ $1$ $-1$ $0 $ $0$
: The coefficients $a_0,...,a_4$ for the expansion of the pull-back of the sections $e_1,e_2,e_6$ and $e_7$ under the action of $t_{e_4}$.[]{data-label="te4exept"}
To determine the remaining coefficients $a_0$, we use the fact that $t_{e_4}({\xi})$ is again a section. Hence $t_{e_4}({\xi})\cdot t_{e_4}({\xi})=-1 $. Using this, and squaring the expansion t\_[e\_4]{}[(e\_i)]{}=e\_i-e\_4+e\_9-n\_5-n\_6+afi=1,2,6, we find $a_0=1$ for $e_1,e_2$ and $e_6$. Hence, t\_[e\_4]{}(e\_i)=f-l+e\_i+e\_4+e\_7+e\_9i=1,2,6, where we used the definitions of $n_5$ and $n_6$ given in (\[def\]). Using the expansion of $t_{e_4}(e_7)$, we find $a_0=0$ for $e_7$ and, hence, that t\_[e\_4]{}(e\_7)=l-e\_3-e\_8. Our results for the coefficients $a_0$ are listed in the last column of Table \[te4exept\]. We can now show
t\_[e\_4]{}(e\_[5]{}) &= t\_[e\_4]{}(e\_[6]{} + n\_[3]{}) = f-l+e\_6+e\_4+e\_7+e\_9 +n\_3= f-l+e\_4+e\_5+e\_7+e\_9.\
Using the transformation laws of $f$ and $e_1,\dots,e_9$ it is easy to read of the transformation of $l$. We find that t\_[e\_4]{}(l)=2f-e\_3+2e\_4+e\_7-e\_8+2e\_9. A general summary of our findings for $t_{e_4}$ are presented in Table \[table-auts\].
The $\tau_{B1}$, $\tau_{B2}$ Action on $H_2(B,{\mathbb Z})$ and Invariant Classes {#comp}
=================================================================================
Having determined the transformation laws of the canonical set of generators $e_1,\dots,e_9$ and $l$ of $H_2(B,{\mathbb Z})$ under $\alpha_B$ in Section \[alpha\] and $t_{e_4}$, $t_{e_6}$, in Section \[trans\], we can now compose them using $\tau_{B1}=t_{e_6}\circ \alpha_B$ and $\tau_{B2}=t_{e_4}\circ \alpha_B$ to determine the action of $\tau_{B1}$ and $\tau_{B2}$. The results are presented in the fifth and sixth columns of Table \[table-auts\] respectively. The table also summarizes our results for $(-1)_B$, $\alpha_B$, $t_{e_6}$ and $t_{e_4}$.
$\hspace{0.5cm}(-1)_{B}$ $\hspace{0.7cm}\alpha_{B}$ $\hspace{0.7cm}t_{e_6}$ $\hspace{0.7cm}t_{e_4}$ $\hspace{0.7cm}\tau_{B1}$ $\hspace{0.7cm}\tau_{B2} $
------- -------------------------- ----------------------------- ---------------------------- ---------------------------- ---------------------------- ----------------------------
$f$ $f$ $f$ $f $ $f$ $f$ $f$
$e_1$ $l-e_7-e_1$ $l -e_7-e_1 $ $f-l+e_1+e_6+e_7+e_9$ $f-l+e_1+e_4+e_7+e_9$ $f-e_1+e_6+e_9$ $f-e_1+e_4+e_9$
$e_2$ $l-e_7-e_2$ $e_2$ $f-l+e_2+e_6+e_7+e_9 $ $f-l+e_2+e_4+e_7+e_9 $ $f-l+e_2+e_6+e_7+e_9$ $f-l+e_2+e_4+e_7+e_9$
$e_3$ $e_3$ $ l-e_3-e_7$ $f-l+e_3+e_6+e_7+e_9 $ $f-e_3+e_4+e_9$ $f-e_3+e_6+e_9$ $f-l+e_3+e_4+e_7+e_9$
$e_4$ $e_4$ $e_4$ $f-l+e_4+e_6+e_7+e_9$ $e_9 $ $f-l+e_4+e_6+e_7+e_9$ $e_9$
$e_5$ $e_5$ $l-e_ 5-e_7$ $f-e_5+e_6+e_9$ $f-l+e_5+e_4+e_7+e_9$ $f-l+e_5+e_6+e_7+e_9$ $f-e_5+e_4+e_9$
$e_6$ $e_6$ $e_6 $ $e_9$ $f-l+e_6+e_4+e_7+e_9$ $e_9$ $f-l+e_6+e_4+e_7+e_9$
$e_7$ $l-e_1-e_2$ $ 2l-(e_1+e_3+e_5+e_7+e_8)$ $l-e_5-e_8$ $l-e_3-e_8$ $l-e_1-e_3$ $l-e_1-e_5$
$e_8$ $e_8$ $l-e_7-e_8 $ $f+e_6-e_8+e_9$ $f+e_4-e_8+e_9$ $f-l+e_6+e_7+e_8+e_9$ $f-l+e_4+e_7+e_8+e_9$
$e_9$ $e_9$ $e_9$ $e_6$ $e_4$ $e_6$ $e_4$
$l $ $2l-(e_1+e_2+e_7)$ $3l-(e_1+e_3+e_5+2e_7+e_8)$ $2f-e_5+2e_6+e_7-e_8+2e_9$ $2f-e_3+2e_4+e_7-e_8+2e_9$ $2f-e_1-e_3+2e_6+e_7+2e_9$ $2f-e_1+2e_4-e_5+e_7+2e_9$
: The action of $(-1)_{B}$, $\alpha_{B}$, $t_{e_6}$, $t_{e_4}$, $\tau_{B1} $ and $\tau_{B2} $ on $H_{2}(B,{\mathbb Z})$. Note that $f$ can be expressed as $f=3l-\sum_{i=1}^{9}e_i$.[]{data-label="table-auts"}
We now want to find all classes in $H_2(B,{\mathbb Z})$ that are invariant under the ${\mathbb Z}_2\times{\mathbb Z}_2 $ automorphism group. Let us denote the subspace of all such invariant classes by $H_2(B,{\mathbb Z})^{inv}$. We will show in [@dopr-ii] that \[rankinv\] H\_2(B,[Z]{})\^[inv]{}=4. Hence, there exist four generators of $H_2(B,{\mathbb Z})^{inv}$. One of these generators is trivial to find, namely f=3l-\_[i=1]{}\^9 e\_i. This follows from the first line of Table \[table-auts\], which reflects the statement, shown in [@opr], that $f$ is preserved by any automorphism of $B$.
A second invariant generator, consisting only of fiber components of $\beta: B \to \cp{1}$, is also easy to find. We see using Table \[tI2\] that n\_1+o\_2 is preserved under the ${\mathbb Z}_2\times{\mathbb Z}_2 $ automorphism group. Using (\[def\]), this can be expressed in terms of the canonical set of generators as n\_1+o\_2=2l-e\_1-e\_2-e\_3-e\_4-e\_5-e\_6+e\_8-e\_9. To find the third invariant generator, we exploit the fact that, for any $\xi \in H_2(B,{\mathbb Z})$, the class \[txi\] +\_[B1]{}()+\_[B2]{}()+\_[B1]{}\_[B2]{}() is invariant under ${\mathbb Z}_2\times{\mathbb Z}_2 $. Choosing $\xi=e_9$, we take as the third generator of $H_2(B,{\mathbb Z})^{inv}$ ie\_9+\_[B1]{}(e\_9)+\_[B2]{}(e\_9)+\_[B1]{}\_[B2]{}(e\_9). Using the results in Table \[table-auts\], we find that \[invi\] i=2e\_6+2e\_4-n\_1+o\_2. This can be expressed in terms of the canonical basis as i=2l-e\_1-e\_2-e\_3+e\_4-e\_5+e\_6-e\_8+e\_9. Finally, we find the fourth invariant generator as a result of enlightened trial and error. The result is \[invM\] M2e\_2-2e\_9-n\_1-n\_2, which, using (\[def\]), can be written as M=-l+2e\_2+e\_7. The invariance of $M$ under ${\mathbb Z}_2\times{\mathbb Z}_2 $ is easily checked using the results in Table \[table-auts\]. It is straightforward to show that $f, n_1+o_2, i$ and $M$ are linearly independent classes. We conclude that \[iM\] f,n\_1+o\_2,i=2e\_6+2e\_4-n\_1+o\_2,M=2e\_2-2e\_9-n\_1-n\_2 form a set of generators [^2] of $H_2(B,{\mathbb Z})^{inv}$. Their intersection numbers are displayed in Table \[inter\].
$i$ $f$ $n_1+o_2$ $M$
----------- ------ ----- ----------- ------
$i$ $-4$ $4$ $4$ $0$
$f$ $4$ $0$ $0$ $0$
$n_1+o_2$ $4$ $0$ $-4$ $0$
$M$ $0$ $0$ $0$ $-4$
: The intersection numbers of the generators of $H_2(B,{\mathbb Z})^{inv}.$[]{data-label="inter"}
This choice of generators will turn out to be very convenient when we solve the numerical conditions on the rank four vector bundles $V$ on the Calabi-Yau space $X$ imposed by particle physics phenomenology [@dopr-iii; @dopr-ii]. Note that it is very easy to find other invariant classes in $H_2(B,{\mathbb Z})$. For example, it is immediately obvious from Table \[tI2\] that $n_2+o_1$ is an invariant class. Furthermore, as we stated, (\[txi\]) is an invariant class for any $\xi \in H_2(B,{\mathbb Z})$. However, it follows from (\[rankinv\]) and the linear independence of $f, n_1+o_2, i$ and $M$ that any other invariant class must be a linear combination of these four generator classes. For example, one easily finds using (\[def\]) that n\_2+o\_1=2f-(n\_1+o\_2). Finally, we denote the set of generators for invariant curve classes on $B^{'} $ by \[iM’\] f\^[’]{},n\^[’]{}\_1+o\^[’]{}\_2,i\^[’]{},M\^[’]{}, where $i^{'}$ and $M^{'}$ are defined similarly to (\[invi\]) and (\[invM\]).
The Homology of $X$ and $Z$ {#XZ}
===========================
Recall from [@opr] and Section \[Z\] that our analysis involves two different types of Calabi-Yau threefolds. The first are the Calabi-Yau spaces $X$ which are simply connected and admit a freely acting automorphism group ${\mathbb Z}_2\times {\mathbb Z}_2$. The second are the smooth quotient spaces \[11.1\] Z=X/([Z]{}\_2\_2) with fundamental group $\pi_1(Z)={\mathbb Z}_2\times {\mathbb Z}_2$. In this section, we will calculate the relevant parts of the homology rings $H_{*}(X,{\mathbb Z})$ and $H_{*}(Z,{\mathbb Z})$ of these two types of threefolds. More precisely, we will first determine \[poinc\] \_[C]{} H\_[i]{}(X,[C]{})=\_[C]{} (H\_[i]{}(X,[Z]{})),i=0,…,6 and then show how to find a set of generators of $H_{4}(X,{\mathbb Z})$. A subset of these generators consists of classes that are invariant under the action of ${\mathbb Z}_2\times {\mathbb Z}_2$. These span the subgroup of invariant homology classes, which we denote by $H_{4}(X,{\mathbb Z})^{inv} \subset H_{4}(X,{\mathbb Z})$. We will explicitly compute a complete set of invariant generators. Using these results, we then determine \_[C]{} H\_[i]{}(Z,[C]{})= \_[C]{}( H\_[i]{}(Z,[Z]{})),i=0,…,6 and explicitly find a set of generators of $H_{4}(Z,{\mathbb Z})$. Both quantities, the dimensions of $H_{i}(Z,{\mathbb C}),\;i=0,\dots,6$ and the generators of $H_{4}(Z,{\mathbb Z})$, are of physical importance. The dimension of the homology groups of $Z$ are a vital ingredient in determining the particle spectrum of string theories compactified on the Calabi-Yau threefold $Z$. The elements of $H_{4}(Z,{\mathbb Z})$ are divisors in $Z$ which clearly descend, via the modding out (\[11.1\]), from the invariant divisors on $X$, that is, the elements of $H_{4}(X,{\mathbb Z})^{inv}$. The classes in $H_{4}(X,{\mathbb Z})^{inv}$ will be used in [@dopr-i] and [@dopr-ii] to construct invariant stable vector bundles on $X$ which, in turn, descend to stable vector bundles on $Z$. These bundles admit connections which correspond to solutions of the hermitian Yang-Mills equations on $Z$.
Let us begin with the Calabi-Yau threefold $X$. It is convenient to use Poincaré duality, that is H\_i(X,[Z]{})H\^[6-i]{}(X,[Z]{}),0=1,…,6. Furthermore, the fact that $X$ is a Kahler manifold allows us to write the Hodge diamond of $X$ as in Figure \[HX\]. Here \[hij\] h\^[a,b]{}\_X=\_[C]{}H\_[|]{}\^[a,b]{}(X),a,b=0,…,3 denotes the dimension of the $a$-th Dolbeaux cohomology of holomorphic $b$-forms on $X$.
$$\xymatrix{
&& & h^{0,0}_X & & & \\
&& h^{1,0}_X & &h^{0,1}_X & & \\
&h^{2,0}_X & & h^{1,1}_X & &h^{0,2}_X & \\
h^{3,0}_X & &h^{2,1}_X & &h^{1,2}_X & &h^{0,3}_X \\
&h^{3,1}_X & & h^{2,2}_X & &h^{1,3}_X & \\
& &h^{3,2}_X & &h^{2,3}_X & & \\
& & & h^{3,3}_X & & & \\
}$$
It is well-known that for Kahler manifolds H\^[k]{}(X,[C]{})=\_[a+b=k]{}H\_[|]{}\^[a,b]{}(X) and, therefore, that \[kahler\] \_[C]{} H\^[k]{}(X,[C]{})=\_[a+b=k]{}h\^[a,b]{}. For simply connected Calabi-Yau threefolds, the Hodge diamond simplifies to the highly symmetric form given in Figure \[hCY\]. Hence, to determine the dimension of the homology groups of $X$, we need to determine $h^{1,1}_X$ and $h^{2,1}_X$ only. To do this, we begin by calculating the elements of $H_{\bar{\partial}}^{1,1}(X)$ explicitly.
$$\xymatrix{
& & & 1 & & & \\
& & 0 & &0 & & \\
&0 & & h^{1,1}_X & &0 & \\
1 & &h^{2,1}_X & &h^{2,1}_X & &1 \\
&0 & & h^{1,1}_X & &0 & \\
& &0 & &0 & & \\
& & & 1 & & & \\
}$$
Since $X$ is simply connected, we have \[12.10\] H\_[|]{}\^[1,1]{}(X)Pic(X)H\^2(X,[Z]{}). Therefore, if we can determine the generators of all line bundles on $X$, we will have determined all generators of $H^2(X,{\mathbb Z})$ and, hence, of $H_{\bar{\partial}}^{1,1}(X)$. Recall that $X$ is the fiber product of two rational elliptic surfaces $B$ and $ B^{'}$. Hence, we have the inclusion map i: B \_ B\^[’]{} B B\^[’]{}. In terms of cohomology, this defines the pull-back map i\^[\*]{}: H\^2(BB\^[’]{},[Z]{} ) H\^2(B \_ B\^[’]{},[Z]{} ). Using the Kunneth formula, we find H\^2(BB\^[’]{},[Z]{} )H\^2(B,[Z]{})H\^2(B\^[’]{},[Z]{}). With the help of the the isomorphisms H\^2(B\_ B\^[’]{},[Z]{} )Pic(B\_ B\^[’]{}),H\^2(B,[Z]{} )Pic(B),H\^2( B\^[’]{},[Z]{} )Pic(B\^[’]{}), we have \[Pic\] i\^[\*]{}:Pic(B) Pic(B\^[’]{}) Pic(B \_ B\^[’]{}). To see how the map $i^{*}$ works, recall that for $X$ we had the natural projections $$\xymatrix{
& X \ar[dl]_-{\pi'} \ar[dr]^-{\pi} && \ar@{}[ddr]|-{.} & \\
B \ar[dr]_-{\beta} & & B' \ar[dl]^-{\beta'}&&& \\
& \cp{1} &&&&
}$$ Consider any line bundle $L$ on $B$ determined by the divisor $D$. That is, L=(D). Then, clearly \^[’\*]{}L=\^[’\*]{}(D)=(\^[’-1]{}(D))=(D\_B\^[’]{}). Similarly, for a line bundle $L^{'}$ on $B^{'}$ determined by the divisor $D^{'}$, that is L\^[’]{}=(D\^[’]{}), we obtain \^[\*]{}L\^[’]{}=\^[\*]{}(D\^[’]{})=(\^[-1]{}(D\^[’]{}))=(B\_D\^[’]{}). Hence, the map (\[Pic\]) is given by \[LL\] i\^[\*]{}:(L,L\^[’]{})\^[’\*]{}L \^[\*]{}L\^[’]{}. It is not to difficult to show that the map $i^{*}$ is surjective, that is, all line bundles on $B\times_{\cp{1}}B^{'}$ arise from line bundles on $B$ and $B^{'}$ via (\[LL\]). But what is the kernel of $i^{*}$? It follows from (\[LL\]) that $(L,L^{'})$ will be in the kernel of $i^{*}$ if and only if \^[’\*]{}L=(\^[\*]{}L\^[’]{})\^[-1]{}. Hence, the divisors $\pi^{'-1}(D)$ and $\pi^{-1}(D^{'})$ associated with $\pi^{'*}L$ and $\pi^{*}L^{'}$ respectively must satisfy \[11.21\] \^[’-1]{}(D)=-\^[-1]{}(D\^[’]{}). This can only happen if $L$ and $L^{'}$ are pull-back bundles from $\cp{1}$ of the form L=\^[\*]{}(a),L\^[’]{}=\^[’\*]{}(-a),a . Then \^[’\*]{}L=(a(ff\^[’]{})),\^[\*]{}L\^[’]{}=(-a(ff\^[’]{})) and, hence, (\[11.21\]) is satisfied. Noting that $\ocp(a),\;a \in {\mathbb Z}$ spans $Pic(\cp{1})\cong H^2(\cp{1},{\mathbb Z})$, it follows that \[gh2\] H\^[2]{}(X,[Z]{})Pic(B \_B\^[’]{}). This result allows us to determine all of the relevant quantities on $X$, namely, the Hodge numbers $h^{1,1}_X, h^{2,1}_X $ and the generators of $H_{4}(X,{\mathbb Z})$.
First consider the Hodge numbers. Using (\[hij\]), (\[12.10\]), (\[gh2\]) and the fact that $\rank H^2(B,{\mathbb Z})=10$, we find \[h11\] h\^[1,1]{}\_X=\_[C]{} H\^[1,1]{}\_[|]{}(X)=\_[C]{} H\^[2]{}(X,[C]{}) =10+10-1=19. To determine $h^{2,1}_X$, note that for complex threefolds the topological Euler characteristic can be calculated as (X)=\_[i=0]{}\^[6]{}(-1)\^i \_[C]{}[H\^i(X,[C]{})]{}. Furthermore, it can be shown that (X)=c\_3(X). But, as discussed in [@opr], $c_3(X)=0$. Using this and the Hodge diamond of Figure \[hCY\], we find 0=(X)=1+h\^[1,1]{}\_X-1-2h\^[2,1]{}\_X-1+h\^[1,1]{}\_X+1. It then follows, using (\[h11\]), that h\^[2,1]{}=19. Note that using (\[kahler\]), the Hodge diamond of Figure \[hCY\] and $h_X^{1,1}=h_X^{2,1}=19$ we can determine $\dim H^i(X,{\mathbb C})$ and, hence, by Poincaré duality $\dim H_i(X,{\mathbb C})$ for all $i=0,\dots,6$.
Now, recall from Poincaré duality (\[poinc\]) that H\_4(X,[Z]{})H\^2(X,[Z]{}). Using this and (\[gh2\]), we can determine the nineteen generators of $H_4(X,{\mathbb Z})$. First, note that if $D$ is a divisor in $B$, that is, $D \in H_2(B,{\mathbb Z})$, then \[12.33\] \^[’-1]{}(D)=D\_ B\^[’]{} H\_4(X,[Z]{}). Similarly, if $D^{'} \in H_2(B^{'},{\mathbb Z})$ then \[12.34\] \^[-1]{}(D\^[’]{})=B\_ D\^[’]{} H\_4(X,[Z]{}). Taking $\{D\}$ to be the canonical set of generators $\{e_1,\dots,e_9,l\}$ of $H_2(B,{\mathbb Z})$ and $\{D^{'}\}$ to be $\{e_1^{'},\dots,e_9^{'},l^{'}\}$ in $H_2(B^{'},{\mathbb Z})$, we see from (\[gh2\]) that the set \[12.35\] {D}\_ B\^[’]{},B\_ {D\^[’]{}} generates $H_4(X,{\mathbb Z})$. Note that there are only nineteen, and not twenty, classes in (\[12.35\]) since \[12.36\] f\_B\^[’]{}=B\_ f\^[’]{}=ff\^[’]{}. This is the reason for the $H^2(\cp{1},{\mathbb Z} )$ denominator in (\[gh2\]). We conclude that (\[12.35\]), subject to (\[12.36\]), gives an explicit set of nineteen generators of $H_4(X,{\mathbb Z})$.
So far, we have analyzed the Hodge diamond and fourth homology group of the Calabi-Yau threefold $X$. We now turn our attention to the Calabi-Yau threefold $Z$ defined in (\[11.1\]). Here, given the previous results, it is easiest to proceed in reverse, first determining the generators of $H_4(Z,{\mathbb Z})$ and then the Hodge numbers. To to this, note that any class in $H_4(Z,{\mathbb Z})$ must descend under modding out from a ${\mathbb Z}_2 \times {\mathbb Z}_2 $ invariant class in $H_4(X,{\mathbb Z})$. If we denote the subspace of invariant classes by $H_4(X,{\mathbb Z})^{inv} \subset H_4(X,{\mathbb Z})$, then there exists an isomorphism H\_4(X,[Z]{})\^[inv]{}H\_4(Z,[Z]{}). What are the generators of $H_4(X,{\mathbb Z})^{inv}$? Recall from (\[invX\]) that the generators of the ${\mathbb Z}_2\times{\mathbb Z}_2 $ automorphism group on $X$ are \_[Xi]{}=\_[Bi]{} \_\_[B\^[’]{}i]{},i=1,2. Acting with these involutions on (\[12.33\]) and (\[12.34\]) we have \[12.38\] \_[Xi]{}(D \_ B\^[’]{})=\_[Bi]{}(D)\_ B\^[’]{} and \[12.39\] \_[Xi]{}(B \_ D\^[’]{})=B \_ \_[B\^[’]{}i]{}(D\^[’]{}) respectively for $i=1,2$. It follows from (\[12.38\]) that $D \times_{\cp{1}} B^{'}$ will be in $H_4(X,{\mathbb Z})^{inv}$ if and only if \_[Bi]{}(D)=D,i=1,2. That is, if and only if D H\_2(B,[Z]{})\^[inv]{}. From the results of the previous section, we know that the classes $f, n_1+o_2, i$ and $M$ given in (\[iM\]) generate $H_2(B,{\mathbb Z})^{inv}$. Similarly, it follows from (\[12.39\]) that $B \times_{\cp{1}} D^{'}$ will be in $H_4(X,{\mathbb Z})^{inv}$ if and only if \_[B\^[’]{}i]{}(D\^[’]{})=D\^[’]{},i=1,2. That is D\^[’]{} H\_2(B\^[’]{},[Z]{})\^[inv]{}, which is generated by the classes $f^{'}, n_1^{'}+o_2^{'}, i^{'}$ and $M^{'}$ given in (\[iM’\]). It is clear, then, that a set of generators of $H_4(X,{\mathbb Z})^{inv}$ is given by \[12.44\] {D\_[inv]{}}\_ B\^[’]{},B \_ {D\_[inv]{}\^[’]{}}, where {D\_[inv]{}}={f, n\_1+o\_2, i, M} and {D\_[inv]{}\^[’]{}}={f\^[’]{}, n\^[’]{}\_1+o\^[’]{}\_2, i\^[’]{}, M\^[’]{}}. Note that it follows from (\[12.36\]) that there are only seven, not eight, classes in (\[12.44\]). Another convenient way of writing the generators (\[12.44\]) is as \^[’-1]{}(f),\^[’-1]{}(n\_1+o\_2),\^[’-1]{}(i),\^[’-1]{}(M),\^[-1]{}(n\_1\^[’]{}+o\_2\^[’]{}),\^[-1]{}(i\^[’]{}),\^[-1]{}(M\^[’]{}). These classes all descend under the modding out of $X$ by ${\mathbb Z}_2 \times {\mathbb Z}_2 $ to form a set of seven generators of $H_4(Z,{\mathbb Z})$. We conclude that H\_4(Z, [Z]{})=7 and, therefore, that $Z$ admits the homology classes required to produce anomaly free, three family particle physics theories.
$$\xymatrix{
& & & 1 & & & \\
& & 0 & &0 & & \\
&0 & & 7 & &0 & \\
1 & &7 & &7 & &1 &. \\
&0 & & 7 & &0 & \\
& &0 & &0 & & \\
& & & 1 & & & \\
}$$
From this result, it is straightforward to compute the Hodge numbers of $Z$. First, note that \[12.47\] H\_[|]{}\^[1,1]{}(Z)H\^2(Z,[Z]{})H\_4(Z,[Z]{}). Then, \[12.48\] h\^[1,1]{}\_Z=\_[C]{} H\_[|]{}\^[1,1]{}(Z)= \_[C]{} H\_4(Z,[C]{})=H\_4(Z,[Z]{})\^[inv]{}=7. The remaining Hodge number, $h_Z^{2,1}$, can be computed using (Z)=\_[i=0]{}\^[6]{} (-1)\^i \_[C]{}H\^i(Z,[C]{}), the fact that (Z)=c\_3(Z)==0 and the structure of the Hodge diamond. We find that 0=1+h\^[1,1]{}\_Z-1-2h\^[2,1]{}\_Z-1+h\^[1,1]{}\_Z+1 which, using (\[12.48\]), implies h\^[2,1]{}\_Z=7. Therefore, the Hodge diamond of $Z$ has the form given in Figure \[hZ\]. Using this, Poincaré duality and (\[kahler\]) it is now easy to read off the dimensions of $H_i(Z,{\mathbb Z})$ for $i=0,\dots,6$.
Acknowledgments {#acknowledgments .unnumbered}
===============
René Reinbacher and Burt Ovrut are supported in part by DOE under contract No. DE-AC02-76-ER-03071 and the NSF Focused Research Grant DMS 0139799. In addition, René Reinbacher wishes to acknowledge partial support from an SAS Dissertation Fellowship and a Daimler-Benz Fellowship. Tony Pantev is supported in part by NSF grants DMS 0099715 and DMS 0139799 and an A.P. Sloan Research Fellowship.
[OPR]{}
P. Hořava and E. Witten, Heterotic and Type I String Dynamics from Eleven Dimensions, (1996) 506.
P. Hořava and E. Witten, Eleven-Dimensional Supergravity on a Manifold with Boundary, (1996) 94.
E. Witten, Strong Coupling Expansion Of Calabi-Yau Compactification, (1996) 135.
A. Lukas, B. A. Ovrut, K.S. Stelle and D. Waldram, The Universe as a Domain Wall, D59 (1999) 086001.
A. Lukas, B. A. Ovrut, K.S. Stelle and D. Waldram, Heterotic M–theory in Five Dimensions, B552 (1999) 246-290.
A. Lukas, B. Ovrut, and D. Waldram, Non-standard embedding and five-branes in heterotic M-Theory, B552 (1999) 246-290.
M. F. Atiyah, N.J. Hitchin, V. G. Drinfeld, Y. I. Manin, Construction of instantons A65 (1996) 185.
R. Friedman, J. Morgan, E. Witten, Vector Bundles And F Theory, 187 (1997) 679-743
R. Friedman, J. Morgan, E. Witten Vector Bundles over Elliptic Fibrations, alg-geom/9709029.
R. Donagi, A. Lukas, B. Ovrut, and D. Waldram, Non-Perturbative Vacua and Particle Physics in M-Theory, 9905 (1999) 018.
R. Donagi, A. Lukas, B. Ovrut, and D. Waldram, Holomorphic Vector Bundles and Non-Perturbative Vacua in M-Theory, , 9906 (1999) 034.
R. Donagi, B. Ovrut, T. Pantev, and D. Waldram, Spectral involutions on rational elliptic surfaces, 5 (2002) 499-561, math.AG/0008011.
R. Donagi, B. Ovrut, T. Pantev, and D. Waldram, Standard-[M]{}odel bundles, 5 (2002) 563-615, math.AG/0008010.
R. Donagi, B. Ovrut, T. Pantev, and D. Waldram, Standard-[M]{}odel bundles on non-simply connected [C]{}alabi-[Y]{}au threefolds, , 0108 (2001) 053, hep-th/0008008.
R. Donagi, B. Ovrut, T. Pantev, and D. Waldram, Standard Models from Heterotic M-theory, , 5 (2002) 93-137.
C. Schoen, On fiber products of rational elliptic surfaces with section, , 197(2):177–199, 1988.
E. Witten, Symmetry Breaking Patterns in Superstring Theory, B258 (1985) 75-100.
E. Witten, private communication.
R. Thomas, Examples of bundles on Calabi-Yau 3-folds for string theory compactifications, 4 (2000) 231-247.
B. Ovrut, T. Pantev, R. Reinbacher, Torus-Fibered Calabi-Yau Threefolds with Non-Trivial Fundamental Group, UPR 1015-T, hep-th/0212221.
R. Donagi, B. Ovrut, T. Pantev, R. Reinbacher, Holomorphic Vector Bundles, Calabi-Yau Spaces with Non-Trivial Homotopy and the Standard Model, .
R. Donagi, B. Ovrut, T. Pantev, R. Reinbacher. Standard-like Models, Nucleon Decay and SU(4) Instantons, .
R. Donagi, A. Lukas, B. Ovrut,, R. Reinbacher. Symmetry breaking pattern by Wilson loops in heterotic M-theory, .
R. Donagi, B. Ovrut, T. Pantev, R. Reinbacher, Stable Vector Bundles on Non-Simply Connected Calabi-Yau Spaces, .
Griffiths & Harris, Principles of Algebraic Geometry, , Published 1994.
K. Kodaira, On compact analytic surfaces, , 78:1–40, 1963.
[^1]: It is helpful at this point to discuss our notation in more detail. On any fibered surface $B$, a section $\xi \in \Gamma(B)$ will determine a homology class, also denoted by $\xi$, in $H_2(B,{\mathbb Z})$. There is a natural notion of addition and inverse in $H_2(B,{\mathbb Z})$ which we denote everywhere in this paper by $+$ and $-$. For generic fibered surfaces, however, there is no group law on $\Gamma(B)$. When $B$ is elliptically fibered, a notion of addition and inverse on $\Gamma(B)$ does exist. To distinguish this from the group law on $H_2(B,{\mathbb Z})$, we denote the group law on $\Gamma(B)$ by ${\stackrel{.}{+}}$ and ${\stackrel{.}{-}}$.
[^2]: To be precise, the set $\{f,n_1+o_2,i,M\}$ generates a sub-lattice of maximal rank in $H_2(B,{\mathbb Z})^{inv}$. They are, in fact, generators of $H_2(B,{\mathbb Q})^{inv}$ and not $H_2(B,{\mathbb Z})^{inv}$. Be that as is may, for our purposes in this paper and in [@dopr-iii; @dopr-ii], these classes are the most convenient. First of all, since they generate a lattice of the same rank as $H_2(B,{\mathbb Z})^{inv}$, they demonstrate that $H_2(B,{\mathbb Z})^{inv}$ is non-empty and of rank four. Secondly, they have convenient intersection numbers which simplify the construction of holomorphic vector bundles in [@dopr-iii; @dopr-ii]. Since this subtlety does not play a role in either this paper or [@dopr-iii; @dopr-ii], here and henceforth we refer to them simply as the generators of $H_2(B,{\mathbb{Z}})^{inv}$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'For $m,n \in {\mathbb{N}}$, $m\geq 1$ and a given function $f : {\mathbb{R}}^m{\longrightarrow}{\mathbb{R}}$, the *polynomial interpolation problem* (PIP) is to determine a *unisolvent node set* $P_{m,n} \subseteq {\mathbb{R}}^m$ of $N(m,n):=|P_{m,n}|=\binom{m+n}{n}$ points and the uniquely defined polynomial $Q_{m,n,f}\in \Pi_{m,n}$ in $m$ variables of degree $\deg(Q_{m,n,f})\leq n \in {\mathbb{N}}$ that fits $f$ on $P_{m,n}$, i.e., $Q_{m,n,f}(p) = f(p)$, $\forall\, p \in P_{m,n}$. For $m=1$ the solution to the PIP is well known. In higher dimensions, however, no closed framework was available. We here present a generalization of the classic Newton interpolation from one-dimensional to arbitrary-dimensional spaces. Further we formulate an algorithm, termed PIP-SOLVER, based on a *multivariate divided difference scheme* that computes the solution $Q_{m,n,f}$ in ${\mathcal{O}}\big(N(m,n)^2\big)$ time using ${\mathcal{O}}\big(N(m,n)\big)$ memory. Further, we introduce unisolvent *Newton-Chebyshev nodes* and show that these nodes avoid *Runge’s phenomenon* in the sense that arbitrary periodic *Sobolev functions* $f \in H^k(\Omega,{\mathbb{R}}) \subsetneq C^0(\Omega,{\mathbb{R}})$, $\Omega =[-1,1]^m$ of regularity $k >m/2$ can be uniformly approximated, i.e., $ \lim_{n\rightarrow \infty}{||\,}f -Q_{m,n,f} {\,||}_{C^0(\Omega)}= 0$. Numerical experiments demonstrate the computational performance and approximation accuracy of the PIP-SOLVER in practice. We expect the presented results to be relevant for many applications, including numerical solvers, quadrature, non-linear optimization, polynomial regression, adaptive sampling, Bayesian inference, and spectral analysis.'
author:
- Michael Hecht
- 'Karl B. Hoffmann'
- 'Bevan L. Cheeseman'
- 'Ivo F. Sbalzarini'
bibliography:
- 'Ref.bib'
title: Multivariate Newton Interpolation
---
Introduction
============
In scientific computing, the problem of interpolating a function $f : {\mathbb{R}}^m {\longrightarrow}{\mathbb{R}}$, $m \in {\mathbb{N}}$, is ubiquitous. Because of their simple differentiation and integration, as well as their pleasant vector space structure, polynomials $Q \in {\mathbb{R}}[x_1,\dots,x_m]$ in $m$ variables of degree $\deg(Q) \leq n$, $m,n \in {\mathbb{N}}$, are a standard choice as interpolants and are fundamental in ordinary differential equation (ODE) and partial differential equation (PDE) solvers. For an overview, we refer to [@NA2] and [@Stoer]. Thus, the *polynomial interpolation problem* (PIP) is one of the most fundamental problems in numerical analysis and scientific computing, formulated as:
\[PolyInt\] Let $m,n \in {\mathbb{N}}$ and $f: {\mathbb{R}}^m{\longrightarrow}{\mathbb{R}}$ be a computable function.
1. Choose $N(m,n)$ nodes $P_{m,n}=\{p_1,\dots,p_{N(m,n)}\} \subseteq {\mathbb{R}}^m$ such that $P_{m,n}$ is unisolvent, i.e., for every $f$ there is exactly one polynomial $Q_{m,n,f} \in \Pi_{m,n}$ fitting $f$ on $P_{m,n}$ as $Q_{m,n,f}(p)=f(p)$ for all $p\in P_{m,n}$.
2. Determine $Q_{m,n,f}$ once a unisolvent node set $P_{m,n}$ has been chosen.
Here, $\Pi_{m,n}$ is the vector space of polynomials $Q \in {\mathbb{R}}[x_1,\dots,x_m]$ in $m$ variables of degree $\deg(Q)\leq n$. Every $Q \in \Pi_{m,n}$ has $N(m,n):=\binom{m+n}{n}$ monomials/coefficients.
The function $f : {\mathbb{R}}^m {\longrightarrow}{\mathbb{R}}$ is assumed to be *computable* in the sense that for any $x \in {\mathbb{R}}^m$ the value of $f(x)$ can be evaluated in ${\mathcal{O}}(1)$ time, where ${\mathcal{O}}(\cdot)$ is the Bachmann-Landau symbol. Note that if $f \in \Pi_{m,n}$, then $Q_{m,n,f}=f$. If $f : {\mathbb{R}}{\longrightarrow}{\mathbb{R}}$ is an arbitrary continuous function in one variable, then the *Weierstrass approximation theorem* [@weier] guarantees that $f$ can be approximated by *Bernstein polynomials*, i.e., there exist polynomials $Q_{n} \in \Pi_{1,n}$ such that $${||\,}f- Q_n {\,||}_{C^0(\Omega)}:= \sup_{x \in \Omega}|f(x) -Q_n(x)| \xrightarrow[n\rightarrow \infty]{} 0$$ for every fixed bounded domain $\Omega\subseteq {\mathbb{R}}$. However, these polynomials $Q_n$ are not necessarily interpolating, i.e., they need not satisfy $Q_{n}(p) = f(p)$ for all $p \in P_{1,n}$. This additional requirement restricts the space of polynomials available to approximate $f$, which is the cause of Runge’s phenomenon [@AT; @faber; @runge]: If the unisolvent nodes $P_{1,n}$ are chosen independently of $f$, the sequence of interpolants $Q_{n,f}\in \Pi_{1,n}$, $n\in {\mathbb{N}}$, can diverge away from $f$, i.e., there exist at least one $f \in C^0(\Omega)$ for which ${||\,}f- Q_{n,f} {\,||}_{C^0(\Omega)} \centernot{\longrightarrow}0 $ for $n{\longrightarrow}\infty$. Therefore, the approximation ability of an interpolation method depends on the choice of $P_{m,n}$ and has to be characterized. More precisely:
\[Task\] Let $m,n \in {\mathbb{N}}$ and $\Omega \subseteq {\mathbb{R}}^m$ be a bounded domain. Further let $S_{m,n}: C^0(\Omega,{\mathbb{R}}){\longrightarrow}\Pi_{m,n}$ be an interpolation solver, i.e., $S_{m,n}(f)=Q_{m,n,f}$ solves the PIP for any given function $f : \Omega \subseteq {\mathbb{R}}^m {\longrightarrow}{\mathbb{R}}$, choosing the unisolvent nodes $P_{m,n} \subseteq \Omega$ independently of $f$.
1. What is the set $\mathcal{A}(S_{m,n}) \subseteq C^0(\Omega,{\mathbb{R}})$ of continuous functions that can be approximated by $S_{m,n}$, i.e., for which ${||\,}f -S_{m,n}(f){\,||}_{C^0(\Omega)} \xrightarrow[n\rightarrow \infty]{}0 , \forall f\in\mathcal{A}$ ?
2. How large is the absolute approximation error $${\varepsilon}(f,S_{m,n}):={||\,}f- S_{m,n}(f) {\,||}_{C^0(\Omega)} \, ?$$
3. How large is the relative approximation error $\mu_{m,n} \geq 1$, such that $${||\,}f- S_{m,n}(f) {\,||}_{C^0(\Omega)} \leq \mu_{m,n}{||\,}f- Q^*_{m,n} {\,||}_{C^0(\Omega)} \quad \forall \, f \in C^0(\Omega,{\mathbb{R}})\,$$ where $Q^*_{m,n} \in \Pi_{m,n}$ is an optimal approximation that minimizes the $C^0$-distance to $f$ ?
The one-dimensional PIP () can be solved efficiently in ${\mathcal{O}}(N(1,n)^2)={\mathcal{O}}(n^2)$ and numerically accurately by various algorithms based on *Newton or Lagrange Interpolation* [@Stoer; @berrut; @gautschi; @LIP]. Though, even in one dimension, there is no efficient general method for finding an optimal node set $P_{1,m}$ that minimizes Runge’s phenomenon; sub-optimal node sets can be generated efficiently. For $\Omega=[-1,1]$, a classic choice of sub-optimal nodes are the roots of the *Chebyshev polynomials* $$\label{cheby}
{\mathrm{Cheb}}_{n}={\left}\{p_k \in {\mathbb{R}}{:}p_k = \cos{\left}(\frac{2k-1}{2(n+1)}\pi{\right})\,, k=1,\dots,n+1{\right}\}\,,$$ which are optimal up to a factor depending on the -th derivative of $f$. Therefore, the approximation ability of *Chebyshev nodes* is characterized by:
\[Cheb\] Let $S_n: C^0(\Omega,{\mathbb{R}}){\longrightarrow}\Pi_n$, $\Omega=[-1,1]$, be an interpolation solver that uses ${\mathrm{Cheb}}_{n}$ as unisolvent nodes, i.e., $S_n(f)=Q_{n,f}$, $Q_{n,f}({\mathrm{Cheb}}_{n})=f({\mathrm{Cheb}}_{n})$.
1. The set of approximable functions satisfies $C^1(\Omega,{\mathbb{R}}) \subsetneq \mathcal{A}(\Omega,{\mathbb{R}}) \subsetneq C^0(\Omega,{\mathbb{R}})$.
2. If $f \in C^{n+1}(\Omega,{\mathbb{R}})$ and $x \in \Omega$ then the absolute approximation error at $x$ can be bounded by $$| f- S_{n}(f)(x) | \leq \frac{1}{(n+1)!}f^{(n+1)}(\xi_x) \prod_{i=1}^{n+1} (x-p_i) \leq \frac{f^{(n+1)}(\xi)}{2^n(n+1)!}\,, \xi_x \in \Omega\,,$$
3. The relative approximation error can be bounded by the *Lebesgue function* $\Lambda({\mathrm{Cheb}}_n)$ as: $${||\,}f- S_{m,n}(f) {\,||}_{C^0(\Omega)} \leq (1+\Lambda({\mathrm{Cheb}}_n) ){||\,}f- Q^*_{n} {\,||}_{C^0(\Omega)} \,,$$ where $\Lambda({\mathrm{Cheb}}_n) = \frac{2}{\pi}\big(\log(n) + \gamma + \log(8/\pi)\big) + {\mathcal{O}}(1/n^2) $ and $\gamma \sim 0.5772$ is Euler’s constant [@brutman].
This provides a pleasing solution to the PIP in one dimension. We refer to [@AT; @gautschi; @brutman; @burden; @Stewart] for further details and proofs.
However, many data sets in scientific computing are functions of more than one variable and therefore require multivariate polynomial interpolation. A solution to the multivariate PIP, complete with a computationally efficient and numerically stable algorithm for computing it, has so far not been available. This is at least partly due to the fact that an efficiently computable characterization of unisolvent nodes in arbitrary dimensions was not known. In one dimension, unisolvent node sets are characterized by the simple requirement that nodes have to be pairwise different, which can obviously be asserted in ${\mathcal{O}}(n^2)$ time. While some unisolvent node sets have been proposed in dimensions [@Bos; @Erb; @Gasca2000; @Chung], generalizations to arbitrary dimensions had a complexity that prohibited their practical implementation [@FAST; @Gasca2000]. Available PIP solvers in higher dimensions therefore use randomly generated node sets and then determine the interpolation polynomial by numerically inverting the resulting *multivariate Vandermonde matrix* $V_{m,n} \in {\mathbb{R}}^{N(m,n)\times N(m,n)}$ in order to compute the coefficients $C_{m,n}$ of $Q_{m,n,f}$ in normal form. Using random node sets is possible due to the famous theorem of Sard, which was later generalized by Smale [@smale]. This theorem states that the superset ${\mathcal{P}}_{m,n}$ of all unisolvent node sets for $m,n \in {\mathbb{N}}$ is a set of second category in the sense of Baire. Therefore, any randomly generated node set is unisolvent with probability 1. Using random nodes, however, can never guarantee numerical stability of the solver, nor can it control Runge’s phenomenon. In addition, numerical inversion of the multivariate Vandermonde matrix $V_{m,n}$ in practice incurs a computational cost larger than that of Newton interpolation. In principle, inverting the Vandermonde matrix should be as complex as solving the PIP, due to the special structure of $V_{m,n}$. However, this structure depends on the choice of unisolvent nodes $P_{m,n}$. Therefore, using random node sets prevents exploiting this structure, so that solving the system of linear equations $$V_{m,n}(P_{m,n})C_{m,n}=F\,, \quad F= (f(p_1),\dots,f(p_{N(m,n)}))^T \in {\mathbb{R}}^{N(m,n)}$$ still requires the same computational time as *general matrix inversion*. A lower complexity bound for inverting general matrices of size $N \times N\,, N\in {\mathbb{N}}$, is given by ${\mathcal{O}}(N^2\log(N))$ [@cormen; @raz; @tveit]. The fastest known algorithm for general matrix inversion is the *Coppersmith-Winograd algorithm* [@COPPER], which requires runtime in ${\mathcal{O}}(N^{2.3728639})$ in its most efficient version [@FAST]. However, the *Coppersmith-Winograd algorithm* is rarely used in practice, because it is only advantageous for matrices so large that memory problems become prevalent on modern hardware [@robinson]. The algorithm that is mostly used in practice is the *Strassen algorithm* [@strassen], which runs in ${\mathcal{O}}(N^{2.807355})$. Alternatively, one can perform *Gaussian elimination* in ${\mathcal{O}}(N^3)$. All of these approaches require ${\mathcal{O}}(N^2)$ memory to store the matrix. Moreover, the numerical robustness and accuracy of these approaches is limited by the condition number of the Vandermonde matrix, which again depends on the choice of $P_{m,n}$ and can therefore not be controlled when using random node sets. Hence, previous approaches to polynomial interpolation become inaccurate or intractable with increasing $N(m,n)$.
Statement of Contribution
-------------------------
Though the relevance of multivariate interpolation is undisputed and feasible interpolation schemes in dimension 1 are known since the 18$^{th}$ century, there was so far no general interpolation scheme for multivariate functions that can guarantee to solve the PIP numerically robustly and accurately, controls Runge’s phenomenon, and is as computationally efficient as Newton interpolation.
We here close this gap by introducing the notion of *multivariate Newton polynomials* and a characterization of *unisolvent nodes* $P_{m,n}$ such that all of the following is true:
1. The unisolvent nodes $P_{m,n}$ generate a well-conditioned Vandermonde matrix such that the interpolant can be computed numerically robustly and accurately.
2. The unisolvent nodes $P_{m,n}$ generate a *lower triangular* Vandermonde matrix with respect to the multivariate Newton polynomials, such that the interpolant can be computed in quadratic time using a *multivariate divided difference scheme*.
3. The unisolvent nodes $P_{m,n}$ control Runge’s phenomenon in the sense that Theorem \[Cheb\] generalizes to dimension $m$ and thereby answers Question \[Task\].
We practically implement the solution as an algorithm, called PIP-SOLVER, which, allows to interpolate arbitrary multivariate functions $ f : {\mathbb{R}}^m {\longrightarrow}{\mathbb{R}}$ numerically accurately. The PIP-SOLVER algorithm is based an a recursive decomposition approach, which yields a *recursive generator* of *unisolvent node sets* and the associated *multivariate divided difference scheme* to determine the interpolant $Q_{m,n,f}$. We show that the PIP-SOLVER requires runtime in ${\mathcal{O}}(N(m,n)^2)$ and memory in ${\mathcal{O}}(N(m,n))$, which matches the performance of Newton interpolation for . Further, the multivariate Newton–form of $Q_{m,n,f}$ allows evaluating and differentiating $Q_{m,n,f}$ in linear time.
Moreover, we show that all Sobolev functions $f \in H^k(\Omega,{\mathbb{R}}) \subseteq C^0(\Omega,{\mathbb{R}})$, $\Omega =[-1,1]^m$ for $k >m/2$ that are periodic on $\Omega$ can be approximated uniformly $${||\,}f- Q_{m,n,f}{\,||}_{C^0(\Omega)} \xrightarrow[n\rightarrow \infty]{}0 \quad \text{for $\Omega=[-1,1]^m$}$$ by using unisolvent nodes of *generalized Newton-Chebyshev type*. Analogous to the one-dimensional case, we provide bounds for the relative and absolute approximation errors. Note that this is probably the best result one can obtain, since uniform approximation of non-continuous functions by polynomials and extrapolating functions $f : \Omega' \supseteq \Omega {\longrightarrow}{\mathbb{R}}$ with $P_{m,n}\subseteq \Omega$ are ill-posed problems. Because $H^k(\Omega,{\mathbb{R}})$ contains non-continuous functions for $k \leq m/2$, the Sobolev functions $ f \in H^k(\Omega,{\mathbb{R}})$ for $k>m/2$ are the largest *Hilbert space* of continuous functions that can be approximated by polynomials in the $C^0$–sense. Further, the Sobolev space, $H^k(\Omega,{\mathbb{R}})$, $k >m/2$, densely contains all *smooth* and thus all *analytical functions*. The periodic boundary condition can always be achieved by rescaling. For example, consider $\widetilde f $ periodic on $\Omega_2=[-2,2]^m$ with $\widetilde f_{|\Omega }=f$ and rescale $\Omega_2$ to $\Omega$. For these and other reasons, $H^k(\Omega,{\mathbb{R}})$, $k >m/2$, is the pivotal analytical choice in scientific computing [@Jost].
Paper Outline
-------------
After stating the main results of this article in section \[main\], we recapitulate classic one-dimensional Newton interpolation in section \[Newt\] and previous multivariate interpolation schemes in section \[FormVander\]. In section \[Decomp\] we provide the mathematical proofs for our results. In section \[RPIP\], we present the *multivariate divided difference scheme* for generalized *multidimensional Newton nodes* as an efficient solver algorithm. In section \[APT\], we show that multivariate Newton-Chebyshev nodes allow proving the uniform approximation result for periodic Sobolev functions $f \in H^k(\Omega,{\mathbb{R}})$, $k>m/2$, and the bounds on the approximation errors. The results are then demonstrated in numerical experiments in section \[EX\]. Finally, we sketch a few of the possible applications in section \[APP\] and conclude in section \[Conc\].
Main Results {#main}
============
We summarize the main results of this article in the following three Theorems. They are based on the realization that the PIP can be decomposed into sub-problems of lesser dimension or lesser degree. Recursion then decomposes the problem along a binary tree whose leafs are associated with constants or zero-dimensional sub-problems, which only require evaluating $f$ to be solved. The decomposition is based on the notion of multivariate Newton polynomials, which we introduce in section \[RPIP\], generalizing Newton nodes to arbitrary dimensions and defining a multivariate Newton basis.
\[I\] Let $m,n \in {\mathbb{N}}$ and $f : {\mathbb{R}}^m {\longrightarrow}{\mathbb{R}}$ be a given function. Then, there exists an algorithm with runtime complexity ${\mathcal{O}}\big(N(\mbox{m,n})^2\big)$ requiring ${\mathcal{O}}(N(\mbox{m,n}))$ memory that computes:
1. A unisolvent node set $P_{m,n} \subseteq {\mathbb{R}}^m$ and the coefficients of the corresponding interpolation polynomial $Q_{m,n,f} \in \Pi_{m,n}$ in normal form.
2. A unisolvent set $P_{m,n} \subseteq {\mathbb{R}}^m$ of multidimensional Newton nodes and the coefficients of the corresponding interpolation polynomial $Q_{m,n,f} \in \Pi_{m,n}$ in multivariate Newton form.
\[M1\]
We generalize classic results for Newton polynomials to show that the present multivariate Newton form is a good choice for multivariate polynomial interpolation. In particular, it enables efficient and accurate numerical computations according to the second main result:
\[II\] Let $m,n \in {\mathbb{N}}$ and a set of multidimensional Newton nodes $P_{m,n}$ be given. Let further $Q \in \Pi_{m,n}$ be a polynomial given in multivariate Newton form. Then, there exist algorithms that compute:
1. The value of $Q(x_0)$ for any $x_0 \in {\mathbb{R}}^m$ in ${\mathcal{O}}(N(m,n))$;
2. The partial derivative $\partial_{x_i} Q|_{x_0}$ for any $i \in \{1,\dots,m\}$ and $x_0 \in {\mathbb{R}}^m$ in ${\mathcal{O}}(nN(m,n))$;
3. The integral $\int_{\Omega}Q(x)\mathrm{d}x$ for any hypercube $\Omega \subseteq {\mathbb{R}}^m$ with runtime complexity in ${\mathcal{O}}(nN(m,n))$. \[M2\]
Studying the approximation properties of multivariate Newton interpolation, we show that periodic Sobolev functions $f \in H^k(\Omega,{\mathbb{R}})$ for $k >m/2$ can be approximated uniformly, and we provide upper bounds for the corresponding absolute and relative approximation errors. To do so let $A_{m,n}\subseteq {\mathbb{N}}^m$ be the set of all multi–indices $\alpha=(\alpha_1,\dots,\alpha_m) \in {\mathbb{N}}^m$ with $|\alpha|=\sum_{i=1}^m\alpha_i \leq n$ .
\[III\] Let $m,n \in {\mathbb{N}}$, $\Omega= [-1,1]^m \subseteq {\mathbb{R}}^m$. Let $S_{m,n}:H^k(\Omega,{\mathbb{R}}) {\longrightarrow}\Pi_{m,n}$, $k>m/2$, denote an interpolation operator with respect to interpolation nodes $P_{m,n}$.
1. If $P_{m,n}$ are of Newton type, then the Lebesgue function $$\begin{aligned}
\Lambda(P_{m,n}, H^k(\Omega)) &= \sup_{f\in H^k(\Omega,{\mathbb{R}}), ||f||_{H^k(\Omega)} =1} ||S_{m,n}(f)||_{C^0(\Omega)}\,, \end{aligned}$$ given by the operator norm of $S_{m,n}$ is bounded by $$\begin{aligned}
\Lambda(P_{m,n}, H^k(\Omega)) &\leq \prod_{i=1}^m\Lambda(P_i,H^{k-(m-1)/2}(\Omega))\,.
\end{aligned}$$
2. If $P_{m,n}$ are of Newton-Chebyshev type, then $\Lambda(P_{m,n}, H^k(\Omega)) \in {\mathcal{O}}(\log(n)^m)$ and every $f \in H^k(\Omega,{\mathbb{R}})$, $k >m/2$, being periodic on $\Omega =[-1,1]^m$ can be approximated uniformly: $${||\,}f -S_{m,n}(f){\,||}_{C^0(\Omega)} \xrightarrow[n\rightarrow \infty]{}0 \,.$$
3. If $f \in C^{n+1}(\Omega,{\mathbb{R}})$ and $P_{m,n}$ are of Newton–type then for every $\alpha \in A_{m,n}\setminus A_{m,n-1}$, $i \in\{1,\dots,m\}$ and every $x \in \Omega$ there is $\xi_x\in \Omega$ such that $$| f(x)- S_{m,n}(f)(x) |\leq \frac{1}{\alpha_i!} \partial^{\alpha_i+1}_{x_i}f(\xi) |N_\alpha(x)|\,,$$ where $N_\alpha(x) = \prod_{i=1}^m\prod_{j=1}^{\alpha_i}(x_i-p_{i,j})$, $x =(x_1,\dots,x_m)$, $p_{i,j}\in {\mathbb{P}}_i$. If $P_{m,n}$ are of Newton-Chebyshev–type then we can further estimate $$| f(x)- S_{m,n}(f)(x) | \leq \frac{1}{2^{\alpha_i}\alpha_i! }\partial^{\alpha_i+1}_{x_i}f(\xi_x)\,.$$
4. For any unisolvent node set $P_{m,n}$ and every $f \in H^k(\Omega,{\mathbb{R}})$ the relative error is bounded by $${||\,}f- S_{m,n}(f) {\,||}_{C^0(\Omega)} \leq \big(1 + \Lambda(P_{m,n}, H^k(\Omega))\big){||\,}f- Q^*_{m,n} {\,||}_{C^0(\Omega)}\,,$$ where $Q^*_{m,n}$ is an optimal approximation that minimizes the $C^0$-distance to $f$.
$\Lambda$ is estimated in $(i)$, but statement $(iv)$ holds for all unisolvent node sets. For Chebyshev nodes, $(iv)$ provides a tight estimate.
Newton Interpolation in Dimension 1 {#Newt}
===================================
Since our $m$-dimensional generalization is a natural extension of *Newton interpolation* in 1D, we first review some classic results in the special case of dimension , with more detailed discussions available elsewhere [@atkinson; @endre; @LIP; @powel; @NA2; @Stewart; @Stoer; @walston]. In one dimension, the Vandermonde matrix $V_{n}(P_n)$ takes the classical form $$V_{n}(P_{n}) = {\left}(\begin{array}{cccc}
1 & p_1 & \cdots & p_1^{ n} \\
\vdots & \vdots & \ddots& \vdots \\
1 & p_{n+1} & \cdots & p_{n+1}^{n}\\
\end{array}{\right})
\,.$$ It is well known that for this matrix to be regular, the nodes $p_1,\dots,p_{n+1}$ have to be pairwise distinct. As we see later, this is also a sufficient condition for the nodes $P_{n}$ to be unisolvent. In light of this fact, we observe that $V_{n}$ induces a vector space isomorphism $\varphi : {\mathbb{R}}^{n+1} {\longrightarrow}\Pi_{n}$, where $\varphi(v)$ is the polynomial with normal-form coefficients $C_n \in {\mathbb{R}}^{n+1}$ such that $V_{n}C_n=v$, $v \in {\mathbb{R}}^{n+1}$. The polynomials $$\label{NewtPoly}
N_i(x) = \prod_{j=1}^{i} (x-p_j) \,, \quad i =0,\dots,n$$ are the *Newton Basis* (NB) of $\Pi_{n}$. When represented with respect to the NB, the Vandermonde matrix becomes a lower triangular matrix of the form $$V_{\mathrm{NB},n}(P_{n}) = {\left}(\begin{array}{cccc}
1 & 0 & \cdots &0 \\
1 & (p_2-p_1) & \cdots &0\\
1 & (p_3 -p_1) & (p_3-p_1)(p_3-p_2)& \vdots \\
\vdots & \vdots & \ddots& \vdots \\
1 & (p_{n+1} -p_1) & \cdots & \prod_{j=1}^n(p_{n+1}-p_j)\\
\end{array}{\right})
\,.$$ Thus, the solution of the PIP in dimension $m=1$ with respect to the NB can directly be obtained as: $$Q_{n,f}(x) = \sum_{i=0}^{n}c_iN_i(x)= \sum_{i=0}^{n}c_i \prod_{j=1}^{i} (x-p_j) \,.$$ More efficiently, the *Aitken-Neville* or *divided difference* scheme determines the coefficients $c_i$ by setting $$[p_1]f:=f(p_1)\,,\quad \quad [p_i,\dots,p_j]f:= \frac{[p_i,\dots,p_{j-1}]f-[p_{i+1},\dots,p_j]f}{x_j-x_i}\,, \,\, j\geq i$$ and proving that $c_{i-1}=[p_1,\dots,p_{i}]f$. Indeed one can verify by induction that $Q_{n,f}$ satisfies the fitting condition $f(P_n)=Q_{n,f}(P_n)$, which uniquely determines $Q_{n,f}$ up to its representation. The induced recursion can be illustrated as follows: $$\label{DDS}
\begin{array}{crcrccrcrc}
[p_1]f \\
& \searrow \\{}
[p_2]f & \rightarrow & [p_1,p_2]f \\
& \searrow & & \searrow \\{}
[p_3]f & \rightarrow & [p_2,p_3]f & \rightarrow & [p_1,p_2,p_3]f \\{}
\vdots & \vdots & \vdots & \vdots & \vdots &\ddots \\{}
& \searrow & & \searrow & & & \searrow \\{}
[p_{n+1}]f & \rightarrow & [p_{n},p_{n+1}]f & \rightarrow & [p_{n-1},p_{n},p_{n+1}]f
& \cdots & \rightarrow & [p_1\ldots p_{n+1}]f . \\
\end{array}$$
We summarize some facts about the classic 1D scheme [@gautschi; @Stoer], as they are important for the generalization to arbitrary dimensions:
\[bN\] Let $n \in {\mathbb{N}}$ and $f \in C^0(\Omega,{\mathbb{R}})\,, \Omega=[-1,1]$
1. The divided difference scheme allows to numerically stable determine the $\mbox{n+1}$ coefficients $c_0,\dots,c_n \in {\mathbb{R}}$ of the interpolant $Q_{n,f}$ with respect to the Newton–basis in ${\mathcal{O}}(n^2)$.
2. Given the interpolant $Q_{n,f}$ in its Newton–form, the *Horner-scheme* allows to compute the value of $Q_{n,f}(x)$ in ${\mathcal{O}}(n)$ for any $x \in \Omega$.
3. Given the interpolant $Q_{n,f}$ in its Newton–form, the value of the derivative $\frac{d}{dx}Q_{n,f}(x)$ can be computed in ${\mathcal{O}}(n)$ for any $x \in \Omega$.
Due to its relationship with *Taylor expansion*, *Newton interpolation* has several advantages over other interpolation schemes. For instance, it easily extends to higher degrees without recomputing the coefficients of lower-order terms, i.e., by incrementally computing higher-order terms. Furthermore, evaluation of the Newton interpolant and its derivatives is straightforward due to their simple form. However, the recursive nature of the divided difference scheme also has certain disadvantages compared to *Lagrange interpolation*, which computes the result at once. We refer to [@berrut] and [@werner] for further discussions of the properties of these schemes.
In any case, due to the uniqueness of the interpolation polynomial, the question how well the interpolant approximates a given function is independent of the specific interpolation scheme, up to numerical rounding errors. Instead, the approximation quality only depends on the choice of interpolation nodes. The approximation quality in the 1D case is given in Theorem .
Multivariate Polynomials {#FormVander}
========================
We follow the notation of P.J. Olver [@MultiVander] to extend our considerations to the general multivariate case. For $m,n\in {\mathbb{N}}$, we denote by $\Pi_{m,n}\subseteq {\mathbb{R}}[x_1,\dots,x_m]$ the vector space of all real polynomials in $m$ variables of bounded degree $n$. While the normal form of a polynomial possesses $N(m,n):=\binom{m+n}{n}$ monomials, the number of monomials of degree $k$ is given by $M(m,k):=\binom{m+k}{m} - \binom{m+k-1}{m}$. We enumerate the coefficients $c_0,\dots,c_{N(m,n)}$ of $Q \in \Pi_{m,n}$ in its normal form as follows: $$\begin{aligned}
Q(x) &= c_0 + c_1 x_1 + \cdots + c_mx_{m}+ c_{m+1}x_1^2+c_{m+2}x_1x_2 \nonumber \\
& + \cdots + c_{2m}x_1x_m + c_{2m+1}x_2^2+ \cdots + c_{M(m,n-1)+1}x_1^n \nonumber \\
& +c_{M(m,n-1)+2}x_1^{n-1}x_2+ \cdots + c_{N(m,n)-2}x_{m-1}x_m^{n-1}\nonumber \\
& + c_{N(m,n)-1}x_m^n\,. \label{C}\end{aligned}$$ We assume that and consider $A_{m,n}:={\left}\{\alpha \in {\mathbb{N}}^m : |\alpha| \leq n{\right}\}$ the set of all multi-indices of order $|\alpha|:= \sum_{k=1}^m\alpha_k \leq n$. The multi-index is used to address the monomials of a multivariate polynomial. For a vector $x=(x_1,\dots,x_m)$ and $\alpha \in A_{m,n}$ we define $$x^\alpha:= x_1^{\alpha_1}\cdots x_m^{\alpha_m}$$ and we order the $M(m,k)$ multi-indices of order $k$ with respect to lexicographical order, i.e., $\alpha_1 =(k,0,\dots,0)$, $\alpha_2=(k-1,1,0\dots,0)$,$\dots$, $\alpha_{M(m,k)} =(0,\dots,0,k)$. The $k$-th *symmetric power* $x^{\odot k }=(x_1^{\odot k },\dots,x_{M(m,k)}^{\odot k }) \in {\mathbb{R}}^{M(m,k)}$ is defined by the entries $$x^{\odot k }_i:= x^{\alpha_i} \,, \quad i=1,\dots,M(m,k)\,.$$ Thus, $x^{\odot 0 }_i =1$, $x^{\odot 1 }_i =x_i$.
For given and a set of nodes $P=\{p_1,\dots,p_{N(m,n)}\} \subseteq {\mathbb{R}}^m$ with $p_i = (p_{1,i}, \dots,p_{m,i})$, we define the *multivariate Vandermonde matrix* $V_{m,n}(P)$ by $$V_{m,n}(P) = {\left}(\begin{array}{ccccc}
1 & p_1 & p_1^{\odot 2} & \cdots & p_1^{\odot n} \\
1 & p_2 &p_2^{\odot 2} & \cdots & p_2^{\odot n}\\
1 & p_3 & p_3^{\odot 2}& \cdots & p_3^{\odot n}\\
1 & \vdots &\vdots & \ddots& \vdots \\
1 & p_{N(m,n)} & p_{N(m,n)}^{\odot 2} & \cdots & p_{N(m,n)}^{\odot n}\\
\end{array}{\right})
\,.$$
We call a set of nodes $P =\{p_1,\dots,p_{N(m,n)}\} \subseteq {\mathbb{R}}^m$ *unisolvent* if and only if the Vandermonde matrix $V_{m,n}(P)$ is regular. Thus, the set of all unisolvent node sets is given by ${\mathcal{P}}_{m,n}={\left}\{ P \subseteq {\mathbb{R}}^m : \det\big(V_{m,n}(P)\big)\not = 0{\right}\}$, which is an open set in ${\mathbb{R}}^m$, since $P \mapsto \det\big(V_{m,n}(P)\big)$ is a continuous function.
Given a real-valued function $f :R^m {\longrightarrow}{\mathbb{R}}$, and assuming that unisolvent nodes $P=\{p_1,\dots p_{N(m,n)}\} \subseteq {\mathbb{R}}^m$ exist, the linear system of equations $V_{m,n}(P)x = F$, with $$x=\big(x_1,\dots,x_{N(m,n)}\big)^\mathsf{T}\,\,\,\text{and}\,\,\, F=\big(f(p_1),\dots,f(p_{N(m,n)})\big)^\mathsf{T} ,$$ has the unique solution $$x = V_{m,n}(P)^{-1}F \,.$$ Thus, by using $c_i:=x_i$ as the coefficients of $Q\in\Pi_{m,n}$, enumerated as in Eq. , we have uniquely determined the solution of Problem \[PolyInt\]. The essential difficulty herein lies in finding a good unisolvent node set posing a well-conditioned problem and in solving $V_{m,n}(P)x =F$ accurately and efficiently. Therefore, we state a well-known result, first mentioned in [@Chung] and then again in [@MultiVander], saying:
\[generic\] Let $m,n \in {\mathbb{N}}$ and $P_{m,n} \subseteq {\mathbb{R}}^m$, $\# P_{m,n} = N(m,n)$. The Vandermonde matrix $V_{m,n}(P_{m,n})$ is regular if and only if the nodes $P_{m,n}$ do not belong to a common algebraic hypersurface of degree $\leq n$, i.e., if there exists no polynomial $Q \in \Pi_{m,n}$, such that $ Q(p) = 0 $ for all $p \in P_{m,n}$.
Indeed $P_{m,n}$ is unisolvent if and only if the homogeneous Vandermonde problem $$V_{m,n}(P)x = 0$$ has no non-trivial solution, which is equivalent to the fact that there is no polynomial $Q \in \Pi_{m,n}\setminus\{0\}$ generating a hypersurface $W =Q^{-1}(0)$ of degree $\deg(W)\leq n$ with $P_{m,n} \subseteq W$.
The geometric interpretation of Proposition \[generic\] is crucial but not constructive, i.e., it classifies node sets but yields no algorithm or scheme to construct unisolvent node sets. In the next section, we therefore further develop our understanding of unisolvent node sets in order to provide a construction algorithm, which is our first main result.
Multivariate Interpolation {#Decomp}
==========================
We provide theorems that allow solving the PIP in a way that fulfills requirements $(R1)$ and $(R2)$ stated in the introduction. We start by considering the multivariate interpolation problem on hyperplanes and then show that the general multivariate PIP can be decomposed into such hyperplane problems.
Interpolation on Hyperplanes {#SC}
----------------------------
We consider the PIP on certain subplanes and therefore define:
Let $\tau : {\mathbb{R}}^m {\longrightarrow}{\mathbb{R}}^m$ be given by $\tau(x) = Ax +b $, where $A \in {\mathbb{R}}^{m\times m}$ is a full-rank matrix, i.e. $\mathrm{rank}(A) =m$, and $b \in {\mathbb{R}}^m$. Then we call $\tau$ an *affine transformation* on ${\mathbb{R}}^m$.
\[proj\] For every ordered tuple of integers $i_1,\dots,i_k \in {\mathbb{N}}$, $i_{q} < i_p$ if $1 \leq q <p \leq k$, we consider $$H_{i_1,\dots, i_k}={\left}\{(x_1,\dots,x_n) \in {\mathbb{R}}^m {:}x_j = 0 \,\, \text{if}\,\, j \not \in \{i_1,\dots,i_k\}{\right}\}$$ the $k$-dimensional hyperplanes spanned by the $i_1,\dots,i_k$-th coordinates. We denote by $ \pi_{i_1,\dots, i_k} : {\mathbb{R}}^m {\longrightarrow}H_{i_1,\dots,i_k}$ and $i_{i_1,\dots, i_k} : {\mathbb{R}}^k \hookrightarrow {\mathbb{R}}^m$, with $i_{i_1,\dots, i_k}({\mathbb{R}}^k)=H_{i_1,\dots,i_k}$, the natural projections and embeddings. We denote by $\pi_{i_1,\dots,i_k}^* : {\mathbb{R}}[x_1,\dots,x_m] {\longrightarrow}{\mathbb{R}}[x_{i_1},\dots x_{i_k}]$ and $i^*_{i_1,\dots, i_k} : {\mathbb{R}}[y_{1},\dots y_{k}] \hookrightarrow {\mathbb{R}}[x_1,\dots,x_m]$ the induced projections and embeddings on the polynomial ring.
\[Hxi\] Let $m,n \in \mathbb{N}$, $\xi_1,\dots,\xi_m \in {\mathbb{R}}^m$ an orthonormal frame (i.e., ${\left}<\xi_i,\xi_j{\right}>=\delta_{ij}$, $ \forall 1\leq i,j\leq m$, where $\delta_{ij}$ denotes the Kronecker symbol), and $b \in {\mathbb{R}}^m$. For ${\mathcal{I}}\subseteq \{1,\dots,m\}$ we consider the hyperplane $$H_{{\mathcal{I}},\xi,b}:= {\left}\{x \in {\mathbb{R}}^m {:}{\left}<x-b,\xi_i{\right}>=0\,, \forall \, i \in {\mathcal{I}}{\right}\}\,.$$ Given a function $f : {\mathbb{R}}^m {\longrightarrow}{\mathbb{R}}$ we say that a set of nodes $P_{k,n} \subseteq H_{{\mathcal{I}},\xi,b}$ and a polynomial $Q \in \Pi_{m,n}$ *solve the PIP* with respect to $k=\dim H_{\mathcal{I}}$, $n\in {\mathbb{N}}$ and $f$ on $H_{\mathcal{I}}$ if and only if $Q(p) =f(p)$ for all $p \in P_{k,n}$ and whenever there is a $Q'\in \Pi_{m,n}$ and $Q'(p)=f(p)$ for all $p \in P_{k,n}$, then $Q'(x)=Q(x)$ for all $x \in H_{{\mathcal{I}},\xi,b}$.
Let $H \subseteq {\mathbb{R}}^m$ be a hyperplane of dimension $k\in {\mathbb{N}}$ and $\tau : {\mathbb{R}}^m{\longrightarrow}{\mathbb{R}}^m$ an affine transformation such that $\tau(H) =H_{1,\dots,k}$. Then we denote by $$\tau^*: {\mathbb{R}}[x_1,\dots,x_m] {\longrightarrow}{\mathbb{R}}[x_1,\dots,x_m]$$ the *induced transformation* on the polynomial ring defined over the monomials as: $$\tau^*(x_i) = \eta_1x_1+\cdots + \eta_mx_{m}\quad \text{with} \quad \eta=(\eta_1,\dots,\eta_m) = \tau(e_i)\,,$$ where $e_1,\dots,e_m$ denotes the standard Cartesian basis of ${\mathbb{R}}^m$.
\[TT\] Let $m,n \in {\mathbb{N}}$ and $P_{m,n}$ be a unisolvent node set with respect to $(m,n)$. Further let $\tau : {\mathbb{R}}^m {\longrightarrow}{\mathbb{R}}^m$, $\tau(x) =Ax +b$ be an affine transformation. Then, $\tau(P_{m,n})$ is also a unisolvent node set with respect to $(m,n)$.
Assume there exists a polynomial such that . Then setting yields a non–zero polynomial of with $\deg(Q_1)\leq n$ and $$Q_1(P_{m,n}) = \tau^*(Q_0)(P_{m,n})
= Q_0(\tau(P_{m,n}))=0\,,$$ which is a contradiction to $P_{m,n}$ being unisolvent. Therefore, $\tau(P_{m,n})$ must be unisolvent.
Next we use the ingredients above to decompose the PIP.
Decomposition of the Multivariate PIP onto Hyperplanes
------------------------------------------------------
Section \[SC\] provides the ingredients to prove our first key result. To avoid confusion, we note that a $0$-dimensional hyperplane $H \subseteq {\mathbb{R}}^m$, $m \geq 1$, is given by a single point $H=\{\mathrm{point}\}$, such that the PIP on $H$ is solved by evaluating $f$ at that point. Vice versa, the zeroth-order PIP (i.e., a constant) with respect to $f : {\mathbb{R}}^m{\longrightarrow}{\mathbb{R}}$ and $n=0$ is solved by evaluating $f$ at some point $q \in {\mathbb{R}}^m$.
Let $m,n \in {\mathbb{N}}$, $m \geq 1$, and $P \subseteq {\mathbb{R}}^m$, such that:
1. There exists a hyperplane $H \subseteq {\mathbb{R}}^m$ of co-dimension 1 such that $P_1:= P \cap H$ satisfies $\#P_1 = N(m-1,n)$ and is unisolvent with respect to $H$, i.e., by identifying $H\cong {\mathbb{R}}^{m-1}$ the Vandermonde matrix $V_{m-1,n}(P_1)$ is regular.
2. The set $P_2= P \setminus H$ satisfies $\#P_2 = N(m,n-1)$ and is unisolvent with respect to $\mbox{(m,n-1)}$, i.e., the Vandermonde matrix $V_{m,n-1}(P_2)$ is regular.
Then $P$ is a unisolvent node set. \[GN\]
Due to Lemma \[TT\], unisolvent node sets remain unisolvent under affine transformation. Thus, by choosing the appropriate transformation $\tau$, we can assume w.l.o.g. that $H= H_{1,\dots, m-1}$. For any polynomial $Q \in {\mathbb{R}}[x_1,\dots,x_m]$ there holds $Q\big(\pi_{1,\dots,m-1}(p)\big) = \pi_{1,\dots,m-1}^*\big(Q(p)\big)$ for all $p \in H$. Thus, by $(i)$ we observe that whenever there is a $Q \in {\mathbb{R}}[x_1,\dots,x_m]$ with $Q(P_1) =0$, then $\deg(Q) \geq \deg\big(\pi_{1,\dots,m-1}^*(Q)\big) > n$. Or, if $\deg\big(\pi_{1,\dots,m-1}^*(Q)\big) \leq \deg(Q) \leq n$, we consider the polynomial $\bar Q_1:= Q - i^*_{1,\dots,m-1}\big(\pi_{1,\dots,m-1}^*(Q)\big)$, which consists of all monomials sharing the variable $x_m$ and $\bar Q_2:= Q -\bar Q_1$ consisting of all monomials not sharing $x_m$. We claim that $\bar Q_2 = 0$. Certainly, $\bar Q_1(x) = 0$ for all $x \in H$. Since $P_1$ is unisolvent there are $p \in P_1$ with $\bar Q_2(p)= \pi_{1,\dots,m-1}^*(\bar Q_2(p)) \not= 0$ implying $Q(p) \not = 0$, which contradicts our assumption on $Q$ and therefore yields $\bar Q_2 = 0$ as claimed. In light of this fact, $Q$ can be decomposed into polynomials $Q=Q_1\cdot Q_2 \quad \text{where} \quad Q_2(x_1,\dots,x_m) =x_m\,.$ Since $\deg(Q_1)\leq n-1$ and $P_2$ is unisolvent we have that $P_2 \not \subseteq Q_1^{-1}(0)$. At the same time, $P_2 \cap H =\emptyset $ implies that $Q_2(p) \not = 0$ for all $p \in P_2$. Hence, there is $p \in P_2$ with $Q(p)\not =0$, proving the theorem due to Proposition \[generic\].
The question arises whether the decomposition of unisolvent nodes given in Theorem \[GN\] allows us to decompose the PIP into smaller, and therefore simpler, sub-problems. Indeed we obtain:
\[SV\] Let $m,n \in {\mathbb{N}}$, $m \geq 1$, $f : {\mathbb{R}}^m{\longrightarrow}{\mathbb{R}}$ be a computable function, $H \subseteq {\mathbb{R}}^m$ a hyperplane of co-dimension 1, $Q_H \in \Pi_{m,1}$ a polynomial satisfying $Q_H^{-1}(0) =H$, and $P_{m,n}=P_1\cup P_2\subseteq {\mathbb{R}}^m$ such that $(i)$ and $(ii)$ of Theorem \[GN\] hold with respect to $H$. Require $Q_1,Q_2 \in \Pi_{m,n}$ to be such that:
1. $Q_1$ solves the PIP with respect to $f$ and $P_1$ on $H$.
2. and $Q_2$ solves the PIP with respect to , $f_1:=(f-Q_1)/Q_H$, and $P_2=P\setminus H$ on ${\mathbb{R}}^m$.
Then, $Q_{m,n,f} =Q_1+Q_HQ_2$ is the uniquely determined polynomial of $\deg(Q)\leq n$ that solves the PIP with respect to $f$ and $P_{m,n}$ on ${\mathbb{R}}^m$.
By our assumptions on $Q_H$ and $Q_1$ we have that $Q(x)=Q_1(x)$ $\forall x \in H$ and therefore $Q(p) =f(p)$ $\forall p \in P_1 = P_{m,n} \cap H$. At the same time, $Q_H(x) \not =0$ for all $x \not \in H$. Therefore, $f_1$ is well defined on $P_2$ and $Q(p)= Q_1(p) + f_1(p)Q_H(p)=f(p)$ $\forall p \in P_2=P_{m,n}\setminus H$. Hence, $\deg(Q)\leq n$, and $Q$ solves the PIP with respect to $f$ and a unisolvent set of nodes $P_{m,n}\subseteq {\mathbb{R}}^m$. Thus, $Q_{m,n,f}$ is the unique solution of the PIP with respect to $P_{m,n}$ and $f$.
Note that $Q_H$ can be constructed by choosing a (usually unit) normal vector $\nu \in {\mathbb{R}}^m$ onto $H$ and a vector $b \in H$, setting $$Q_H(x) = \nu\cdot(x - b)\,.$$ Indeed, this guarantees that $Q_H(x)=0$ for all $x \in H$ and $Q_H(x)\not =0$ for all $x \in {\mathbb{R}}^m\setminus H$.
As a consequence of Theorems \[GN\] and \[SV\], we can close this section by stating the first part of Main Result I and delivering its proof.
Let $m,n \in {\mathbb{N}}$, and $f : {\mathbb{R}}^m {\longrightarrow}{\mathbb{R}}$ be a computable function. Then, there exists an algorithm with runtime complexity in ${\mathcal{O}}\big(N(m,n)^2\big)$, requiring storage in ${\mathcal{O}}\big(N(m,n)\big)$, that computes:
1. A unisolvent node set $P_{m,n} \subseteq {\mathbb{R}}^m$.
2. The normal form coefficients $c_0,\dots,c_{N(m,n)-1}$ of the interpolation polynomial $Q_{m,n,f} \in \Pi_{m,n}$ with $Q_{m,n,f}(p) = f(p)\,, \,\forall p \in P_{m,n}$.
\[mainT\]
We start by proving $(i)$ and $(ii)$ with respect to the runtime complexity. To do so, we claim that there is a constant $C\in {\mathbb{R}}^+$ and an algorithm computing $(i)$ and $(ii)$ in less than computation steps. To prove this claim, we argue by induction on $N(m,n)$. If $N(m,n)=1$, then $m=0$ or $n=0$. Thus, interpolating $f$ is given by evaluating $f$ at one single node, which can be done in ${\mathcal{O}}(1)$ by our assumption on $f$.
Now let $N(m,n)>1$. Then $m>0$ and we choose $\nu,b\in {\mathbb{R}}^m$ with ${||\,}\nu {\,||}=1$ and consider the hyperplane $H=Q_H^{-1}(0)$, $Q_H(x)=\nu(x-b)$. By identifying $H\cong {\mathbb{R}}^{m-1}$, induction yields that we can determine a set of unisolvent nodes $P_1\subseteq H$ in less than computation steps. Induction also yields that a set $P_2 \subseteq {\mathbb{R}}^m$ of unisolvent nodes can be determined with respect to $n-1$ in less than computation steps. By translating $P_2$ with $\lambda \nu$, i.e., setting $P_2'=P_2 + \lambda \nu $, $\lambda \in {\mathbb{R}}$, we can guarantee that $P_2 \cap H = \emptyset$. Hence, the union $P_{m,n}=P_1 \cup P_2'$ of the corresponding unisolvent sets of nodes is also unisolvent due to Theorem \[GN\], proving $(i)$.
To show $(ii)$ we have to compute the coefficients of $Q_{m,n,f}$ in normal form. By induction, a polynomial $Q_1 \in \Pi_{m,n}$ that solves the PIP with respect to $f$ and $P_1$ on $H$ can be determined in normal form in less than $CN(m-1,n)^2$ steps. We consider $f_1' : {\mathbb{R}}^m {\longrightarrow}{\mathbb{R}}$ with $f_1'(x):= \big(f(x-\lambda \nu) - Q_1(x-\lambda \nu)\big)/Q_H(x-\lambda \nu)$. By induction, we can compute $Q_2\in\Pi_{m,n-1}$ in less than $CN(m,n-1)^2$ computation steps, such that $Q_2$ solves the PIP with respect to $f_1'$ and $P_2$, . Thus, $Q_2$ solves the PIP with respect to $f_1(x)= \big(f(x) - Q_1(x)\big)/Q_H(x) = f_1'(x+\lambda \nu)$, $P_2'$, and . Due to Theorem \[SV\], we have that $Q_1+Q_HQ_2$ solves the PIP with respect to $f$ and $P_{m,n}$. It remains to bound the steps required for computing the normal form of $Q_1+Q_HQ_2$. The bottleneck herein lies in the computation of $Q_HQ_2$, which requires $C_2(m+1)N(m,n-1)$, $C_2 \in {\mathbb{R}}^+$ computation steps. Observe that $m \leq N(m-1,n)$ for $n\geq 1$, which shows that $Q_1+Q_HQ_2$ can be computed in less than $C_3N(m-1,n)N(m,n-1)$, $C_3 \in {\mathbb{R}}^+$ computation steps. It is $N(m,n)=N(m-1,n)+N(m,n-1)$. Hence, by assuming $2C\geq C_3$, we have that $$C\big(N(m-1,n)^2 + N(m,n-1)^2\big) + C_3N(m-1,n)N(m,n-1) \leq CN(m,n)^2 \,,$$ proving $(ii)$.
We obtain the storage complexity by using an analogous induction argument. If $N(m,n)=1$ then $m=0$ or $n=0$ and $f: \{\mathrm{point}\}{\longrightarrow}{\mathbb{R}}$ is interpolated by evaluating $f$ at one point. Therefore we have to store that point yielding storage complexity ${\mathcal{O}}(1)$. If $N(m,n)>1$, using the same splitting of the problem as above, induction yields that there is $D \in {\mathbb{R}}^+$ such that we have to store at most $DN(m-1,n)$ and $DN(m,n-1)$ numbers for each of the two sub-problems, respectively. Altogether we need to store $D\big(N(m-1,n) +N(m,n-1)\big) = DN(m,n)$ numbers, proving the storage complexity.
So far, we have provided an existence result stating that the PIP can be solved efficiently in ${\mathcal{O}}(N(m,n)^2)$. The derivation of an actual algorithm based on the recursion implicitly used in Theorem \[mainT\] is given in the next section.
Recursive Decomposition of the PIP {#RPIP}
==================================
Based on the recursion expressed in Theorems \[GN\] and \[SV\], we can derive an efficient algorithm to compute solutions to the PIP in a numerically robust and computationally efficient way. The resulting algorithm derived hereafter, called PIP-SOLVER is based on a binary tree $T_{m,n}$ as a straightforward data structure resulting from the recursive problem decomposition. It also uses a *multivariate divided difference scheme* to solve the sub-problems, leading to better numerical robustness and accuracy than other approaches (comparisons will be given in Section \[EX\]).
Multivariate Newton Polynomials
--------------------------------
The essential data structure required to formulate multivariate Newton interpolation is given by the following binary tree:
Let $m,n \in \mathbb{N}$. We define a binary tree $T_{m,n}=({\mathcal{V}}_{m,n},{\mathcal{E}}_{m,n})$ with vertex labeling $\sigma : {\mathcal{V}}_{m,n}{\longrightarrow}{\mathbb{N}}\times {\mathbb{N}}$, $\sigma(v)=(\dim(v),\deg(v))$ as follows: We start with a root $\rho$ and set , . For a vertex $v \in {\mathcal{V}}_{m,n}$ with and we introduce a left child $l$ with label and a right child $r$ with label .
Furthermore, we consider the set $\Gamma_{m,n}$ of *leaf paths* $\gamma$ starting at $\rho$ and ending in a leaf $v_\gamma \in L_{m,n} \subset {\mathcal{V}}_{m,n}$ of $T_{m,n}$. For every leaf path $\gamma \in \Gamma_{m,n}$ we define $$[\gamma]=(k_1,\dots,k_m)^\mathsf{T} \in {\mathbb{N}}^{m}\,, \quad k_i = \# {\left}\{v \in \gamma{:}\dim(v)=i{\right}\}$$ to be its *descent vector*, which uniquely identifies the leaf path $\gamma$. \[TMN\]
Figure \[TMN\] illustrates the tree $T_{m,n}$ and the concept of a leaf path $\gamma$. As one can easily verify, the depth of $T_{m,n}$ is given by $\mathrm{depth}(T_{m,n})=m+n-1$, while the total number $\# L_{m,n}$ of leaves of $T_{m,n}$ is given by $N(m,n)$.
Next we introduce the multidimensional generalization of Newton nodes, given by a *non-uniform, affine-transformed, sparse T-grid*:
Given $m,n \in {\mathbb{N}}$ and a set of nodes $P_{m,n}\subseteq {\mathbb{R}}^m$ with $\#P_{m,n}=N(m,n)$, we say $P_{m,n}$ are [*multidimensional Newton nodes*]{} generated by $${\mathbb{P}}_{m,n}: = \oplus_{i=1}^m{\mathbb{P}}_i\,, \quad {\mathbb{P}}_i=\{p_{i,1},\dots,p_{i,n+1}\} \subseteq {\mathbb{R}}$$ if and only if there exists an affine transformation $\tau : {\mathbb{R}}^m {\longrightarrow}{\mathbb{R}}^m$ such that for every $p \in P_{m,n}$ there is exactly one $\gamma \in \Gamma_{m,n}$ with $[\gamma]=(k_1,\dots,k_m)^T\in {\mathbb{N}}^m$ such that: $$\tau(p)= (\bar p_{1,k_1},\dots,\bar p_{m,k_{m}})^\mathsf{T}\,, \,\text{where we set } \,\, \bar p_{k_l}= {\left}\{\begin{array}{ll}
p_{l,n+1} &\text{if}\,\, k_{l'}=0 \,,\,\,\forall \, l'<l\\
p_{l,k_l} & \text{else}\,.
\end{array}{\right}.$$ If $\tau={\mathrm{id}}_{{\mathbb{R}}^m}$, we call $P_{m,n}$ *canonical multidimensional Newton nodes*. \[NTN\]
Considering the special case of canonical multidimensional Newton nodes, and assuming ${\mathbb{P}}_i ={\mathbb{P}}_{i'}$ for all occurring $1 \leq i,i' \leq m$, an intuition of the definition in terms of sparse $T$-grids reveals. Indeed, $T$-grids are based on decomposing space along a binary tree, as we do here. Furthermore, the tree $T_{m,n}$ is a sparse binary tree, i.e., the full tree depth is only reached by few of the leaves.
Note that the $P_{m,n}$ can be recursively split into subsets $P_1,P_2 \subseteq P_{m,n}$ with $P_1\subseteq H$ for some hyperplane $H\subseteq {\mathbb{R}}^m$, such that the assumptions of Theorem \[GN\] are satisfied for every recursion step. Even more, the recursion satisfies the *defining property of Newton nodes*, which is that the projection $$\label{parallel}
\pi_H(P_2) \subseteq H \,\,\, \text{of $P_2$ onto}\,\,\, H\,\,\, \text{satisfies}\,\,\, \pi_H(P_2)\subseteq P_1\,.$$ This fact is going to be the key ingredient for the proof of Theorem \[DDST\].
Given $m,n \in {\mathbb{N}}$, $T_{m,n}$, $\Gamma_{m,n}$, and multidimensional Newton nodes $P_{m,n}$ generated by ${\mathbb{P}}_{m,n}$. For $\gamma \in \Gamma_{m,n}$ with descent vector $[\gamma]=(k_1,\dots,k_m)^\mathsf{T}\in {\mathbb{N}}^{m}$ and $x=(x_1,\dots,x_m)^\mathsf{T}\in {\mathbb{R}}^m$, we call $$N_{[\gamma]}(x)= \prod_{i=1}^m\prod_{l=1}^{k_i-1}(x_i-p_{i,l})$$ the [*multivariate Newton polynomials*]{} with respect to $P_{m,n}$.
Let $m,n \in {\mathbb{N}}$, $f: {\mathbb{R}}^m {\longrightarrow}{\mathbb{R}}$ be a function, and $P_{m,n}$ a set of multidimensional Newton nodes generated by ${\mathbb{P}}_{m,n}$. Then, there exist unique coefficients $c_{[\gamma]}\in {\mathbb{R}}$, $\gamma \in \Gamma_{m,n}$, such that: $$Q(x) = \sum_{\gamma \in \Gamma(m,n)} c_{[\gamma]} N_{[\gamma]}(x)$$ satisfies $Q(p)=f(p)$ for all $p \in P_{m,n}$. Since $Q \in \Pi_{m,n}$, $Q$ is the unique solution of the PIP with respect to $(m,n,f,P_{m,n})$, implying that $P_{m,n}$ is unisolvent. \[UniN\]
We argue by induction on $N(m,n)$. If $N(m,n)=1$ then $P_{m,n}=\{p\}$ consists of one point. Therefore, $Q(x)=f(p)$ and the claim holds. If $N(m,n)>1$, then we can assume w.l.o.g. that $P_{m,n}$ are canonical, i.e., that the affine transformation $\tau$ in Definition \[NTN\] is the identity, $\tau = {\mathrm{id}}_{{\mathbb{R}}^m}$ (see Lemma \[TT\] and Definition \[NTN\]). We consider $$P_1={\left}\{p=(p_1,\dots,p_{m})^\mathsf{T} \in P_{m,n} \subseteq {\mathbb{R}}^m {:}p_m=p_{m,1}\in {\mathbb{P}}_{m}{\right}\}\,,$$ $P_2 =P_{m,n}\setminus P_1$, and $H= {\left}\{x \in {\mathbb{R}}^m {:}x_m =p_{m,1}{\right}\}$. Now note that $P_1$ is a multidimensional Newton node set on $H$ w.r.t. and $P_2$ is a multivariate Newton node set w.r.t. on ${\mathbb{R}}^m$. Hence, by induction, $P_1,P_2$ are unisolvent. Thus, $P_1,P_2$, and $H$ satisfy the assumptions of Theorem \[GN\], which completes the proof.
Indeed, in dimension $m=1$ the $N_{[\gamma]}(x)$ correspond to the classical Newton polynomials from Eq. . Given canonical multidimensional Newton nodes $P_{m,n}$, and ordering the paths $\Gamma_{m,n}$ from left to right with respect to the leaf ordering of $T_{m,n}$, we observe that for the multidimensional Newton nodes $p_{[\gamma]}$ corresponding to $\gamma$ there holds $$N_{[\gamma]}(p_{[\gamma']})= 0 \quad \text{if}\,\,\, \gamma < \gamma'\,.$$ Thus, like in the 1D case, the associated Vandermonde matrix $V_{m,n}(P_{m,n})$ is of lower triangular form, yielding an alternative proof of Proposition \[UniN\] and a possibility of computing the coefficients $c_{[\gamma]}$ in ${\mathcal{O}}(N(m,n)^2)$ steps. Just like in the 1D case, a divided difference scheme can be formulated for computing the coefficients $c_{[\gamma]}$ even more efficiently.
Alternative Formulation of the 1D Divided Differences Scheme
------------------------------------------------------------
We consider the case of dimension $m=1$. In this case, the tree $T_{1,n}$ has the special form illustrated in Figure \[T1D\]. Choosing pairwise disjoint nodes $P_n=\{p_1,\dots,p_{n+1}\}\subseteq {\mathbb{R}}$ and enumerating the leaf paths $\gamma _i \in \Gamma_{1,n}$ from left to right, we observe that $$N_{[\gamma_i]}=N_{i}(x)=\prod_{j=1}^i (x_j-p_j) \qquad i =0,\dots,n\,,$$ where the Newton polynomials $N_i$ were defined in Eq. . Though it is trivial, the nodes $P_n$ can be seen as lying on the $0$-dimensional planes $H_i=\{x \in {\mathbb{R}}^1 {:}x= p_i\}=\{p_i\}$. While Theorem \[GN\] is fulfilled anyhow in this case, this interpretation allows to recursively set up the statement of Theorem \[SV\]. That is, for a given computable function $f: {\mathbb{R}}{\longrightarrow}{\mathbb{R}}$ and $k=1,\dots,n$ we set: $$\begin{aligned}
f_0(x)&=f(x)\,, & c_0&=f_0(p_1) \,, \nonumber \\
f_k(x)&=\frac{f_{k-1}(x)-f_{k-1}(p_{k})}{x-p_k} \,, & c_k&=f_k(p_{k+1})\,. \label{DDS2}\end{aligned}$$ Indeed the computation of the $c_k$ is similar to the divided difference scheme illustrated in Eq. . However, the two schemes are not identical. In our alternative formulation, the differences and divisors appear in a different order. Still, both schemes compute the same result, as can easily be verified by hand.
Let $f : {\mathbb{R}}{\longrightarrow}{\mathbb{R}}$ be a computable function, and $T_{1,n}$, $\Gamma_{1,n}$ be given. Further let $P_n=\{p_1,\dots,p_{n+1}\} \subseteq {\mathbb{R}}$ be a set of pairwise disjoint nodes. Then, the polynomial $$Q(x)=\sum_{k=0}^{n}c_kN_{[\gamma_k]}(x)$$ that solves the 1-dimensional PIP with respect to can be determined in ${\mathcal{O}}(n^2)$ computation steps. \[DDSC\]
By induction on $n$, the statement follows directly from Proposition \[UniN\] and Theorem \[SV\]. The runtime complexity can be shown analogously to the proof of Theorem \[mainT\].
The Multivariate Divided Differences Scheme
-------------------------------------------
In the general case of dimension $m \in {\mathbb{N}}$, we introduce the following notions:
Let $m,n \in {\mathbb{N}}$, and $T_{m,n}$, $\Gamma_{m,n}$ be given. Let $\gamma \in \Gamma_{m,n}$ with descent vector $[\gamma]=(k_1,\dots,k_m)\in {\mathbb{N}}^m$. We consider $1\leq i\leq m$ with $k_i >1$. If the leaf $l_\gamma$ where the path encoded by $\gamma$ ends has degree $\deg(l_\gamma)>0$, then we define $$\begin{aligned}
\Delta_i(\gamma):=
{\left}\{\delta \in \Gamma_{m,n} {:}[\delta]=(l_1,\dots,l_m) \,\,\text{with} \,\,\begin{array}{ll}
1\leq l_i < k_i \,, & \\
l_j = k_j\,, & \text{for} \,\, j\not = i
\end{array}\,\, {\right}\} .
\end{aligned}$$ If the leaf degree $\deg(l_\gamma)=0$ then for $i=1$ we consider $$\Delta_1^0(\gamma):=
{\left}\{\delta \in \Gamma_{m,n} {:}[\delta]=(l_1,k_2\dots,k_m) \,, \quad l_1 =1,\dots,k_1-1{\right}\}$$ and for $i >1$ we define $i_0=\min {\left}(\{j < i {:}k_j >1\} \cup \{1\}{\right})$ and $$\begin{aligned}
\Delta_i^0(\gamma):=
{\left}\{\delta \in \Gamma_{m,n} {:}[\delta]=(l_1,\dots,l_m) \,\text{with} \,\begin{array}{ll}
l_{i} < k_{i}\,, & \\
l_j = k_j\,, & j\not = i,i_0 \\
l_{i_0} = k_{i}-l_i+k_{i_0}\,, &
\end{array}\,\, {\right}\}\,.
\end{aligned}$$ Finally, we order the paths in $\Delta_i(\gamma),\Delta_i^0(\gamma)$ with respect to the leaf ordering of $T_{m,n}$ from left to right.
Indeed, the $\Delta_i(\gamma),\Delta^0_i(\gamma)$ correspond to all multidimensional Newton nodes that are required to compute the coefficient of $N_{[\gamma]}$. For instance, the leafs $l_\delta$ of $ \delta \in \Delta_{i}^0(\gamma)$, $i>1$, have degree $\deg(l_\delta)=0$ and are given by parallel translation of $l_\gamma$ along the multidimensional Newton grid $P_{m,n}$ in dimension $i$. More precisely, we define:
Let $m,n \in {\mathbb{N}}$, $f : {\mathbb{R}}^m {\longrightarrow}{\mathbb{R}}$ be a computable function, $P_{m,n}$ be a set of multidimensional Newton nodes, and $T_{m,n},\Gamma_{m,n}$ be given. For $\gamma \in \Gamma_{m,n}$ with $[\gamma]=(k_1,\dots,k_m)$, we consider $\Delta_i(\gamma),\Delta_i(\gamma)^0=\{\delta_{1,i},\dots,\delta_{k_i,i}\}$, and $m_\gamma \in {\mathbb{N}}$ such that $J_{\gamma}=\{ j_1,\dots,j_{m_{\gamma}}\} \subseteq \{1,\dots,m\}$ contains all indices with $k_{i} >1$ for $i \in J_{\gamma}$. By reordering if necessary we assume that $J_\gamma$ is ordered in reverse direction, i.e., $ j_{l} > j_{l'}$ if $l<l'$ for all $l,l' \in \{1,\dots,m_{\gamma}\}$. We denote by $p_{h,i}\in P_{m,n}$ the nodes corresponding to $\delta_{h,i}$ and define $$\begin{aligned}
F_{\gamma,1}^0 & =f(p_{j_1,1}) \,, & F_{\gamma,1}^k &:=\frac{F_{\gamma,1}^{k-1}(p_{j_1,k}) - F_{\gamma,1}^{k-1}(p_{j_1,k-1})}{(p_{j_1,k-1}-p_{j_1,k})_{j_1} }\,, \,\,\,1\leq k\leq k_{j_1}\,, \\
F_{\gamma,i}^0 & :=F_{\gamma,i-1}^{k_{j_i}} \,, & F_{\gamma,l}^k &:=\frac{F_{\gamma,l}^{k-1}(p_{j_l,k}) - F_{\gamma,l}^{k-1}(p_{j_l,k-1})}{(p_{j_l,k-1}-p_{j_l,k})_{j_l} }\,,
\begin{array}{l}
1\leq k\leq k_{j_l}\,,\\
1\leq l \leq m_\gamma\,,
\end{array} \end{aligned}$$ where $(p_{j_l,k-1}-p_{j_l,k})_{j_l}$ denotes the $j_l$-th coordinate/component of $(p_{j_l,k-1}-p_{j_l,k})$. \[MVDD\]
This definition yields the alternative divided difference scheme given in Eq. for the special case $m=1$.
Let $m,n \in {\mathbb{N}}$, $f : {\mathbb{R}}^m {\longrightarrow}{\mathbb{R}}$ be a computable function, $T_{m,n}$, $\Gamma_{m,n}$ be given, and $P_{m,n}\subseteq {\mathbb{R}}^m$ be a set of canonical multidimensional Newton nodes generated by ${\mathbb{P}}_{m,n}$. Then, the polynomial $$Q(x)=\sum_{\gamma \in \Gamma_{m,n}}c_{[\gamma]}N_{[\gamma]}(x)\,, \quad c_{[\gamma]}= F_{\gamma,m_\gamma}^{k_{j_{m_\gamma}}}\,,$$ with $[\gamma]=(k_1,\dots,k_{j_{m_\gamma}},\dots,k_{j_1},\dots,k_m)$ solves the PIP with respect to $(m,n,f,$ $P_{m,n})$ and can be computed in ${\mathcal{O}}(N(m,n)^2)$ operations. \[DDST\]
We argue by induction on . For $m=1, n\in {\mathbb{N}}$, the statement is already proven in Corollary \[DDSC\]. Thus, we proceed by induction on $m$. If $m>1$, then we consider $\Gamma_1=\{\gamma_1,\dots,\gamma_{N(m-1,n)} \}\subseteq \Gamma_{m,n}$ with $[\gamma_h]=(k_1,\dots,k_m)\in {\mathbb{N}}^{m} $ and $k_{m}=1$, $h=1,\dots,N(m-1,n)$. Furthermore, we denote by $P_1\subseteq P_{m,n}$ the corresponding nodes and $\Gamma_2 = \Gamma_{m,n}\setminus \Gamma_1$, $P_2 = P_{m,n}\setminus P_1$. In addition, we consider the sub-trees $T_1\cong T_{m-1,n}$, $T_2\cong T_{m,n-1}$ spanned by $\Gamma_1$, $\Gamma_2$, respectively. Note that the $m$-th coordinate is constant, i.e., for $p,p'\in P_1$ we have $p_m=p_m'=p_{m,1} \in {\mathbb{P}}_{m,n}$. Since the hyperplane $H= {\left}\{x \in {\mathbb{R}}^m {:}x_m =p_{m,1}{\right}\}$ can be identified with ${\mathbb{R}}^{m-1}$ by shifting the last coordinate to $0$, induction yields that $$c_{[\gamma_1]} = F_{\gamma_1,m_{\gamma_1}}^{k_{j_{m_{\gamma_1}}}},\dots,c_{[\gamma_{N(m-1,n)}]}= F_{\gamma_{N(m-1,n)},m_{\gamma_{N(m-1,n)}}}^{k_{j_{m_{\gamma_{N(m-1,n)}}}}}\,.$$ These are the uniquely determined coefficients solving the PIP with respect to on $H$. We further follow Theorem \[SV\] and set $f_1(x)=(f(x)-Q_1(x))/(x_m-p_{m,1})$ with $Q_1(x) = \sum_{\gamma \in \Gamma_1}c_{[\gamma]}N_{[\gamma]}(x)$. The defining property of multidimensional Newton nodes is their parallelism, i.e., $\pi_H(P_{2}) \subseteq P_1$, where $\pi_H$ denotes the orthogonal projection onto $H$. Furthermore, we observe that $Q_1(x)$ is constant in direction perpendicular to $H$. Therefore, $$Q_1(p) = Q_1(\pi_H(p))=f(\pi_H(p)) \quad \text{for all}\quad p \in P_2\,.$$ Thus, $$\begin{aligned}
f_1(p) &= (f(p)-f(\pi_H(p)))/(\pi_H(p)_m - p_{m,1})\\
&=(f(p)-f(p_1,\dots, p_{m-1},p_{m,1}))/(p_m - p_{m,1})\,.\end{aligned}$$ Following Definition \[MVDD\], we observe that $
d_{[\delta_1]} = F_{\delta_1,K_{\delta_1}}^{k_{j_{m_{\delta_1}}}},\dots,d_{[\delta_{N(m-1,n)}]}$ $= F_{\delta_{N(m,n-1)},K_{\delta_{N(m,n-1)}}}^{k_{j_{m_{\delta_{N(m,n-1)}}}}}
$ yielding $$d_{[\delta_1]}=c_{[\gamma_{N(m-1,n)+1}]}\,\dots,d_{[\delta_{N(m-1,n)}]}= c_{[\gamma_{N(m,n)}]}\label{ce}$$ Hence, by induction, the $d_{[\delta]}$, $\delta\in \Gamma_{m,n-1}$, are the uniquely determined coefficients solving the PIP w.r.t. on ${\mathbb{R}}^m$. Moreover, due to Eq. , we have: $$Q_2(x) = \sum_{\delta\in T_{m-1,n}}d_{[\delta]}N_{[\delta]}(x) = \frac{1}{(x_m-p_{m,1})}\sum_{\gamma\in \Gamma_{2}}c_{[\gamma]}N_{[\gamma]}(x)\,.$$ Now Theorem \[SV\] yields that $$Q(x)= Q_1(x)+Q_H(x)Q_2(x)= \sum_{\gamma \in \Gamma_{m,n}}c_{[\gamma]}N_{[\gamma]}(x)\,, \quad Q_H(x) = x_m-p_{m,1},$$ solves the PIP w.r.t. , which proves the statement. The runtime complexity follows from the proof of Theorem \[mainT\].
By considering $f'=f \circ \tau^{-1}$ with $\tau$ from Definition \[NTN\], Theorem \[DDST\] also extends to the case of non-canonical multidimensional Newton nodes, i.e., to affine transformations of canonical multidimensional Newton nodes. Together, Theorems \[SV\] and \[DDST\] and Proposition \[UniN\] prove Main Result I as stated in Theorem \[I\] in Section \[main\]. We next characterize some basic properties of multivariate Newton polynomials and how they can be exploited for basic computations.
Properties of Multivariate Newton Polynomials
---------------------------------------------
Analogously to its 1D version, the multivariate divided difference scheme provides an efficient and numerical robust method for interpolating functions by polynomials. We generalize some classical facts of the 1D case, yielding Main Result \[II\], here restated with a bit more detail as:
Let $m,n \in {\mathbb{N}}$, $T_{m,n}$, $\Gamma_{m,n}$, and a set of canonical multidimensional Newton nodes $P_{m,n}$ be given. Let further $Q \in \Pi_{m,n}$, $Q(x)= \sum_{\gamma \in \Gamma_{m,n}}c_{[\gamma]}N_{[\gamma]}(x)$ be a polynomial given in multivariate Newton form w.r.t. $T_{m,n}$, $P_{m,n}$. Then, there exist algorithms with runtime complexity in ${\mathcal{O}}(N(m,n))$ that compute:
1. The value of $Q(x_0)$ for any $x_0 \in {\mathbb{R}}^m$ in ${\mathcal{O}}(N(m,n))$;
2. The partial derivative $\partial_{x_i} Q|_{x_0}$ for any $i \in \{1,\dots,m\}$ and $x_0 \in {\mathbb{R}}^m$ in ${\mathcal{O}}(nN(m,n))$;
3. The integral $\int_{\Omega}Q(x)\mathrm{d}x$ for any hypercube $\Omega \subseteq {\mathbb{R}}^m$ with runtime complexity in ${\mathcal{O}}(nN(m,n))$. \[M2\]
\[EVN\]
We argue by induction on $m,n \in {\mathbb{N}}$. For $m=1, n\in {\mathbb{N}}$ all statements are known to be true [@Stoer; @gautschi; @atkinson; @endre]. If $m>1$ then, by assumption, the affine transformation $\tau : {\mathbb{R}}^m {\longrightarrow}{\mathbb{R}}^m$ from Definition \[NTN\] is the identity, i.e., $\tau={\mathrm{id}}_{{\mathbb{R}}^m}$. Let ${\mathbb{P}}_{m,n}:=\oplus_{i=1}^m {\mathbb{P}}_i$, ${\mathbb{P}}_i=\{p_{i,1},\dots,p_{i,n+1}\}$ be the generating nodes of $P_{m,n}$, see Definition \[NTN\]. We consider the hyperplane $H={\left}\{x \in {\mathbb{R}}^m {:}x_m = p_{m,1}{\right}\}$ and, for $\Gamma_1=\{\gamma\in \Gamma_{m,n}{:}[\gamma]_{m}=1\}$, we denote by $P_1 \subseteq P_{m,n}$ the multidimensional Newton nodes corresponding to the leaves of $\gamma \in \Gamma_1$. Further, we set $Q_1(x)= \sum_{\gamma \in \Gamma_1}c_{[\gamma]}N_{[\gamma]}(x)$. By identifying $H$ with ${\mathbb{R}}^{m-1}$ induction yields that $(i),(ii),(iii)$ can be computed in ${\mathcal{O}}(N(m-1,n))$. Setting $Q_2(x) = Q(x)-Q_1(x)$, we have $Q_2(x) = (x_m-p_{m,1})Q_3(x)$, where $Q_3 \in \Pi_{m,n-1}$ is a polynomial in Newton form w.r.t. $T_{m,n-1}$ and $P_2=P_{m,n}\setminus P_1$ (see Theorem \[DDST\]). Thus, by induction, $(i),(ii)$ can be computed in ${\mathcal{O}}(N(m,n-1))$, ${\mathcal{O}}(nN(m,n-1))$ for $Q_3$, which proves $(i),(ii)$ because $N(m-1,n)+N(m,n-1)=N(m,n)$. Claim $(iii)$ follows from $(i)$ and $(ii)$ by applying partial integration to $Q(x) = Q_1(x) + (x_m-p_{m,1})Q_3(x)$, i.e., $$\begin{aligned}
\int_{\Omega} Q(x)dx & =\int_{\Omega} Q_1(x)dx -\int_{\Omega} Q_3(x)dx \\
& + \int_{\Omega \cap H_{1,\dots,m-1}}(x_m-p_{m,1})\partial_{x_m} Q_3(x)\big|_{x_m=\pm 1}dx
\end{aligned}$$
and using induction.
The recursive subdivision of the problem into sub-problems of lower dimension or degree, as also used in the proof of Theorem \[EVN\], can be used to implement a generalization of the classical *(inverse) Horner scheme* [@Stoer; @gautschi; @atkinson; @endre]. In the case of arbitrary multidimensional Newton nodes $P_{m,n}$ with $\tau(P_{m,n}) = \bar P_{m,n}$ for canonical nodes $\bar P_{m,n}$, we can evaluate $Q\circ \tau^{-1}(x)$ with respect to $\bar P_{m,n}$. The runtime complexity then increases by adding the cost of inverting the affine transformation $\tau$ from Definition \[NTN\], which is given by the cost of inverting an $m \times m$ matrix.
Approximation Theory {#APT}
====================
Studying how well arbitrary continuous functions can be approximated by polynomials, a fundamental observation was made. The observation is that even though the Weierstrass Theorem states that every function $f : I\subseteq {\mathbb{R}}{\longrightarrow}{\mathbb{R}}$, $I=[a,b]$ can be approximated by Bernstein polynomials in the $C^0$-sense [@weier], for every choice of interpolation nodes $P_n$ there exists a continuous function for which the interpolant $Q_{n,f}$ does not converge, i.e., ${||\,}f - Q_{n,f}{\,||}_{C^0(\Omega)} \centernot{\longrightarrow}0$ for $n \rightarrow \infty$ [@faber]. This observation is known as [*Runge’s phenomenon*]{}. However, when choosing *Chebyshev nodes* the class of functions that cannot be approximated (i.e., for which Runge’s phenomenon occurs) seem to be pathological and extremely unlikely to occur in any “real world” application or data set. We here specify these facts and provide generalized statements for the multidimensional case.
Sobolev Theory
--------------
We start by providing the analytical setting required to formulate our results. In [@Sob] an excellent overview of Sobolev theory is given. Here, we simply recall that for every domain $\Omega \subseteq {\mathbb{R}}^m$ the functional space $C^k(\Omega,{\mathbb{R}})$ denotes the real vector space of all functions $f : \Omega {\longrightarrow}{\mathbb{R}}$ that are $k$ times differentiable in the interior of $\Omega$ and are continuous on the closure $\overline \Omega = \Omega \cup \partial \Omega$. Further, by equipping $C^k(\Omega,{\mathbb{R}})$ with the norm $${||\,}f{\,||}_{C^k(\Omega)} = \sum_{l=0}^k\sup_{x \in \Omega}\sum_{\alpha \in A_{m,l}}{\left}|\partial^\alpha f(x){\right}|\,, \quad \partial^\alpha f(x) =
\partial^{|\alpha_1|}_{x_1}\cdots \partial^{|\alpha_n|}_{x_n}f(x),$$ we obtain a Banach space. We consider the Sobolev space of $L^p$–functions with well-defined weak derivatives up to order $k$, i.e., in general for $1\leq p < \infty$ $$\label{Wkp}
W^{k,p}=\Big\{ f \in L^p(\Omega,{\mathbb{R}}) {:}{||\,}f {\,||}_{W^{k,p}(\Omega)}^p:=\sum_{\alpha\in A_{m,k}}{||\,}\partial^\alpha f{\,||}_{L^p(\Omega)}^p < \infty \Big\}\,.$$ The *Hilbert space* $H^k(\Omega,{\mathbb{R}}):=W^{k,2}(\Omega,{\mathbb{R}})$ is of special interest. In the following, we assume that $\Omega = [-1,1]^m$ is the standard hypercube and denote by ${\mathbb{T}}^m = {\mathbb{R}}^m / 2{\mathbb{Z}}^m$ the torus with fundamental domain $\Omega$. Then, $H^k(\Omega,{\mathbb{R}}) \subseteq L^2(\Omega,{\mathbb{R}})$, $0 \leq k \leq \infty$, implies that every $f \in H^k(\Omega,{\mathbb{R}})$ can be written as a power series $$f(x)= \frac{1}{|\Omega|}\sum_{\alpha \in {\mathbb{N}}^m} d_\alpha x^\alpha\,, \quad d_\alpha \in {\mathbb{R}}$$ almost everywhere, i.e., the identity is violated only on a set $\Omega'\subseteq \Omega$ of Lebesgue measure zero. Further, for every function $\widetilde f : {\mathbb{T}}^m {\longrightarrow}{\mathbb{R}}$, there is a uniquely determined function $f : \Omega {\longrightarrow}{\mathbb{R}}$ such that $\widetilde f(x + 2{\mathbb{Z}}^m)=f(x)$. Therefore, $f$ can obviously be extended to a periodic function on ${\mathbb{R}}^m$ and is called the [*periodic representative*]{} of $\widetilde f$. Thus, we can write $f$ as a Fourier series $$f(x)= \frac{1}{|\Omega|}\sum_{\alpha \in {\mathbb{N}}^m} c_\alpha e^{\pi i {\left}<\alpha, x{\right}> }\,, \quad c_\alpha \in {\mathbb{C}}$$ and observe that the space $H^k({\mathbb{T}}^m,{\mathbb{R}})$ can be defined as $$H^k({\mathbb{T}}^m,{\mathbb{R}})=\overline{C^{\infty}({\mathbb{T}}^m,{\mathbb{R}})}^{{||\,}\cdot{\,||}_{H^k(\Omega)}} ,
\label{Compl}$$ which is the completion of $C^{\infty}({\mathbb{T}}^m,{\mathbb{R}})$ with respect to the $H^k$-norm [@Sob]. Therefore, $$\begin{aligned}
\label{frac}
{||\,}f {\,||}_{H^k(\Omega)}^2:= &\sum_{\beta\in A_{k,m}}{\left}<\partial^\beta f,\partial^\beta f {\right}>_{L^2(\Omega)} \nonumber \\
= & \sum_{\beta \in A_{m,k}, \alpha \in {\mathbb{N}}^m}\big(\pi^{\beta_{i_1}+\cdots+\beta_{i_l}}\alpha_{i_1}^{\beta_{i_1}}\cdots\alpha_{i_l}^{\beta_{i_l}}|c_{\alpha}|\big)^2 \,, \end{aligned}$$ where the $\alpha_{i_j}$, $j =1,\dots,l \in {\mathbb{N}}$, are the non-vanishing entries of $\alpha \in {\mathbb{N}}^m$.
Thus, $C^{\infty}({\mathbb{T}}^m,{\mathbb{R}})\subseteq H^k({\mathbb{T}}^m,{\mathbb{R}})$ is a dense subset and, due to the right-hand side of Eq. , a notion of fractal derivatives can be given, i.e., $H^k(\Omega,{\mathbb{R}})$ with $k \in {\mathbb{R}}^+$ and $\alpha \in {\mathbb{R}}^m$, $\alpha_i \geq 1$, $|\alpha|\leq k$, can be considered. Vice versa, due to the *Sobolev and Rellich-Kondrachov embedding Theorem* [@Sob], we have that whenever $k > m/2$ then $H^k({\mathbb{T}}^m,{\mathbb{R}}) \subseteq C^0({\mathbb{T}}^m,{\mathbb{R}})$ and the embedding $$i : H^k({\mathbb{T}}^m,{\mathbb{R}}) \hookrightarrow C^0({\mathbb{T}}^m,{\mathbb{R}})$$ is well defined, continuous, and compact. Thus, for $m \in {\mathbb{N}}$ and all periodic representatives $f$ of $ \widetilde f\in {\mathbb{C}}^0({\mathbb{T}}^m,{\mathbb{R}})$, there exists a constant $c=c(m,\Omega) \in {\mathbb{R}}^+$ such that $$\label{norm}
{||\,}f {\,||}_{C^0(\Omega)} \leq c\,{||\,}f {\,||}_{H^k(\Omega)}$$ and for every $B \subseteq H^k({\mathbb{T}}^m,{\mathbb{R}})$ we have that $i(B) \subseteq C^0({\mathbb{T}}^m,{\mathbb{R}})$ is precompact whenever $B$ is bounded in $H^k({\mathbb{T}}^m,{\mathbb{R}})$. By the trace Theorem [@Sob], we observe furthermore that whenever $H \subseteq {\mathbb{R}}^m$ is a hyperplane of co-dimension $1$, then the induced restriction $$\label{trace}
\varrho: H^k(\Omega,{\mathbb{R}}) {\longrightarrow}H^{k-1/2}(\Omega \cap H,{\mathbb{R}})$$ is continuous, i.e., ${||\,}f_{| \Omega \cap H} {\,||}_{H^{k-1/2}(\Omega\cap H)} \leq d{||\,}f {\,||}_{H^k(\Omega)}$ for some $d=d(m,\Omega) \in {\mathbb{R}}^+$. We consider $$\Theta_{m,n}= \Big\{\widetilde f\in H^k({\mathbb{T}}^m,{\mathbb{R}}){:}f(x) = \frac{1}{|\Omega|}\sum_{\alpha \in A_{m,n}} c_\alpha e^{\pi i {\left}<\alpha, x{\right}> }\,, c_{\alpha} \in {\mathbb{C}}\Big\}$$ the space of all finite Fourier series of bounded frequencies and denote by $$\begin{aligned}
\theta_n : H^k({\mathbb{T}}^m,{\mathbb{R}}){\longrightarrow}& \Theta_{m,n} \subseteq C^0({\mathbb{T}}^m,{\mathbb{R}})\,, \nonumber\\
\tau_n : H^k(\Omega,{\mathbb{R}}){\longrightarrow}& \Pi_{m,n} \subseteq C^0(\Omega,{\mathbb{R}})\label{pro}\end{aligned}$$ the corresponding projections onto $\Theta_{m,n}$ or onto the space of polynomials $\Pi_{m,n}$. Further, we denote by $\theta_n^\perp = I -\theta_n$, $\tau_n^\perp=I-\tau_n$ the complementary projections. Then, we have:
Let $k,m \in {\mathbb{N}}$, $\Omega =[-1,1]^m \subseteq {\mathbb{R}}^m$ be the standard hypercube, $\widetilde f \in H^{k}({\mathbb{T}}^m,{\mathbb{R}})$, and $f$ its periodic representative. Then:
1. ${||\,}f {\,||}_{C^0(\Omega)} \leq {||\,}f {\,||}_{H^k(\Omega)}$ for all $\widetilde f \in H^k({\mathbb{T}}^m,{\mathbb{R}})$.
2. ${||\,}\theta_n^\perp(f){\,||}_{C^0(\Omega)} \in o\big((m/n)^{k}\big)$, i.e. $$(n/m)^{k} {||\,}\tau_n^\perp(f){\,||}_{C^0(\Omega)} \xrightarrow[n\rightarrow \infty]{} 0\quad \text{for every}\,\, m \in {\mathbb{N}}\,.$$
3. For $n,l \in {\mathbb{N}}$, the operator norm of $\tau_{n}^\perp \theta_l: H^k({\mathbb{T}}^m,{\mathbb{R}}) {\longrightarrow}C^0({\mathbb{T}}^m,{\mathbb{R}})$ is bounded by ${||\,}\tau_{n}^\perp \theta_l {\,||}\leq N(m,l)^{1/2}\frac{(\pi l^2)^{n+1}}{(n+1)!}$.
\[shrink\]
To show $(i)$, we approximate $f$ by a finite Fourier series, i.e., we assume that $f(x) = \frac{1}{|\Omega|}\sum_{\alpha \in A_{m,n}}c_\alpha e^{\pi i {\left}<\alpha,x{\right}>}$. The Fourier basis is orthonormal, i.e., $$\frac{1}{|\Omega|}\int_\Omega e^{\pi i {\left}<\alpha,x{\right}>}\cdot e^{-\pi i {\left}<\beta,x{\right}>} \mathrm{d}\Omega = {\left}\{ \begin{array}{ll}
1 \,, & \text{if} \,\,\, \alpha =\beta\\
0\, & \text{else.}
\end{array}{\right}.$$ Thus, we compute $$\begin{aligned}
{||\,}f {\,||}^2_{C^0(\Omega)} =&\sup_{x \in \Omega}\Big|\sum_{\alpha \in A_{m,n}}c_\alpha e^{\pi i {\left}<\alpha,x{\right}>}\Big|^2 \leq \sum_{\alpha \in A_{m,n}}|c_\alpha|^2
={||\,}f {\,||}^2_{L^2(\Omega)} \leq {||\,}f {\,||}_{H^k(\Omega)}^2 \,.\end{aligned}$$ Now $(i)$ follows from the density of the approximation given in Eq. and the continuity of the norm ${||\,}\cdot {\,||}_{H^k(\Omega)}$. To show $(ii)$, we denote by $\alpha_{i_1}, \dots, \alpha_{i_l}$, $l\leq m$, all non-vanishing entries of $\alpha \in {\mathbb{N}}^m\setminus\{0\}$ and observe that $$\partial^\beta(c_\alpha e^{\pi i {\left}<\alpha,x{\right}>})=(i\pi)^{\beta_{i_1}+\cdots+\beta_{i_l}}\alpha_{i_1}^{\beta_{i_1}}\cdots\alpha_{i_l}^{\beta_{i_l}}c_{\alpha} e^{\pi i {\left}<\alpha,x{\right}>} \,, \,\,\,\beta\in A_{m,k}\,.$$ If $|\alpha|>m $ then at least one $\alpha_{i_h} > n/m$ for some $1 \leq h \leq l$. Thus, by choosing $\beta \in A_{m,k}$ with $\beta_{h}=k$, we obtain $$\big|\partial^\beta(c_\alpha e^{\pi i {\left}<\alpha,x{\right}>} )\big| \geq (\pi n/m)^{k}|c_\alpha|\,, \quad \text{for} \,\,\,|\alpha|\geq m\,.$$ Hence, for $n >m$ $$\begin{aligned}
(n/m)^{2k} ||\theta_n^\perp f||_{C^0(\Omega)}^2 \leq ||\theta_n^\perp f||_{H^k(\Omega)}^2 \,.\end{aligned}$$ Due to Eq. we have $||\theta_n^\perp f||_{H^k(\Omega)} \xrightarrow[n\rightarrow \infty]{} 0$, which yields $||\theta_n^\perp f||_{C^0(\Omega)} \in o\big((m/n)^{k}\big)$ for every fixed $m \in {\mathbb{N}}$, proving $(ii)$. To show $(iii)$, we assume ${||\,}f{\,||}_{H^k(\Omega)} \leq 1$, write $$\theta_l(f) = \sum_{\alpha \in A_{m,l}} c_{\alpha}e^{\pi i {\left}<\alpha,x{\right}>}\,, c_\alpha \in {\mathbb{C}},$$ and recall that for every $x \in {\mathbb{R}}^m$ there exists $\xi_x\in \Omega$ such that $$e^{i\pi{\left}<\alpha,x{\right}>} = \sum_{n \in {\mathbb{N}}}\frac{(i\pi{\left}<\alpha,x{\right}>)^n}{n!} = \sum_{h \leq n }\frac{(i\pi{\left}<\alpha,x{\right}>)^h}{h!} + \frac{\partial^{n+1}_ve^{i\pi{\left}<\alpha,\xi_x{\right}>}}{(n+1)!}(i\pi{\left}<\alpha,x{\right}>)^{n+1}\,,$$ where $\partial_v$ denotes the partial derivative in direction $v = \frac{x}{|x|} \in {\mathbb{R}}^m$. Therefore, the last term is the *Lagrange remainder*. Now $$\partial^{n+1}_ve^{i\pi{\left}<\alpha,\xi_x{\right}>}= \partial^{n}_v\big<\nabla e^{i\pi{\left}<\alpha,\xi_x{\right}>},v\big> = \big<\alpha,v\big>\partial^{n}_ve^{i\pi{\left}<\alpha,\xi_x{\right}>}= \big<\alpha,v\big>^{n+1}e^{i\pi{\left}<\alpha,\xi_x{\right}>}$$ and $|{\left}<\alpha,v{\right}>|, |{\left}<\alpha,x{\right}>| \leq |\alpha|$ imply that $${\left}|\frac{\partial^{n+1}_ve^{i\pi{\left}<\alpha,\xi_x{\right}>}}{(n+1)!}(i\pi{\left}<\alpha,x{\right}>)^{n+1}{\right}| \leq \frac{ (\pi|\alpha|)^{2n+2}}{(n+1)!} .$$ Hence, due to $|\alpha|\leq l$, $|c_\alpha| \leq 1$, we bound $$\begin{aligned}
{||\,}\tau_{n}^\perp \theta_l(f){\,||}_{C^0(\Omega)} & \leq {\left}(\sum_{\alpha \in A_{m,l}} |c_{\alpha}|^2 {\left}(\frac{ (\pi|\alpha|)^{2n+2}}{(n+1)!}{\right})^2{\right})^{1/2}\\
&\leq
\frac{\pi^{n+1}}{(n+1)!}{\left}(\sum_{\alpha \in A_{m,l}}|\alpha|^{4(n+1)}{\right})^{1/2}\\
&\leq \frac{\pi^{n+1} l^{2n+2}}{(n+1)!}N(m,l)^{1/2} ,
\end{aligned}$$ which yields $(iii)$.
Lebesgues Functions
-------------------
Lebesgue functions measure the relative approximation error of an interpolation scheme. More precisely: Let $m,n \in {\mathbb{N}}$, $\Omega= [-1,1]^m \subseteq {\mathbb{R}}^m$, and $S_{m,n}:C^0(\Omega,{\mathbb{R}}) {\longrightarrow}\Pi_{m,n}$ denote an interpolation scheme with respect to unisolvent interpolation nodes $P_{m,n}$. That is, for $f \in C^0(\Omega,{\mathbb{R}})$, $S_{m,n}(f) \in \Pi_{m,n}$ solves the PIP with respect to $(m,n,f,P_{m,n})$. Since $P_{m,n}$ is unisolvent and independent of $f$, it is readily verified that $S_{m,n}$ is a linear operator. Therefore, the following is well defined:
Let $m,n \in {\mathbb{N}}$, $\Omega=[-1,1]^m$, and $P_{m,n}\subseteq \Omega$ be unisolvent interpolation nodes. Consider the interpolation operator $S_{m,n}:C^0(\Omega , {\mathbb{R}}) {\longrightarrow}\Pi_{m,n}$ and its restriction $S_{m,n | H^k(\Omega)} : H^k(\Omega,{\mathbb{R}}) {\longrightarrow}\Pi_{m,n}$. Then we define by $$\begin{aligned}
\Lambda(P_{m,n},C^0(\Omega)): = & \sup_{ f\in C^0(\Omega,{\mathbb{R}}), ||f||_{C^0(\Omega)} =1} ||S_{m,n}(f)||_{C^0(\Omega)} \\
\Lambda(P_{m,n},H^k(\Omega)): = & \sup_{ f\in H^k(\Omega,{\mathbb{R}}), ||f||_{H^k(\Omega)} =1} ||S_{m,n | H^k(\Omega)}(f)||_{C^0(\Omega)}\end{aligned}$$ the [*operator norms*]{} of $S_{m,n}$ and $S_{m,n | H^k(\Omega)}$, respectively. Denote with ${\mathcal{P}}_{m,n}$ the set of all unisolvent node sets with respect to $m,n \in {\mathbb{N}}$ then $$\begin{aligned}
\Lambda_{m,n,C^0(\Omega)} : {\mathcal{P}}_{m,n}{\longrightarrow}{\mathbb{R}}\,, & \quad P_{m,n} \mapsto \Lambda(P_{m,n},C^0(\Omega))\,,\\
\Lambda_{m,n,H^k(\Omega)} : {\mathcal{P}}_{m,n}{\longrightarrow}{\mathbb{R}}\,, & \quad P_{m,n} \mapsto \Lambda(P_{m,n},H^k(\Omega)) \end{aligned}$$ are called the *Lebesgue functions* with respect to the considered norms.
Due to Lemma \[shrink\], $${\left}\{f\in H^k(\Omega,{\mathbb{R}}) {:}||f||_{H^k(\Omega)} =1{\right}\} \subseteq {\left}\{f\in C^0(\Omega,{\mathbb{R}}) {:}||f||_{C^0(\Omega)} =1{\right}\},$$ implying that $$\label{LEB}
\Lambda(P_{m,n},H^k(\Omega)) \leq \Lambda(P_{m,n},C^0(\Omega)) \,.$$
Understanding the behavior of Lebesgue functions in 1D is crucial for extending their definition to arbitrary dimensions. In particular, the following observation is key to our further considerations:
\[PartLEB\] Let $l,n \in {\mathbb{N}}$, $l,n \geq 1$, $\Omega=[-1,1]^m$, $P_n= \{p_1,\dots,p_{n+1}\}$ be a set of pairwise disjoint nodes, and $P_l= \{p_1,\dots,p_{l+1}\} \subseteq P_n$. Then $$\Lambda(P_l,C^0(\Omega)) \leq \Lambda(P_n,C^0(\Omega))\,, \quad \Lambda(P_l,H^k(\Omega)) \leq \Lambda(P_n,H^k(\Omega)) \text{ for } k > m/2.$$
We choose small intervals $I_{h,{\varepsilon}}=[p_{h}-{\varepsilon},p_{h}+{\varepsilon}]$, $I_{h,\delta}=[p_{h}-\delta,p_{h}+\delta]$, $h >l+1$ with ${\varepsilon}> \delta >0$ small enough so that $I_{h,{\varepsilon}}\cap P_n = \{p_h\}$. Further, we consider smooth cut–off functions $\beta_h : [p_h -{\varepsilon},p_h + {\varepsilon}] {\longrightarrow}[0,1]$, with $\beta_h(p_h\pm {\varepsilon})=1$, $\beta_h^\pm(p_h\pm \delta)=0$. We denote by $S_{1,k}$, $S_{1,n}$ the interpolation schemes with respect to $P_k$, $P_n$, respectively, and set $$\tilde f(x)= {\left}\{\begin{array}{ll}
f(x) \,, & \text{if} \,\,\, x \not \in I_{h,{\varepsilon}}, \,\,\, h>k+1 \\
\beta_h^\pm(x)f(x) + (1- \beta_h^\pm(x))f(p_{l+1})\,, & \text{if} \,\,\, x \in [p_h\pm\delta,p_h\pm{\varepsilon}] \,, \,\,\, h>l+1 \\
f(p_{l+1})\,, & \text{if} \,\,\, x \in I_{h,\delta} \,\,\, h>k+1 .
\end{array}{\right}.\,$$ Obviously, it is ${||\,}\tilde f{\,||}_{C^0(\Omega)}, {||\,}\tilde f{\,||}_{H^k(\Omega)}\leq 1$. Moreover, following the alternative divided difference scheme from Eq. , we find $f_j(p_j)=0$ for all $j>l+1$. Thus, the coefficients $c_j$, $j >l$, of the polynomial $S_{1,n}(\tilde f)$ in Newton form vanish, while the $c_0,\dots,c_{l}$ are given by $\tilde f_0(p_1) = f_0(p_1),\dots, \tilde f_l(p_{l+1}) =f_l(p_{l+1})$. Hence, $S_{1,n}(\tilde f) =S_{1,l}(f) = S_{1,l}(\tilde f)$. Therefore, $${||\,}S_{1,n}(\tilde f){\,||}_{C^0(\Omega)}={||\,}S_{1,l}(\tilde f){\,||}_{C^0(\Omega)} ={||\,}S_{1,l}(f){\,||}_{C^0(\Omega)} \,.$$ Observing that $f$ was arbitrarily chosen, this completes the proof.
Newton-Chebyshev Nodes
----------------------
In order to extend the study of Runge’s phenomenon to multiple dimensions, we introduce a multidimensional notion of *Chebyshev nodes* and provide the essential approximation results.
Let $m,n \in {\mathbb{N}}$, $T_{m,n}$, and $\Gamma_{m,n}$ be given. Let $P_{m,n}$ be the canonical multidimensional Newton nodes generated by $${\mathbb{P}}_{m,n}= \oplus_{l=1}^m {\mathrm{Cheb}}_n\,,$$ where ${\mathrm{Cheb}}_n$ was defined in Eq. . Then, we call $P_{m,n}$ [*canonical multidimensional Newton-Chebyshev nodes*]{}, and we call every affine transformation $\tau(P_{m,n})$ of $P_{m,n}$ [*multidimensional Newton-Chebyshev nodes*]{}.
Let $m,n,k \in {\mathbb{N}}$, $k >m/2$, and $S_{m,n}$ be an interpolation operator with respect to multidimensional Newton nodes $P_{m,n}$ generated by ${\mathbb{P}}_{m,n}= \oplus_{l=1}^m P_l$, $\#P_l=n+1$. Then $$\Lambda(P_{m,n},H^k(\Omega)) \leq \prod_{l=1}^m \Lambda(P_l, H^{k-(m-1)/2}([-1,1]))\,.$$ If $P_{m,n}$ are multidimensional Newton-Chebyshev nodes, then in particular $$\label{MC}
\Lambda(P_{m,n},H^k(\Omega)) \leq \Lambda({\mathrm{Cheb}}_n,C^0([-1,1]))^m \in {\mathcal{O}}(\log(n)^m) \,.$$ \[NCL\]
We argue by induction on $m$. For $m=1$ the claim directly follows from Eq. . If $m>1$ then we can assume w.l.o.g. (i.e., by changing coordinates if necessary) that $P_{m,n}$ are canonical Newton-Chebyshev nodes, and we can consider the hyperplanes $H_1,\dots,H_{n}$ given by $H_i = \{x \in {\mathbb{R}}^m {:}x_m =p_{m,i} \in P_{m}\}$, $i =1,\dots,n$. Denote by $Q_{H_i}(x)=x_m -p_{m,i}$ the linear polynomial defining $H_i = Q_{H_i}^{-1}(0)$ and by $\pi_{H_i}: {\mathbb{R}}^m {\longrightarrow}H_i$, $ \pi_{H_i}(x) = (x_1,\dots,x_{m-1},p_{m,i})$, the corresponding projections. Then, by Theorems \[GN\], \[SV\], and Proposition \[UniN\], we have $$\begin{aligned}
S_{m,n}(f) =& S_{m-1,n,H_1}(f) + Q_{H_1}\Big( S_{m-1,n-1}(f_1) + Q_{H_2}\big( S_{m-1,n-2}(f_2) + \cdots \nonumber\\
&Q_{H_n}\big(S_{m-1,1}(f_{n-1}) + S_{m,0}(f_n) \big) \dots \Big)\,, \label{NI}\end{aligned}$$ where $$f_0 =f\,, \quad f_k = \frac{f_{k-1}(x)-f_{k-1}(\pi_{H_k}(x))}{Q_H(x)}$$ and the $S_{m-1,n-i}$ interpolate with respect to $P_{m-1,n-i}$ generated by $${\mathbb{P}}_{m-1,n-i} = \{p_{m,i+1}\}\times \oplus_{l=1}^{m-1} P_l \,.$$ By observing that the interpolant $S_{m-1,n-i}(f_i)$ is constant along directions normal to $H_i$, and by identifying $H_i \cong {\mathbb{R}}^{m-1}$, induction and Lemma \[PartLEB\] yield $$\Lambda(P_{m-1,n-i},H^k(\Omega)) \leq \prod_{l=1}^{m-1} \Lambda(P_l,H^{k-(m-2)/2}([-1,1]^{m-1})) =:\Lambda^*\,.$$ Though the $f_i$ are multivariate functions, the remaining interpolation is done with respect to $x_m$ in only $1$ variable. Hence, recalling the 1D estimation, we get $$\begin{aligned}
{||\,}S_{m,n}(f) {\,||}_{C^0(\Omega)} & \leq \Lambda^*{||\,}f_0 + Q_{H_1}\big( f_1 + \cdots +Q_{H_n}\big(f_{n-1} + f_n \big) \dots \big){\,||}_{C^0(\Omega)}\\
& \leq \Lambda^*\Lambda(P_m,H^{k-(m-1)/2}([-1,1]))\\
&\leq \prod_{l=1}^{m} \Lambda(P_l,H^{k-(m-1)/2}([-1,1]))\,.\end{aligned}$$ This proves the first statement. The proof of Eq. follows from Theorem \[Cheb\].
Let $m,n,k \in {\mathbb{N}}$, $k>m/2$, $\Omega= [-1,1]^m \subseteq {\mathbb{R}}^m$, ${\mathbb{T}}^m = {\mathbb{R}}^m/2{\mathbb{Z}}^m$, and $\widetilde f \in H^k({\mathbb{T}}^m,{\mathbb{R}})$ with periodic representative $f$. Consider an interpolation operator $S_{m,n}:H^k(\Omega,{\mathbb{R}}) {\longrightarrow}\Pi_{m,n}$ with respect to multidimensional Newton-Chebyshev nodes $P_{m,n}$. Then $${||\,}f -S_{m,n}(f){\,||}_{C^0(\Omega)} \xrightarrow[n\rightarrow \infty]{}0 \,.$$
The proof is based on balancing the approximation of $f$ in Fourier basis and by polynomials. To do so, we let $n \in {\mathbb{N}}$, $D,K_1,K_0 \in {\mathbb{R}}^+ $, $ D\geq 1$, $K_1>K_0\geq 2 $, and $l=l(n) \in {\mathbb{N}}$ such that $$\label{bal}
\frac{(Dl)^{K_1}}{n} \xrightarrow[n\rightarrow \infty]{} 0 \quad \text{and} \quad \frac{\log(n)^{K_0}}{l} \xrightarrow[n\rightarrow \infty]{} 0 \,,$$ i.e., $(Dl)^{K_1} \in o(n)$ and $\log(n)^{K_0}\in o(l)$. We consider the projections $\tau_n,\tau_n^\perp,\theta_l,\theta_l^\perp$ from Eq. and use the fact that $S_{m,n}$ is a linear operator in order to bound $$\begin{aligned}
{||\,}f -S_{m,n}(f){\,||}_{C^0(\Omega)} & \leq {||\,}\theta_l(f) -S_{m,n}(\theta_l(f)){\,||}_{C^0(\Omega)} \\
&+ {||\,}\theta_l^\perp(f) -S_{m,n}(\theta_l^\perp(f)){\,||}_{C^0(\Omega)} \,.\end{aligned}$$ We then use Theorem \[NCL\] and Lemma \[shrink\]$(i)$ to observe that there exists a constant $c\in {\mathbb{R}}^+$ so that the second term on the right-hand side satisfies $$\begin{aligned}
{||\,}\theta_l^\perp(f) -S_{m,n}(\theta_l^\perp(f)){\,||}_{C^0(\Omega)} & \leq {||\,}\theta_l^\perp(f) -S_{m,n}(\theta_l^\perp(f)){\,||}_{H^k(\Omega)} \\
&\leq \big(1+\Lambda(S_{m,n},H^k(\Omega))\big) {||\,}\theta_l^\perp(f){\,||}_{H^k(\Omega)} \\
&\leq c\big(1+\log(n)^m)\big){||\,}\theta_l^\perp(f){\,||}_{C^0(\Omega)} \,.\end{aligned}$$ By Lemma \[shrink\]$ii)$, we find that ${||\,}\theta_l^\perp(f){\,||}_{C^0(\Omega)} \in o((m/l)^k)$ for $k>m/2$. Since $K_0 \geq 2$, we obtain $${||\,}\theta_l^\perp(f) -S_{m,n}(\theta_l^\perp(f)){\,||}_{C^0(\Omega)} \leq \frac{1+\log(n)^m}{l^{m/2}}{||\,}f{\,||}_{C^0(\Omega)} \xrightarrow[n\rightarrow \infty]{} 0 \,.$$ Similarly, we bound the remaining term by $$\begin{aligned}
{||\,}\theta_l(f) -S_{m,n}(\theta_l(f)){\,||}_{C^0(\Omega)} & \leq {||\,}\pi_n\theta_l(f) -S_{m,n}(\pi_n\theta_l(f)){\,||}_{C^0(\Omega)} \\
&+ {||\,}\pi_n^\perp\theta_l(f) -S_{m,n}(\pi_n^\perp\theta_l(f)){\,||}_{C^0(\Omega)}. \end{aligned}$$ Since $\pi_n\theta_l(f) \in \Pi_{m,n}$, we have $S_{m,n}(\pi_n\theta_l(f))=\pi_n\theta_l(f)$, implying that the first term vanishes. Now we again use Theorem \[NCL\] and Lemma \[shrink\]$(i)$ and \[shrink\]$(iii)$ to observe that there exists a constant $c\in {\mathbb{R}}^+$ so that $$\begin{aligned}
{||\,}\pi_n^\perp\theta_l(f) -S_{m,n}(\pi_n^\perp\theta_l(f)){\,||}_{C^0(\Omega)} & \leq {||\,}\pi_n^\perp\theta_l(f) -S_{m,n}(\pi_n^\perp\theta_l(f)){\,||}_{H^k(\Omega)} \\
& \leq c(1 + \log(n)^m){||\,}\pi_n^\perp\theta_l(f){\,||}_{H^k(\Omega)}\\
& \leq c(1 + \log(n)^m)N(m,l)^{1/2}\frac{(\pi l^2)^{n+1}}{(n+1)!} {||\,}f{\,||}_{C^0(\Omega)}. \end{aligned}$$ Since $N(m,l)=\frac{(m+l)!}{n!l!} = \frac{(m+1)\cdots(m+l)}{l!} \in {\mathcal{O}}(l^m)$ and because $n! >(n/2)^{n/2}$ for all $n \in {\mathbb{N}}$, there exists a contant $d\in {\mathbb{R}}^+$ so that $$\begin{aligned}
c(1 + \log(n)^m)N(m,l)^{1/2}\frac{(\pi l^2)^{n+1}}{(n+1)!} & \leq d(1+l^m)l^{m/2}{\left}(\frac{(\pi l^2)}{n+1} {\right})^{\!\! n+1} .\end{aligned}$$ Choosing $D, K_1 \in {\mathbb{R}}^+$ large enough we can further use Eq. to bound $$\begin{aligned}
{||\,}\pi_n^\perp\theta_l(f) -S_{m,n}(\pi_n^\perp\theta_l(f)){\,||}_{C^0(\Omega)} & \leq d(1+l^m)l^{m/2}\frac{(\pi l^2)^{n+1}}{n^{(n+1)/2}} {||\,}f {\,||}_{C^0(\Omega)}\\
& \leq {\left}(\frac{(Dl)^{K_1}}{n+1}{\right})^{\!\!(n+1)/2} {||\,}f {\,||}_{C^0(\Omega)}\xrightarrow[n\rightarrow \infty]{} 0 \,.\end{aligned}$$ Hence, all terms converge to zero as $n \rightarrow \infty$, proving the theorem.
Approximation Errors
---------------------
We generalize the classic estimates of approximation errors in 1D to arbitrary dimensions $m \in {\mathbb{N}}$.
Let $m,n \in {\mathbb{N}}$ and $\Omega= [-1,1]^m \subseteq {\mathbb{R}}^m$. Let $S_{m,n}:H^k(\Omega,{\mathbb{R}}) {\longrightarrow}\Pi_{m,n}$, $k>m/2$, denote an interpolation operator with respect to canonical multidimensional Newton nodes $P_{m,n}$ generated by ${\mathbb{P}}_{m,n}=\oplus_{i=1}^m P_i$, $P_i={\left}\{p_{i,1},\dots,p_{i,n+1}{\right}\}$.
1. If $f \in C^{n+1}(\Omega,{\mathbb{R}})$ then for every $\alpha \in A_{m,n}\setminus A_{m,n-1}$, $i \in\{1,\dots,m\}$ and every $x \in \Omega$ there is $\xi_x\in \Omega$ such that $$\label{AE}
| f(x)- S_{m,n}(f)(x) |\leq \frac{1}{\alpha_i!} \partial^{\alpha_i+1}_{x_i}f(\xi_x) |N_\alpha(x)|\,,$$ where $N_\alpha(x) = \prod_{i=1}^m\prod_{j=1}^{\alpha_i}(x_i-p_{i,j})$, $x =(x_1,\dots,x_m)$, $p_{i,j}\in P_i$. If $P_{m,n}$ are multidimensional Newton-Chebyshev nodes, then we can further bound $$\label{AEE}
| f(x)- S_{m,n}(f)(x) | \leq \frac{1}{2^{\alpha_i}\alpha_i! }\partial^{\alpha_i+1}_{x_i} f(\xi_x)\,.$$
2. For any unisolvent node set $P_{m,n}$ the relative interpolation error is $${||\,}f- S_{m,n}(f) {\,||}_{C^0(\Omega)} \leq (1+\Lambda(P_{m,n},H^k(\Omega)){||\,}f- Q^*_{m,n} {\,||}_{C^0(\Omega)}$$ for all $f \in H^k(\Omega,{\mathbb{R}})$, where $Q^*_{m,n}$ is an optimal approximation that minimizes the $C^0$-distance to $f$.
Changing coordinates if necessary, we can assume w.l.o.g. that $P_{m,n}$ are canonical nodes. To show $(i)$, we follow the argumentation of the classical proof in 1D [@gautschi]. We consider the line $$L_{\alpha, i}= {\left}\{x \in {\mathbb{R}}^m {:}x_j = p_{j,1+\alpha_j}\,\,\text{for}\,\, 1 \leq j< i \,, \quad x_j = p_{j,1+\alpha_j} \,\,\text{for}\,\, i<j\leq m {\right}\}$$ and choose $\bar x \in (\Omega \cap L_{\alpha,i})\setminus P_{m,n}$. The function $g_{\alpha,i} : {\mathbb{R}}^m {\longrightarrow}{\mathbb{R}}$ given by $$g_{\alpha,i}(x) = f(x) -Q_{m,n,f}(x) - G(\bar x) N_\alpha(x) \,, \,\,\, G(\bar x)= \big(f(\bar x) -Q_{m,n,f}(\bar x)\big)/N_\alpha(\bar x)$$ is of class $C^{n+1}$ and possesses roots, namely $\{p_{i,1},\dots,p_{i,\alpha_i}, \bar x\}$, on $L_{\alpha, i}$. Recursively applying Rolle’s Theorem, this implies that $\partial^{k}_{x_i}g_{\alpha,i}$ possesses roots on $L_{\alpha,i}$, $0 \leq k \leq \alpha_i$. Hence $$\partial^{\alpha_i+1}_{x_i} g_\alpha(\xi)= 0 \,, \quad \text{for some}\,\,\, \xi=\xi_{\bar x} \in L_{\alpha,i} \cap \Omega \,.$$ Theorems \[GN\] and \[SV\] yield Eq. , which implies that the restriction of $Q_{m,n,f}$ to $L_{\alpha,i}$ is of degree $\alpha_i$. Thus, $\partial^{\alpha_i +1}_{x_i} Q_{m,n,f\,| L_{\alpha,i}}= 0$. Since $\partial^{\alpha_i}_{x_i} N_\alpha(x) = \alpha_i!$, this yields $$G(\bar x) = \frac{1}{\alpha_i! }\partial^{\alpha_i+1}_{x_i} f(\xi),$$ implying Eq. . Combining Lemma \[PartLEB\] with the classic error estimation for Chebyshev nodes in 1D [@gautschi] yields Eq. . To show $(ii)$, we recall that $S_{m,n}$ is a projection, i.e., $S_{m,n}(S_{m,n})=S_{m,n}$ implying that $S_{m,n}(Q) = Q$ for any $Q \in \Pi_{m,n}$. Thus, for every $f \in H^k(\Omega,{\mathbb{R}})$, we bound $$\begin{aligned}
{||\,}f- S_{m,n}(f) {\,||}_{C^0(\Omega)} & = {||\,}f- Q_{m,n}^* + Q_{m,n}^* - S_{m,n}(f) {\,||}_{C^0(\Omega)} \\
& \leq {||\,}f- Q_{m,n}^*{\,||}_{C^0(\Omega)} + {||\,}Q_{m,n}^* - S_{m,n}(f) {\,||}_{C^0(\Omega)} \\
& = {||\,}f- Q_{m,n}^*{\,||}_{C^0(\Omega)} + {||\,}S_{m,n}(Q_{m,n}^*) - S_{m,n}(f) {\,||}_{C^0(\Omega)} \\
& \leq (1+\Lambda(S_{m,n}),H^k(\Omega)){||\,}f- Q_{m,n}^*{\,||}_{C^0(\Omega)} .\end{aligned}$$
In summary, we have established all approximation results of Main Result III as stated in Theorem \[III\] in Section \[main\], thereby extending the well-known classic results from 1D to arbitrary dimensions. This answers Question \[Task\] set out in the problem statement.
Numerical Experiments {#EX}
=====================
In order to illustrate our approach and demonstrate its performance in practice, we implement a prototype MATLAB (R2018a (9.4.0.813 654) version of the PIP-SOLVER running on an Apple MacBook Pro (Retina, 15-inch, Mid 2015) with a 2.2GHz Intel Core i7 processor and 16GB 1600MHz DDR3 memory unser macOS Sierra (version 10.12.6.). The following numerical experiments demonstrate the computational performance and approximation accuracy of our solver in comparison with the classic numerical approaches.
![Numerical error for dimension $m=5$.[]{data-label="ACC2"}](Acc_test_I.pdf)
![Numerical error for dimension $m=5$.[]{data-label="ACC2"}](Acc_test_II.pdf)
For given $m,n \in {\mathbb{N}}$ and function $f: {\mathbb{R}}^m {\longrightarrow}{\mathbb{R}}$, we compare the following methods in terms of accuracy and runtime:
1. The *PIP-SOLVER* generates multidimensional Newton-Chebyshev nodes and determines the coefficients $C \in {\mathbb{R}}^{N(m,n)}$ of the interpolant $Q_{m,n,f}$ in multivariate Newton form.
2. The *Linear Solver* uses the multidimensional Newton-Chebyshev nodes $P_{m,n}$ generated by the PIP-SOLVER and then applies the MATLAB linear system solver to solve $$V_{m,n}(P_{m,n})C=F\,, \,\, F=\big(f(p_1),\dots,f(p_N(m,n))\big)^\mathsf{T} \in {\mathbb{R}}^{N(m,n)}$$ for the coefficients $C \in {\mathbb{R}}^{N(m,n)}$ of the interpolant $Q_{m,n,f}$ in normal form.
3. The *Linear Random Solver* uses nodes $P_{m,n}$ placed uniformly at random and then applies the MATLAB linear system solver to solve $$V_{m,n}(P_{m,n})C=F\,, \,\, F=\big(f(p_1),\dots,f(p_N(m,n))\big)^\mathsf{T} \in {\mathbb{R}}^{N(m,n)}$$ for the coefficients $C \in {\mathbb{R}}^{N(m,n)}$ of the interpolant $Q_{m,n,f}$ in normal form.
4. The *Inversion* method uses the multidimensional Newton-Chebyshev nodes $P_{m,n}$ generated by the PIP-SOLVER and then inverts the Vandermonde matrix $V_{m,n}(P_{m,n})$ using LU-decomposition to compute the coefficients $C \in {\mathbb{R}}^{N(m,n)}$ of the interpolant $Q_{m,n,f}$ in normal form.
We first compare the accuracy of the three approaches, which also serves to validate our method. To do so, we choose uniformly-distributed random numbers $c_1,\dots,c_{N-1} \in [-1,1]^{N(m,n)}$ to be the coefficients of a polynomial $Q \in \Pi_{m,n}$ in normal or multivariate Newton form. We then set $f = Q$ and measure the maximum absolute error in any coefficient, i.e. ${||\,}c- \widetilde{c} {\,||}_\infty$, when recovering $\widetilde Q$ by solving the PIP with respect to $(m,n,f)$.
Figures \[ACC1\] and \[ACC2\] show the average and min-max span of the numerical errors (over 5 repetitions with different $i.i.d.$ random coefficients; same 5 for every approach) for fixed degree $n=3$ and dimensions $m=2,\dots,35$, as well as for fixed dimension $m=5$ and degree $n=1,\dots,15$, with logarithmic scale in the $y$-axis.
The case $n=3$ is of high practical relevance, e.g., when interpolating cubic splines. In both cases, all methods show high accuracy, which reflects the fact that Newton-Chebyshev nodes yield well-conditioned PIPs. This is confirmed by the *Linear Solver* and the *Inversion* method showing comparable accuracy, while the *Linear Random Solver* is less accurate. The error of the PIP-SOLVER is almost constant on the level of the machine accuracy (double-precision floating-point number types). Especially in high dimensions, the PIP-SOLVER is several orders of magnitude more accurate than the other approaches.
![Runtimes for $n=3$, $1\leq m\leq 100$.[]{data-label="time2"}](time_3_35.pdf)
![Runtimes for $n=3$, $1\leq m\leq 100$.[]{data-label="time2"}](Time_high_v2.pdf)
We compare the computational runtimes of the approaches. To do so, we choose uniformly-distributed random function values $f_1,\dots,$ $f_{N(m,n)} \in [-1,1]$, $m,n \in {\mathbb{N}}$ as interpolation targets. Then, we measure the time required to generate the unisolvent interpolation nodes $P_{m,n}$ and add the time taken to solve the PIP with respect to $f : {\mathbb{R}}^m {\longrightarrow}R$ with $f(p_i)=f_i$, $p_i \in P_{m,n}$, $i =1,\dots,N(m,n)$ by each approach.
Algorithm Intervals Degree Pre-factor $p$ Exponent $q$
------------------------ ------------------- -------- ----------------- --------------
*Inversion* $m =15,\dots,35 $ $n=3$ $p =0.010737 $ $q=2.2982 $
*Linear Solver* $m =15,\dots,35 $ $n=3$ $p=0.0076072 $ $q=2.2907 $
*Linear Random Solver* $m =15,\dots,35 $ $n=3$ $p=0.012009 $ $q=2.3289 $
*PIP-SOLVER* $m=15,\dots,35$ $n =3$ $p=0.0076964 $ $q=1.2006 $
*PIP-SOLVER* $m=15,\dots,100$ $n =3$ $p=0.0034101 $ $q=1.2258 $
: Scaling of the computational cost by fitting the cost model $pN(m,n)^q$.
\[Tab\]
The average and min-max span of the runtimes (over 10 repetitions with different $i.i.d.$ random function values; same 10 for every approach) are shown in Figures \[time1\] and \[time2\] versus the dimension $m$ for fixed degree $n=3$.
While the actual problem size is $N(m,n)$, the dimension $m$ or the degree $n$ are more intuitive when characterizing a problem of fixed degree or fixed dimension, respectively. In low dimensions (inset figure) the *Linear Random Solver* performs best due to its low overhead for generating unisolvent nodes. However, at about $m=9$ there is a cross-over above which the PIP-SOLVER is much faster than the other methods. The absolute runtimes are below 0.05 seconds at the cross-over point, even though our prototype implementation of the PIP-SOLVER is not optimized. As Figure \[time2\] shows, even our simple implementation of the PIP-SOLVER can handle instances of dimension $m=100$ in the same time as the other methods require for $m=35$. Since $N(100,3)/N(35,3) \approx 20 $ the PIP-SOLVER outperforms the other approaches.
The scaling of the computational cost with respect to problem size $N(m,n)$ is reported in Table \[Tab\] for $n=3$. We fit all measurements with the cost model $pN(m,n)^q$. All fits show an *R-square* of 1. We observe that the exponent of the cost scaling of the PIP-SOLVER is more than $1$ less than the exponents of the other methods with pre-factors that are never larger. The quadratic upper bound we have proven in this paper for the PIP-SOLVER holds in all tested cases. The other approaches roughly scale with an exponent of 2.3, as expected.
In addition to having a lower time complexity, the PIP-SOLVER also requires less memory than the other approaches. Indeed, the PIP-SOLVER requires only ${\mathcal{O}}(N(m,n))$ storage, whereas all other approaches require ${\mathcal{O}}(N(m,n)^2)$ storage to hold the Vandermonde matrix. Due to this lower space complexity, we could solve the PIP for large instances, i.e., for $m >80$, where $N(m,3)\geq 10^5$ in less than 2 minutes, see Figure \[time2\], while classical approaches failed to solve such large instances due to insufficient memory on the computer used for the experiment.
We measure the runtime of the PIP-SOLVER for different polynomial degrees $n$. Again, we choose uniformly-distributed random function values $f_1,\dots,f_{N(m,n)} \in [-1,1]$, $m,n \in {\mathbb{N}}$, as interpolation targets. Then, we measure the time required to generate the multidimensional Newton-Chebyshev nodes $P_{m,n}$ and add the time taken to solve the PIP with respect to $f : {\mathbb{R}}^m {\longrightarrow}R$ with $f(p_i)=f_i$, $p_i \in P_{m,n}$, $i =1,\dots,N(m,n)$ for different $n$.
Figure \[degree\] shows the average and min-max span of the runtimes (over 10 repetitions with different $i.i.d.$ random function values) versus the dimension $m \in {\mathbb{N}}$ with logarithmic scale on the $y$-axis. We again fit the curves in the admissible intervals with the cost model $pN(m,n)^q$. Again, the goodness of fit as measured by the *R-square* is 1 in all cases. As expected, the exponent does not change much with degree. Just as in 1D, the almost linear scaling for low degrees reflects the computational power of the multivariate divided difference scheme.
![Runtimes of the PIP-SOVER for degrees $n=1,\dots,6$.[]{data-label="degree"}](Time_Degree.pdf)
Next, we test the approximation properties of the PIP-SOLVER. A classic test case in approximation theory is *Runge’s function* $$f_R(x)=\frac{1}{1+25x^2}\,.$$ One can easily verify that $\left|\frac{\mathrm{d}^k f_R}{\mathrm{d}x^k}(1/5)\right| {\longrightarrow}\infty$, for $k \rightarrow \infty$. Thus, $f_R$ is smooth but unbounded with respect to ${||\,}\cdot {\,||}_{C^\infty(\Omega)}$, which means that $f_R \not \in (C^\infty(\Omega),||~\cdot~||_{C^{\infty}(\Omega)})$. This is the reason, why $f_R$ can not be approximated by interpolation with *equidistant nodes* [@runge]. However, as long as $k$ remains bounded, we have ${||\,}f_R {\,||}_{C^k(\Omega)} \leq C_K$ for some $C_K \in {\mathbb{R}}^+$. Thus, for all $m \geq 1$, one verifies that the multivariate analog $f_R : {\mathbb{R}}^m {\longrightarrow}{\mathbb{R}}$ with $f_R(x)= \frac{1}{1+25{||\,}x{\,||}^2}$ satisfies $f_R \in H^k(\Omega,{\mathbb{R}})$ for $k >m/2$. By Theorem \[NCL\], all Sobolev functions can be approximated when using Newton-Chebyshev nodes. Therefore, we consider the multidimensional $f_R$ when testing the approximation abilities of the PIP-SOLVER.
We consider $m=5$, $\Omega=[-1,1]^m$ and use the PIP-SOLVER to compute the interpolant $Q_{m,n,f_R,CN}$ with respect to multidimensional Newton-Chebyshev nodes and the interpolant $Q_{m,n,f_R,EN}$ with respect to multidimensional Newton nodes $P_{m,n}= \oplus_{i=1}^m{\mathbb{P}}_i$ with $\#{\mathbb{P}}_i=n+1$ equidistant on $[-1,1]$. For technical resons we consider only even degrees $n \in 2 {\mathbb{N}}$ allowing to choose $0$ as the center of the Chebyshev nodes. To estimate the $C^0$ distance between the interpolants, we generate 400 uniformly random points $P \subseteq \Omega$ once and measure $|Q_{m,n,f_R,CN}(p) - f_R(p)|$, $|Q_{m,n,f_R,EN}(p)-f_R(p)|$ for each $p\in P$ and $n =2,4,\dots,24$.
Figure \[Runge\] plots the maximum and the mean of the distances $|Q_{m,n,f_R,CN}(p) - f_R(p)|$, $|Q_{m,n,f_R,EN}(p)-f_R(p)|$ over the 400 randomly chosen but fixed points, with logarithmic scale in the $y$-axis. Though the interpolant $Q_{m,n,f_R,EN}$ with equidistant nodes approximates $f_R$ for low degrees, it diverges with increasing degree $n \geq 6$. In contrast, the interpolant $Q_{m,n,f_R,CN}$ continuous to converge to $f_R$ uniformly on $P$. Thus, we confirm that also in higher dimensions equidistant Newton nodes are infeasible for approximating Runge’s function, while Newton-Chebyshev nodes result in uniform convergence.
![Interpolating Runge’s function in fixed dimension $m=5$ with degrees $n=2,4,\dots,24$.[]{data-label="Runge"}](Runge_v4.pdf)
Potential Applications {#APP}
======================
We highlight several potential applications of the PIP-SOLVER in scientific computing and computational science. This list is by no means exhaustive, as PIPs are a fundamental component of many numerical methods. However, the following applications may not be obvious:\
[**A1)**]{} *Basic numerics:* Given $m,n\in {\mathbb{N}}$, $m\geq 1$, and a function $f : {\mathbb{R}}^m {\longrightarrow}{\mathbb{R}}$. It is classical in numerical analysis to determine the integral $\int_{\Omega}f \,\mathrm{d}\Omega$, $\Omega \subseteq {\mathbb{R}}^m$, and the partial derivatives $\partial_{x_i}f(x) $, $1 \leq i \leq m$ of $f$. Upon the PIP, these desired quantities can easily be computed for the interpolation polynomial $Q_{m,n,f}$ of $f$. Due to our Main Result III (Theorem \[III\]), $Q_{m,n,f} \xrightarrow[n\rightarrow\infty]{} f$ uniformly. Thus, $\int_{\Omega}Q_{m,n,f} \,\mathrm{d}\Omega \xrightarrow[n\rightarrow\infty]{} \int_{\Omega}f \,\mathrm{d}\Omega$ and by strengthen the conditions on $f$ also $\partial_{x_i}Q_{m,n,f}(x) \xrightarrow[n\rightarrow\infty]{} \partial_{x_i}f(x) $ uniformly. A comparison with other approaches from numerical analysis is worth considering.\
[**A2)**]{} *Gradient descent* over multivariate functions is often used to (locally) solve (non-convex) optimization problems, where $\Omega \subseteq \mathbb{R}^m$ models the space of possible solutions for a given problem and $f: {\mathbb{R}}^m {\longrightarrow}{\mathbb{R}}$ is interpreted as an objective function. Thus, one wants to minimize $f$ on $\Omega$. Often, the function $f$ is not explicitly known $\forall x \in \Omega$, but can be evaluated point-wise. Due to the Infeasibility of previous interpolation methods for , direct interpolation of $f$ on $\Omega$ was often not possible or not considered. Therefore, classical numerical methods like the *Newton-Raphson iteration* could not be used if analytical gradients were not available. Instead, *discrete or stochastic gradient descent* methods were usually applied. However, these methods converge slowly and are potentially inaccurate. The PIP-SOLVER allows (locally) interpolating $f$ even for and enables (locally) applying classic *Newton-Raphson methods*. Consequently, local minima could be found faster and more accurately. Moreover, the analytical representation of $Q$ potentially allows determining the global optimum if $f$ can be uniformly approximated [@PolyMin; @Min].\
[**A3)**]{} *ODE & PDE solvers:* A core application of numerical analysis is the approximation of the solution of Ordinary Differential Equations (ODE) or Partial Differential Equations (PDE). This always involves a (temporal and/or spatial) discretization scheme and a solver for the resulting equations. There are three classes of methods: collocation schemes, Galerkin schemes, and spectral methods. Spectral methods are based on Fourier transforms. Since FFTs are only efficient on regular Cartesian grids, spectral methods are hard to apply in complex geometries and on adaptive-resolution discretizations. The present PIP-SOLVER, however, is not limited to polynomial bases and could enable spectral methods on arbitrary distributions of discretization points in arbitrary geometries by interpolating with respect to a Fourier basis [@IEEE]. Collocation methods can generally be understood as PIPs, as becomes obvious in the generalized formulation of finite-difference schemes and mesh-free collocation methods [@Schrader]. The inversion of the Vandermonde matrix implied in mesh-free methods, or the choice of mesh nodes in compact finite-difference schemes, could benefit from the algorithms presented here. Finally, Galerkin schemes are based on expanding the solution of the differential equation in some basis functions, which is essentially what the PIP does for the orthogonal basis of monomials. Since the theory presented here is not restricted to this particular choice of basis, it is conceivable that similar algorithms can be formulated for other bases as well, potentially even for non-orthogonal ones.\
[**A4)**]{} *Adaptive sampling* methods aim to explore a domain $\Omega \subseteq {\mathbb{R}}^m$ such that the essential information of $f : \Omega {\longrightarrow}{\mathbb{R}}$ is recovered. A classic example from statistics is multidimensional *Bayesian inference* [@bayesian], which relies on adaptive sampling methods. Mostly, these methods are based on *Markov-Chain Monte-Carlo* or *sequential Monte-Carlo* sampling. The notion of unisolvent node sets in high dimensions, as we have provided here, potentially helps design sampling proposals that explore $\Omega$ in a more controlled, complete, or more efficient way.\
[**A5)**]{} *Spectral analysis:* The foundation of spectral analysis is to represent a function $ f : \Omega \subseteq {\mathbb{R}}^m {\longrightarrow}{\mathbb{R}},{\mathbb{C}}$ with respect to some functional basis $f(x)= \sum_{i=1}^\infty c_ib_i(x)$, $c_i \in {\mathbb{R}},{\mathbb{C}}$, ${\mathrm{span}}\{b_i\}_{i\in{\mathbb{N}}}=L^2(\Omega,{\mathbb{R}})$, e.g., with respect to Zernike or Chebyshev polynomials, linear (Fourier) harmonics, spherical harmonics, etc. This allows analyzing and understanding the essential character $f_0 = \sum_{i\in I} c_ib_i(x)$ of $f$, where the finite set $I$ is chosen such that the $c_i$ with $i \in I$ cover the most relevant amplitudes. Interpolating $f$ with respect to the specified basis $b_i$ is the classic method of computing the coefficients $c_i$. In [@IEEE] we have already described how to extend the PIP-SOLVER to Fourier basis. An adaption to other functional bases can be done analogously, which potentially improves numerical spectral analysis.\
[**A6)**]{} *Cryptography:* A maybe surprising application is found in cryptography. There, the PIP is used to “share a secret” by choosing a random polynomial $Q \in {\mathbb{Z}}[x]$ with integer coefficients in dimension $m=1$. Knowing the values of $Q$ at $n+1$ different nodes (keys) enables one to determine for some large prime number $p \in {\mathbb{N}}$. However, knowing only $n$ “keys” (nodes) prevents one for opening the “door”. Certainly, this method can be generalized to arbitrary dimensions using our approach [@Shamir]. Since the PIP-SOLVER performs with machine accuracy, it can also prevent the reconstructed message from being corrupted by numerical noise.\
There are many other computational schemes that require interpolation or are closely related to the PIP *linear or polynomial regression* in machine learning. We therefore close with a qualitative discussion of the PIP-SOLVER and state open questions, which potentially yield generalizations and further improvements in the future.
Discussion and Conclusions {#Conc}
==========================
Even though *Newton interpolation* in one dimension has been known since the 18$^\text{th}$ century, this may be the first generalization of this fundamental algorithm to arbitrary dimensions. We have provided a complete characterization of the polynomial interpolation problem (PIP) in arbitrary dimensions and polynomial degrees. We have provided an algorithm called PIP-SOLVER (see Algorithm \[alg:PIPSOLVER\]) that computes the solution to generalized PIPs in ${\mathcal{O}}(N(m,n)^2)$ time and ${\mathcal{O}}(N(m,n))$ space, where $N(m,n)$ is the number of unknown coefficients of the interpolation polynomial with $m$ variables of degree $n$. We have shown that the algorithm generates unisolvent node sets for which the Vandermonde matrix is not only well conditioned, but has triangular form, enabling efficient and accurate numerical solution using a multivariate divided difference scheme also presented here. We have further provided the corresponding extensions of the Horner scheme, enabling evaluating the polynomial in ${\mathcal{O}}(N(m,n))$ and its integral and derivatives in ${\mathcal{O}}(nN(m,n))$ time. Lastly, we have studied the approximation properties of multivariate Newton interpolation polynomials and derived the notion of multidimensional Newton-Chebyshev nodes, showing that any Sobolev function of sufficient regularity can be uniformly approximated, and we have provided the corresponding upper bounds on the approximation errors. Taken together, these contributions solve Problem \[PolyInt\] and answer Question \[Task\] for arbitrary dimensions and degrees.
Choose admissible plane $H$ according to Theorem \[GN\] **return** $f_{|H}$ **return** $f(0)$ Choose $Q_H \in \Pi_{m,1}$ with $Q_H(H)=0$ $Q_1 = \text{PIP-SOLVER}(f_{|H},m-1,n)$ $\widehat f = (f - Q_1)/Q_H $ on ${\mathbb{R}}^m\setminus H$ $Q_{m,n,f} = Q_1 + Q_H\cdot \text{PIP-SOLVER}(\widehat f,m,n-1)$ **return** $Q_{m,n,f}$
The problem statement and questions considered here are not new. Nor is the idea of decomposing the PIP w.r.t. $m,n \in {\mathbb{N}}$ into sub-problems of dimension and degree and , respectively, which has already been mentioned in [@Guenther; @2000]. In [@Gasca], the cases $m=2,3$ were treated explicitly, while a generalization to arbitrary dimension was sketched and some characterizations of unisolvent nodes in arbitrary dimensions were given. However, the problem of computing the interpolation polynomial $Q_{m,n,f}$ efficiently and accurately remained unsolved. Indeed, all previous decomposition approaches were limited to relatively low dimensions and to nodes on pre-defined grids or meshes [@Bos; @Erb; @FAST; @Gasca2000; @Chung], not providing a general algorithm for solving the PIP for arbitrary $m,n\in {\mathbb{N}}$.
Here, we were able to provide such a general algorithm and prove bounds on its time and space complexity. The achievement that made this possible was the introduction of unisolvent multidimensional Newton nodes $P_{m,n}$, which, combined with the multivariate Newton basis of $\Pi_{m,n}$, yielded a form of the Vandermonde matrix $V_{m,n}(P_{m,n})$ that allows solving the system of linear equations $V_{m,n}(P_{m,n})C=F$ in less time than what is required for general matrix inversion.
We demonstrated and validated our results in a practical software implementation of the PIP-SOLVER, showing scaling to high-dimensional spaces as common in applications ranging from machine learning to computational statistics. Owing to the elegance of the theory, the implementation of the PIP-SOLVER is straightforward and results in a simple code.
Our simple reference implementation is not tuned for efficiency at the time of writing. In the future, we foresee distributed- and shared-memory parallel implementations in compiled programming languages to further reduce runtimes and enable even larger problems to be solved. This is possible since the recursive decomposition yields sub-problems that can be processed in parallel, using inter-process communication to ensure correct decomposition of the problem and synthesis of the final solution.
A possible extension of the presented theory is to also cover interpolation in other bases, such as the Fourier basis [@IEEE], spherical harmonics, or Zernike polynomials. We also believe that our approach can be extended to multivariate barycentric or Lagrange Interpolation. In 1D, it is well known that Newton and Lagrange polynomials are related through the *barycentric weights* [@berrut; @werner]. Precomputing these weights, barycentric Lagrange interpolation only requires linear time ${\mathcal{O}}(N(1,n))$ to compute the interpolant of degree $n\in {\mathbb{N}}$ in 1D. The present approach to multivariate Newton polynomials could lead to multivariate barycentric Lagrange interpolation schemes running in ${\mathcal{O}}(mN(m,n))$ for arbitrary $m,n \in {\mathbb{N}}$. Similarly, multivariate Hermite Interpolation could also be considered, since efficient realizations of this concept are closely related to Newton and Lagrange interpolation. In 1D, Hermite interpolation is a classic concept [@gautschi] that requires one to know the function $f : [-1,1]{\longrightarrow}{\mathbb{R}}$ and its derivatives on less than $n+1$ nodes in order to compute the interpolant $Q_{f,n}\in \Pi_{1,n}$.
A case of special interest is spline interpolation [@unserSplines]. Fast implementations of spline interpolation are available [@fspline; @unserFast], making them a powerful and popular tool. Spline interpolation is based on decomposing the domain $\Omega =[-1,1]^m$ into smaller, shifted hypercubes $\Omega_i=[-{\varepsilon},{\varepsilon}]^m + p_i$, $p_i \in \Omega$, $ i \in I$, ${\varepsilon}>0$, and “gluing” the interpolants $Q_i : \Omega_i {\longrightarrow}{\mathbb{R}}$ to a $k$-times differentiable global function $Q$, $k \in {\mathbb{N}}$. Therefore, $\#I$ increases exponentially with dimension $m$. Tensorial formulations [@tensor; @tensor2] are available for efficient local spline interpolation. However, the exponential scaling of the number of hypercubes, $\#I$ in which this has to be done cannot be overcome. This is why spline interpolation is mostly used for lower-dimensional problems. Further, the mathematical character of $f$ is not recovered in the spline basis and the approximation quality depends on the spline degree and on the choice of node conditions [@schoen; @unserWiener; @Unser:2005]. Therefore, spline interpolation is well suited to signal and image processing in low dimensions. The $L^2$-bases we proposed in $A5)$ above provide a potentially interesting choice in high dimensions. In principle, Hermite interpolation could also be used to glue spatially decomposed interpolants to a global $k$-times differentiable function. The notion of a globally unisolvent node set $P_{m,n}$ could then provide a way of spatially decomposing $\Omega$ such that the resulting global Hermite interpolant is of high approximation quality and can be computed efficient. We expect that this hybrid Hermite-spline interpolation method would relax some of the issues with splines in high dimensions.
The main practical limitation of our approach is that it requires the function $f : \Omega {\longrightarrow}{\mathbb{R}}$ to be computable in constant time, which means that the algorithm is free to choose the interpolation nodes. In many problems, however, $f$ is only known on a previously fixed node set ${\mathbb{P}}\subseteq {\mathbb{R}}^m$ (i.e., the data given). In the case where ${\mathbb{P}}$ is a (regular) grid, we can choose multivariate Newton nodes $P_{m,n}\subseteq {\mathbb{P}}$ and our approach works. However, if ${\mathbb{P}}$ is arbitrarily scattered, our approach does not directly apply. While resampling/reorganizing the data points can sometimes be an option, a general solution is outstanding. In particular, the optimal ordering of the nodes yielding the best numerical approximation in the sense of minimal rounding errors remains to be investigated. While this does not matter in infinite-precision arithmetic, finite-precision floating-point arithmetic accuracy, as well as algorithm speed, can be improved by appropriately ordering the points [@tal1988high].
The main theoretical limitation of our approach is that we bounded the Lebesgue functions with respect to the $H^k$-norm for $k>m/2$. Additionally, we assumed that the considered functions $f \in C^0(\Omega,{\mathbb{R}})$, $\Omega=[-1,1]^m$, $m \in {\mathbb{N}}$ are periodic. While these assumptions match the requirements of many practical applications, the classic Lebesgue function estimates just require $f$ to be continuous. Hence, a deeper study of Lebesgue functions with respect to the powerful Sobolev analysis of periodic functions might improve the bounds presented here and might provide a way of controlling the convergence rate of $Q_{m,n,f}{\longrightarrow}f$.
Notwithstanding these open questions, we suspect that our concepts could provide a general perspective for considering multivariate interpolation problems, since, for dimension $m=1$, our concepts include the classic Newton interpolation scheme. We thus hope that the concepts and algorithms presented here will be useful to the community across application domains.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'This paper presents a novel approach to wave propagation inside the Fabry-Perot framework. It states that the time-averaged Poynting vector modulus could be nonequivalent with the squared-field amplitude modulus. This fact permits the introduction of a new kind of nonlinear medium whose nonlinearity is proportional to the time-averaged Poynting vector modulus. Its transmittance is calculated and found to differ with that obtained for the Kerr medium, whose nonlinearity is proportional to the squared-field amplitude modulus. The latter emphasizes the nonequivalence of these magnitudes. A space-time symmetry analysis shows that the Poynting nonlinearity should be only possible in noncentrosymmetric materials.'
author:
- Alberto Lencina
- Pablo Vaveliuk
title: 'Squared-field amplitude modulus and radiation intensity nonequivalence within nonlinear slabs'
---
Introduction
============
A classical topic in electromagnetism is the study of wave transmission in a finite parallel-plane faces medium, know as Fabry-Perot resonator. When the medium presents nonlinear behavior, bistability appears [@gibbs]. To explain this phenomenon, the nonlinear Fabry-Perot resonator (NLFP) was modelled by a third order susceptibility or Kerr-type nonlinearity [@marburger]. At monochromatic plane wave excitation, the NLFP stationary regime is summarized in a non-time-dependent nonlinear wave equation for the complex field amplitudes, the nonlinear Helmholtz equation (NLHE). Hence, the reflectance and transmittance problem reduces to finding the NLHE solution with appropriate boundary conditions for the field amplitude modulus and phase.
The NLHE complexity led to approximate methods of resolution. Many approaches consider two counter-propagating waves in the medium and the analysis is done by separately considering the effects on each wave [@marburger; @miller; @danckaert; @biran]. Unfortunately, the linear superposition principle is no longer valid in nonlinear media and the separation of the electromagnetic field in these back and forth waves is meaningless. As a result, the NLHE separation into two equations, one for each wave, is only possible by neglecting various coupling nonlinear terms that would give an important contribution to the accuracy of the transmittance results. Moreover the Slowly Varying Envelope Approximation (SVEA) is often applied to these waves [@marburger; @miller; @danckaert] when, yet within the counter-propagating wave approach, its validity was questioned [@biran]. Also, the boundary conditions were simplified rather than rigorously treating them [@marburger; @fobelets]. The above facts suggest that all these approximated approaches could not physically be equivalents to the exact problem.
The work done by Chen and Mills exactly solve the NLFP for an absorptionless Kerr-type medium [@mills1]. The proper of its resolution method was to assume a general complex field within the medium, disregarding the concept of counter-propagating waves. Chen and Mills derive a two coupled equation system for the field amplitude modulus and phase together with general boundary conditions, thus obtaining an analytic-transcendental solution for the transmittance of the NLFP.
On the other hand, their work permitted us to note an implicit difference between the time-averaged Poynting vector modulus, i.e. the electromagnetic radiation intensity $I$, and the squared-field amplitude modulus $(|E|^{2})$ inside the nonlinear medium. If the nonequivalence of these magnitudes were true, it could change certain well-established fundamental concepts in classical electrodynamics. This fact motivated us to develop a novel approach to wave propagation in nonlinear media inside the Fabry-Perot framework called *S-Formalism*. It introduces a new variable related to the time-averaged Poynting vector which states that its magnitude could be nonequivalent with the squared-field amplitude modulus contrary to commonly accepted. Furthermore, the S-Formalism presents two important advantages: it permits to directly monitor the radiation intensity within the medium, and it avoids approximations, such as the SVEA, simplification of boundary conditions, and so on.
The fact that the time-averaged Poynting vector modulus be nonequivalent with the squared-field amplitude modulus, as the S-Formalism will show, implies that the nonlinearity of Kerr-type media is not proportional to the intensity which is contrary to what has been established to-date. This assertion leads to the following question regarding the modelling of the NLFP: is it a Kerr-type nonlinearity, or does it vary proportionately to the intensity? As this question does not have a definitive answer, the existence of the latter cannot be denied. Then, we define the *Poynting medium* as the medium where nonlinearity is proportional to intensity. Thus, our objective is to solve the *Poynting-NLFP* through the S-Formalism comparing the resultant transmittance with that obtained for a *Kerr-NLFP* to remark the nonequivalence between squared-field amplitude modulus and radiation intensity.
![An harmonic plane wave strikes a nonlinear Fabry-Perot resonator, to be reflected and transmitted. The Regions I and III constitute, for simplicity, the same linear dielectric medium (e.g., air).\
[]{data-label="fig=problema"}](Figura1.eps){width="6.5cm" height="4.5cm"}
In section II, we derive the S-Formalism in the following form: first the time-averaged Poynting vector assuming harmonic fields is calculated. Then, a dimensionless variable which is proportional to intensity is introduced. Thus, a set of field evolution differential equations and the general boundary condition equations on the new field variable are obtained, which constitutes the S-Formalism. In section III, the new approach is applied to derive the transmittance for the Poynting-NLFP and the results compared with that obtained for a Kerr-NLFP. A brief discussion about the possibily of existence and observation of Poynting media is given. Finally, in section IV, we conclude.
The S-Formalism approach
========================
Refering to Figure \[fig=problema\], we start writing the linearly polarized transversal harmonic electromagnetic fields of frequency $\omega$ as
$$\begin{aligned}
\mathbf{E}_{\ell}(z,t)=\frac{1}{2}\left(E^{\omega}_{\ell}(z)e^{-i\omega
t}+c.c.
\right)\mathbf{\hat{i}},\\[3pt]
\mathbf{H}_{\ell}(z,t)=\frac{1}{2}\left(H^{\omega}_{\ell}(z)e^{-i\omega
t}+c.c. \right)\mathbf{\hat{j}},\end{aligned}$$
where $E^{\omega}_{\ell}(z)$ and $H^{\omega}_{\ell}(z)$ are the non-time-dependent complex amplitudes for $\ell= $I, II, III. From region I, a plane wave of amplitude $E_{0}$ and wave vector $k_{0}$ impinges perpendicularly on a nonmagnetic, isotropic and spatially nondispersive medium of thickness $d$ (region II). The optical field is assumed to maintain its polarization along this region so that a scalar approach is valid. The reflected and transmitted plane waves have amplitudes $rE_{0}$ and $tE_{0}$ with $r$ and $t$ the complex reflection and transmission coefficients, respectively. Then, at region I and III the spatial-dependent complex amplitudes are given by $$\begin{aligned}
E_{I}^{\omega}(z)&=&E_{0}\left(e^{i k_{0}z}+r e^{-i k_{0}z}
\right)\label{eq=fielI},\\[3pt]
E_{III}^{\omega}(z)&=&E_{0}t e^{i k_{0}z}\label{eq=fielIII}.\end{aligned}$$ Similarly to Ref. [@mills1], at region II, we write down the following *ansatz* for the spatial-dependent complex amplitude of the electric field: $$E^{\omega}_{II}(z)\:=\:E_0 \mathcal{E}(z) e^{i
\phi(z)},\label{eq=camporegII}$$ where the dimensionless amplitude $\mathcal{E}(z)$, and phase $\phi(z)$, are both real functions of $z$.
The time-averaged Poynting vector $\langle\:\mathbf{E}_{\ell}(z,t)\times\mathbf{H}_{\ell}(z,t)\:\rangle$ can be easily calculated with the aid of Faraday’s law, giving $$\begin{aligned}
\langle\mathbf{S}\rangle_{\ell}=\frac{1}{2 \mu_{0}\omega}Im
\left\{\left[E^{\omega}_{\ell}(z)\right]^{*}\frac{\partial
E^{\omega}_{\ell}(z)}{\partial z}\right\}\mathbf{\hat{k}},\end{aligned}$$ where $\mu_{0}$ is the vacuum permeability. From this expression we calculate the intensity for the three regions:
\[poyting\] $$\begin{aligned}
\langle S\rangle_{I}&=&I_{0}\left(1-|r|^{2}\right),\\[3pt]
\langle S\rangle_{II}
&=&I_{0}\:k_{0}^{-1}
\mathcal{E}^{2}(z)\frac{\partial\phi(z)}{\partial
z} \equiv I_{0}S(z)\label{eq=defS}\label{eq=SregII},\\[3pt]
\langle S\rangle_{III}&=&I_{0}|t|^{2},\label{transmitancia}\end{aligned}$$
where $I_{0}={k_{0}E^{2}_{0}}/({2 \mu_{0}\omega})$ is the incident intensity. In region II, Eq. (\[eq=defS\]) defines the dimensionless field variable $$\label{S}
S\equiv k_{0}^{-1} \mathcal{E}^{2}\frac{\partial\phi}{\partial z},$$ directly related with the intensity inside the medium and which will characterize the S-Formalism. From Eq. (\[S\]) it is clear that if $\phi$ is not a linear function on $z$, as is often happens in nonlinear media, then $S$ and $\mathcal{E}^{2}$ are nonequivalents.
The next step is to derive the NLHE in terms of the classical field variables ($\mathcal{E},\phi$), and transform it into a set of equivalent equations in terms of ($\mathcal{E},S$). The NLHE is derived from the macroscopic Maxwell equations complemented by appropriate constitutive relations. We assume that the polarization $\mathbf{P}$, and current density $\mathbf{J}$, vary only in the electric field direction with frequency $\omega$, neglecting higher harmonics, and their spatial-dependent complex amplitudes satisfy the following constitutive relations: $$\begin{aligned}
P^{\omega}_{II}(z)&=&\epsilon_{0}\chi_{gen}[z,E^{\omega}_{II},H^{\omega}_{II}]\:
E^{\omega}_{II}(z),\\[3pt]
J^{\omega}_{II}(z)&=&\sigma_{gen}
[z,E^{\omega}_{II},H^{\omega}_{II}]\:E^{\omega} _{II} (z),\end{aligned}$$ where $\epsilon_0$ is the vacuum permittivity and, $\chi_{gen}$ and $\sigma_{gen}$ are the generalized susceptibility and conductivity respectively, that are real and contain the linear as well as a possible nonlinear medium response. Note that the constitutive relations are not explicitly written since cases could exist where it is not possible to describe the nonlinear polarization and current density by the classical electric field power expansion. Thereby, the scalar NLHE is $$\begin{aligned}
\Big[\frac{d^2}{dz^2}
+k_{0}^2(1+\chi_{gen}) +i \omega \mu_{0}\:\sigma_{gen}\Big]
E^{\omega}_{II}(z)=0. \label{eq=nlhe}\end{aligned}$$ This equation constitutes the starting point to study several linear and nonlinear monochromatic wave propagation phenomena within the Fabry-Perot framework. Substituting Eq. (\[eq=camporegII\]) into (\[eq=nlhe\]) and by using Eq. (\[S\]), we derive the following set of spatial evolution equations for the field variables $\mathcal{E}(z)$ and $S(z)$:
$$\begin{aligned}
&&\frac{d^2\mathcal{E}}{d z^2}
+k_{0}^{2}\left((1+\chi_{gen}[z,\mathcal{E},S])\:\mathcal{E}
-\frac{S^{2}}{\mathcal{E}^{3}}\right)=0,
\\[3pt]
&&\frac{d S}{d z}+\frac{\omega}{k_{0}}\mu_{0}\:\sigma_{gen}
[z,\mathcal{E},S]\:\mathcal{E}^{2}\label{eq=poyntinhtheo}=0.\end{aligned}$$
\[eq=formalismoS\]
To guarantee the physical content of the solution, these equations must be necessarily complemented with the following boundary conditions: the continuity of the tangential components of the electric and magnetic field at the interfaces. The general boundary conditions were rigourously derived in Ref. [@mills1]: four equations as functions of $(\mathcal{E},\phi)$ at $z=0$ and $z=d$ which, by using Eq. (\[S\]), are transformed into three equations in terms of the variables $(\mathcal{E},S)$ to give
$$\begin{aligned}
&&\left(\mathcal{E}(0)+\frac{S(0)}{\mathcal{E}(0)} \right)^{2}+
\left(\frac{1}{k_{0}} \frac{d\mathcal{E}}{dz} \Big|_{z=0} \right)^{2}=4,\\[3pt]
&&S(d)-\mathcal{E}^{2}(d)=0,\\[3pt]
&&\frac{d\mathcal{E}}{dz}\Big|_{z=d}=0.\end{aligned}$$
\[eq=condecont\]
From Eqs. (\[poyting\]) and (\[eq=condecont\]), the transmittance is obtained as $$T=|t|^2=S(d),$$ and the energy conservation is guaranteed thorough the expression: $$|r|^2+|t|^2=1-\left(S(0)-S(d)\right).$$ This equation establishes that the reflectance and transmittance are limited by the boundary values of the time-averaged Poynting vector. For a nonabsorbent medium $S(d)=S(0)$, then $|r|^{2}+|t|^{2}=1$.
Equations (\[eq=formalismoS\]) and (\[eq=condecont\]) represent the “S-Formalism” which were derived without assuming approximations such as the counter-propagating waves, SVEA, simplifications on the boundary conditions, and so on. Also, note that Eq. (\[eq=poyntinhtheo\]) represents the time-averaged Poynting Theorem applied to the problem of harmonic fields simplifying the interpretation of $\sigma_{gen}$ as the dissipation properties of the medium. In particular, when $\sigma_{gen}=0$, the dimensionless intensity $S$ is a constant fixed by the boundary conditions. Furthermore, through $S(z)$ it is possible to monitor directly the intensity along the medium as a function of the spatial coordinate, as opposed to using the conventional formalism.
The S-Formalism is useful to analyze the linear case as well as the nonlinear one. Before to study the latter, i.e. the comparison between the Poynting and Kerr media in an effort to show the explicit difference between $S$ (or $I$) and $\mathcal{E}^2$ (or $|E^{2}|$) in nonlinear media, we refer to the linear case. According to our analysis [@ajp], there are only two situations where the relationship $I=cte |E^2|$ holds true: firstly, a single plane wave propagates in a infinite or semi-infinite linear dielectric characterized by $\sigma_{gen}=0$ and $\chi_{gen}=\chi^{(1)}$ where $\chi^{(1)}$ is the linear susceptibility. Under these conditions, Eqs. (\[eq=formalismoS\]) relate the constants $S$ and $\mathcal{E}$ by $S=(1+\chi^{(1)})^{1/2}\:
\mathcal{E}^2$; secondly, a single plane wave propagates in a semi-infinite linear absorber characterized by $\chi_{gen}=\chi^{(1)}$ and $\sigma_{gen}=\sigma$ where $\sigma$ is the ohmic conductivity and such that $S(z)\propto \mathcal{E}^2(z)$, being both proportional to a decreasing exponential function of $z$. On the contrary, when the medium is finite, e.g. a Fabry-Perot with boundary conditions at interfaces, $S$ is no longer equivalent to $\mathcal{E}^2$, not even for the linear dielectric case because $S$ is a constant and $\mathcal{E}^2$ is an oscillating function of $z$ [@ajp].
The Poynting medium
===================
Constitutive relations and transmittance results
------------------------------------------------
At this point, we introduce the Poynting medium by the following constitutive relations: $$\begin{aligned}
\chi_{gen}&=&\chi^{(1)}+\gamma I_0 S(z),\label{constitutive}\\[3pt]
\sigma_{gen}&=&0,\end{aligned}$$ where $\gamma$ is the nonlinear coefficient. Eqs. (\[eq=formalismoS\]) have a simple analytical solution given by $$\begin{aligned}
S(z)&=&S_{0},\\[3pt]
\mathcal{E}(z)&=&\sqrt{\frac{S_{0}}{2}\left[
\left(1-\frac{k_{0}^{2}}{k_{1}^{2}}\right)\cos \left[2 k_{1}
(z-d)\right]+1+\frac{k_{0}^{2}}{k_{1}^{2}}\right]},\:\:\:\:\:\:\:
\label{eq=epsPoy}\end{aligned}$$ where $\gamma>0$ and $k_{1}^{2}=k_{0}^{2}(1+\chi^{(1)}+\gamma
I_{0} S_{0})$. The constant $S_{0}$ is fixed by $$\left(1-\frac{k_{1}^{2}}{k_{0}^{2}}\right)\mathcal{E}^{2}(0)+
\left(3+\frac{k_{1}^{2}}{k_{0}^{2}}\right)S_{0}-4=0.
\label{eq=bcPoy}$$ Combining Eqs. (\[eq=epsPoy\]) and (\[eq=bcPoy\]), the transmitance can be expressed in a similar fashion as the linear Fabry-Perot resonator as $$\label{airy}
T=\frac{1}{1+F\sin^{2}\left(k_1 d\right)},$$ where $F=k_0(1-k_1/k_0)^2/(4k_1)$. Carefully noting that Eq. (\[airy\]) is a transcendental expression since $k_{1}$ depends on $S_{0}$.
![Transmittance against nonlinear parameter for $(a_i)$ Poynting medium, $(b_i)$ Kerr medium with $k_{0}\:d=2
\pi$. For $i=1$, $\chi^{(1)}=1.25$; and $i=2$, $\chi^{(1)}=5.25$.\
[]{data-label="fig=curvas"}](Figura2.eps){width="8.5cm" height="5cm"}
Now, we compare the transmittance results of the Poynting and Kerr media. The latter defined by $$\begin{aligned}
\chi_{gen}&=&\chi^{(1)}+\gamma I_0 \mathcal{E}^2(z),\\[3pt]
\sigma_{gen}&=&0.\end{aligned}$$
![Transmittance against dimensionless thickness for $(a)$ Poynting medium, $(b)$ Kerr medium with $\chi^{(1)}=5.25$ and $\gamma I_{0}=2$.\
[]{data-label="fig=curvas"}](Figura3.eps){width="8.5cm" height="3cm"}
The Kerr-NLFP transmittance results were taken from Ref. [@mills1]. Figure 2 shows $T$ against the nonlinear parameter $\gamma I_{0}$ for two different values of $\chi^{(1)}$. Figures 2($a_i$) and 2($b_i$) correspond to the Poynting and Kerr medium, respectively. From these figures, it is apparent that the transmittance of the Poynting-NLFP as well as the Kerr-NLFP are multistable. However, for increasing values of $\chi^{(1)}$, the peak transmittance separation diminishes for the Kerr medium while it increases for the Poynting medium. Also, the Kerr multistability appears for smallest values of the nonlinear parameter $\gamma I_0$. The transmittance difference of both media emphasizes the $I$ and $|E|^2$ nonequivalence. Figure 3 depicts $T$ on the dimensionless thickness $k_0d(1+\chi^{(1)})^{1/2}/(2\pi)$ enhancing the nonlinearity difference of Poynting and Kerr media. Note that the departure from Airy-type function for the Kerr medium is stronger than for the Poynting medium.
On the other hand, Figure 4 shows that the Poynting nonlinear susceptibility, $\chi_{gen}-\chi^{(1)}$, has a constant value along the medium ($z$ coordinate). In return, the Kerr nonlinear susceptibility varies periodically. This fact implies the formation of a phase grating in the Kerr medium on the contrary to the Poynting medium. Perhaps, this substantial difference can be measured, and this could be the starting point to experimentally identify a Poynting medium.
Transformation properties under spatial inversion and time reversal
-------------------------------------------------------------------
It is a fact that several unusual types of nonlinearities were predicted before its experimental observation, as it was remarked, for example, in the pioneer works of Baranova et al. [@baranova]. With the aim to elucidate the isotropic medium requirements to observe the new phenomena, those authors pointed out the necessity of a analysis about transformation properties of electromagnetic quantities under rotations, spatial inversion and time reversal. Therefore, this symmetry analysis is also necessary to delimit the Poynting medium requirements.
![Nonlinear susceptibility against dimensionless spatial coordinate for each of the three solutions compatible with the boundary conditions. Continuous line: Poynting medium. Broken line: Kerr medium. The parameter values are $\gamma I_0=9$, $\chi^{(1)}=1.25$, and $k_0\:d$ as defined in Fig. 2.\
[]{data-label="fig=indices"}](Figura4.eps){width="6cm" height="3.5cm"}
The magnitude that characterizes the electromagnetic response of a Poynting medium is their nonlinear susceptibility $\chi^{(P)}$ which is linear on the time-averaged Poynting vector $\langle\textbf{S}\rangle$, as follows from the constitutive relations \[Eq. (\[constitutive\])\]. In a general form, it can be written as $$\chi^{(P)}_{ij}=\gamma_{ijk}
\:\big\langle\textbf{S}(\textbf{r},t)\big\rangle_k,\label{relconst}$$ with $i,j,k = x,y,z$ and $$\big\langle\textbf{S}(\textbf{r},t)\big\rangle_k=\frac{1}{T}
\int^{t+T}_t
\big[\textbf{E}(\textbf{r},t')\times
\textbf{H}(\textbf{r},t')\big]_k\:
dt',\label{chivss}$$ where $\textbf{E}$ and $\textbf{H}$ are harmonics of period $\tau=2\pi/\omega$ and time interval $T\gg \tau$. The susceptibility tensor transforms as even under spacial inversion $(r \rightarrow -r)$ and time reversal $(t \rightarrow -t)$, contrary to the Poynting vector and its time-averaged which transforms odd under spatial inversion and time reversal, i.e. $\big\langle\textbf{S}(\textbf{r},t)\big\rangle\rightarrow
-\big\langle\textbf{S}(-\textbf{r},t)\big\rangle$ and $\big\langle\textbf{S}(\textbf{r},t)\big\rangle\rightarrow
-\big\langle\textbf{S}(\textbf{r},-t)\big\rangle$, respectively [@jackson]. Then, a medium possessing a linear connection between $\chi_{ij}$ and $\langle S\rangle_z$ should be noninvariant with respect to spatial inversion and time reversal. Otherwise, the space-time symmetry will be violated in the constitutive relation \[Eq. (23)\].
The lack of parity symmetry under inversion of coordinates is proper of materials without inversion center, i. e. Poynting nonlinearity should be only possible in *noncentrosymmetric* materials. There are several materials candidates to possess a Poynting nonlinearity, as for example the cubic crystals with zincblended structure such as GaAs, InSb and others. In these materials intensity-dependent transmission and bistability was experimentally observed [@InSb]. Also, isotropic homogeneous liquids formed by nonracemic mixtures or solutions of mirror-asymmetric (chiral) molecules with strong nonlinear optical susceptibility, as product of several nonlinear processes [@korotev], are also feasible of posses a Poynting nonlinearity. In addition, parity under time reversal should be violated in Poynting media. This means that any weak dissipative process, that converts field energy into heat, is necessary to remove the rule relating to the $t\rightarrow -t$ transformation. For example, either very weak absorbtion or current flow by external quasi-static field, that basically do not affect the wave propagation at light frequency $\omega$, would ensure the medium non-invariance under time reversal.
We believe that, in spite of experimental works are required, the above preliminary analysis could stimulate further discussions regarding the existence of Poynting media.
Conclusions
===========
In summary, we derive a new formalism in terms of dimensionless variables related with the time-averaged Poynting vector and field amplitude modulus within the Fabry-Perot framework. The S-Formalism shows explicitly that the energy intensity and squared-field amplitude modulus are only equivalents for a single plane wave propagating in a linear infinite or semi-infinite medium. Otherwise, they are nonequivalents. Besides, the S-Formalism presents two important advantages: it permits to directly monitor the time-averaged Poynting vector in the medium and it avoids approximations, such as SVEA, simplification of the boundary conditions, and so on. To emphasize this nonequivalence we introduce the Poynting medium, whose nonlinearity is proportional to the intensity instead of the electric squared-field amplitude modulus such as in the Kerr medium. We find marked disagreement in the transmittance of both media, which support the differences between I and $|E|^2$. Also, a space-time symmetry analysis shows that the Poynting nonlinearity should be only possible in noncentrosymmetric materials.
The statements and analysis pointed out here constitute an advance on theoretical views of basic concepts in electrodynamics. The S-Formalism could be important in problems where the time-averaged Poynting vector must be rigourously monitored like in photoconductor or photorefractive materials. Further to this particular case studied here, this new approach leaves open the possibility of new physical results in actual topics on nonlinear wave propagation such as spatial solitons, wave mixing and others. Finally, we leave open the possibility that experimental techniques, based on intensity dependent phase changes of a Gaussian beam such as *Z-Scan Technique* [@z-scan], could not truly measure Kerr-type nonlinearity. On the contrary, they could be measuring a Poynting-type instead.
The authors thank Prof. Boris Ya. Zel’dovich and anonymous Referee for the suggestions about symmetry properties of Poynting media. The authors also thank Victor Waveluk for valuable advices. A. L. thanks to CLAF-CNPq fellowship.
[xxxx]{} H. M. Gibbs, S. L. McCall and T. N. C. Venkatesan, Phys. Rev. Lett **36**, 1135 (1976). J. H. Marburger and F. S. Felber, Phys. Rev. A **17**, 335 (1978). D. A. B. Miller, IEEE J. Quantum Electron.**QE-17**, 306 (1981). J. Danckaert *et al.*, Opt. Commun. **71**, 317 (1989). B. Biran, Opt. Commun. **78**, 183 (1990). K. Fobelets and K. Thielemans, Phys. Rev. A **53**, 4400 (1996). W. Chen and D.L. Mills, Phys. Rev. B **35**, 524 (1987); **38**, 12814 (1988). A. Lencina, B. Ruiz and P. Vaveliuk (unpublished). N. B. Baranova, Yu. V. Bogdanov and B. Ya. Zel’dovich, Sov. Phys. Usp. **20**, 870 (1977); Opt. Commun. **22**, 243 (1977). J. D. Jackson, *Classical Electrodynamics*, (Wiley, New York 1999), 3rd. ed., p. 271. H. M. Gibbs *et al.*, Phylos. Trans. R. Soc. London A **313**, 245 (1984); D. A. B. Miller, S. D. Smith and A. Johnston, Appl. Phys. Lett. **35**, 658 (1979). N. I. Koroteev, Kvantovaya Elektron. **21**, 1063 (1994). M. Sheik-Bahae *et al.*, IEEE J. Quantum Electron. **QE-26**, 760 (1990).\
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The problem of statistical learning is to construct a predictor of a random variable $Y$ as a function of a related random variable $X$ on the basis of an i.i.d. training sample from the joint distribution of $(X,Y)$. Allowable predictors are drawn from some specified class, and the goal is to approach asymptotically the performance (expected loss) of the best predictor in the class. We consider the setting in which one has perfect observation of the $X$-part of the sample, while the $Y$-part has to be communicated at some finite bit rate. The encoding of the $Y$-values is allowed to depend on the $X$-values. Under suitable regularity conditions on the admissible predictors, the underlying family of probability distributions and the loss function, we give an information-theoretic characterization of achievable predictor performance in terms of conditional distortion-rate functions. The ideas are illustrated on the example of nonparametric regression in Gaussian noise.'
author:
-
bibliography:
- 'rate\_constrained\_learning.bib'
title: Learning From Compressed Observations
---
Introduction and problem statement {#sec:intro}
==================================
Let $X$ and $Y$ be jointly distributed random variables, where $X$ takes values in an [*input space*]{} $\cX$ and $Y$ takes values in an [*output space*]{} $\cY$. The problem of statistical learning is about constructing an accurate predictor of $Y$ as a function of $X$ on the basis of some number of independent copies of $(X,Y)$, often with very little or no prior knowledge of the underlying distribution. A very general decision-theoretic framework for learning was proposed by Haussler [@Hau92]. In a slightly simplified form it goes as follows. Let $\cP$ be a family of probability distributions on $\cZ = \cX \times \cY$. Each member $P$ of $\cP$ represents a possible relationship between $X$ and $Y$. Also given are a [*loss function*]{} $\map{\ell}{\cY \times \cY}{\R^+}$ and a set $\cF$ of functions ([*hypotheses*]{}) from $\cX$ into $\cY$. For any $f \in \cF$ and any $P \in \cP$ we have the [*expected loss*]{} (or [*risk*]{}) $$L(f,P) = \E\ell(f(X),Y) \equiv \int_\cZ \ell(f(x),y)dP(x,y),
$$ which expresses quantitatively the average performance of $f$ as a predictor of $Y$ from $X$ when $(X,Y) \sim P$. Let us define the minimum expected loss $$L^*(\cF,P) \deq \inf_{f \in \cF} L(f,P)
$$ and assume that the infimum is achieved by some $f^* \in \cF$. Then $f^*$ is the best predictor of $Y$ from $X$ in the hypothesis class $\cF$ when $(X,Y) \sim P$. The problem of statistical learning is to construct, for each $n$, an approximation to $f^*$ on the basis of a [*training sequence*]{} $\{Z_i\}^n_{i=1}$, where $Z_i = (X_i,Y_i)$ are i.i.d. according to $P$, such that this approximation gets better and better as the sample size $n$ tends to infinity. This formulation of the learning problem is referred to as [*agnostic*]{} (or [*model-free*]{}) learning, reflecting the fact that typically only minimal assumptions are made on the causal relation between $X$ and $Y$ and on the capability of the hypotheses in $\cF$ to capture this relation. It is general enough to cover such problems as classification, regression and density estimation.
Formally, a [*learning algorithm*]{} (or [*learner*]{}, for short) is a sequence $\{ \wh{f}_n \}^\infty_{n=1}$ of maps $\map{\wh{f}_n}{\cZ^n \times \cX}{\cY}$, such that $\wh{f}_n(Z^n,\cdot) \in \cF$ for all $n$ and all $Z^n \in \cZ^n$. Let $Z = (X,Y) \sim P$ be independent of the training sequence $Z^n$. The main quantity of interest is the [*generalization error*]{} of the learner, $$\begin{aligned}
L(\wh{f}_n, P) &\deq& \E \Big[ \ell \big( \wh{f}_n(Z^n,X),Y \big) \Big | Z^n \Big] \\
&\equiv& \int_\cZ \ell(\wh{f}_n(Z^n,x),y)dP(x,y).
$$ The generalization error is a random variable, as it depends on the training sequence $Z^n$. One is chiefly interested in the asymptotic probabilistic behavior of the [*excess loss*]{} $L(\wh{f}_n, P) - L^*(\cF,P)$ as $n \to \infty$. (Clearly, $L(\wh{f}_n, P) \ge L^*(\cF,P)$ for every $n$.) Under suitable conditions on the loss function $\ell$, the hypothesis class $\cF$, and the underlying family $\cP$ of probability distributions, one can show that there exist learning algorithms which not only [*generalize*]{}, i.e., $\E L(\wh{f}_n, P) \to L^*(\cF,P)$ as $n \to \infty$ for every $P \in \cP$ (which is the least one could ask for), but are also [*probably approximately correct*]{} (PAC), i.e. $$\lim_{n \to \infty} P \left( Z^n : L(\wh{f}_n, P) > L^*(\cF,P) + \epsilon \right) = 0
\label{eq:universal_consistency}$$ for every $\epsilon > 0$ and every $P \in \cP$. (See, e.g., Vidyasagar [@Vid03].)
This formulation assumes that the training data are available to the learner with arbitrary precision. This assumption may not always hold, however. For example, the location at which the training data are gathered may be geographically separated from the location where the learning actually takes place. Therefore, the training data may have to be communicated to the learner over a channel of finite capacity. In that case, the learner will see only a quantized version of the training data, and must be able to cope with this to the extent allowed by the fundamental limitations imposed by rate-distortion theory. In this paper, we consider a special case of such learning under rate constraints, when the learner has perfect observation of the input part $X^n = (X_1,\ldots,X_n)$ of the training sequence, while the output part $Y^n = (Y_1,\ldots,Y_n)$ has to be communicated via a noiseless digital channel whose capacity is $R$ bits per sample. This situation, shown in Figure \[fig:rate\_limited\_learning\], may arise, for example, in remote sensing, where the $X_i$’s are the locations of the sensors and the $Y_i$’s are the measurements of the sensors having the form $f_0(X_i) + Z_i$, where $\map{f_0}{\cX}{[0,1]}$ is some unknown function and the $Z_i$’s are i.i.d. zero-mean Gaussian random variables with variance $\sigma^2$. Assuming that the sensors are dispersed at random over some bounded spatial region $\cX$ and the location of each sensor is known following its deployment, the task of the sensor array is to deliver, over a rate-limited channel, an approximation $\wh{Y}^n$ of the measurement vector $Y^n = (Y_1,\ldots,Y_n)$ to some central location, where the vector $X^n$ of the sensor locations and the compressed version $\wh{Y}^n$ of the sensor measurements will be fed into a learner that will approximate $f_0$ by some function $\wh{f}_n(X^n,\wh{Y}^n,\cdot)$ from a given hypothesis class $\cF$.
![The set-up for learning from compressed data with side information.[]{data-label="fig:rate_limited_learning"}](rate_limited_learning){width="\columnwidth"}
In this paper, we establish information-theoretic upper bounds on the achievable generalization error in this setting. In particular, we relate the problem of agnostic learning under (partial) rate constraints to conditional rate-distortion theory [@Ber71 Section 6.1], [@Gra72], [@Wyn78 Appendix A], which is concerned with lossy source coding in the presence of side information both at the encoder and at the decoder. In the set-up shown in Figure \[fig:rate\_limited\_learning\], the input part $X^n = (X_1,\ldots,X_n)$ of the training sequence, which is available both to the encoder and to the decoder (hence to the learner), plays the role of the side information, while the output part $Y^n = (Y_1,\ldots,Y_n)$ is to be coded using a lossy source code operating at the rate of $R$ bits per symbol. Furthermore, because the distribution of $(X,Y)$ is known only to be a member of some family $\cP$, the lossy codes must be robust in the presence of this uncertainty.
Let us formally state the problem. Let $\cP,\cF,\ell$ be given. A scheme for agnostic learning under partial rate constraints (from now on, simply a [*scheme*]{}) operating at rate $R$ is specified by a sequence of triples $\{ (e_n, d_n, \wh{f}_n) \}^\infty_{n=1}$, where $\map{e_n}{\cX^n \times \cY^n}{\{1,\ldots,2^{nR}\}}$ is the encoder, $\map{d_n}{\cX^n \times \{1,\ldots,2^{nR}\}}{\cY^n}$ is the decoder, and $\map{\wh{f}_n}{\cX^n \times \cY^n}{\cF}$ is the learner. We shall often abuse notation and let $\wh{f}_n$ denote also the function $\wh{f}_n(X^n,\wh{Y}^n,\cdot)$. For each $n$, the output of the learner is a hypothesis $\wh{f}_n(X^n,\wh{Y}^n,\cdot) \in \cF$, where $\wh{Y}^n = d_n(X^n,e_n(X^n,Y^n))$ is the reproduction of $Y^n$ given the side information $X^n$. For any $P \in \cP$, the main object of interest associated with the scheme is the generalization error $$L(\wh{f}_n,P) \deq \E \Big[ \ell \big( \wh{f}_n(X^n,\wh{Y}^n,X),Y \big) \Big| X^n, Y^n \Big],$$ where $(X,Y) \sim P$ is assumed independent of $\{(X_i,Y_i)\}^n_{i=1}$ (to keep the notation simple, we suppress the dependence of the generalization error on the encoder and the decoder). In particular, we are interested in the achievable values of the asymptotic expected excess risk. We say that a pair $(R,\Delta)$ is [*achievable for $(\cF,\cP,\ell)$*]{} if there exists a scheme $\{(e_n,d_n,\wh{f}_n)\}^\infty_{n=1}$ operating at rate $R$, such that $$\limsup_{n \to \infty} \E L(\wh{f}_n,P) \le L^*(\cF,P) + \Delta
$$ for every $P \in \cP$. After listing the basic assumptions in Sec. \[sec:assumptions\], we derive in Sec. \[sec:results\] sufficient conditions for $(R,\Delta)$ to be achievable. We then apply our results to the setting of nonparametric regression in Sec. \[sec:regression\]. Discussion of results and an outline of future directions are given in Sec. \[sec:discussion\].
Related work
------------
Previously, the problem of statistical estimation from compressed data was considered by Zhang and Berger [@ZhaBer88], Ahlswede and Burnashev [@AhlBur90] and Han and Amari [@HanAma98] from the viewpoint of multiterminal information theory. In these papers, the underlying family of distributions of $(X,Y)$ is parametric, i.e., of the form $\cP = \{P_\theta\}_{\theta \in \Theta}$, where $\Theta$ is a subset of $\R^k$ for some finite $k$, and one wishes to estimate the “true" parameter $\theta^*$. The i.i.d. observations $\{(X_i,Y_i)\}^n_{i=1}$ are drawn from $P_{\theta^*}$, and the input part $X^n$ is communicated to the statistician at some rate $R_1$, while the output part $Y^n$ is communicated at some rate $R_2$. The present work generalizes to the nonparametric setting the case considered by Ahlswede and Burnashev [@AhlBur90], namely when $R_1 = \infty$. To the best of the author’s knowledge, this paper is the first to consider the problem of nonparametric learning from compressed observations with side information.
Assumptions {#sec:assumptions}
===========
We begin by stating some basic assumptions on $\cF$, $\cP$ and $\ell$. Additional assumptions will be listed in the sequel as needed.
The input space $\cX$ is taken to be a measurable subset of $\R^d$, while the output space is either a finite set (as in classification) or the set of reals $\R$ (as in regression or function estimation). We assume throughout that the family $\cP$ of distributions on $\cX \times \cY$ is such that the mutual information $I(X;Y) < \infty$ for every $P \in \cP$. All information-theoretic quantities will be in bits, unless specified otherwise.
We assume that there exists a learning algorithm which generalizes optimally in the absence of any rate constraints. Therefore, our standing assumption on $(\cF,\cP,\ell)$ will be that the induced function class $\cL_\cF = \{\ell_f : f \in \cF\}$, where $\ell_f(z) \deq \ell(f(x),y)$ for all $z = (x,y) \in \cZ$, satisfies the [*uniform law of large numbers*]{} (ULLN) for every $P \in \cP$, i.e., $$\sup_{f \in \cF} \left| \frac{1}{n}\sum^n_{i=1} \ell_f(Z_i) - \E \ell_f(Z) \right| \to 0, \qquad \rm{a.s.}
\label{eq:ULLN}$$ where $Z,Z_1,Z_2,\ldots$ are i.i.d. according to $P$. Eq. (\[eq:ULLN\]) implies that, for any sequence $\{f_n\} \subset \cF$, $$\left| \frac{1}{n}\sum^n_{i=1} \ell_{f_n}(Z_i) - \E\ell_{f_n}(Z)\right| \to 0, \qquad \rm{a.s.}$$ This holds even in the case when each $f_n$ is random, i.e., $f_n(\cdot) = f_n(Z^n,\cdot)$. The ULLN is a standard ingredient in proofs of consistency of learning algorithms: if $(\cF,\cP,\ell)$ are such that (\[eq:ULLN\]) holds, then the [*Empirical Risk Minimization*]{} algorithm (ERM), given by $$\wh{f}_n = \argmin_{f \in \cF} \frac{1}{n}\sum^n_{i=1} \ell_f(Z_i),
$$ is PAC in the sense of (\[eq:universal\_consistency\]) [@Vid03 Theorem 3.2].
Next, we assume that the loss function $\ell$ has the following “generalized Lipschitz" property: there exists a concave, continuous function $\map{\eta}{\R^+}{\R^+}$, such that for all $f \in \cF$, $x \in \cX$ and $u,u' \in \cY$ $$\left| \ell(f(x),u) - \ell(f(x),u') \right| \le \eta(\ell(u,u')).
\label{eq:loss_function_smoothness}$$ This holds, for example, in the following cases:
- Suppose that $\ell$ is a metric on $\cY$. Then, by the triangle inequality we have $\ell(y,u) \le \ell(y,u') + \ell(u',u)$ for all $y,u,u' \in \cY$, so (\[eq:loss\_function\_smoothness\]) holds with $\eta(t) = t$.
- Suppose that $\cY = [0,1]$ and $\ell(u,u') = |u-u'|^p$ for some $p \ge 1$. Then one can show that $$\left| \ell(f(x),u) - \ell(f(x),u') \right| \le p | u - u'|$$ for all $\map{f}{\cX}{\cY}$, $x \in \cX$ and $u,u' \in \cY$, so (\[eq:loss\_function\_smoothness\]) holds with $\eta(t) = p t^{1/p}$.
Finally, we need to pose some assumptions on the metric structure of the class $\cP$ with respect to the [*variational distance*]{} [@Gra90a Sec. 5.2], which for any two probability distributions $P_1,P_2$ on a measurable space $(\cZ,\cA)$ is defined by $$d_V(P_1,P_2) \deq 2\sup_{A \in \cA} |P_1(A) - P_2(A)|.
$$ A finite set $\{P_1,\ldots,P_M\} \subset \cP$ is called an [*(internal) $\epsilon$-net*]{} for $\cP$ with respect to $d_V$ if $$\sup_{P \in \cP} \min_{1 \le m \le M} d_V(P,P_m) \le \epsilon.
$$ The cardinality of a minimal $\epsilon$-net, denoted by $N(\epsilon,\cP)$, is called the [*$\epsilon$-covering number*]{} of $\cP$ w.r.t. $d_V$, and the [*Kolmogorov $\epsilon$-entropy*]{} of $\cP$ is defined as $H(\epsilon,\cP) \deq \log N(\epsilon,\cP)$ [@KolTih61]. We assume that the class $\cP$ satisfies [*Dobrushin’s entropy condition*]{} [@Dob70], i.e., for every $c > 0$ $$\lim_{\epsilon \to 0} \frac{H(\epsilon,\cP)}{2^{c/\epsilon}} = 0.
\label{eq:dobrushin_condition}$$ This condition is satisfied, for example, in the following cases: (1) $\cX$ and $\cY$ are both finite sets; (2) $\cP$ is a finite family; (3) $\cZ$ is a compact subset of a Euclidean space, and all $P \in \cP$ are absolutely continuous with densities satisfying a uniform Lipschitz condition [@KolTih61; @Dob70].
The results {#sec:results}
===========
To state our results we shall need some notions from conditional rate-distortion theory [@Ber71 Sec. 6.1], [@Gra72], [@Wyn78 Appendix A]. Fix some $P \in \cP$. Given a pair $(X,Y) \sim P$ and a nonnegative real number $D$, define the set $\cM(D)$ to consist of all $\cY$-valued random variables $\wh{Y}$ jointly distributed with $(X,Y)$ and satisfying the constraint $\E \ell(Y,\hat{Y}) \le D$, where the expectation is taken with respect to the joint distribution of $X,Y,\wh{Y}$. Then the [*conditional rate-distortion function*]{} of $Y$ given $X$ w.r.t. $P$ is defined by $$R_{Y|X}(D,P) \deq \inf \left\{ I(Y; \wh{Y} | X) : \wh{Y} \in \cM(D) \right\},
$$ where $I(Y; \wh{Y} | X)$ is the conditional mutual information between $Y$ and $\wh{Y}$ given $X$. Our assumption that $I(X;Y) < \infty$ ensures the existence of $R_{Y|X}(D,P)$ [@Wyn78]. In operational terms, $R_{Y|X}(D,P)$ is the minimum number of bits needed to describe $Y$ with expected distortion of at most $D$ given perfect observation of a correlated random variable $X$ (the side information) when $(X,Y) \sim P$. As a function of $D$, $R_{Y|X}(D,P)$ is convex and strictly decreasing everywhere it is finite, hence it is invertible. The inverse function is called the [*conditional distortion-rate function*]{} of $Y$ given $X$ and is denoted by $D_{Y|X}(R,P)$. Finally, let $$\bbD_{Y|X}(R,\cP) \deq \sup_{P \in \cP} D_{Y|X}(R,P).
$$ We assume that $\bbD_{Y|X}(R,\cP) < \infty$ for all $R \ge 0$.
We shall also need the following lemma, which can be proved by a straightforward extension of Dobrushin’s random coding argument from [@Dob70] to the case of side information available to the encoder and to the decoder:
\[lm:robust\_codes\] Let $\cP$ satisfy Dobrushin’s entropy condition (\[eq:dobrushin\_condition\]). Assume that the loss function $\ell$ either is bounded or satisfies a uniform moment condition $$\sup_{P \in \cP} \E[\ell(Y,y_0)^{1+\delta}] < \infty
\label{eq:moment_condition}$$ for some $\delta > 0$ with respect to some fixed reference letter $y_0 \in \cY$. Then for every rate $R \ge 0$ there exists a sequence $\{(e_n,d_n)\}^\infty_{n=1}$ of encoders $\map{e_n}{\cX^n \times \cY^n}{\{1,\ldots,2^{nR}\}}$ and decoders $\map{d_n}{\cX^n \times \{1,\ldots,2^{nR}\}}{\cY^n}$, such that $$\limsup_{n \to \infty} \sup_{P \in \cP} \E \ell_n(Y^n,\wh{Y}^n) \le \bbD_{Y|X}(R,\cP),$$ where $\wh{Y}^n = d_n(X^n,e_n(X^n,Y^n))$ and $\ell_n(Y^n,\wh{Y}^n) = n^{-1}\sum^n_{i=1} \ell(Y_i,\wh{Y}_i)$ is the normalized cumulative loss between $Y^n$ and $\wh{Y}^n$.
Our main result can then be stated as follows:
\[thm:upper\_bound\] Under the stated assumptions, for any $R \ge 0$ there exists a scheme $\{ (e_n,d_n,\wh{f}_n)\}$ operating at rate $R$, such that $$\limsup_{n \to \infty} \E L(\wh{f}_n,P) \le L^*(\cF,P) + 2\eta(\bbD_{Y|X}(R,\cP)).$$ Thus, $(R,2\eta(\bbD_{Y|X}(R,\cP)))$ is achievable for every $R \ge 0$.
Given $n$, $Z^n \in \cZ^n$ and $f \in \cF$, define the [*empirical risk*]{} $$\wh{L}_{Z^n}(f) \deq \frac{1}{n} \sum^n_{i=1} \ell_f(Z^n)$$ and the minimum empirical risk $$\wh{L}^*_{Z^n}(\cF) \deq \Inf_{f \in \cF} \wh{L}_{Z^n}(f).$$ We shall write $\wh{L}_{X^n,Y^n}(f)$ and $\wh{L}^*_{X^n,Y^n}(\cF)$ whenever we need to emphasize separately the roles of $X^n$ and $Y^n$.
Suppose that the encoder $e_n$ and the decoder $d_n$ are given. Let $\wh{Y}^n$ denote the reproduction of $Y^n$ given the side information $X^n$, i.e., $\wh{Y}^n = d_n(X^n,e_n(X^n,Y^n))$. We then define our learner $\wh{f}_n$ by $$\wh{f}_n = \argmin_{f \in \cF} \wh{L}_{X^n,\wh{Y}^n}(f).
\label{eq:quantized_ERM}$$ In other words, having received the side information $X^n$ and the reproduction $\wh{Y}^n$, the learner performs ERM over $\cF$ on $\{(X_i,\wh{Y}_i)\}^n_{i=1}$. Using the property (\[eq:loss\_function\_smoothness\]) of the loss function $\ell$ and the concavity of $\eta$, we have the following estimate: $$\begin{aligned}
&& \sup_{f \in \cF} \big| \wh{L}_{X^n,Y^n}(f) - \wh{L}_{X^n,\wh{Y}^n}(f) \big| \nonumber \\
&& \quad \le \sup_{f \in \cF} \frac{1}{n}\sum^n_{i=1} \big| \ell(f(X_i),Y_i) - \ell(f(X_i),\wh{Y}_i) \big| \nonumber \\
&& \quad \le \frac{1}{n}\sum^n_{i=1} \eta (\ell(Y_i,\wh{Y}_i)) \nonumber \\
&& \quad \le \eta \big( \ell_n(Y^n,\wh{Y}^n )\big). \label{eq:loss_approximation}\end{aligned}$$ In particular, this implies that $$\big|\wh{L}_{X^n,Y^n}(\wh{f}_n) - \wh{L}_{X^n,\wh{Y}^n}(\wh{f}_n) \big| \le \eta\big( \ell_n(Y^n, \wh{Y}^n) \big)
\label{eq:approximation_bound_1}$$ and $$\big| \wh{L}^*_{X^n,Y^n}(\cF) - \wh{L}^*_{X^n,\wh{Y}^n}(\cF) \big| \le \eta\big( \ell_n(Y^n,\wh{Y}^n) \big).
\label{eq:approximation_bound_2}$$ We then have $$\begin{aligned}
\wh{L}_{X^n,Y^n}(\wh{f}_n) &\stackrel{{\rm (a)}}{\le}& \wh{L}_{X^n,\wh{Y}^n}(\wh{f}_n) + \eta \big( \ell_n(Y^n, \wh{Y}^n) \big) \\
&\stackrel{{\rm (b)}}{=}& \wh{L}^*_{X^n,\wh{Y}^n}(\cF) + \eta \big( \ell_n(Y^n,\wh{Y}^n) \big) \\
&\stackrel{{\rm (c)}}{\le}& \wh{L}^*_{X^n,Y^n}(\cF) + 2\eta\big( \ell_n(Y^n,\wh{Y}^n) \big), \end{aligned}$$ where (a) follows from (\[eq:approximation\_bound\_1\]), (b) from the definition of $\wh{f}_n$, and (c) from (\[eq:approximation\_bound\_2\]). Suppose that the data are distributed according to a particular $P \in \cP$. Taking expectations and using the concavity of $\eta$ and Jensen’s inequality, we obtain $$\E \wh{L}_{Z^n}(\wh{f}_n) \le \E \wh{L}^*_{Z^n}(\cF) + 2\eta \big( \E \ell_n(Y^n,\wh{Y}^n) \big).$$ Using this bound and the continuity of $\eta$, we can write $$\begin{aligned}
&& \limsup_{n \to \infty} \E L(\wh{f}_n,P) - L^*(\cF,P) \nonumber \\
&& \qquad \le \lim_{n \to \infty} \E \big [ L(\wh{f}_n,P) - \wh{L}_{Z^n}(\wh{f}_n) \big ] \nonumber \\
&& \qquad\quad + \lim_{n \to \infty} \E \big [ \wh{L}^*_{Z^n}(\cF) - L^*(\cF,P)\big] \nonumber \\
&& \qquad\quad + 2\eta \Big( \limsup_{n \to \infty} \E \ell_n(Y^n,\wh{Y}^n) \Big). \label{eq:excess_loss_bound}\end{aligned}$$ The two leading terms on the right-hand side of this inequality are zero by the ULLN. Moreover, given $R$, Lemma \[lm:robust\_codes\] asserts the existence of a sequence $\{(e_n,d_n)\}^\infty_{n=1}$ of encoders $\map{e_n}{\cX^n \times \cY^n}{\{1,\ldots,2^{nR}\}}$ and decoders $\map{d_n}{\cX^n \times \{1,\ldots,2^{nR}\}}{\cY^n}$, such that $$\limsup_{n \to \infty} \E\ell_n(Y^n,\wh{Y}^n) \le \bbD_{Y|X}(R,\cP), \qquad \forall P \in \cP.$$ Substitution of this into (\[eq:excess\_loss\_bound\]) proves the theorem.
All pairs $(R,\Delta)$ with $\Delta \ge 2\eta(\bbD_{Y|X}(R,\cP))$ are achievable.
\[rem:lower\_bound\] In the Appendix, we show that a corresponding lower bound derived by the usual methods for proving converses in lossy source coding is strictly weaker than the “obvious" lower bound based on the observation that $\E L(\wh{f}_n,P) \ge L^*(\cF,P)$ for any $\wh{f}_n$. It may be possible to obtain nontrivial lower bounds in the minimax setting, which we leave for future work (see also Sec. \[sec:discussion\]).
\[rem:finite\_sample\_bound\] Under some technical conditions on the function class $\{ \ell_f: f \in \cF\}$ (see, e.g., [@Men03]), one can show that $$\E \sup_{f \in \cF} \Big| \wh{L}_{Z^n}(f) - L(f,P) \Big| \le C/\sqrt{n}, \qquad \forall P \in \cP$$ for some constant $C$ that depends on $\cF,\ell$. Using this fact and the same bounding method that led to Eq. (\[eq:excess\_loss\_bound\]), but without taking the limit superior, we can get the following finite-sample bound for every scheme $\{(e_n,d_n,\wh{f}_n)\}^\infty_{n=1}$ with $\wh{f}_n$ given by (\[eq:quantized\_ERM\]) and [*arbitrary*]{} $e_n,d_n$: $$\E L(\wh{f}_n,P) \le L^*(\cF,P) + 2\eta\big(\E \ell_n(Y^n,\wh{Y}^n) \big) + C'/\sqrt{n},$$ where $C' = 2C$.
The following theorem shows that we can replace condition (\[eq:loss\_function\_smoothness\]) with the requirement that $\ell$ be a power of a metric:
\[thm:metric\_power\] Suppose that the loss function $\ell$ is of the form $\ell(y,u) = d(y,u)^r$ for some $r \ge 1$, where $d$ is a metric on $\cY$. Then for any rate $R \ge 0$ the scheme constructed in the proof of Theorem \[thm:upper\_bound\] is such that $$\limsup_{n \to \infty} \E \Big[L(\wh{f}_n,P)^{1/r}\Big] \le L^*(\cF,P)^{1/r} + 2 \bbD_{Y|X}(R,\cP)^{1/r}$$ holds for every $P \in \cP$.
We proceed essentially along the same lines as in the proof of Theorem \[thm:upper\_bound\], except that the bound (\[eq:loss\_approximation\]) is replaced with an argument based on Minkowski’s inequality to yield $$\E\Big[\wh{L}_{Z^n}(\wh{f}_n)^{1/r}\Big] \le \E \Big[\wh{L}^*_{Z^n}(\cF)^{1/r}\Big] + 2\Big(\E \ell_n(Y^n,\wh{Y}^n) \Big)^{1/r}.$$ The rest is immediate using the ULLN as well as concavity and continuity of $t \mapsto t^{1/r}$ for $t \ge 0$.
Example: nonparametric regression {#sec:regression}
=================================
As an example, let us consider the setting of nonparametric regression. Let $\cX$ be a compact subset of $\R^d$ and $\cY = \R$. The training data are of the form $$Y_i = f_0(X_i) + Z_i, \qquad 1 \le i \le n
\label{eq:nonparametric_regression}$$ where the regression function $f_0$ belongs to some specified class $\cF$ of functions from $\cX$ into $[0,1]$, the $X_i$’s are i.i.d. random variables drawn from the uniform distribution on $\cX$, and the $Z_i$’s are i.i.d. zero-mean normal random variables with variance $\sigma^2$, independent of $X^n$. We take $\ell(y,u) = |y-u|^2$, the squared loss. Note that $\ell$ satisfies the condition of Theorem \[thm:metric\_power\] with $r = 2$.
Because $f_0$ is unknown, we take as the underlying family $\cP$ the class of all absolutely continuous distributions with densities of the form $p_f(x,y) = V^{-1} \cN(y; f(x),\sigma^2)$, $f \in \cF$, where $V$ is the volume of $\cX$ and $\cN(y; f(x),\sigma^2)$ is the one-dimensional normal density with mean $f(x)$ and variance $\sigma^2$. Because the functions in $\cF$ are bounded between $0$ and $1$, it is easy to show that the uniform moment condition (\[eq:moment\_condition\]) of Lemma \[lm:robust\_codes\] is satisfied with $\delta = 1$ and $y_0 = 0$.
We suppose that $\ell$ and $\cF$ are such that the function class $\cL_\cF$ satisfies the ULLN.[^1] Let $Q$ denote the uniform distribution on $\cX$ and for any square-integrable function $f$ on $\cX$ define the $L_2$ norm by $$\| f \|^2_{2,Q} \deq \int_\cX f^2(x) dQ(x) \equiv \frac{1}{V}\int_\cX f^2(x) dx.$$ Let us denote by $N_{2,Q}(\epsilon,\cF)$ the $\epsilon$-covering number of $\cF$ w.r.t. $\| \cdot \|_{2,Q}$, i.e., the smallest number $M$ such that there exist $M$ functions $\{ f_m \}^M_{m=1}$ in $\cF$ satisfying $$\sup_{f \in \cF} \min_{1 \le m \le M} \| f - f_m \|_{2,Q} \le \epsilon.$$ We assume that $\cF$ is such that for every $c > 0$ $$\lim_{\epsilon \to 0} \frac{\log N_{2,Q}(\epsilon,\cF)}{2^{c/\epsilon}} = 0.
\label{eq:function_covering_condition}$$ This condition holds, for example, if the functions in $\cF$ are uniformly Lipschitz or if $\cX$ is a bounded interval in $\R$ and $\cF$ consists of functions satisfying a Sobolev-type condition [@KolTih61].
If $\cF$ satisfies (\[eq:function\_covering\_condition\]), then $\cP$ satisfies Dobrushin’s entropy condition (\[eq:dobrushin\_condition\]).
Given $f \in \cF$, let $P_f$ denote the distribution with the density $p_f$. It is straightforward to show that $$I(P_f \| P_g) = \frac{1}{2\sigma^2} \| f - g \|^2_{2,Q}, \qquad \forall f,g \in \cF
$$ where $I(\cdot \| \cdot)$ is the relative entropy (information divergence) between two probability distributions, in nats. Using Pinsker’s inequality $d_V(P_1,P_2) \le \sqrt{2 I(P_1 \| P_2)}$ [@Gra90a Lemma 5.2.8], we get $$d_V(P_f \| P_g) \le \frac{1}{\sigma} \| f - g \|_{2,Q}, \qquad \forall f,g \in \cF.
\label{eq:dvar_bound}$$ Given $\epsilon > 0$, let $\{f_m \}^M_{m=1} \subset \cF$ be a $\sigma\epsilon$-net for $\cF$ w.r.t. $\|\cdot\|_{2,Q}$. Then from (\[eq:dvar\_bound\]) it follows that $$\sup_{f \in \cF} \min_{1 \le m \le M}d_V(P_f,P_{f_m}) \le \sup_{f \in \cF} \min_{1 \le m \le M} \frac{\| f - f_m \|_{2,Q}}{\sigma} \le \epsilon,$$ i.e., $\{P_{f_m}\}^M_{m=1}$ is an $\epsilon$-net for $\cP$ w.r.t. $d_V$. This implies, in particular, that $N(\epsilon,\cP) \le N_{2,Q}(\sigma\epsilon,\cF)$ for every $\epsilon > 0$. This, together with (\[eq:function\_covering\_condition\]), proves the lemma.
\[lm:sup\_DRF\] For any $R \ge 0$, $\bbD_{Y|X}(R,\cP) = \sigma^2 2^{-2R}$.
Fix some $f \in \cF$ and consider a pair $(X,Y) \sim P_f$. Then $Y = f(X) + Z$, where $Z \sim {\rm Normal}(0,\sigma^2)$ is independent of $X$. Because $\ell$ is a difference distortion measure, Theorem 7 of [@Gra72] says that, for any measurable function $\map{\psi}{\cX}{\cY}$, $$D_{Y|X}(R,P_f) = D_{Y - \psi(X)|X}(R,P_{f-\psi}),$$ where $P_{f-\psi}$ is the distribution of $$Y - \psi(X) \equiv f(X) - \psi(X) + Z;$$ furthermore, if $Y - \psi(X)$ is independent of $X$, then $D_{Y|X}(R,P_f) = D_{Y - \psi(X)}(R)$, the (unconditional) distortion-rate function of $Y - \psi(X)$. Taking $\psi = f$, we get $D_{Y|X}(R,P_f) = D(R,\sigma^2)$, the distortion-rate function of a memoryless Gaussian source with variance $\sigma^2$ w.r.t. squared error loss, which is equal to $\sigma^2 2^{-2R}$ [@Ber71 Theorem 9.3.2]. Hence $D_{Y|X}(R,P_f)$ is independent of $f$. Taking the supremum over $\cF$ finishes the proof.
Now we can state and prove the main result of this section:
Consider the regression setting of (\[eq:nonparametric\_regression\]). Under the stated assumptions, for any $R \ge 0$ there exists a scheme $\{ (e_n,d_n,\wh{f}_n )\}^\infty_{n=1}$, such that $$\limsup_{n \to \infty} \E \left[ L(\wh{f}_n,P_f)^{1/2} \right] \le \sigma(1 + 2^{-R+1}) \label{eq:regression_bound_2}$$ holds for every $f \in \cF$.
As follows from the above, the triple $(\cF,\cP,\ell)$ satisfies all the assumptions of Theorem \[thm:metric\_power\]. Therefore for any $R \ge 0$ there exists a scheme $\{ (e_n,d_n,\wh{f}_n ) \}^\infty_{n=1}$ operating at rate $R$, such that $$\limsup_{n \to \infty} \E \left[ L(\wh{f}_n,P_f)^{1/2} \right] \le L^*(\cF,P_f)^{1/2} + 2^{-R+1}\sigma, \label{eq:regression_bound_2a}$$ holds for every $f \in \cF$ (we have also used Lemma \[lm:sup\_DRF\]). It is not hard to show that $$L(g,P_f) = \| f - g \|^2_{2,Q} + \sigma^2, \qquad \forall f,g \in \cF,$$ whence it follows that $L^*(\cF,P_f) = \sigma^2$ for every $f \in \cF$. Substituting this into (\[eq:regression\_bound\_2a\]), we get (\[eq:regression\_bound\_2\]).
Discussion and future work {#sec:discussion}
==========================
We have derived information-theoretic bounds on the achievable generalization error in learning from compressed data (with side information). There is a close relationship between this problem and the theory of robust lossy source coding with side information at the encoder and the decoder. A major difference between this setting and the usual setting of learning theory is that the techniques are no longer [*distribution-free*]{} because restrictions must be placed on the underlying family of distributions in order to guarantee the existence of a suitable source code. The theory was applied to the problem of nonparametric regression in Gaussian noise, where we have shown that the penalty incurred for using compressed observations decays exponentially with the rate.
We have proved Theorems \[thm:upper\_bound\] and \[thm:metric\_power\] by adopting ERM as our learning algorithm and optimizing the source code to deliver the best possible reconstruction of the training data. In effect, this imposes a [*separation structure*]{} between learning and source coding. While this “modular" approach is simplistic (clearly, additional performance gains could be attained by designing the encoder, the decoder and the learner jointly), it may be justified in such applications as remote sensing. For instance, if the source code and the learner were designed jointly, then any change made to the hypothesis class (say, if we decided to replace the currently used hypothesis class with another based on tracking the prior performance of the network) might call for a complete redesign of the source code and the sensor network, which may be a costly step. With the modular approach, no such redesign is necessary: one merely makes the necessary adjustments in the learning algorithm, while the sensor network continues to operate as before.
Let us close by sketching some directions for future work. First of all, it would be of interest to derive information-theoretic lower bounds on the generalization performance of rate-constrained learning algorithms. In particular, just as Ahlswede and Burnashev had done in the parametric case [@AhlBur90], we could study the asymptotics of the [*ninimax excess risk*]{} $$\delta_n(R) \deq \inf_{(e_n,d_n,\wh{f}_n)} \sup_{P \in \cP} \Big[\E L(\wh{f}_n,P) - L^*(\cF,P) \Big],$$ where the infimum is over all encoders, decoders and learners operating on a length-$n$ training sequence at rate $R$. Secondly, we could dispense with the assumption that the learner has perfect observation of the input part of the training sample, in analogy to the situation dealt with by Zhang and Berger [@ZhaBer88]. Finally, keeping in mind the motivating example of sensor networks, it would be useful to replace the block coding approach used here with an efficient distributed scheme.
Acknowledgments {#acknowledgments .unnumbered}
===============
Discussions with Todd Coleman are gratefully acknowledged. This work was supported by the Beckman Fellowship.
Let us assume for simplicity that $\cP$ is a singleton, $\cP = \{P\}$, and that $\cY$ is a finite set. Consider a scheme $\{(e_n,d_n,\wh{f}_n)\}$ operating at rate $R$. Fix $n$ and define the $n$-tuple $W^n$ via $$W_i \deq \wh{f}_n(X^n,\wh{Y}^n,X_i), \qquad 1 \le i \le n.$$ Also, let $J = e_n(X^n,Y^n)$. Then we can write $$\begin{aligned}
nR &\ge& H(J|X^n) \nonumber \\
&\ge& H(\wh{Y}^n|X^n) \nonumber \\
&\ge& I(\wh{Y}^n; Y^n | X^n) \nonumber \\
&=& H(Y^n | X^n) - H(Y^n | X^n, \wh{Y}^n) \nonumber \\
&=& H(Y^n | X^n) - H(Y^n | X^n, \wh{Y}^n, W^n) \label{eq:fctn}\end{aligned}$$ $$\begin{aligned}
&=& \sum^n_{i=1} [H(Y_i | X_i) - H(Y_i | X^n, \wh{Y}^n, W^n, Y^{i-1} ) ] \nonumber \\
&\ge& \sum^n_{i=1} [H(Y_i | X_i) - H(Y_i | X_i, W_i)] \nonumber \\
&=& \sum^n_{i=1} I(Y_i; W_i | X_i) \nonumber \\
&\ge& \sum^n_{i=1} R_{Y|X} (\E \ell(W_i,Y_i),P) \nonumber \\
&\ge& nR_{Y|X}(\E \ell_n(W^n,Y^n),P),\nonumber\end{aligned}$$ where (\[eq:fctn\]) follows from the fact that $W^n$ is a function of $\wh{Y}^n$ and $X^n$. The remaining steps follow from standard information-theoretic identities and from convexity. Therefore, $$\liminf_{n \to \infty} \E\ell_n(W^n,Y^n) \ge D_{Y|X}(R,P).$$ Because $\E L(\wh{f}_n,P) = \E \ell_n(W^n,Y^n) + o(1)$ by the ULLN, $$\liminf_{n \to \infty} \E L(\wh{f}_n,P) \ge D_{Y|X}(R,P).
\label{eq:info_lower_bound}$$ Now, given any $f \in \cF$, we can interpret $f(X)$ as a zero-rate approximation of $Y$ (using only the side information $X$), so $L(f,P) \ge D_{Y|X}(0,P) \ge D_{Y|X}(R,P)$ for any $R \ge 0$. In particular, $L^*(\cF,P) \ge D_{Y|X}(R,P)$ for all $R$, and $$\liminf_{n \to \infty} \E L(\wh{f}_n,P) \ge L^*(\cF,P) \ge D_{Y|X}(R,P)$$ for all $R$. Thus, the information-theoretic lower bound (\[eq:info\_lower\_bound\]) is weaker than the bound $\Liminf_{n \to \infty} \E L(\wh{f}_n,P) \ge L^*(\cF,P)$.
[^1]: See Györfi et al. [@GKKW02] for a detailed exposition of the various conditions when this is true.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Supervised deep learning techniques have achieved great success in various fields due to getting rid of the limitation of handcrafted representations. However, most previous image retargeting algorithms still employ fixed design principles such as using gradient map or handcrafted features to compute saliency map, which inevitably restricts its generality. Deep learning techniques may help to address this issue, but the challenging problem is that we need to build a large-scale image retargeting dataset for the training of deep retargeting models. However, building such a dataset requires huge human efforts.
In this paper, we propose a novel deep cyclic image retargeting approach, called Cycle-IR, to firstly implement image retargeting with a single deep model, without relying on any explicit user annotations. Our idea is built on the reverse mapping from the retargeted images to the given input images. If the retargeted image has serious distortion or excessive loss of important visual information, the reverse mapping is unlikely to restore the input image well. We constrain this forward-reverse consistency by introducing a cyclic perception coherence loss. In addition, we propose a simple yet effective image retargeting network (IRNet) to implement the image retargeting process. Our IRNet contains a spatial and channel attention layer, which is able to discriminate visually important regions of input images effectively, especially in cluttered images. Given arbitrary sizes of input images and desired aspect ratios, our Cycle-IR can produce visually pleasing target images directly. Extensive experiments on the standard RetargetMe dataset show the superiority of our Cycle-IR. In addition, our Cycle-IR outperforms the Multiop method and obtains the best result in the user study. Code is available at <https://github.com/mintanwei/Cycle-IR>.
author:
- 'Weimin Tan, Bo Yan, Chumin Lin, Xuejing Niu [^1] [^2][^3]'
title: 'Cycle-IR: Deep Cyclic Image Retargeting'
---
image retargeting, deep learning, cycle consistency.
Introduction
============
The popularity of mobile devices has greatly improved the quality, efficiency, and convenience of people life. However, the diverse display size of mobile devices leads to a media display problem that due to the different size between the input media and the device screen, media may not be perfectly suitable for full-screen display. With the explosive growth of media content on social networks, this problem is further aggravated. Traditional image resizing methods often cause shrinking, stretching, or clipping. To solve this problem, content-aware image retargeting approaches are proposed to manipulate media content to make it adapt to different aspect ratios of device screen intelligently[@Liang2017ObjectiveQP; @Dong2016ImageRB; @Zhou2017PerceptuallyAI; @Shao2017QoEGuidedWF; @Lau2018ImageRV; @Li2019LearningAC; @Lin2014ObjectCoherenceWF; @Lei2017DepthPreservingSI; @Li2015DepthPreservingWF]. Recently, image retargeting algorithms are also extended to various fields such as aesthetic enhancement[@Xiang2010VideoRF], video synopsis[@Li2016SurveillanceVS; @Nie2013CompactVS], and low bit-rate image retrieval[@Tan2016ImageRF; @Tan2018BeyondVR; @Yan2017CodebookGF], *etc*.
![Overview of the proposed Cycle-IR framework. By cyclically utilizing the retargeting results of the forward inference of deep image retargeting network (IRNet), our IRNet can be trained in an unsupervised way, without the requirement of any manual label.[]{data-label="FigOverFramework"}](OverFramework.pdf){width="8.5"}
A good content-aware image retargeting approach is able to preserve visually important object details, while removing unimportant pixels and preventing obvious distortions. Numerous works have been proposed and achieved great success in recent years[@avidan2007seam; @fang2011saliency; @mansfield2010scene; @qi2012seam; @noh2012seam; @dong2014summarization; @grundmann_2010; @panozzo2012robust; @zhang2015retargeting]. Typically, these image retargeting methods follow a conventional pipeline. Firstly, they calculate a saliency map to measure the importance of each pixel in the input image. Then, several constraints such as content and structure similarity or symmetry preserving are defined. Finally, based on the calculated saliency map and the defined preserving constraints, they define a manipulating way to remove or insert pixels in the input image, yielding the desired target image. Such a strategy is limited by the fixed design principle. For example, in traditional retargeting algorithms, saliency map is widely employed and calculated based on the low-level features of input image such as image edge, color contrast, depth information, or handcrafted features. The failure of saliency map estimation commonly results in poor quality of retargeting results, which greatly limits the generality of these approaches.
Deep learning techniques have demonstrated outstanding performance in various fields. However, few studies have attempted to solve the image retargeting problem using deep learning techniques. The reason is that image retargeting task belongs to an uncertain problem, *i.e.*, the way to produce the ideal retargeting results is uncertain. In addition, the best way to evaluate retargeting results is subjective evaluation. These reasons make the image retargeting task quite different from those tasks with a definite goal or objective evaluation criteria, such as image super-resolution, image classification, object detection, and semantic segmentation, *etc*. Furthermore, these reasons also make it difficult to build a large-scale image retargeting dataset that contains input images and corresponding target images with different aspect ratios.
Without such a dataset, we cannot accurately train an image retargeting model that is based on conventional deep learning techniques. Therefore, using deep learning techniques to solve image retargeting problem becomes an extremely challenging problem. Cho $et~al$. [@cho2017weakly] make an attempt to use deep learning technique to addressing image retargeting problem. They propose a weakly and self-supervised deep network to learn a shift map for manipulating the aspect ratio of input images. In order to obtain a good shift map, they use two loss terms of structural similarity and content consistency to optimize the deep network. Despite the encouraging progress, this method requires the input image to be of fixed size, which greatly limits the practicability of this method.
In this study, we solve the aforementioned problems by introducing a deep cyclic image retargeting approach (**Cycle-IR**). Our key idea is that whether the size of the input image is reduced or increased, the target image generated by an ideal image-retargeting model should be highly consistent with the visual perception of the input image, free of distortions and artifacts, and complete preserving of important visual information, *etc*. Therefore, if the retargeted image is fed back to the ideal image-retargeting model as input, the mapped-back image should be similar to the original input image in terms of visual perception and structural similarity. The coming questions are ***how to establish the forward-backward mapping consistency?*** and ***how to effectively evaluate the visual consistency between the input image and the mapped-back image?***
In light of the above discussions, this paper introduces a novel Cycle-IR approach based on the characteristics of image retargeting that is able to generate target images with different resolutions. Accordingly, we propose a deep image retargeting network (**IRNet**) to implement the retargeting process. In order to form the cycle relation, our IRNet is designed to output a low-resolution and a high-resolution target image, simultaneously, which is different from conventional retargeting methods that output only a single target image. Fig. \[FigOverFramework\] shows the basic idea of our Cycle-IR, which consists of a two-stage inference process. The first stage is to produce two target images with different resolutions given an input image and the target aspect ratio. Then, at the second stage, the image retargeting model (sharing network parameters at the first stage) uses the retargeted image as input to reconstruct the original input image. Specifically, the IRNet maps the input images to the retargeted images, and then maps them back. In order to measure the forward-backward mapping coherency effectively, we propose a cyclic perception coherence loss. This cycle coherence loss can encourage the IRNet to discover the visual importance of the input image and reduce the distortions and artifacts of the target image. In addition, this loss prevents the training process from requiring any explicit user annotations. In the entire training process of IRNet, we only need unlabeled color images and the pre-trained VGG16 model[@Simonyan2015VeryDC].
We present extensive experiments to evaluate the effectiveness of our Cycle-IR on the public RetargeMe dataset. Numerous representative image retargeting methods are compared with the proposed approach explicitly in terms of visual quality and user study. To compare with the weakly supervised deep retargeting method[@cho2017weakly], we implement it using Tensorflow platform. Note that our Cycle-IR is an unsupervised deep retargeting approach and is trained on the RGB images from HKU-IS dataset[@li2016deep] that have no overlap with RetargeMe. Therefore, all images in RetargeMe can be used as test set. Experiment results show that our Cycle-IR can implicitly learn a good mapping from input images to target images, where the mapping is able to make the input and output follow the same distribution. Besides, we conduct extensive experiments to analyze the performance of our Cycle-IR. Benefiting from the automated learning ability of deep networks, our Cycle-IR produces the target image with better visual quality than other retargeting methods, and obtains the best result in the objective evaluation of user study.
In summary, the main contributions of this work are:
- This work proposes a novel unsupervised deep image retargeting approach by leveraging cyclic visual perception consistency between the input images and the reverse-mapped results. This is the first attempt to address the problem of image retargeting in an unsupervised deep learning way. Our Cycle-IR does not require any labeled data, additional parameter settings, or human assistance. It performs favorably against the state-of-the-art image retargeting approaches.
- Different from previous retargeting methods that output one retargeted image, our deep image retargeting model outputs a pair of retargeted images. This special design is helpful for obtaining a stable retargeting model by exploiting a pair of cycle constraints.
- A cyclic perception coherence loss is proposed to evaluate the cycle coherence between the forward-backward mapping results. This cycle loss allows our IRNet to train in an end-to-end manner and enforces IRNet to implicitly learn a mapping that the input images and the target images follow the same distribution. Besides, the advantage of our cycle loss makes it possible to incorporate it into other image retargeting methods, improving the perceptual quality of the target images they generate.
- We design a spatial and channel attention layer, which is able to discover the visually interesting areas of input images effectively, especially in cluttered images. We exploit this attention layer as a learnable network component to help assist the IRNet in learning accurate attention map.
Related work
============
In this section, we review previous works related to this paper. Over the past decade, numerous image retargeting works have been proposed [@avidan2007seam; @fang2011saliency; @mansfield2010scene; @qi2012seam; @noh2012seam; @dong2014summarization; @grundmann_2010; @panozzo2012robust; @zhang2015retargeting; @Zhang2015RetargetingSP; @Qu2013ContextAwareVR; @Gallea2014PhysicalMF]. The majority of image retargeting methods are based on hand-crafted visual attention map such as gradient map, color contrast, object segmentation, face detection, *etc*. We refer the readers to survey literature [@Pal2016ContentAwareIR] for more details. Here, we mainly discuss representative image retargeting methods and a recent deep learning based method. Besides, we will review the successful applications of cycle constraint in other fields.
**Conventional retargeting**: Most of conventional image retargeting approaches can directly retarget input images to target aspect ratios according to the previous calculated visual attention map, in which the bright areas indicate the interesting areas in the input image. These approaches commonly exploit traditional methods to extract pixel saliency information, and then define a manipulating way to insert or remove pixels in order to obtain the target image. Seam carving [@avidan2007seam] is a good example, which uses three ways ($e.g.$, pixel gradient, entropy energy, visual saliency) to measure the importance of pixels in the input image. Then, it defines a way of seam carving to change the aspect ratio of the input image. A seam is an 8-connected path of pixels from each column or row in the source image. The retargeted image is obtained by iteratively removing or inserting seams.
Different from the seam carving method in a discrete way, Panozzo $et~al.$ [@panozzo2012robust] propose to continuously transform the input image into the target image. Specifically, they overlay a uniform grid on the input image and assign different scaling factors to each grid cell. The scaling factors of important regions are large, and conversely, small. As a result, those important regions are preserved as much as possible, while less important regions are shrinking obviously. Wang $et~al.$ [@wang2008optimized] propose a scale-and-stretch warping method. They calculate optimal scaling factors for image regions based on the edge and saliency map. Jin $et~al.$ [@jin2010nonhomogeneous] cast image retargeting problem into a quadratic program based on a triangular mesh. All these conventional approaches need to calculate an importance map based on low-level visual cues or other human priors. The poor quality of importance map significantly degrades the performance of these methods, which seriously limits their generality.
**Weakly supervised retargeting**: Because the visual quality of retargeted images highly depends on the subjective assessment, it is hard to build a large-scale image retargeting dataset for supervisory learning a deep model. Thus, there are few supervised retargeting methods. Cho $et~al$. [@cho2017weakly] introduce deep learning technique into the image retargeting task. Their deep model learns a shift map to implement pixel-wise mapping from the source image to the target image. They define a loss function containing two terms: content loss and structure loss. The content loss enables the retargeted image outputted by the deep network to have the same class as the source image. Therefore, their approach requires image-level annotations to construct the content loss. This approach may fail to deal with those images that do not belong to any class in the training set. Besides, in order to reduce distortions in the target image, the method proposes to use 1D duplicate convolution to smooth the learned shift map. This operation is helpful for reducing visual artifacts in the target image but requires the input image to be a fixed size.
**Cycle Constraint**: Cycle constraints have been explored in other fields in order to regularize model predictions and improve its stability. For language translation, Richard shows the translation quality can be improved effectively by using a back translation and reconciliation strategy[@Brislin1970]. For computer vision, high-order and multimodal cycle consistency have been used to improve the stability and quality of model predictions such as co-segmentation[@Huang2013ConsistentSM], motion prediction[@Zach2010DisambiguatingVR], 3D shape matching[@Huang2013ConsistentSM], semantic alignment[@Zhou2015FlowWebJI], depth estimation[@Godard2017UnsupervisedMD], image-to-image translation, video frame interpolation[@Liu2019DeepVF]. For deep learning, researchers have incorporated the concept of cycle consistency into the regularization of optimizing deep networks[@Zhou2016LearningDC]. In this work, we demonstrate a novel and feasible way of exploiting the concept of circle consistency to address the image-retargeting problem. To the best of our knowledge, we are the first to improve image retargeting by using circle consistency. Our Cycle-IR is capable of producing high-quality target images and achieves the state-of-the-art result.
Deep Cyclic Image Retargeting
=============================
Our goal is to develop a deep image-retargeting model that can directly generate high-quality target images when users provide input images with arbitrary sizes and desired aspect ratios. In this section, firstly, we present an overview of the Cycle-IR framework in Section \[OverviewCyclicIR\]. Then, we introduce the cyclic perception coherence loss in Section \[CyclicPerceptionCoherence\]. Furthermore, we introduce implementation details of the proposed IRNet in Section \[IRNetImplementation\]. Finally, we justify our design strategy and describe the training process in Section \[Justifications\] and \[TrainingandInference\], respectively.
Cycle-IR Framework {#OverviewCyclicIR}
------------------
An ideal image retargeting model should have the ability to preserve visually interesting areas in the input image and prevent significant distortions and artifacts in the target image when changing the aspect ratio of the input image. That is to say, the removed or inserted information in the target image is trivial and can be easily restored or eliminated. Inspired by this observation, we propose a cycle image-retargeting framework. The key idea of our Cycle-IR is that when the target image is obtained by eliminating the content of input image, the removed information can be effectively restored by re-feeding the target image into the retargeting model. Similarly, when the target image is obtained by inserting the content of the input image, the inserted information can be easily removed by re-feeding the target image into the retargeting model. Both cases are required to satisfy the condition that the reverse mapping result of the target image has the same pixel distribution as the input image, *i.e.*, cyclic perception coherence. This cycle coherence is also helpful for obtaining target images that are free of distortions and artifacts.
{width="18"}
The proposed cyclic image retargeting framework is illustrated in Fig. \[FigOverFramework\]. We design a two-stage inference procedure so that the retargeting model shared by both forward and reverse mapping directions in the cycle constraint can be learned stably. Consider an input image $I_0$ ($H_{I_0}\times W_{I_0}$) and desired aspect ratio $\phi_h \in (0,1]$ and $\phi_w \in (0,1]$ that are passed through the IRNet model (see Section \[IRNetImplementation\] for detail). At the first stage (forward retargeting), the IRNet model outputs two target images $I^{LR}$ and $I^{HR}$ in order to establish the cycle relation between the forward and backward retargeting procedure. Note that this is different from previous image retargeting methods that yield only one target image in a single inference.
$$I^{LR}, I^{HR} = IRNet_{FWD}(I_0)
\label{equationFWD}$$
The resolution of $I^{LR}$ is defined as follows:
$$\begin{aligned}
H_{I^{LR}} = \phi_h * H_{I_0} \\ \nonumber
W_{I^{LR}} = \phi_w * W_{I_0}
\label{equationSizeOfLR}\end{aligned}$$
Different from conventional retargeting methods, we set the resolution of $I^{HR}$ as follows: $$\begin{aligned}
H_{I^{HR}} = \frac{\mu_h}{\phi_h} * H_{I_0} \\ \nonumber
W_{I^{HR}} = \frac{\mu_w}{\phi_w} * W_{I_0}
\label{equationSizeOfHR}\end{aligned}$$ where $\mu_h$ and $\mu_w$ are used to control the size of $I^{HR}$ and are set to 1 in the implementation. We design such resolutions for $I^{LR}$ and $I^{HR}$ in order to form the circle constraint for optimizing the IRNet.
At the second stage (reverse retargeting), we re-feed the target image obtained in the first stage into the IRNet to obtain the reverse mapping results. Note that the IRNet in this stage shares the same network parameters as the first stage. The IRNet outputs $I^{top}_{LR}$ and $I^{top}_{HR}$ from the input of $I^{LR}$ and $I^{bottom}_{LR}$ and $I^{bottom}_{HR}$ from the input of $I^{HR}$, respectively.
$$\begin{aligned}
I^{top}_{LR}, I^{top}_{HR} = IRNet_{REV}(I^{LR}) \\ \nonumber
I^{bottom}_{LR}, I^{bottom}_{HR} = IRNet_{REV}(I^{HR})\end{aligned}$$
where $I^{top}_{HR}$ ($H_{I_0}\times W_{I_0}$) and $I^{bottom}_{LR}$ ($H_{I_0}\times W_{I_0}$) are used to compute the cyclic coherency loss. Note that $I^{top}_{LR}$ ($H_{I_0}*\phi_h^2\times W_{I_0}*\phi_w^2$) and $I^{bottom}_{HR}$ ($H_{I_0}*\mu_h^2/\phi_h^2\times W_{I_0}*\mu_w^2/\phi_w^2$) are not used to compute the cycle loss because their resolutions are different from the size of input image.
Finally, we can establish the cyclic perception coherence for $I_0$ and $I^{top}_{HR}$, and $I_0$ and $I^{bottom}_{LR}$. The following section will dedicate to state the cyclic perception coherence loss.
Cyclic Perception Coherence Loss {#CyclicPerceptionCoherence}
--------------------------------
These widely used loss functions, such as the mean absolute error (MAE) and mean squared error (MSE), are not fully consistent with human visual perception when evaluating the quality of retargeted images. Minimizing the MSE loss often results in over-smoothed results because among all possible results, generative models will produce an average result to satisfy the minimum value of MSE. Besides, the pixel-level MSE considers that all pixels in the input image have the same visual importance. It will force the IRNet model to learn a uniform scaling, rather than the non-uniform resizing we desire. To solve this problem, we present the cyclic perception coherence loss $\mathcal{L}_{pair}$ to assess the perceptual consistency between the input image and the reverse-mapped image. This cycle loss enables our IRNet to implicitly discover the visually interesting areas in the input image and effectively learn a mapping that the input images and the target images follow the same distribution. In this way, we can obtain favorable retargeting results by using $\mathcal{L}_{pair}$ to optimize our IRNet.
{width="15.5"}
The perceptual loss has demonstrated superior performance in various fields. Different from previous approaches, we propose a pair of cyclic perception coherence loss $\mathcal{L}_{pair}$ to provide a more stable constraint for optimizing IRNet. $\mathcal{L}_{pair}$ is defined as.
$$\begin{aligned}
\label{LossFuntion}
\mathcal{L}_{pair}=\frac{ 1}{L } \sum_{l=4}^L[(f_l(I_0)-f_l(I_{HR}^{top})) \times\beta_l]^2 + \\ \nonumber \frac{ 1}{L } \sum_{l=4}^L[(f_l(I_0)-f_l(I_{LR}^{bottom})) \times\beta_l]^2\end{aligned}$$
where $f$ represents the pre-trained VGG16 model and $L=5$ (corresponding to the layers of VGG16 model). Considering image retargeting needs to focus not only on the semantic region of objects, but also on the overall structure of the image, we use multiple deep representations to evaluate the perceptual consistency instead of a single deep representation. $\beta_4$ and $\beta_5$ are set to 1 and 3, respectively. We empirically set $\beta_5$ to 3 because representations in deeper layer contain more semantic information. Eq. (\[LossFuntion\]) implies that these high-level deep representations of the input image and the reverse-mapped image should be consistent.
Implementation of IRNet {#IRNetImplementation}
-----------------------
The proposed Cycle-IR framework in Fig. \[FigOverFramework\] is flexible, allowing users to freely design the required network structure to implement it. Fig. \[FigNetworkArchitecture\] shows the implementation of our image retargeting network (IRNet), which is a fully convolutional architecture. The IRNet consists of a backbone (Conv4-1 of VGG16), three convolutional layers, and a spatial and channel attention layer. Despite the simplicity of our IRNet, it achieves excellent performance in manipulating the aspect ratio of input images. More advanced networks can be easily incorporated to achieve better performance.
**Obtaining Visual Attention Map**. Given an input image $I_0$ and a target aspect ratio $\phi_h$ and $\phi_w$, a deep representation $F_{map}$ is obtained by passing through the backbone and three convolutional layers that implement a non-linear transformation. Since the activation values of neurons in $F_{map}$ are often scattered in space and channel, we need an effective way to obtain the visual attention map corresponding to the input image. Therefore, we feed the $F_{map}$ into the spatial and channel attention layer to obtain the visual attention map $I_{attn}$.
$$I_{attn}=\Gamma_{attn}(F_{map})
\label{VisualAttentionMap}$$
where $\Gamma_{attn}$ denotes the non-linear transformation of the spatial and channel attention layer. See below for details.
**Spatial and Channel Attention Layer.** Fig. \[SpatialChannelAttention\] shows the network architecture of the proposed spatial and channel attention layer, which consists of two components of spatial attention and channel attention. The input of channel attention component is the output of the second convolutional layer in the spatial attention component instead of the extracted deep representation $F_{map}$. This architecture design helps to utilize differentiated information from spatial attention components. $I_{attn}$ can also be calculated as follows:
$$I_{attn}= F_{map} * \omega_s * \omega_c
\label{SCALayer}$$
where $\omega_s$ and $\omega_c$ represent the spatial and channel weights produced by the spatial and channel attention components, respectively.
**Generating desired target image**. To obtain the desired target image, we employ a continuous method to deform the resolutions of input images with the guidance of the obtained visual attention map $I_{attn}$. Note that the entire deformation process is integrated into the optimization of IRNet model, which is conductive to allowing the IRNet to decide on its own way to generate the best target image. Specifically, we cover the input image with a uniform grid of M-column and N-row. The size of each grid cell is $H_{I_0}$/M $\times$ $W_{I_0}$/N. Based on the learned attention map $I_{attn}$, we calculate the scaling factor of each grid cell for reconstructing $I^{LR}$ as follows.
$$\begin{aligned}
S^h_{i}(I^{LR})= \frac{1}{N}\sum_{j=1}^{N} \frac{1}{1+e^{-I_{attn}(i,j)}} \\ \nonumber
S^w_{j}(I^{LR})= \frac{1}{M}\sum_{i=1}^{M} \frac{1}{1+e^{-I_{attn}(i,j)}}
\label{DesiredTargetImage}\end{aligned}$$
Based on the calculated $S^h_{i}(I^{LR}) \in (0,1]$ and $S^w_{j}(I^{LR}) \in (0,1]$, we can easily obtain the scaling factor of each grid cell for reconstructing $I^{HR}$ as follows.
$$\begin{aligned}
S^h_{i}(I^{HR})= 1-S^h_{i}(I^{LR})+\psi_h \\ \nonumber
S^w_{j}(I^{HR})= 1-S^w_{j}(I^{LR})+\psi_w
\label{DesiredTargetImage2}\end{aligned}$$
where $\psi_h$ and $\psi_w$ are the adjustment factors and set to 1 in the implementation. Generally, deep network has the ability to learn suitable adjustment factors (*i.e.*, $S^h(I^{LR})$ and $S^w(I^{LR}))$ for different scenarios.
Afterwards, we deform each grid based on the calculated $S(I^{LR})=\{S^h(I^{LR}),S^w(I^{LR})\}$ and $S(I^{HR})=\{S^h(I^{HR}),S^w(I^{HR})\}$ and obtain the target images $I^{LR}$ and $I^{HR}$. It turns out that our Cycle-IR can greatly improve the quality of image retargeting without incurring extra human assistance or weak tag. Furthermore, the concept of cycle coherency is extended by considering task-specific knowledge, and can automatically decide the areas to be deleted or reserved in the input image.
{width="18"}
Justifications {#Justifications}
--------------
The proposed Cycle-IR framework is motivated by the following ideas:
- Building a large-scale image retargeting dataset for training a deep retargeting network is a challenging work. It not only requires finding an effective way to generate target images, but also needs huge human efforts to assess their quality. In addition, to improve the practicability of image retargeting, it is necessary to get rid of the limitation of handcrafted design principles. Image retargeting algorithm needs to obtain the generalization ability of dealing with different complex scenarios by learning. Hence, we avoid the supervised and weakly supervised methods and present an unsupervised way to learn a deep retargeting model effectively.
- Using cycle constraint to optimize the retargeting model makes deep learning based image retargeting possible. Furthermore, through cyclic perception coherence, the IRNet model can stably learn a favorable retargeting mapping, yielding visually pleasing retargeting results.
- Visual attention areas in the input image should be located accurately, especially in cluttered images. To this end, we design a spatial and channel attention layer to find visually important areas in a learnable way. We leverage this attention layer to help assist the IRNet in learning accurate attention map. Developing such attention layer has the additional advantages of avoiding using large-scale networks to achieve similar performance, thereby reducing computational cost and memory footprint.
Training and Inference {#TrainingandInference}
----------------------
To train the IRNet model, we use the RGB images from HKU-IS dataset [@li2016deep] as the training set. This dataset contains 4,447 RGB images. Note that saliency labels in the HKU-IS dataset [@li2016deep] are discarded during training, *i.e.*, any RGB image without user annotations can be exploited as our training set. The RGB images from HKU-IS dataset [@li2016deep] contains diverse scenes, which helps to make the trained model have good generality. During training, the input aspect ratios are randomly generated for each training batch within $(H_{I_0}/4 \sim H_{I_0}/2) \times (W_{I_0}/4 \sim W_{I_0}/2)$, similar to [@cho2017weakly]. The training process is fast, which takes around 1 hour for 50 epochs on a machine with an Nvidia GPU Quadro M4000. Generally, after training 50 epochs, the IRNet is capable of producing satisfactory results. The loss function of Eq. (\[LossFuntion\]) is optimized using Adam algorithm with an initial learning rate of $1\times e^{-3}$. The batch size is set to 4. During inference, given an input image and a desired target size, the IRNet model can directly output the target image with the desired aspect ratio, so no pre-processing (*e.g.*, computing saliency map, detecting faces, semantic segmentation, *etc.*) or post-processing (*e.g.*, stitching, zooming, etc.) is necessary.
{width="18"}
Experiment
==========
For a clear understanding of the proposed approach, we conduct extensive evaluations to investigate the performance of the proposed Cycle-IR in Sections \[PixelVsPerceptual\] to \[NetworkAndEfficiency\]. Specifically, we analyze the retargeting quality of Cycle-IR under different loss functions such as pixel-wise loss vs. perceptual loss (see Section \[PixelVsPerceptual\]) and single cycle loss vs. pair cycle loss (see Section \[SingleCycleVsDoubleCycle\]), which help to understand the impact of different losses. In addition, we discuss the effect of human annotated saliency map on the training IRNet model in Section \[SaliencyGuidance\]. This discussion will help us understand the role of incorporating object location information to help the IRNet model distinguish the visually important areas in the input image. Furthermore, we demonstrate several visual examples to evaluate the retargeting quality of IRNet model under different input aspect ratios in Section \[AdjustmentOfAspectRatio\]. Finally, we introduce the network size and computational efficiency of IRNet in Section \[NetworkAndEfficiency\]. Afterward, in Section \[ComparisonWithPriorArt\], we explicitly compare our model with current state-of-the-art methods on the standard RetargetMe [@rubinstein2010comparative] dataset in terms of visual quality and user study. Finally, in Section \[FailureCaseAndFutureWork\], we show several failure cases of our IRNet and provide some insights to improve it in future work.
Pixel-Wise Loss vs. Perceptual Loss {#PixelVsPerceptual}
-----------------------------------
To exhibit the difference between the pixel-wise loss and the perceptual loss, where both of them can be used to measure the difference between the input images and the reverse-mapped images, we select multiple representative images from RetargetMe dataset and show the visual comparisons in Fig. \[FigPixelVsPerceptual\]. To demonstrate their differences in optimizing IRNet intuitively, we also visualize the visual attention map produced by the spatial and channel attention layer. As Fig. \[FigPixelVsPerceptual\] shows, the pixel-wise loss cannot encourage the IRNet to learn a good retargeting model because it regards all pixels in the input image as the same importance. The resulting attention map is sparse and often focuses on the edges of objects. In contrast, the perceptual loss enables the IRNet to focus on integral objects, yielding high-quality target images. Note that the entire training and testing process of our IRNet do not introduce any human annotations or assistance, which is a completely unsupervised learning procedure. Thus, the experiment result well demonstrates the effectiveness of our Circle-IR. In addition, we also demonstrate the results of pixel-wise loss plus perceptual loss. We observe that the pixel-wise loss cannot further improve the performance of perceptual loss. Therefore, in the following experiments, if there is no explicit statement, the used retargeting results are the outputs of the IRNet model optimized by perceptual loss. For convenience, we refer to our approach as Cycle-IR.
![Visual comparison between the annotated saliency map and the learned attention map[]{data-label="FigSaliencyGuidance1"}](SaliencyGuidance1.pdf){width="8.5"}
![Demonstration of integrating saliency maps as guidance with pixel-wise loss or perceptual loss.[]{data-label="FigSaliencyGuidance2"}](SaliencyGuidance2.pdf){width="8.5"}
{width="18"}
Single Cycle Loss vs. Pair Cycle Loss {#SingleCycleVsDoubleCycle}
-------------------------------------
The proposed cyclic perception coherence loss $\mathcal{L}_{pair}$ (see Eq. (\[LossFuntion\])) has two terms that correspond to two reverse mappings, respectively. In this section, we experiment with different loss terms to understand the function of each reverse mapping. Here, we use “Single Cycle Loss (top)” and “Single Cycle Loss (bottom)” to denote the reverse mapping loss term in the top and bottom (see Fig. \[FigNetworkArchitecture\]), respectively. “Pair Cycle Loss” means that both reverse mapping terms are used. Fig. \[FigdoubleCycleLoss\] illustrates their differences in optimizing IRNet model with several visual examples. Compared with “Single Cycle Loss (bottom)” , “Single Cycle Loss (top)” plays a more important role in the optimization of IRNet model. This is reasonable because the top reverse mapping focuses on restoring the information removed in the forward retargeting, while the bottom reverse mapping aims at removing the information inserted in the forward retargeting. Therefore, the top branch poses stronger constraints than the bottom one. Furthermore, by integrating both branches, more advanced performance is achieved by “Pair Cycle Loss”.
Saliency Guidance {#SaliencyGuidance}
-----------------
In this section, we discuss the effect of saliency guidance in optimizing IRNet model. Salient object detection has been studied over the past two decades. Accordingly, a large number of saliency detection datasets are built to promote the development of this field. Here, we carry out an experiment in which the saliency maps provided by HKU-IS dataset are embedded as a priori information in our cycle consistency loss. Figure \[FigSaliencyGuidance1\] shows the visual comparison between the annotated saliency map and the learned attention map (pixel-wise loss + saliency map). It turns out that with the guidance of annotated saliency map, the IRNet tends to output sparse attention map, resulting in degrading the retargeting performance. To further understand the effect of the saliency map as guidance, we also incorporate it with cyclic perception loss, as shown in Fig. \[FigSaliencyGuidance2\]. We observe that using saliency maps as guidance makes no contribution to both pixel-wise loss and perceptual loss. The possible reason is that the IRNet is not large enough, so it has not learned to detect salient objects in images completely.
Zoom In and Out {#AdjustmentOfAspectRatio}
---------------
We employ a fully convolutional framework to implement our Cycle-IR approach. Thus, our Cycle-IR is able to deal with input images with arbitrary sizes and generate target images with arbitrary aspect ratios. Figure \[FigAdjustmentOfAspectRatio\] shows some visual examples of the input aspect ratio $\phi_w$ that equals to 0.5, 0.75, 1.25, 1.5 and 1.75, respectively. Despite the large scale span (from 0.5 to 1.75), our Cycle-IR can well preserve the important areas (such as people, dog, and football) and the overall structure (the white line on the football ground in the fifth row) of the input image. These visual examples demonstrate the great ability of our Cycle-IR to zoom in and out.
Network Size and Computational Efficiency {#NetworkAndEfficiency}
-----------------------------------------
As Fig. \[FigNetworkArchitecture\] shows, our IRNet model consists of a backbone, three convolutional layers, and a specially designed attention layer. The network parameters of IRNet are about 3.164 M. Such a small amount of network parameters is helpful for fast convergence of the IRNet model. In addition, with our un-optimized TensorFlow implementation on an Nvidia GPU Quadro M4000, our cycle-IR takes 0.365 s for yielding a single image with 1024 $\times$ 768 resolution for resizing to half the original width.
Comparison with prior art {#ComparisonWithPriorArt}
-------------------------
In the past decade, numerous representative retargeting approaches have been proposed such as simple scaling operator (SCL), cropping windows (CR), nonhomogeneous warping (WARP [@wolf2007nonhomogeneous]), seam carving (SC) [@avidan2007seam], scale-and-stretch (SNS) [@wang2008optimized], Multiop [@rubinstein2009multioperator], shift maps (SM) [@pritch2009shiftmap], streaming video (SV) [@krahenbuhl2009a], and energy-based deformation (LG) [@karni2009energybased]. These approaches have published retargeting results on the standard RetargetMe dataset. Thus, we can easily compare with them by evaluating on the RetargetMe dataset. Besides, we also compare with a new retargeting approach [@cho2017weakly], which employs deep network to learn a shift map for manipulating the positions of pixels in the input image. We evaluate the performance of all retargeting approaches in terms of visual quality and user study.
![Visual comparison of our Cycle-IR with representative retargeting approaches.[]{data-label="FigComparisonWithPriorArt"}](ComparisonWithPriorArt.pdf){width="8.5"}
### Visual Quality
In this section, we provide a subjective comparison with several representative retargeting approaches and a deep learning based weakly supervised retargeting method. Through the subjective comparison of retargeting results, we will analyze the advantages and disadvantages of each approach in detail.
**Comparing with Representative Retargeting Approaches**. Fig.\[FigComparisonWithPriorArt\] shows the visual comparison of our Cycle-IR with several representative retargeting approaches by resizing input images to its half width on RetargetMe [@rubinstein2010comparative] dataset. The CR method directly removes the content of input images, resulting in losing important information in the input images. The Multiop [@rubinstein2009multioperator] approach achieves good results on most scenes, which benefits from combining the merits of different retargeting operates. The QP method [@Chen2010ContentawareIR] can preserve the most important information in the input image, but it introduces distortions and artifacts in the target image. The typical SC method[@avidan2007seam] may deform important objects when those seams carved cross over the important objects. The SCL just merges adjacent pixels, which results in over reducing important objects. Benefiting from the powerful ability of deep learning and the effectiveness of cycle consistency, our Cycle-IR is able to produce high-quality target images.
![From left to right is the original image, Cho $ et~al $. [@cho2017weakly] method, our Cycle-IR approach. The original images are retargeted to their half width.[]{data-label="FigWeaklyDL"}](WeaklyDL.pdf){width="8"}
**Comparing with Weakly Supervised Deep Retargeting Approach**. Recently, Cho $ et~al $. [@cho2017weakly] propose a weakly supervised image retargeting approach. Their deep network is trained on Pascal VOC 2007 dataset [@everingham2010the] with image-level annotations. Fig. \[FigWeaklyDL\] demonstrates the comparison result of two images, which is quoted from their published paper. From Fig. \[FigWeaklyDL\] we can observe that comparing with the method [@cho2017weakly], our Cycle-IR can better preserve important objects in the input images. For example, in [@cho2017weakly] method, the men’s hair in “glass” image is reduced too much, while our approach preserves hair better. Besides, the men’s forehead has some distortion in [@cho2017weakly] method. For “eagle” image, our approach preserves better for hawk’s wings than[@cho2017weakly] method.
For comprehensively comparing with Cho $et~al$. [@cho2017weakly] method, we implement it using TensorFlow platform. Fig. \[FigWeaklyDL2\] shows more visual comparisons on RetargetMe dataset. It intuitively shows that significant distortions often appear in the retargeted results of Cho $et~al$. [@cho2017weakly] method. In contrast, benefiting from the constraint of cyclic perception consistency, our Cycle-IR avoids this serious problem.
{width="18"}
**Method** **$P_1$** **$P_2$** **$P_3$** **$P_4$** **$P_5$** **$P_6$** **$P_7$** **$P_8$** **$P_9$** **$P_{10}$** **$P_{11}$** **$P_{12}$** **$P_{13}$** **$P_{14}$** **Total** **Prefer**
--------------------------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -------------- -------------- -------------- -------------- -------------- ----------- ------------
Warp[@wolf2007nonhomogeneous] 15 21 24 13 17 16 19 13 17 15 14 19 10 15 228 12.52%
SCL 30 15 17 15 13 23 18 26 27 25 21 18 22 28 298 16.37%
SC [@avidan2007seam] 24 27 30 30 32 30 25 14 29 10 30 28 16 22 347 19.06%
Multiop[@rubinstein2009multioperator] 30 32 31 37 33 29 36 36 26 37 34 31 37 28 457 25.10%
**Cycle-IR** **31** **35** **28** **35** **35** **32** **32** **41** **31** **43** **31** **34** **45** **37** **490** **26.92%**
### User Study
In this part, we conduct a user study to evaluate the preference of retargeted results objectively. Representative retargeting approaches including SC[@avidan2007seam], SCL, Multiop[@rubinstein2009multioperator], and Warp[@wolf2007nonhomogeneous] are selected to participate in this user study, because these approaches have robust retargeting performance in most cases. Besides, we invite 14 participants to evaluate the retargeting performance of each approach in terms of visual quality (small distortions and artifacts) and information preservation. We randomly select 13 images from RetargetMe benchmark. All images are retargeted to half width. To improve the efficiency of user study, we developed an evaluating tool using Python. During the evaluation, one original image and two retargeted images produced by 2 out of 5 approaches (including our Cycle-IR) are randomly shown to participants. Afterward, participants choose one retargeted image they prefer.
This user study requires 1,820 comparisons in total. Each image requires 10 comparisons. Each participant needs to evaluate 10 $\times$ 13 = 130 times. Table \[TableUserStudy\] shows the statistical result of user study. Among the representative retargeting approaches, the Multiop method obtains the most votes and is preferred in 25.10% in total comparisons, which is consistent with the evaluation results of most methods. Our Cycle-IR obtains 490 votes, which account for 26.92% in total comparisons. The user preference of our approach is higher than that of all test methods. This objective evaluation results in user study are consistent with the subjective comparisons.
![Some failure cases.[]{data-label="FigFailureCases"}](FailureCases.pdf){width="8.3"}
Failure Case and Future Work {#FailureCaseAndFutureWork}
----------------------------
Although the proposed Cycle-IR is capable of producing compelling target images in many cases, the results are far from uniformly encouraging. Some failure retargeting results of our approach have been shown in Fig. \[FigFailureCases\]. In general, these failure cases can be classified into two circumstances. The first one is that the background areas are regarded as the visual importance regions. Typical examples are the images shown in the first and second rows of Fig. \[FigFailureCases\]. The “obama” image is mostly caused by very low contrast, which easily leads to the Cycle-IR failing to distinguish the difference between people (“children” and “obama”) and cars. In addition, for the “surfer” case, Cycle-IR has a similar retargeting result to SCL method because the sea waves are considered an important area to be preserved. The second type of failure cases is due to the lack of complete attention to important visual areas, as shown in the bottom two rows of Fig. \[FigFailureCases\]. The important areas of these two images are too large or scattered. Though our Cycle-IR is able to detect some parts of the visually interesting regions, it is still difficult to segment these regions out completely.
There are several possible ways to improve the performance of Cycle-IR. Firstly, a more reasonable loss can make IRNet better optimized to deal with different complex scenes. To better measure the difference of the input images and the reverse-mapped images, we can use a deep network to assess their differences such as using a discriminator. Besides, we can replace the pixel-manipulating step with a learnable component. We think meta-learning is a good choice to get rid of manually designed manipulation. By feeding the desired aspect ratio and the coordinate information of target image, the learnable component outputs a set of parameters such as convolutional kernels. These learned parameters can be used to reconstruct the target images with arbitrary sizes. Finally, deep learning based aesthetic assessment has achieved great success in recent years. Thus, introducing the aesthetic assessment into the optimization of Cycle-IR may help the model to achieve better performance in human visual perception.
Conclusion
==========
In this paper, we have presented a deep cyclic image retargeting approach that is capable of producing visually pleasing target images given the input images with arbitrary sizes and the desired aspect ratios. We also present a simple yet effective IRNet model to implement the proposed Cycle-IR. Our IRNet model outputs a pair of retargeted images instead of a single one, which is helpful for obtaining a stable retargeting model by exploiting a pair of cycle constraints. To help assist the IRNet in learning accurate attention map, we design a spatial and channel attention layer to discriminate the visual attention of input images effectively. Besides, a cyclic perception coherence loss is introduced to optimize the IRNet model. Extensive experiments including investigation of our Cycle-IR, comparison of other retargeting approaches, and user study demonstrate the superior performance of our approach. More importantly, this work pushes the boundaries of what is possible in the unsupervised setting for content-aware image retargeting.
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}
Y. Liang, Y.-J. Liu, and D. Gutierrez, “Objective quality prediction of image retargeting algorithms,” *IEEE Transactions on Visualization and Computer Graphics*, vol. 23, pp. 1099–1110, 2017.
W. Dong, F. Wu, Y. Kong, X. Mei, T.-Y. Lee, and X. Zhang, “Image retargeting by texture-aware synthesis,” *IEEE Transactions on Visualization and Computer Graphics*, vol. 22, pp. 1088–1101, 2016.
Y. Zhou, L. Zhang, C. Zhang, P. Li, and X. Li, “Perceptually aware image retargeting for mobile devices,” *IEEE Transactions on Image Processing*, vol. 27, pp. 2301–2313, 2017.
F. Shao, W. Lin, W. Lin, Q. Jiang, and G. Jiang, “Qoe-guided warping for stereoscopic image retargeting,” *IEEE Transactions on Image Processing*, vol. 26, pp. 4790–4805, 2017.
C. P. Lau, C. P. Yung, and L. M. Lui, “Image retargeting via beltrami representation,” *IEEE Transactions on Image Processing*, vol. 27, pp. 5787–5801, 2018.
Y. Li, D. Liu, H. Li, L. Li, Z. Li, and F. Wu, “Learning a convolutional neural network for image compact-resolution,” *IEEE Transactions on Image Processing*, vol. 28, pp. 1092–1107, 2019.
S.-S. Lin, C.-H. Lin, S.-H. Chang, and T.-Y. Lee, “Object-coherence warping for stereoscopic image retargeting,” *IEEE Transactions on Circuits and Systems for Video Technology*, vol. 24, pp. 759–768, 2014.
J. Lei, M. Wu, C. Zhang, F. Wu, N. Ling, and C. Hou, “Depth-preserving stereo image retargeting based on pixel fusion,” *IEEE Transactions on Multimedia*, vol. 19, pp. 1442–1453, 2017.
B. Li, L. yu Duan, C.-W. Lin, T. Huang, and W. Gao, “Depth-preserving warping for stereo image retargeting,” *IEEE Transactions on Image Processing*, vol. 24, pp. 2811–2826, 2015.
Y. Y. Xiang and M. S. Kankanhalli, “Video retargeting for aesthetic enhancement,” in *ACM Multimedia*, 2010.
X. Li, Z. Wang, and X. Lu, “Surveillance video synopsis via scaling down objects,” *IEEE Transactions on Image Processing*, vol. 25, pp. 740–755, 2016.
Y. Nie, C. Xiao, H. Sun, and P. Li, “Compact video synopsis via global spatiotemporal optimization,” *IEEE transactions on visualization and computer graphics*, vol. 19 10, pp. 1664–76, 2013.
W. Tan, B. Yan, K. Li, and Q. Tian, “Image retargeting for preserving robust local feature: Application to mobile visual search,” *IEEE Transactions on Multimedia*, vol. 18, pp. 128–137, 2016.
W. Tan, B. Yan, and C. Lin, “Beyond visual retargeting: A feature retargeting approach for visual recognition and its applications,” *IEEE Transactions on Circuits and Systems for Video Technology*, vol. 28, pp. 3154–3162, 2018.
B. Yan, W. Tan, K. Li, and Q. Tian, “Codebook guided feature-preserving for recognition-oriented image retargeting,” *IEEE Transactions on Image Processing*, vol. 26, pp. 2454–2465, 2017.
S. Avidan and A. Shamir, “Seam carving for content-aware image resizing,” in *ACM Transactions on Graphics (TOG)*, vol. 26, no. 3, 2007, p. 10.
Y. Fang, Z. Chen, W. Lin, and C. Lin, “Saliency-based image retargeting in the compressed domain,” in *19th ACM International Conference on Multimedia*, 2011, pp. 1049–1052.
A. Mansfield, P. Gehler, L. Van Gool, and C. Rother, “Scene carving: Scene consistent image retargeting,” in *Computer Vision–ECCV 2010*, 2010, pp. 143–156.
S. Qi and J. Ho, “Seam segment carving: retargeting images to irregularly-shaped image domains,” in *Computer Vision–ECCV 2012*, 2012, pp. 314–326.
H. Noh and B. Han, “Seam carving with forward gradient difference maps,” in *Proceedings of the 20th ACM international conference on Multimedia*, 2012, pp. 709–712.
W. Dong, N. Zhou, T.-Y. Lee, F. Wu, Y. Kong, and X. Zhang, “Summarization-based image resizing by intelligent object carving,” *IEEE Transactions on Visualization and Computer Graphics*, vol. 20, no. 1, pp. 1–1, 2014.
M. Grundmann, V. Kwatra, M. Han, and I. Essa, “Discontinuous seam-carving for video retargeting,” in *2010 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)*, 2010, pp. 569–576.
D. Panozzo, O. Weber, and O. Sorkine, “Robust image retargeting via axis-aligned deformation,” in *Computer Graphics Forum*, vol. 31, no. 2pt1, 2012, pp. 229–236.
L. Zhang, M. Wang, L. Nie, L. Hong, Y. Rui, and Q. Tian, “Retargeting semantically-rich photos,” *IEEE Transactions on Multimedia (TMM)*, vol. 17, no. 9, pp. 1538–1549, 2015.
D. Cho, J. Park, T.-H. Oh, Y.-W. Tai, and I. S. Kweon, “Weakly-and self-supervised learning for content-aware deep image retargeting,” in *2017 IEEE International Conference on Computer Vision (ICCV)*, 2017, pp. 4568–4577.
K. Simonyan and A. Zisserman, “Very deep convolutional networks for large-scale image recognition,” *CoRR*, vol. abs/1409.1556, 2015.
G. Li and Y. Yu, “Deep contrast learning for salient object detection,” in *Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)*, 2016, pp. 478–487.
L. Zhang, M. Wang, L. Nie, L. Hong, Y. Rui, and Q. Tian, “Retargeting semantically-rich photos,” *IEEE Transactions on Multimedia*, vol. 17, pp. 1538–1549, 2015.
Z. Qu, J. Wang, M. Xu, and H. Lu, “Context-aware video retargeting via graph model,” *IEEE Transactions on Multimedia*, vol. 15, pp. 1677–1687, 2013.
R. Gallea, E. Ardizzone, and R. Pirrone, “Physical metaphor for streaming media retargeting,” *IEEE Transactions on Multimedia*, vol. 16, pp. 971–979, 2014.
R. Pal and P. C. Tripathi, “Content-aware image retargeting: A survey,” 2016.
Y.-S. Wang, C.-L. Tai, O. Sorkine, and T.-Y. Lee, “Optimized scale-and-stretch for image resizing,” in *ACM Transactions on Graphics (TOG)*, vol. 27, no. 5, 2008, p. 118.
Y. Jin, L. Liu, and Q. Wu, “Nonhomogeneous scaling optimization for realtime image resizing,” *The Visual Computer*, vol. 26, pp. 769–778, 2010.
R. W. Brislin, “[Back-translation for cross-cultural research]{},” *Journal of Cross-Cultural Psychology*, 1970.
Q.-X. Huang and L. J. Guibas, “Consistent shape maps via semidefinite programming,” *Comput. Graph. Forum*, vol. 32, pp. 177–186, 2013.
C. Zach, M. Klopschitz, and M. Pollefeys, “Disambiguating visual relations using loop constraints,” *2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition*, pp. 1426–1433, 2010.
T. Zhou, Y. J. Lee, S. X. Yu, and A. A. Efros, “Flowweb: Joint image set alignment by weaving consistent, pixel-wise correspondences,” *2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)*, pp. 1191–1200, 2015.
C. Godard, O. M. Aodha, and G. J. Brostow, “Unsupervised monocular depth estimation with left-right consistency,” *2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)*, pp. 6602–6611, 2017.
Y.-L. Liu, Y.-T. Liao, Y.-Y. Lin, and Y.-Y. Chuang, “Deep video frame interpolation using cyclic frame generation,” in *AAAI 2019*, 2019.
T. Zhou, P. Kr[ä]{}henb[ü]{}hl, M. Aubry, Q.-X. Huang, and A. A. Efros, “Learning dense correspondence via 3d-guided cycle consistency,” *2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)*, pp. 117–126, 2016.
M. Rubinstein, D. Gutierrez, O. Sorkine, and A. Shamir, “A comparative study of image retargeting,” in *ACM transactions on graphics (TOG)*, vol. 29, no. 6, 2010, p. 160.
L. Wolf, M. Guttmann, and D. Cohenor, “Non-homogeneous content-driven video-retargeting,” pp. 1–6, 2007.
M. Rubinstein, A. Shamir, and S. Avidan, “Multi-operator media retargeting,” *international conference on computer graphics and interactive techniques*, vol. 28, no. 3, p. 23, 2009.
Y. Pritch, E. Kavvenaki, and S. Peleg, “Shift-map image editing,” pp. 151–158, 2009.
P. Krahenbuhl, M. Lang, A. Hornung, and M. H. Gross, “A system for retargeting of streaming video,” *international conference on computer graphics and interactive techniques*, vol. 28, no. 5, p. 126, 2009.
Z. Karni, D. Freedman, and C. Gotsman, “Energy-based image deformation,” *symposium on geometry processing*, vol. 28, no. 5, pp. 1257–1268, 2009.
R. Chen, D. Freedman, Z. Karni, C. Gotsman, and L. Liu, “Content-aware image resizing by quadratic programming,” *2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition - Workshops*, pp. 1–8, 2010.
M. Everingham, L. Van Gool, C. K. I. Williams, J. M. Winn, and A. Zisserman, “The pascal visual object classes (voc) challenge,” *International Journal of Computer Vision*, vol. 88, no. 2, pp. 303–338, 2010.
[^1]:
[^2]:
[^3]:
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present observations with VLT and HST of the broad emission lines from the inner ejecta and reverse shock of SN 1987A from 1999 Feb. until 2012 Jan. (days 4381 – 9100 after explosion). We detect broad lines from $\Ha$, $\Hb$, Mg I\], Na I, \[O I\], \[Ca II\] and a feature at $\sim$ 9220 Å. We identify the latter line with Mg II [$\lambda \lambda$ ]{}9218, 9244, which is most likely pumped by Ly$\alpha$ fluorescence. $\Ha$, and $\Hb$ both have a centrally peaked component, extending to $\sim 4500$ and a very broad component extending to $\ga 11,000$ , while the other lines have only the central component. The low velocity component comes from unshocked ejecta, heated mainly by X-rays from the circumstellar environment, whereas the broad component comes from faster ejecta passing through the reverse shock, created by the collision with the circumstellar ring. The flux in $\Ha$ from the reverse shock has increased by a factor of $4-6$ from 2000 to 2007. After that there is a tendency of flattening of the light curve, similar to what may be seen in soft X-rays and in the optical lines from the shocked ring. The core component seen in $\Ha$, \[Ca II\] and Mg II has experienced a similar increase, which is consistent with that found from HST photometry. The ring-like morphology of the ejecta is explained as a result of the X-ray illumination, depositing energy outside of the core of the ejecta. The energy deposition of the external X-rays is calculated using explosion models for SN 1987A and we predict that the outer parts of the unshocked ejecta will continue to brighten because of this. We finally discuss evidence for dust in the ejecta from line asymmetries.'
author:
- 'Claes Fransson, Josefin Larsson, Jason Spyromilio, Roger Chevalier, Per Gröningsson, Anders Jerkstrand, Bruno Leibundgut, Richard McCray, Peter Challis, Robert P. Kirshner, Karina Kjaer, Peter Lundqvist and Jesper Sollerman'
bibliography:
- 'sn1987a\_broad\_lines\_v12.bib'
title: Late Spectral Evolution of the Ejecta and Reverse Shock in SN1987A
---
Introduction {#sec_introd}
============
During the first decade after the explosion the spectrum of SN 1987A was dominated by lines from newly synthesized metals in the inner ejecta, powered by the decay of [${}^{56}$]{}Co, [${}^{57}$]{}Co and [${}^{44}$]{}Ti. Now, more than twenty years after explosion the ejecta are involved in an increasingly intense collision with the circumstellar ring. This is seen in both optical/UV, radio and X-rays [see e.g., @McCray2007 for a review]. In the optical the collision manifests itself most clearly as a forest of increasingly bright emission lines with velocities of $\sim 300$ [@Pun2002; @Groningsson2008a]. However, a number of very broad lines, in particular Ly$\alpha$ and $\Ha$ with velocities of $\ga 10^4$ , are also evident in the spectrum. Broad H$\alpha$ and Ly$\alpha$ lines from the reverse shock were first seen by [@Sonneborn1998]. It is likely that most of that emission is coming from collisional excitation of the neutral expanding H I by shocked electrons, giving an extremely broad component [@Michael1998b]. [@Michael1998a; @Michael2003] found from modeling of narrow slit observations with HST that this emission was concentrated to a region of $\sim \pm 30\degr$ from the equatorial ring, with much less emission from higher latitudes.
The evolution of the broad H$\alpha$ was discussed by [@Smith2005] from a combination of observations with STIS and the Magellan 6.5 m telescope. An interesting prediction was that the ionizing photons from the reverse shock would pre-ionize the high velocity ejecta gas and thereby quench the broad Ly$\alpha$ and $\Ha$ emission from the reverse shock. Because of the limited signal to noise (S/N) and spectral resolution in the STIS spectra and differences in the slit width, they could, however, not make any firm statement on the evolution of the $\Ha$ flux.
Both the Ly$\alpha$ and H$\alpha$ evolution from 1999 to 2004 were discussed by [@Heng2006] from STIS observations, who found an increase in both the Ly$\alpha$ and H$\alpha$ fluxes by factors 5.7 to 9.4 and 2 to 3, respectively. The larger increase of Ly$\alpha$ was explained as a result of resonance back-scattering of the Ly$\alpha$ photons. They also discussed what they called the ‘interior emission’, i.e., line emission. This could either originate in low velocity ejecta or from charge transfer in the post-shock gas of slow H II with high velocity H I. A careful analysis of this mechanism by [@Heng2007], however, found that this ‘interior emission’, corresponding to the ‘broad component’ in Balmer dominated shocks, was too strong to be explained by the charge transfer model.
[@France2010] discussed the Ly$\alpha$ and H$\alpha$ emission based on STIS observations from 2010. They find a further increase in the Ly$\alpha$ and H$\alpha$ fluxes, for the latter by a factor of $\sim
1.7$ from 2004 to 2010. Most interesting, from the large extent of the blue wing of Ly$\alpha$ they propose that Ly$\alpha$ photons from the hot-spots are boosted in energy by scattering against the expanding ejecta. This works for Ly$\alpha$ since it is a resonance line but not for H$\alpha$.
Recently, @Larsson2010 [in the following L11] found from photometry of the ejecta images from HST that while the flux from the ejecta decayed as expected from the [${}^{44}$]{}Ti decay until 2001 (day $\sim$ 5000), it has increased since then. By 2010 the flux was $\sim 3$ times higher in the B and R-bands than in 2001. While L11 mainly used the HST photometry, the flux increase was supported by observations of the flux of the \[Ca II\] [$\lambda \lambda$ ]{}7292, 7324 lines, that will be discussed in this paper. In L11 it was proposed that the increase in the optical flux is caused by reprocessing of the X-ray and EUV emission from the ejecta - ring collision. This is further strengthened by the changing morphology of the ejecta as seen in the HST imaging, where Larsson et. al. (2012, in prep., in the following L13) find that most of the $\Ha$ emission from the inner ejecta has a ring-like morphology. This is in contrast to the emission in the 1.644 $\mu$m \[Si I\]/\[Fe II\] emission, which mainly comes from the core [see also @Kjaer2010].
In this paper we discuss ground based high S/N observations with high spectral resolution from the Very Large Telescope (VLT), complemented by STIS observations from HST. A high S/N is especially important in order to trace the faint line wings to high velocity and also to detect additional, weaker lines in addition to the ones discussed by [@Smith2005] and [@Heng2006]. In particular, we discuss the evolution of the reverse shock in time and the evolution of the lines from the inner regions of the ejecta. These are particularly interesting to monitor since they may signal the re-ionization of the inner ejecta by the hard radiation from the ring collision.
In Sect. \[sec\_obs\] we discuss the observations and reductions, and in Sect. \[sec\_results\] we detail the results. In Sect. \[sec\_discuss\] we discuss the implications for the reverse shock and for the emission from the ejecta. Summary and conclusions are given in Sect. \[sec\_summary\].
Observations {#sec_obs}
============
VLT observations {#sec_vlt}
----------------
The ground based observations were performed with the VLT at ESO, using the FORS2 and UVES instruments. SN 1987A has been monitored regularly with these instruments since 1999. Table \[tab:obslog\_uves\] and Table \[tab:obslog\_fors\] summarize the UVES and FORS2 observations. Primarily, the purpose has been to follow the evolution of the narrow lines from the ring collision. We have therefore mainly used the UVES high resolution spectrograph. To maximize the spectral resolution we have chosen a comparatively narrow slit, 0.8 wide.
In most seasons, observations with seeing of order 0.6were obtained, just enough to spatially resolve the ring into the northern and southern parts [see e.g., @Groningsson2008a]. The position angle of the slit was 30 in all observations. In Fig. \[fig\_slits\] we show the slit orientations and widths for the UVES observations at two epochs.
The FORS2 observations were obtained with a wide 1.6 slit and are accompanied by a local spectrophotometric standard. To estimate slit losses of the UVES spectra we have also obtained low resolution FORS2 observations with a narrow slit (see Table \[tab:uves\_fors\]) Additionally, the slit acquisition images of the UVES spectrograph have been inspected to ensure that the alignment of the instrument slit to the supernova was consistent throughout the data set. This, as described in [@Groningsson2008a], gives us confidence that the error in the absolute flux calibration is between not larger than 30%. Furthermore, the UVES observations employ a 12 long slit. This provides a clean observation of H$\alpha$ emission from the LMC at the edges of the slit, which it is plausible to assume is invariant over the period of our observations and can be expected to be extended with respect to the slit (therefore insensitive to seeing). The variation of the flux in the LMC lines is well within 20%. The uncertainties are considered to be almost exclusively due to the position of the supernova with respect to the slit and the seeing rather than throughput of the instrument or the atmosphere.
The reductions of the FORS2 data followed classical long slit techniques using the [FIGARO]{} data reduction package. The UVES reductions used the ESO pipeline. All wavelength settings were processed using both optimal extraction and a manual extraction along the slit. For the H$\alpha$ and H$\beta$ lines the strong emission from the narrow circumstellar lines poses a challenge for optimal extraction algorithms, and the 2-dimensional manual extraction and sky subtraction was preferred. The difference between the two methods only affects the narrow components which do not form part of this work. However, for the broad H$\alpha$ and H$\beta$ lines we have used the robust 2-D extractions here.
[l l c c c c c c c c]{} 1&1999 Oct 16&4618&346+580&303 – 388&1.0&40,000&1,200&1.0&1.4\
&&&&476 – 684&&&&\
&&&&&&&&\
2&2000 Dec 10 – 14&5040&346+580&303 – 388&0.8&50,000&10,200&0.4–0.8&1.4\
&&&&476 – 684&&&&\
&Dec 09 – 10&5038&390+860&326 – 445&&&9,360&0.4&1.4–1.6\
&&&&660 – 1060&&&&\
&&&&&&&&\
3&2002 Oct 06 – Dec 14&5738&346+580&303 – 388&0.8&50,000&10,200&0.7–1.0&1.5–1.6\
&&&&476 – 684&&&&\
&Dec 13 - 14&5772&390+564&326 – 445&&&9,360&0.4–1.1&1.4–1.5\
&&&&458 – 668&&&&\
&Oct 04 - 05&5702&437+860&373 – 499&&&9,360&0.4–1.1&1.4–1.5\
&&&&660 – 1060&&&&\
&&&&&&&&\
4&2005 Mar 21 - Apr 12&6611&346+580&303 – 388&0.8&50,000&9,200&0.6–0.9&1.5–1.8\
&&&&476 – 684&&&&\
&Apr 09 - 11&6621&437+860&373 – 499&&&4,600&0.5&1.6–1.7\
&&&&660 – 1060&&&&\
&&&&&&&&\
5&2005 Nov 01&6826&346+580&303 – 388&0.8&50,000&2,300&0.9&1.4\
&&&&476 – 684&&&\
&Oct 20 - Nov 15&6825&437+860&373 – 499&&&9,200&0.5–1.0&1.4–1.6\
&&&&660 – 1060&&&&\
&&&&&&&&\
6&2006 Oct 01 – 21&7170&346+580&303 – 388&0.8&50,000&9,000&0.5–0.9&1.4–1.5\
&&&&476 – 684&&&\
&Oct 29 - Nov 15&7196&437+860&373 – 499&&&9,000&0.5–1.0&1.4\
&&&&660 – 1060&&&&\
&&&&&&&&\
7&2007 Oct 23 - Nov 28 &7565&346+580&303 – 388&0.8&50,000&11,250&0.8–1.4&1.4–1.6\
&&&&476 – 684&&&\
&Oct 23&7547&437+860&373 – 499&&&9,000&1.1–1.4&1.4–1.6\
&&&&660 – 1060&&&&\
&&&&&&&&\
8&2008 Nov 23 - 2009 Feb 08&7982&346+580&303 – 388&0.8&50,000&9,000&0.8–1.0&1.4–1.5\
&&&&476 – 684&&&\
&2009 Jan 09 – 25&7998&437+860&373 – 499&&&11,250&0.8–1.3&1.4–1.5\
&&&&660 – 1060&&&&\
&&&&&&&&\
9&2010 Nov 06 – 15&8661&346+580&303 – 388&0.8&50,000&9,000&0.7–1.1&1.4–1.7\
&&&&476 – 684&&&\
&2010 Oct 19 – Nov 15&8652&437+860&373 – 499&&&11,250&0.6–1.1&1.4\
&&&&660 – 1060&&&&\
&&&&&&&&\
10&2011 Nov 06 – Dec 03&9019&346+580&303 – 388&0.8&50,000&9,000&0.8–1.1&1.4–1.5\
&&&&476 – 684&&&\
&2011 Nov 05 – Dec 04&9035&437+860&373 – 499&&&9,000&0.6–1.4&1.4–1.7\
&&&&660 – 1060&&&&\
\[tab:obslog\_uves\]
Average epoch of spectrum since explosion, 1987 Feb. 23 .
[l l c c c c c c c c]{} STIS&1999 Feb 21 – 27& 4384&750L&524 – 636&$0.5$&666&$10,500$&&\
STIS&1999 Aug 30 – 31& 4571&750M&630 – 687&$3 \times 0.1$&6,000&$3 \times 7,804$&&\
&&&&&&&&\
FORS1&2002 Dec 30&5788&600R&514 – 730&0.70&1,660&5,400&0.7–0.9&1.4–1.6\
FORS1&2002 Dec 30&5788&600B&336 – 576&0.70&1,110&7,200&0.7–1.0&1.4\
&&&&&&&&\
STIS&2004 Jul 18 – 23& 6358&750L&524 – 636&$3 \times 0.2$&666&$3 \times 5,468$&&\
&&&&&&&&\
FORS2&2006 Nov 24&7213&600RI&512 – 845&0.7&620&1,200&0.7&1.5\
FORS2&2006 Nov 24&7213&600RI&512 – 845&1.62&620&1,200&0.7&1.5\
FORS2&2006 Dec 21&7240&1028Z&786 – 962&1.62&1,580&1,320&0.8&1.4\
FORS2&2006 Dec 21&7240&1200R&590 – 740&1.62&1,320&1,320&0.9&1.4\
FORS2&2006 Dec 21&7240&1200R&590 – 740&0.70&3,060&1,380&1.1&1.4\
FORS2&2006 Dec 21&7240&1400V&465 – 596&1.62&1,300&1,380&0.7&1.4\
&&&&&&&&\
FORS2&2007 Nov 07&7561&600RI&537 – 870&1.62&620&600&0.9–1.1&1.4\
FORS2&2007 Nov 07&7561&1028Z&786 – 962&1.62&1,580&1,740&1.0–1.2&1.4–1.5\
FORS2&2007 Nov 07&7561&1200R&590 – 740&1.62&1,320&600&1.3–1.4&1.4\
FORS2&2007 Nov 07&7561&1200R&590 – 740&0.70&3,060&600&0.9–1.0&1.4\
FORS2&2007 Nov 07&7561&1400V&465 – 596&1.62&1,300&900&1.0–1.2&1.4\
&&&&&&&&\
FORS2&2008 Nov 04 – Dec 26&7950&600RI&537 – 870&1.62&620&1,200&0.9–1.0&1.4\
FORS2&2008 Nov 28 – Dec 03&7951&1400V&465 – 596&1.62&1,300&1,800&0.6–1.0&1.4–1.5\
FORS2&2008 Dec 26&7976&1200R&590 – 740&0.70&3,060&600&0.9&1.4\
FORS2&2008 Dec 26&7976&1200R&590 – 740&1.62&1,320&600&0.8–1.0&1.4\
FORS2&2008 Dec 26&7976&1028Z&786 – 962&1.62&1,580&1,740&0.8–0.9&1.4\
&&&&&&&&\
STIS&2010 Jan 01& 8378&750L&524 – 636&$0.2$&666&$14,200$&&\
&&&&&&&&\
FORS2&2011 Nov 25&9040&600RI&537 – 870&1.62&620&1,200&0.9–1.0&1.41\
FORS2&2011 Dec 03&9049&1400V&465 – 596&1.62&1,300&1,800&0.6–1.0&1.55\
FORS2&2012 Jan 14&9090&1200R&590 – 740&1.62&1,320&1,400&0.8–1.0&1.41\
FORS2&2012 Jan 14&9090&1028Z&786 – 962&1.62&1,580&3,480&0.6–0.9&1.41\
FORS2&2012 Jan 23&9099&1200R&590 – 740&0.70&3,060&1,200&0.9&1.55\
\[tab:obslog\_fors\]
Average epoch of spectrum since explosion, 1987 Feb. 23.
In this paper we concentrate on the evolution of the broad lines from the ejecta and reverse shock. To show this more clearly we subtract the narrow and intermediate velocity lines by a cubic spline interpolation between the extremes of the narrow lines. Figure \[fig2\] shows the result before and after subtraction for $\Ha$ for the day 7976 FORS spectrum. Even if the resulting line has a smooth shape, one should note that especially the peak of this line is uncertain because of the narrow line contamination. In some cases comparison with HST spectra helps in this respect, even if their spectral resolution is much lower (Sect. \[sec\_stis\]).
The UVES slit only covers part of the ejecta and this affects the flux and the line profiles we observe. The major axis of the ring is 1.6 and the minor axis 1.1, so only part of the outer ejecta, which now fills most of the ring, is within the UVES slit (Fig. \[fig\_slits\]). Because most of the flux in the high velocity wings come from the central parts of the projected image along the line of sight (LOS), these are likely to be less affected than the low velocity parts of the line. These come both from the supernova core and the outer parts of the projected image. Part of the latter emission may be missed by using a narrow slit.
Fig. \[fig1\] shows a comparison between one of the slit integrated spectra obtained with UVES with the 0.8 slit and the FORS2 spectrum with the 1.6 slit for $\Ha$. The FORS2 slit covers most of the ejecta, while the UVES slit only covers the central fraction of it.
The exact fraction of the total flux depends on the seeing and the spatial origin of the emission. In Fig. \[fig1\] we have subtracted the continuum, which mainly comes from the ring collision (Sect. \[sec\_results\]). Most of the difference in the FORS2 and UVES line profiles comes from the parts of the ejecta outside the slit, i.e., in the NW and SE directions. In the ring plane, where most of the emission from the reverse shock originates, this corresponds to velocities in the range -5000 – +5000 , while higher velocities, coming mainly from material expanding in our direction, should fall within the narrow UVES slit.
In Table \[tab:uves\_fors\] we give the ratio of the H$\alpha$ fluxes with the different slit widths at the epochs when we have observations with both instruments. This indicates that the fraction lost stays nearly constant with time and is close to proportional to the width of the slit. We note that inner ejecta, moving across the line of sight at $\la 4000$ for 20 years will still be within 0.33 of the remnant’s center. Although slightly spread out by the seeing, most of this emission should then fall within the 0.8slit of UVES. Most observations were made with a seeing $\la 0.8\arcsec$ and the comparison with the FORS observations indicate that the slit losses are small for the core component. (See Fig. \[fig\_core\_rev\], below.)
[l c c]{} 7240 / 7170&1.9&0.96\
7561 / 7565&1.8&1.18\
7976 / 7982&1.9&1.24\
9090 / 9019&2.0&\
\[tab:uves\_fors\]
HST STIS Observations {#sec_stis_obs}
---------------------
The HST STIS observations used in this paper were carried out in 1999 (day 4381, G750L grating, and days 4571-4572, G750M grating), 2004 (days 6355-6360, G750L grating) and 2010 (day 8378, G750L grating). The G750L grating has a spectral resoltion of $\sim 450\ $ and covers the wavelength interval between 5240 – 10270 Å, which includes H$\alpha$ and \[Ca II\]. The spectral resolution of the G750M grating is significantly better ($\sim 50\ $), but in this observation the wavelength coverage was reduced to 6295 – 6867 Å, which means that only the H$\alpha$ line is included.
For the 1999 observation G750L a wide slit of $0.5$ was used [@Michael2003] at a position angle 25.6. In addition, spectra with the G750M grism with three parallel slits of $0.1$ width at a position angle 27 were taken. For the 2004 observation [@Heng2006] a $0.2$ slit with a position angle of 0 was placed in three different locations, thus covering all of the inner ejecta as shown in Fig. \[fig3f\]. Finally, for the 2010 observation [@France2010] one 0.2 slit with position angle 0 was used, covering only the central part of the ejecta perpendicular to the slit. Details of all observations are summarized in Table \[tab:obslog\_fors\] and in more detail in Table 2 in L13 and we refer the reader to that paper for details regarding the data reduction.
Results {#sec_results}
=======
In Fig. \[fig\_ejecta\_spectra\] we show a subset of the full UVES spectra from Dec. 2002 until 2011 Nov., together with the STIS spectra from 1999 Feb. and 2004 Jul. (Sects. \[sec\_stis\_obs\] and \[sec\_stis\]). Also here we have subtracted the many narrow lines originating from the unshocked and the shocked ring from the UVES spectra. These spectra have all been de-reddened by E$_{B-V} = 0.19$ mag using the [@Cardelli1989] reddening law. To show the relative energy distribution we plot in this figure $\lambda F_\lambda$ .
Most of the rising continuum towards the UV is Paschen and two-photon emission from the ring collision, which is apparent when we compare the continuum with that in the STIS observations, which isolate the ejecta better (Sect. \[sec\_stis\]). The Balmer continuum below 3646 Å from the same source is also prominent.
Superimposed on this are several broad lines, in particular $\Ha,
\Hb$, Mg I\] [$\lambda$ ]{}4571 and \[Ca II\] [$\lambda \lambda$ ]{}7292, 7324. We identify an interesting feature at $\sim$ 9220 Å as emission by Mg II [$\lambda \lambda$ ]{}9218, 9244. In the blue wing of $\Ha$ there is most likely also a contribution from \[O I\] [$\lambda \lambda$ ]{}6300, 6364. We discuss these and some weaker features further in Sect. \[sec\_FeII\].
{width="16cm"}
We have fit the total continuum spectrum with the sum of free-bound and two-photon continua from H I and He I [e.g., @Ercolano2006]. The two-photon contribution will be suppressed at higher densities, and its relative contribution will therefore depend on the density. To include this effect we have employed a model atom for H I similar to that used in [@Kozma1998I].
Our best fits correspond to a temperature of $\sim 3 {\times 10^{4}}$ K and a density of $\sim 10^6 \ \ccm$, although the density could be higher without affecting the results appreciably. These numbers are reasonable for the conditions in the shocked gas of the ring.
The first thing to note from Fig. \[fig\_ejecta\_spectra\] is the increase in the flux of all broad lines during this period. A good example is the \[Ca II\] [$\lambda \lambda$ ]{}7292, 7324 doublet, which shows a steady increase in the flux. We also note that the strength of the \[Ca II\] lines in the STIS spectrum are in between the UVES 2002 Oct. (day 5702) and 2005 Nov. (day 6825) fluxes, which shows that we include most of the flux from the central ejecta also in the UVES observations, despite the narrow slit. There is, however, a large difference in the line profile of the H I lines between the UVES and STIS observations, with much stronger ’shoulders’ in the ground based observations. This is caused by the larger extraction region along the slit for the UVES spectra compared to the STIS spectra (see Sect. \[sec\_stis\]). This results in a larger contribution from the reverse shock in the UVES spectra,
Because the hydrogen lines and the other lines differ substantially in their properties and origin we discuss the results of these separately.
Hydrogen lines {#sec_hydrogen}
--------------
Figure \[fig4\] shows the evolution of the slit integrated $\Ha$ line profile with time from 2000 Dec. to 2011 Nov. from our UVES observations. In addition to removing the narrow lines, we have here subtracted the continuum level by simply subtracting a constant flux for each date. Because of the slow variation of the continuum level this is a reasonable approximation, without introducing additional parameters.
Extracting different regions along the slit we can separate the northern and southern parts of the ejecta as shown in Fig. \[fig3\]. The northern component is clearly blueshifted, while the southern is redshifted. This was found already by [@Smith2005], and is discussed in detail in L13.
We have also measured the evolution of the flux in the two regions, and find this to be very similar, with roughly equal fluxes in the two.
From Figs. \[fig1\] - \[fig4\] one can distinguish two components to the line profile, as was done in [@Smith2005]. One velocity component reaching $\sim 4500$ from the core, and one high velocity component reaching velocities $\ga 10,000$ . These evolve fairly independently of each other, and to determine the flux in each of these components we have assumed the broad component to include the part of the line profile with velocity higher than $\sim 2500$ on the blue and red sides. Between these velocities we make a polynomial interpolation between the red and blue parts of the profile of the broad component, and assume that the flux above this represents the contribution from the core and that below from the reverse shock (see Fig. \[fig\_core\_split\] for the Dec. 2011 (day 9019) spectrum). This is a reasonable procedure since a radially thin shell, a fair approximation to the geometry of the reverse shock, should have a flat line profile [see e.g., @Michael2003 for simulations with different geometries].
However, given the limitations of ground based observations, there may be substantial systematic errors in the fluxes determined. This applies especially to the core component, where the interpolation of the line profile in the red part of the line is uncertain. Also the division between the reverse shock and the core component introduces an additional uncertainty in the core flux. The much higher contribution to the flux by the reverse shock implies that it is less affected by this procedure.
In Fig. \[fig\_core\_rev\] we show the resulting light curves for the core component and the reverse shock component. Note that the UVES slit only covers 0.8of the ejecta and to get the total flux from the reverse shock we therefore multiply these by the correction factor shown in Table \[tab:uves\_fors\]. The flux from the core component should fall mainly within the slit, although part of this may also fall outside due to the broadening of the PSF because of the seeing. We have in the same way determined the fluxes from the dates where FORS observations are available. This is an important check because the 1.6 slit should cover the full ejecta and reverse shock. This is shown as the filled squares in Fig. \[fig\_core\_rev\].
As an additional check of the uncertainty introduced by the interpolation we also determine the core flux in the part of the line between $-2500$ and $- 1000$ not affected by the narrow lines, shown as open circles in Fig. \[fig\_core\_rev\] .
A third way of determining the evolution is to plot the monochromatic flux at velocities characteristic of the central ejecta and reverse shock, respectively. In Fig. \[fig5b\] we show the flux at -1300 , which is as close to the peak as one can safely trace H$\alpha$ without interference with the narrow lines, and at -3000 , which is at the plateau, as well as their difference. We believe that, although this only gives the monochromatic flux at these velocities, it gives a more accurate representation of the flux evolution than the total flux.
For the reverse shock we see that, including the correction for the slit width, there is good agreement between the UVES and FORS fluxes, which shows that the more complete UVES set gives a good representation of the time evolution.
The core fluxes are considerably more uncertain, both in terms of the absolute flux and time evolution. In Sect. \[sec\_stis\] we estimate the flux from the core from high spatial resolution STIS spectra. We there find a flux nearly a factor of two higher, which gives an estimate of the systematic errors in the ground based estimates, mainly caused by the contamination by the narrow lines. The fact that the total interpolated flux and the directly observable flux between $-2500$ to $-1000 $ show a very similar increase, however, indicates that the [*relative*]{} flux evolution is reasonably accurate. We discuss the implications of the light curves further in Sect. \[sec\_rev\_shock\].
We can compare the H$\alpha$ flux with that determined by [@Smith2005] on 2005 Feb. 25 (day 6577). On this day they find a total flux from the reverse shock component of $1.37(\pm 0.15) {\times 10^{-13}} \ \ergs \rm
cm^{-2}$. This is approximately a factor of two higher than what we determine from UVES for 2005 Apr. The slit they use is similar to ours, 0.8, and the seeing is also similar. The position angle is different from ours, P.A. $-10$, compared to 30, and may affect the flux somewhat. However, they calibrate their flux with the part of the HST F658N narrowband image of the ring that falls within the slit. This should give a reasonable calibration, which also compensates for the fraction of the light from this region that is scattered outside the slit because of the seeing in the ground based image. In addition, they add a correction from the HST image for the fraction of the flux which falls outside of the slit. After the correction applied to our data for the UVES slit from Table \[tab:uves\_fors\], we find a fair agreement between our flux and that of [@Smith2005]. To compare the FORS2 & UVES data we have applied this correction to our light curve. Although this does not include all flux, it does result in a consistent light curve. The total flux in the reverse shock at each time can be found by multiplying the flux in Figure \[fig\_core\_rev\] by the factor in Table \[tab:uves\_fors\].
The maximum velocity of $\Ha$ is not well defined on either the red or the blue side. On the blue side it is difficult to establish the true extent of the line due to blending with the broad \[O I\] [$\lambda \lambda$ ]{}6300, 6364 lines from the ejecta around 9096 and 12,022 , respectively. On the red side narrow and intermediate velocity lines from He I [$\lambda$ ]{}6855 and Fe II [$\lambda$ ]{}6872, interfere. We can therefore only establish lower limits to the maximum expansion velocities on each side, $\sim 9000$ on the blue side while on the red side the line extends at least to 11,000 , where the line is still substantially above the ’continuum’ level, and could extend to considerably higher velocities, up to $\sim 13,000$ .
Because of the difficulty in determining the exact continuum level the evolution of the flux at velocities $\ga 10,000$ is uncertain. At lower velocities the flux is increasing monotonously up to $\sim 7500$ days, after which it flattens. The gaps in the line profile at high velocities make it difficult to establish whether the maximum velocity is affected by the interaction with the external medium.
The maximum velocity can be compared to the width of the red wing of $\Ha$ at 7.87 years, which [@Chugai1997] found to extend to $\sim 10,800$ . Although the S/N in their observations was considerably lower, our velocity measurements are consistent with theirs. [@Michael2003] report diffuse $\Ha$ emission out to $\sim
\pm 12,000$ from the central direction of the ejecta in their Oct. 1999 observation, presumably the same emission as is giving rise to the high velocity wing in our observations. This is also supported by the fact that this part of the line is similar in the northern and southern part of the spectrum (Fig. \[fig3\]), as expected from a direction in the projected center. As we show in Sect. \[sec\_stis\], also the STIS 2004 and 2010 spectra give a velocity close to this.
Most of the change in flux of H$\alpha$ takes place at low and intermediate velocities. A clear break in the profile can be seen at $\pm 4000$ (Fig. \[fig4\]). Above this velocity the line profile shows a gradual decrease to the maximum velocity. The main change in flux of the UVES profiles occurs below $\sim 6000 \ \kms$ for both the blue and red wings, as can be seen in Figs. \[fig4\] and \[fig5b\]. At higher velocities the profile is nearly unchanged. This can be investigated in somewhat more detail from the FORS spectra in Fig. \[fig4a\]. When comparing the spectra from days 7240 and 7976 we see that the increase in flux only occurs for velocities below $\sim 5000$ . If we now compare the spectra from days 7976 and 9090 the change only occurs for velocities below $\sim 4600$ . Although these velocities are uncertain, and depend on e.g., the exact continuum level, we therefore note that the velocity below which the flux increases is getting smaller with time.
Below $\sim 2500 \ \kms$ there is a linear rise of the profile with time, which is likely to be caused by a superposition of the core component and the flat reverse shock profile. This can clearly be seen in the STIS observations discussed in Sect. \[sec\_stis\]. On the red side no flat section is seen, which is explained by the fact that the region between $3300 - 6000 \ \kms$ is blocked by lines from the ring. We also note that there is an asymmetry between the red and blue sides, with a considerably lower flux on the red (southern) side.
When comparing the line profiles from the southern and northern parts (Fig. \[fig3\]) it is clear that the blue wing of the ’flat’ part of the line originates from the northern half of the ejecta/ring, while the opposite is the case for the red part, as is expected from the geometry. The higher flux from the NE side of the debris, which produces the blue-shifted side of the line profile, is consistent with the stronger ejecta-ring interaction seen on that side [e.g., @Racusin2009].
Figure \[fig7b\] shows $\Hb$ together with $\Ha$ for days 7982 – 7998. While the continuum level of $\Ha$ is well determined, $\Hb$ is in a crowded region and the continuum is more difficult to define, which introduces some uncertainty. By scaling the $\Hb$ flux by a factor 3.0, as in Fig. \[fig7b\], we find good agreement between the two lines for the blue wing. The red wing of $\Hb$ is, however, considerably stronger compared to $\Ha$. This is most likely a result of blending with other lines. In their 1995 observations [@Chugai1997] identify two Fe II lines at 4889 Å and 5018 Å. In addition, the ‘bump’ at $\sim
4000$ can be identified with the strong Fe II a 6S – z 6P${}^o$ [$\lambda$ ]{}4924 transition. The velocities of these lines are marked in Fig. \[fig7b\]. We also note the lower velocity of the blue wing of $\Hb$ compared to $\Ha$, which confirms that this part of ‘$\Ha$’ is caused by blending with the \[O I\] [$\lambda \lambda$ ]{}6300, 6364 lines. The lines therefore complement each other in filling several of the ‘gaps’ caused by the lines from the ring collision, as well as blending with other ejecta lines.
Including a correction for reddening, the observed $\Ha/\Hb$ ratio of 3.0 corresponds to $\Ha/\Hb = 2.5$. There is no indication that this ratio varies over the line profile. The measured ratio is, however, sensitive to the chosen level of the continuum, which is especially problematic for $\Hb$, where there are few regions free from other lines (see Fig. \[fig7b\]). Given this uncertainty, our line ratio is similar to that found by [@Chugai1997] at 7.87 years, $\Ha/\Hb = 3.8$. At that epoch the flux was, however, dominated by the inner ejecta, with little contribution from the reverse shock. Because the emission from the reverse shock and from the ejecta are produced by different mechanisms, there is no reason to expect these ratios to remain the same (Sect. \[sec\_rev\_shock\]). This ratio can be compared to modeling of the Balmer emission in supernova remnants, where [@Chevalier1980] find an H$\alpha$/H$\beta $ ratio of 3 – 5 depending on the optical depth in the Lyman lines. The uncertainty in the value determined here is large enough that it is unclear if there is any diagnostic value in this ratio.
In addition to these lines one can in Fig. \[fig\_ejecta\_spectra\] identify a strong line coinciding with H$\gamma$, which is clearly blended with other lines. Based on the observations in [@Chugai1997] and modeling in [@Jerkstrand2011] these are mainly Fe I, and in some cases also Fe II, lines. The Ca I [$\lambda$ ]{}4226 line, found by [@Jerkstrand2011] to be strong in the 8 year spectrum, may also be present. Higher members of the Balmer series may contribute below $\sim 4200 \ \AA$.
Comparison with STIS observations of H$\alpha$ {#sec_stis}
----------------------------------------------
HST narrow slit spectroscopy of Ly$\alpha$ and $\Ha$ between 1999 – 2004 has been discussed extensively by [@Michael2003] and [@Heng2006], and recent observations from 2010 have been discussed by [@France2010]. These papers were focused on the reverse shock properties, while we here emphasize the core component and the connection to our ground-based observations. Although the spectral resolution of the G750L grating is only $\sim 450$ (FWHM), which is more than one order of magnitude less than UVES, it is adequate for the broad lines. The most important advantage of the STIS data is that it is possible to directly relate different spatial regions to their velocities. In the following discussion we will focus on the observations from 1999 and 2004, where the slits covered the full central region of the ejecta. The 2010 observation only covered the central $0.2$ and is therefore difficult to compare with the other observations.
The last ejecta spectra not strongly affected by the ring collision are the STIS G750L and G750M spectra from 1999. The full G750L spectrum from 1999 Feb. (day 4381) is shown in Fig. \[fig\_ejecta\_spectra\]. This spectrum was taken with a 0.5 slit, which covered the full inner ejecta at this epoch. The spectra taken with the G750M grating (days 4571-4572) only cover the $\Ha$ line, but offer a much better resolution of $\sim 50 \ \kms$. The observations were performed with three 0.1 slits, which together cover most of the inner ejecta. In the upper panel of Fig. \[fig\_ha\_1999\_2004\] we show a comparison of the $\Ha$ profiles from the G750L and G750M gratings. Both spectra were extracted from the central $\pm 0.3$ along the slits. There is clearly a good agreement between the two spectra, with the only notable difference being that the medium resolution profile contains narrow lines from $\Ha$, \[N II\] [$\lambda \lambda$ ]{}6548, 6584 due to contamination from the outer ring, passing through the ejecta. We will discuss the line profile more below.
In the later 2004 spectrum, the ring interaction is much more prominent. Figure \[fig3f\] shows an ACS image of SN 1987A taken in the F625W filter on 2003 Nov. 28 (day 6122) with the three slit positions, each 0.2 wide, as well as the 2-dimensional spectrum from the central slit. The latter is similar to Fig. 4 in [@Heng2006]. We include it here to illustrate the area of extraction for the spectra.
{width="17cm"}
The full STIS spectrum from the central area indicated in Fig. \[fig3f\] is shown in Fig. \[fig\_ejecta\_spectra\]. The degree of contamination from the outer ring and scattered light in the central part of the line is difficult to estimate. We plot the 2004 spectrum together with the 1999 low resolution spectrum in the lower panel of Fig. \[fig\_ha\_1999\_2004\]. The 2004 spectrum is the total over all three slits, scaled to the same level as the 1999 spectrum outside of $\pm 2000 \ \kms$. We first note the similar shape of the wings between $2000 - 5000 \ \kms$ on both the red and blue sides. There is little evolution between these epochs in these velocity intervals. The central part of the 2004 spectrum is dominated by scattered light from the shocked ring emission as evidenced by the fact that the $\Ha$/\[N II\] ratio is similar to that of the shocked ring [@Groningsson2008b].
The 2-dimensional STIS spectra clearly display a high velocity component close to the ring on both the red and blue sides. This is the freely expanding, neutral reverse shock component. Most of this emission comes from $\sim 3000 - 8000$ on the blue side and from $\sim 4000 - 7000$ on the red side. The blue component is fairly uniformly distributed over the three slit positions close to the northern part of the ring, while the fainter component on the red side mainly comes from the south-eastern side of the ring. [@Heng2006] find that the flux from the reverse shock component in $\Ha$ increased by a factor $2 - 3$ from 1999 Sep. 18 to 2004 Jul. 18 (days 4589 and 6355, respectively).
The increase in the northern component seen by [@Heng2006] can also be seen in our UVES and FORS spectra in Fig. \[fig4\] on the blue side at $\sim -5000$ . A similar increase on the red side from the southern part of the reverse shock can also be seen. The high spatial resolution observations by HST are therefore consistent with our VLT observations.
We extracted one-dimensional spectra from the regions marked by red and black lines in Fig. \[fig3f\] and added the resulting spectra from all three slit positions. Figure \[fig3d\] shows the resulting $\Ha$ line profiles with different sampling in the N – S direction from the center. The similarity of the $\Ha$ line from the full area (area within the black lines in Fig. \[fig3f\]) with the UVES 2005 Mar. spectrum in Fig. \[fig4\] is noteworthy. Although the S/N, as well as the spectral resolution, of the STIS spectrum is much worse than the UVES spectrum, one can recognize several similarities. In particular, the flat regions at 3000 – 8000 on both the red and blue sides are apparent in this integration. The low velocity part of the line is completely dominated by the narrow lines from the ring collision.
In the next extraction from the inner region between -0.3 and +0.3 (red lines) the high velocity component of the line decreases by a factor $\ga 3$. This is clear evidence that this spectral feature originates in the reverse shock, appearing as a thin streak in Fig. \[fig3f\] (right panel). The central region is dominated by the ejecta component. There is, however, still a flat region at both high positive and negative velocities, originating from the reverse shock in the line of sight towards the center of the ejecta.
The spatial resolution of STIS allows us to isolate the core component, and in this way check the VLT observations. Assuming that the emission from the core has the same profile as in 1999 (Fig. \[fig\_ha\_1999\_2004\]) we find for the $\Ha$ flux on day 6355 that the flux is nearly a factor of two larger than the flux from the UVES measurement that is closest in time (from day 6622) (Fig. \[fig\_core\_rev\]). A comparison between Fig. \[fig4\] and the red spectrum in Fig. \[fig3d\] shows that there are two main reasons for this. Firstly, the blue wing of the UVES core component is lost due to the interpolation between $-1000$ and $2000$ and, secondly, some of the UVES low-velocity component (especially on the red side) has been lost in the process of removing the narrow lines.
The line profile from the low resolution STIS spectra of the ejecta shows some indication of an asymmetric velocity line profile, with more emission on the blue side. Unfortunately, the low velocity part of the line profiles in the G750L spectra is severely affected by scattered ring emission. We therefore concentrate on the medium resolution spectrum from 1999. To study the asymmetry of the line we have reflected the line profile around zero velocity, and compared this with the original in Fig. \[fig\_ha\_symm\].
When we compare the red and blue line profiles we see that for $\ga 2000 \kms$ the blue and red wings are nearly identical. At lower velocities the blue wing is $\sim 15 \%$ stronger. We therefore find that solely on the integrated line profile there is only weak evidence for an asymmetry from the $\Ha$ line.
The 2D spectra in Fig. \[fig3f\], on the other hand, show strong blue emission from especially the northern side, while there is little emission from the southern side. This originates in the inner ejecta and extends from zero velocity to $\sim -5000$ . This red/blue asymmetry is consistent with dust absorption of the emission from the far side of the ejecta.
The message we get from the integrated line profiles and the 2D spectra may seem somewhat contradictory. It does, however, show that even strong asymmetries may cancel in the integrated spectra and ideally one needs the full 2D (or even better 3D) information. This is discussed in more detail in L13.
Metal lines from the inner core
-------------------------------
Although not covering the blue part, the STIS 1999 (day 4384) spectrum in Fig. \[fig\_ejecta\_spectra\] shows the most important metal lines, as well as H$\alpha$. The distribution of the different elements can be inferred from their line profiles. In Fig. \[fig\_stis\_profiles\] we show these, centered on the blue doublet components. When we compare the blue wings of these lines we see that the Na I and \[Ca II\] lines have very similar profiles to H$\alpha$, while the \[O I\] [$\lambda$ ]{}6300 line has a smaller blue extent. This is consistent with the results in [@Jerkstrand2011] where the Na I and \[Ca II\] lines are dominated by emission in the H envelope, while the \[O I\] lines arise mainly in the O rich core.
Note that the \[O I\] [$\lambda$ ]{}6364 line sits on top of the flat reverse shock component from the H$\alpha$ line (Fig. \[fig\_ha\_1999\_2004\]), which partly explains its comparable strength to the \[O I\] [$\lambda$ ]{}6300 component (usually $1/3$ of the latter). In addition, there may be a contribution to these lines from Fe I [$\lambda \lambda$ ]{}6280, 6350 [see Fig. 4 in @Jerkstrand2011].
Except for H$\alpha$, easily identified broad lines in the UVES spectra are \[Ca II\] [$\lambda \lambda$ ]{}7292, 7324 and Mg I\] [$\lambda$ ]{}4571. There is also a clear feature coinciding with the \[O I\] [$\lambda \lambda$ ]{}6300, 6364 doublet. In addition, Fig. \[fig\_ejecta\_spectra\] shows an increasingly strong line feature at $\sim 9210$ Å. In Sect. \[sec\_FeII\] we identify this with Mg II [$\lambda \lambda$ ]{}9218, 9244. Of these the \[Ca II\] lines, and at the last epochs the Mg II lines, are both the strongest and best defined.
As mentioned in the introduction, and discussed in Section \[sect\_xrays\], the energy input after $\sim 5000$ days comes from X-rays produced by the collision with the ring more than from radioactivity. To compare the line profiles at the latest stages when X-rays dominate, we show in Fig. \[fig6\] H$\alpha$, \[Ca II\] [$\lambda \lambda$ ]{}7292, 7324 and the Mg II [$\lambda \lambda$ ]{}9218, 9244 lines from day 9019 - 9935. We have here removed the narrow lines, as before, and also subtracted a continuum level from each line. To more easily compare the core component of the lines we show in the lower panel the same lines normalized to the same level for the blue wing, which is well defined for all three lines. To isolate the core component we have for H$\alpha$ put the ’continuum’ level close to the ’shoulder’ on the red wing (at $\sim 3.5 \times 10^{-16} \ \ergs \ {\rm cm}^{-2} \ \AA^{-1}$ in the upper panel).
When comparing the \[Ca II\] and $\Ha$ lines in Fig. \[fig6\] the former lacks the flat, boxy part of the line profile. As the lower panel shows, the central part of $\Ha$ and the blue wing of the two other lines, below $\sim
4000 - 5000$ are, however, very similar and argue for a similar origin. The full extent on the blue side is difficult to estimate because of contaminating lines from the ring, but reaches at least 4000 . On the red side the lines are less well-defined, and also affected by the blue components of the \[Ca II\] and Mg II doublets. The Mg I\] [$\lambda$ ]{}4571 line has a lower S/N, but its profile is consistent with that of the \[Ca II\] line.
Similarly to the other broad lines the line profile of the \[Ca II\] line is contaminated by several strong lines from the ring collision. We can nevertheless trace most of it by interpolation (Fig. \[fig6b\]). This provides a good representation of the true line profile as confirmed by the comparison of the \[Ca II\] line profiles from the Jul. 2004 STIS spectrum and the 2004 Apr. UVES spectrum, shown as the dashed line in the figure.
Figure \[fig6b\] shows that the general line profile remains roughly the same over the whole period. In particular, the maximum velocities of the red and blue wings remain at $\sim \pm 5000$ . This and the very different shape of the line in comparison with $\Ha$ argue strongly for this line originating in the core and not from the reverse shock region. This is also confirmed by the spatial location in the STIS spectrum of the \[Ca II\] region (Fig. \[fig6bb\]) which shows a central component extending to $\sim
5000$ .
The absence of a reverse shock component in the \[Ca II\] lines (as well as other ionized species) is mainly a result of the low abundance in the envelope. The combination of a forbidden transition and a low ionization potential also means that the number of excitations per ionization will be low. This is in contrast to the Li-like ions, like C IV, N V and O VI, which have resonance lines far below the ionization level. This results in a large number of excitations for each ionization [@Laming1996]. This compensates for the low abundance and produces lines that are as strong as those of H and He. This is, however, not the case for the \[Ca II\] and the other forbidden lines in the optical, like the \[O I\] and Mg I\] lines. We also note that because of the positive charge the Ca II ions will, in contrast to H I, be affected by the magnetic field in the shock. Their velocity will therefore quickly be isotropised and the velocity broadening will therefore be characteristic of the post-shock velocity.
The upper panel of Fig. \[fig7\] shows the evolution of the total flux of the \[Ca II\] lines from the UVES observations. The squares show the flux determined from the interpolated line profile where the missing sections of the line in Fig. \[fig6b\] have been replaced by spline interpolations. To estimate the uncertainty introduced by this interpolation we have also calculated the flux from the line, omitting these regions. These are shown as triangles. For a comparison with the interpolated line we have multiplied these fluxes by a constant factor 2.5. As an extra check we also show the flux from the STIS 2004 observation measured from the central area shown in Fig. \[fig3f\]. This flux ($7.4 \times
10^{-15}\ \rm{erg\ cm^{-2}\ s^{-1}}$ on day 6355 after the explosion) is similar to that found from UVES, and well within the error bars.
We see in Fig. \[fig7\] that the flux increased monotonically by a factor of $\sim 4-6$ up to day $\sim$ 7000. At later epochs it is nearly constant. This increase in flux is an important indication that there is additional energy input to the inner ejecta in addition to the radioactive energy source. In L11 these lines were used as an independent confirmation of the HST photometry, where the spatial information was combined with the photometry to show that the increased flux was coming from the inner ejecta, and not from the reverse shock region.
At these stages, the \[Ca II\] [$\lambda \lambda$ ]{}7292, 7324 lines are mainly excited by fluorescence of UV emission through the H & K lines [@Li1993; @Kozma1998II; @Jerkstrand2011]. Therefore the Ca II triplet [$\lambda \lambda$ ]{}8498, 8542, 8662 is expected to have the same total flux as the [$\lambda \lambda$ ]{}7292, 7324 lines, unless collisional de-excitation is important. The STIS spectrum in Fig. \[fig\_ejecta\_spectra\] shows that there indeed is a line feature at this wavelength range, and with a flux consistent with that of the [$\lambda \lambda$ ]{}7292, 7324 lines, although it is more smeared out in wavelength than the [$\lambda \lambda$ ]{}7292, 7324 lines.
We also note the presence of an additional broad line with a peak wavelength close to that of He I [$\lambda$ ]{}5876/Na I [$\lambda \lambda$ ]{}5890, 5896. [@Chugai1997] claim that this is Na I and based on modeling by [@Jerkstrand2011] this is a likely identification.
The [$\lambda$ ]{}9220 Å feature {#sec_FeII}
--------------------------------
The broad line at $\sim$ 9220 Å has no obvious identification. The full extent is $\pm 3500$ , indicating an origin in the core of the supernova. It is unlikely that this is a blend of lines from the shocked ring, which have a width of $\pm
300$ , since the individual lines would then be resolved. This is also confirmed from a direct inspection of the STIS spectra, which shows that the [$\lambda$ ]{}9220 feature has the same spatial distribution as the \[Ca II\] [$\lambda \lambda$ ]{}7292, 7324 lines. Possible identifications include the Paschen 9 line at 9229.0 Å. In this case we would, however, expect to see P10 at 9015 Å nearly as strong and P8 9546 a factor $\sim 1.5$ stronger. Neither of these are seen at this level. In addition, for pure Case B recombination this line is expected to have a flux of $\sim 3 \%$ of $\Hb$ [@Hummer1987], which is too weak to explain the line.
Another possible identification is a blend of Fe II lines, powered by Ly$\alpha$ fluorescence. This is the mechanism which produces the Fe II lines in SN 1995N , but in that case, the Ly$\alpha$ was produced in the circumstellar interaction [@Fransson2002]..
[@Sigut1998; @Sigut2003] have calculated Fe II spectra including Ly$\alpha$ excitation from the first excited state, a${}^4$D, to higher levels, in particular u${}^4$P, u${}^4$D and v${}^4$F. The cascade from these to lower levels result in a number of strong Fe II lines, with the strongest being the 9122.9, 9132.4, 9175.9, 9178.1, 9196.9, 9203.1 and 9204.6.
Observationally, [@Hamann1994] see a strong feature at 9216.5 in the spectrum of Eta Carinae. They identify this mainly with Fe II [$\lambda \lambda$ ]{}9203, 9204. Based on their observation, also Fe II 8490.1 is then expected to be strong. These authors claim all these lines to be pumped by Ly$\alpha$. A weaker broad feature at this position is probably present in the STIS spectrum in Fig. \[fig\_ejecta\_spectra\], so this may be consistent. The same feature is also seen in the symbiotic nova PU Vul by [@Rudy1999] and again identified as Fe II [$\lambda \lambda$ ]{}9176–9205.
For the Ly$\alpha$ pumping mechanism to work the Ly$\alpha$ flux needs to be strong and the population of the Fe II a${}^4$D level large enough for fluorescence to be efficient. In addition, the width of the Ly$\alpha$ line should be large enough to have a substantial flux at the Fe II line. The first and third requirements are certainly fulfilled. Both the Ly$\alpha$ flux from the ring collision [e.g., @Heng2006 and references therein] and from the radioactive excitation of the ejecta [@Jerkstrand2011] are strong. A major problem is, however, to understand the excitation to the a${}^4$D level from which the fluorescence takes place. The excitation energy corresponds to $\sim
11,000$ K, which is much larger than the temperature expected in the radioactively powered core, 100 – 200 K [@Jerkstrand2011]. The level is powered by non-thermal excitations and recombinations, but these are unlikely to be sufficient to make the line optically thick. We therefore conclude that the Ly$\alpha$/ Fe II fluorescence is unlikely to work here.
A more plausible identification of the [$\lambda$ ]{}9220 feature is Mg II [$\lambda \lambda$ ]{}9218, 9244 from the 4p ${}^2$P${}^o$ level. This line can arise either as a result of fluorescence by Ly$\alpha$ or Ly$\beta$. The 5p ${}^2$P${}^o$ level is connected to the Mg II ground state by [$\lambda \lambda$ ]{}1026.0, 1026.1, in nearly perfect resonance with Ly$\beta$ at 1025.7 Å. This decays to 4p ${}^2$P${}^o$ via either the 5s ${}^2$S or 4d ${}^2$D level, with emission at 8214, 8235 and 7877, 7896, respectively. This has earlier been suggested to explain UV and IR lines seen in QSOs and Seyferts by [@Grandi1978] and [@Morris1989]. In our case this has a problem in that the [$\lambda \lambda$ ]{}7877, 7896 and [$\lambda \lambda$ ]{}8214, 8234 lines are expected to have similar strengths as the [$\lambda \lambda$ ]{}9218, 9244 lines, but are not seen in the spectra. The efficiency of the Ly$\beta$ / Mg II fluorescence compared to branching into $\Ha$ is also expected to be low.
Instead, we propose that the lines arise as a result of Ly$\alpha$ fluorescence directly to the 4p ${}^2$P${}^o$ level. The wavelength differences to the multiplet levels are 1239.9, 1240.4 Å. A velocity shift of $\sim 6000$ will therefore redshift the Ly$\alpha$ photons into these lines, unless other optically thick lines are present at shorter wavelength. A velocity difference of this magnitude can e.g., arise from a photon emitted on the far side of the core and absorbed on the near side,
In the analysis of the 8 year spectrum in [@Jerkstrand2011] it was indeed found that the Mg II line dominates the production of the [$\lambda$ ]{}9220 emission. Magnesium is mainly in the form of Mg II in the hydrogen rich regions, and the optical depth of the [$\lambda \lambda$ ]{}1239.9, 1240.4 lines are larger than unity in the central regions within the supernova core, i.e. within $3000-4000$ . This remains true also at $\sim 20$ years and we therefore expect the fluorescence to be effective also at these epochs. We also find that the lines at [$\lambda \lambda$ ]{}10,914 –10,952 should have a strength a factor $\sim 2$ fainter than the 9220 feature. Inspecting one of the few existing NIR spectra of the ejecta in [@Kjaer2010] there is a strong feature at this wavelength, although this region is noisy and contaminated by emission from the ring. This therefore supports the identification above. Other lines expected are the [$\lambda \lambda$ ]{}2929 , 2937 lines in the UV. These are, however, likely to be scattered by the many optically thick resonance lines in the UV.
As is apparent from Fig. \[fig\_ejecta\_spectra\], the flux of this line increases rapidly with time. In Fig. \[fig7\] we show the total line flux from the spline interpolated line profile as squares. As in the previous section, we also show the flux from the unblocked sections of the line as triangles. Because a smaller fraction of the line is affected by narrow lines, we only have to multiply this by a factor of 1.4 to bring it to the same level as the full, interpolated flux. The first point at 5038 days shows a negative flux due to a combination of the noise and level of the subtracted continuum. It is therefore an estimate of the errors in the flux determination.
When we compare it with the evolution of the \[Ca II\] lines in the upper panel we see the same basic evolution. The scatter of the points is, however, smaller due to the smaller fraction of line blocked by narrow lines.
Discussion {#sec_discuss}
==========
The broad lines extracted from the spectra of SN 1987A in this paper represent observations of a supernova ejecta in the transition from radioactively powered supernova to one powered by the interaction with the CS environment. Only a few supernovae dominated by circumstellar interaction already after the first year, like SN 1979C [@Milisavljevic2009], can compete with this. Compared to these we have in the case of SN 1987A also spatial information, which is crucial in interpreting the observations.
The emission from the ejecta can clearly be separated into two components. One of these comes from the inner regions of the supernova ejecta, well represented by the \[Ca II\] [$\lambda \lambda$ ]{}7292, 7324 lines. This is the same component as was seen in the spectrum at 5 – 6 years and 7.8 years by [@Wang1996] and [@Chugai1997], respectively. The other component, seen in the Balmer lines, comes from the reverse shock, resulting from the interaction with the circumstellar ring. The Balmer lines have, however, also a lower velocity component from the inner parts of the ejecta. Because of the different origins of the two components we discuss them one by one in the next sections.
Reverse shock {#sec_rev_shock}
-------------
The reverse shock in SN 1987A has been discussed by e.g., [@Michael1998a; @Michael2003; @Smith2005; @Heng2006] and [@France2010]. The most important results in this paper regarding the reverse shock are the evolution of the line profiles, as well as fluxes.
As mentioned in Sect. \[sec\_introd\], [@Smith2005] discuss the evolution of the flux from the reverse shock extensively. A particularly interesting point was their prediction that, around 2012–14, the H$\alpha$ flux from the reverse shock should reach a maximum and then gradually vanish. The rationale is that the EUV and soft X-ray flux from the shocks ionizes the neutral hydrogen before it reaches the reverse shock. The protons are then accelerated in the shock and do not emit any line radiation, except by charge transfer with any unshocked H I still present.
From the H$\alpha$ luminosity and the estimate that 0.2 H$\alpha$ photons are emitted for each hydrogen ionization [@Smith2005] estimate a flux of $8.9{\times 10^{46}} \ \rm s^{-1}$ H I atoms crossing the shock per second at an age of 18 years, corresponding to a density of $n({\rm H I}) \sim 60-70 \ \ccm$. From the X-ray luminosity they estimate an ionizing luminosity of $3.7{\times 10^{45}} \ \rm s^{-1}$ by number, corresponding to $\sim 8{\times 10^{34}} \ \ergs $. They therefore estimate that $\sim 4 \%$ of the incoming hydrogen atoms become ionized as they cross the shock.
The observations by [@Smith2005] only extended to 2005 Feb. (day 6577), and had rather large observational uncertainties arising from the different instruments and lines used, limiting the extent to which firm conclusions with regard to this prediction could be made. [@Heng2006] used available STIS observations and showed that H$\alpha$ had increased by a factor of $\sim 4$ from 1997 to 2004. Unfortunately, there was little spatial overlap between the various epochs and the uncertainties in the flux were therefore large.
In spite of not covering the full ejecta because of the narrow slit, our UVES observations in Fig. \[fig4\] have the advantages of being obtained with the same instrumental setup, as well as monitoring the same area of the ejecta and ring. In addition, our time coverage is a factor of two longer. Further, the FORS2 observations in Fig. \[fig4a\] cover the full ring (except for minor slit losses due to the seeing). These provide a good estimate of the total flux, as well as a check of the evolution as determined from UVES.
The light curve of the reverse shock (Fig. \[fig\_core\_rev\]) exhibits a steady increase in the flux by a factor of $\sim 3.5$ between days 5000 and day 7500. However, after $\sim $7500 days there is a clear flattening of the light curve. The total increase from $5000 - 9000$ days is a factor $4-5$. We note that also the monochromatic flux of $\Ha$ at $-$3000 in Fig. \[fig5b\] shows a similar flattening. This velocity is dominated by the reverse shock and is subject to less systematic errors from the interpolation between the lines.
A similar flattening as seen in the reverse shock flux evolution has been seen in other aspects of the ring collision. In the optical range the flux evolution of the narrow lines from the shocked ring show a flattening of most emission lines from day $\sim 7000$ [@Groningsson2008b]. These lines are formed behind the slow shocks transmitted into the ring, mainly as a result of the photoionization by the soft X-rays from these shocks. Their flux is therefore a measure of the ejecta – ring interaction.
In the X-ray range, [@Park2011] find a similar flattening in the 0.5 – 2 keV X-ray flux.The hard 3 – 10 keV flux, however, continues to increase. Decomposing the soft flux within the reflected shock model by [@Zhekov2010], Park et al. find a find a dramatic decrease of the X-ray flux produced by the shock that is transmitted into the ring. It is not clear that this is consistent with the evolution of the optical lines. The degree of this flattening has, however, been challenged by [@Maggi2012]. The radio emission, correlated with the hard X-rays, continues to increase up to at least 2010 (day 8014) [@Zanardo2010].
The ionization of the hydrogen close to the reverse shock depends on both the ionizing flux and the density. The latter can be estimated from models of the ejecta used for light curve calculations and spectra during the first years after explosion. From the [@Shigeyama1990] 14E1 model we can approximate the density profile at 20 years with $\rho(V) = 2.0{\times 10^{-23}} (t/20 \ \rm yrs)^{-3} \ V_4^{-8.6} \ \gccm$. [@Michael2003] fit the 10H model of [@Woosley1988] with a similar power law, but the density is a factor of 3.0 higher. The uncertainty in the density is considerable, both from differences in the explosion models and also from the variations in the exact position of the reverse shock. We therefore introduce a factor $k$ to take this into account, which is expected to be in the range $1 \la k \la 3$. With an H : He abundance of 1 : 0.25 by number we get for the envelope density $$n_{\rm H \ I}(r)= 71 k \left({r \over R_{\rm s}}\right)^{-8.6} \left({t \over 20 \ {\rm yrs}}\right)^{5.6} \ \ccm.
\label{eq_hiden}$$ where $R_{\rm s}$ is the radius of the reverse shock. The hydrogen density at the reverse shock therefore increases rapidly with time as long as the ejecta are in the steep part of the density profile (i.e. $V \ga 4000$ ). At lower velocity the density gradient flattens considerably, and consequently the density at the reverse shock will increase more slowly and ultimately decrease as the flat part of the density profile is encountered for $V \la 3000 - 4000$ . This will, however, only occur at $\ga 40$ years. This assumes an ejecta structure similar to that of the 1D models discussed above, and instabilities may greatly change this (see below).
From 2005 Jan. (day 6533), which was the epoch of the observations by [@Smith2005], to 2010 Sept. (day 8619) the 0.5 - 2 keV X-ray flux has increased by a factor of $\sim 3.6$ [@Park2011]. From Eq. (\[eq\_hiden\]) we, however, find that the density at the reverse shock during this period has increased by a factor of $\sim 4.7$. The ionization of the pre-shock gas should therefore be nearly constant, or even decreasing in the last epochs. This applies especially to the period when the X-ray flux levels off. We do therefore not expect a rapid decrease of the flux from the reverse shock as was predicted on the basis of a steadily increasing X-ray flux.
In the case of a stationary reverse shock, as may be the case for the shock in the ring plane, we expect the intensity of the H$\alpha$ line to be $$I_{\nu} \approx {\epsilon \over 4 \pi} n_{\rm H I}(R_s) V_{\rm ejecta}(\rm R_s)
\label{eq_inu}$$ where $\epsilon = 0.2 - 0.25$ is the number of $\Ha$ photons per ionization between the shocked electrons and the free-streaming H I and $V_{\rm ejecta}(\rm R_s) $ is the ejecta velocity at the shock [e.g., @Michael2003]. With the density from Eq. (\[eq\_hiden\]) we find that $I_{\nu} \propto t ^{5.6} V_{\rm ejecta} \propto t ^{4.6}$ [see e.g., @Heng2006]. From Fig. \[fig\_core\_rev\] we find that the flux of the reverse shock has increased by a factor $\sim 4.5$ from day 5040 to day 7170. After this it stays nearly constant. From the above scaling we would expect it to have increased by a factor of $\sim 5.1$, i.e., close to what is observed.
The scaling above only applies to a stationary shock of constant surface area. In reality, the reverse shock is expected to expand (in the observer frame) above the ring, where it sweeps up the constant, low density H II region. In this case [@Heng2006] find that the flux should increase as $F \propto t^{(2n-9+4s-ns)/(n-s)}$, where $n$ is the power law index of the ejecta density and $s$ that of the circumstellar medium. For $n=8.6$ and a constant density medium, we find $F \propto t^{0.95}$. The observed flux should therefore evolve somewhere between these limits, although the ring plane should dominate and a flux evolution closer to $F \propto t^{4.6}$ is expected, which also gives the best agreement with the observations up to $\sim 7100$ days.
The nearly constant flux after $\sim$ 7000 days may be a result of an early encounter with the higher density inner hydrogen region, close to the core. The flux evolution will then be determined by the area involved in this and the density, $F \propto \Omega(t) \ \rho_{0}(V_{\rm ejecta}(R_s)) \ t^{-3} $, where $\Omega(t)$ is the solid angle of the ’high density’ encounter and $\rho_{0}$ is the density at a reference time $t_0$. From the HST observations in L13 it is indeed found that there is especially in the southern part of the ring a bright region in H$\alpha$, which may come from such a high density region of the ejecta. At the last HST observation discussed in L13 at 8328 days (22.5 yrs) this feature is close to the projected position of the reverse shock. Based on its low radial velocity it should be close to the plane of the sky, rather than the ring plane. Taking a maximum radial velocity of 1500 , the position corresponds to an ejecta velocity of $\sim 4500 - 5000$ . It is therefore likely that the ejecta has a density at lower velocities considerably different from the 1D models. This is also confirmed by the simulations by [@Hammer2010]. An evolution of the reverse shock flux departing from the above simple scalings should therefore not be surprising.
The line profiles combined with the spatial information provide important constraints on the reverse shock geometry, as was demonstrated by [@Michael2003]. This applies especially to the maximum velocity, which is limited by the equatorial ring in the ring plane.
Our late spectra can give additional information on this from the high velocity wings of H$\alpha$ . [@Sugerman2005] find a semi-major axis of the ring of 0.82 and an inclination of 43. This gives a radius of the ring equal to $6.1\times10^{17}$ cm for a distance of 50 kpc. Assuming a radius for the reverse shock equal to $\sim 80$% of the blast wave (ring) [@Chevalier1982] the radius of the reverse shock is $\sim 4.9\times10^{17}$ cm, and the ejecta velocity at the shock therefore $$V_{\rm ejecta}(R_{\rm s}) = 7740 \left({t \over 20 \ {\rm yrs}}\right)^{-1} \ \rm km \ s^{-1} \ ,$$ where $t$ is the time since explosion in years. The maximum LOS velocity of the ejecta in the ring plane is therefore expected to be $$V_{\rm max} =
5,660~ \left({t \over 20 \ {\rm yrs}}\right)^{-1} \kms.
\label{eq_vmax}$$
The fact that we observe velocities up to $\sim
11,000$ for H$\alpha$ even in our last 2009 observations (Fig. \[fig4\]), suggests that the expansion of the ejecta must now be anisotropic, with a larger extension above the ring and near stagnation in the plane of the ring.
It is interesting to compare our observed $\Ha$ expansion velocity with the morphology found from the modeling of radio observations, as well as $\Lya$ observations. [@Ng2008] find from the radio emission that the best fit to the morphology is provided by a torus model with a thickness of $\pm 40 \degr$from the equatorial plane, which is similar to the thickness found by [@Michael1998a; @Michael2003] from the $\Lya$ modeling, $\sim \pm 30\degr$. With an inclination of $\sim
43 \degr$ of the ring, this means that the reverse shock extends up to nearly the direct LOS to the center of the supernova. A maximum observed velocity of $11,000$ therefore also corresponds to a similar radial velocity in this direction, and an extension of the ejecta to $\sim 6.9
{\times 10^{17}} (t/20 \ \rm yrs)$ cm, or $\sim 15 \%$ larger than the radius of the ring and $\sim 40$ % larger than the reverse shock location in the ring plane. That the maximum velocity comes from the part of the reverse shock expanding in the LOS is confirmed by the STIS observations in Fig. \[fig6bb\], which shows the highest velocity to occur close to the center of the ejecta.
We find little change in either the maximum velocity or the flux of the red and blue wings above $\sim 8000$ (Figs. \[fig4\] and \[fig4a\]). The maximum velocity is, however, difficult to determine accurately because of blending with the \[O I\] lines from the ejecta on the blue side and narrow lines on the red. The STIS observations in Fig. \[fig6bb\], however, show that there does seem to be a decrease of the reverse shock velocity in H$\alpha$ close to the LOS of the center of the ring. Although contaminated by the \[O I\] emission from the ejecta especially in 2004, we find in the 2004 spectrum a maximum blue velocity of $ -12,200$ , while in 2010 this velocity has decreased to $-9100$ . The corresponding values on the red side are $+8300$ , and $+7400$ . The velocities on the red side are lower than the maximum measured with UVES and FORS2, which are likely to be a result of the limited S/N of the STIS spectra.
The steepening of the line profile with time is roughly consistent with the maximum LOS velocity of the ejecta in the plane of the equatorial ring (see Sect. \[sec\_rev\_shock\]). This is the point of maximum interaction, and it is therefore not surprising to find that the most dramatic change occurs around this velocity, as was concluded already by [@Michael2003]. If the interaction takes place in a thin cylinder concentric to the ring with small extent in the polar direction, and with roughly equal strength along the rim, it will result in a double peaked line profile between $\pm V_{\rm max}$ (Eq. \[eq\_vmax\]). As the thickness of the shell increases in the polar direction the line approaches a flat, boxy profile [see @Michael2003 for a discussion]. The increasing flux of the central peak of the line is therefore partly a result of the increase of this component. As Figs. \[fig\_core\_rev\] and \[fig5b\] show, there is, however, also a substantial increase of the flux from the inner ejecta component on top of this.
The fact that there is emission above $\sim 8000$ may require the ejecta to interact with a medium of comparable density to that in the plane. At 20 years and 11,000 we get from Eq. (\[eq\_hiden\]) a number density $n \approx 3$ cm$^{-3}$. With this density profile we find a ratio between the ejecta density in the ring plane, where $V_{\rm ej} \approx
7740$ , and the density at 11,000 equal to $\sim 20$. With an H II density of 100 cm$^{-3}$ [@Chevalier1995] an estimate based on $V_{\rm s} \approx (\rho_{\rm ej}/\rho_{\rm H II})^{1/2} V_{\rm
ej}$ gives a shock velocity of $\sim 1900$ into the H II region above the ring plane.
According to [@Mattila2010], 100 cm$^{-3}$ H II region gas can, however, not fill the full volume from the ring plane up to our LOS, as this would violate observational constraints on the narrow line emission. It is therefore likely that the H II region is less dense along our LOS than 100 cm$^{-3}$, which results in a higher shock velocity than 1900 km/s. This is also indicated by [@Ng2008] who find from the radio a mean expansion speed of $4000 \pm
400$ over the period 1992 – 2008. However, X-ray observations by Chandra indicate a considerably lower expansion speed of $\sim
1625$ [@Park2011].
The high velocity emission from the ejecta in the LOS could also be a result of the X-ray ionization from the ring. If a large fraction of the X-ray flux is below $\sim 0.1$ keV these X-rays may be absorbed in the outer ejecta and at high altitudes from the ring (see Fig. \[fig\_endep\_2d\] below). They can there provide the necessary energy for the H$\alpha$ emission. The fact that there is little change in the flux in the high velocity wing of H$\alpha$, however, argues against this explanation.
Adopting a similarity solution we expect $V_{\rm ej} \propto
t^{-1/3}$, giving a decrease in the ejecta velocity by $\sim 20 \%$ over 10 years. There is reason to doubt that the similarity solution applies over the small time range since the impact on the H II region. With a maximum velocity of 11,000 this may be in conflict with the observations (Fig. \[fig4\]). As pointed out in Sect. \[sec\_hydrogen\], the maximum velocity could be higher and a decrease would then be ’hidden’ in the many lines from the ring collision.
Ejecta component
----------------
Based on photometry of the ejecta with the WFPC2, ACS and WFPC3 on HST, L11 find that the flux has increased by a factor of 2 – 3 in the R- and B-like bands from day 5000 to day 8000. The R-band is dominated by $\Ha$, while the B-band is a mix of $\Hb$ and Fe I-II lines [@Jerkstrand2011].
Our light curve of $\Ha$ (Figs. \[fig\_core\_rev\] and \[fig5b\]) shows an increase by a factor of $4-6$, while the \[Ca II\] [$\lambda \lambda$ ]{}7292, 7324 lines (Fig. \[fig7\]) increased by a factor of $5 - 8$ over the same period. Given that the HST observations cover a wide spectral range for each filter and that there is a considerable uncertainty in the line fluxes because of blending with the narrow lines, this is consistent with the increase found from the HST broad band photometry. This therefore confirms spectroscopically what was concluded based mainly on imaging in L11. In addition, we see that other lines, in particular the Mg II [$\lambda \lambda$ ]{}9218, 9244 lines (see Fig. \[fig\_ejecta\_spectra\]), show a similar, or even larger increase.
Later than $\sim $ 7000 days the $\Ha$, \[Ca II\] and Mg II fluxes level off, similarly to the reverse shock lines and the X-rays, discussed in the previous section. As we discuss in next section (see also L11), there is a close connection between the X-rays from the ring collision and the ejecta flux. It is therefore expected that the ejecta flux follows the flattening seen in X-rays.
As for the line profiles from the ejecta, $\Ha$ is severely contaminated by the narrow lines, as well as the reverse shock emission. The \[Ca II\] lines are less affected by these and also lack a reverse shock component (Fig. \[fig6b\]). The line shape on the blue side is well defined and approximately triangular, similar to that of the $\Ha$ line in the early STIS observation in Fig. \[fig\_ha\_1999\_2004\]. The blue side reaches $\sim 4500$ , and does not change appreciably from 2002 to 2012 (days 5738 – 9019). This velocity is similar to $\Ha$, as measured from the 1999 STIS G750M spectrum in Fig. \[fig\_ha\_1999\_2004\]. Both the similar line profiles and the similar maximum velocities indicate that these lines arise from the same region.
L13 show that the changing morphology results from the same cause as the increase observed in the light curve. They find that the $\Ha$ emission from the inner ejecta has changed from a centrally dominated, elliptical profile before year $2000$ to one dominated by a ring-like structure at later stages. Given this complex morphology it is somewhat surprising that the line profiles from the core component change relatively little, except in flux (see Figs. \[fig4\] and \[fig\_ha\_1999\_2004\]). As noted above, a reason for this may be that most of the change occurs for LOS velocities less than $\sim 1500$ , a spectral region which as discussed earlier is blocked by the narrow lines.
There is also little evolution of the \[Si I\]/\[Fe II\] line from the VLT/SINFONI observations (L13). This emission is, however, centrally peaked and is likely to come from more central, processed material, protected from most of the X-rays, and does not display any major change in morphology between 2005 and 2010.
The energy deposition by the X-rays from the ring collision {#sect_xrays}
-----------------------------------------------------------
L11 show that up to day 5000 the light curves in the R and B-bands are compatible with the radioactive decay of [${}^{44}$]{}Ti. They concluded that to explain the increasing ejecta flux an additional energy source is needed. The most likely such source is the soft X-rays emitted by shocks resulting from the ejecta - ring interaction. We now discuss the details of the deposition of these and their conversion into optical/UV radiation.
The observed X-ray luminosity above 0.5 keV is $\sim 4 \times 10^{36} \ergs$ at the last observation at 8619 days by [@Park2011]. Based on the spectra, [@Zhekov2010] has modeled the X-ray emission with three components, one fast blast wave, one soft component from the transmitted shocks into the dense clumps in the ring and one from the reflected shock component. The soft radiation from the transmitted shocks into the ring, with $k T_e \sim
0.35$ keV, dominates the total flux.
After $\sim 10$ years the hydrogen envelope was mostly transparent to X-rays with energy $\ga 1$ keV (L11). The inner core region, containing both mixed-in hydrogen and synthesized heavy elements, which increase the opacity, is, however, still opaque up to $\sim 8$ keV. The X-rays will there be thermalized into optical, IR and UV radiation. Only after $\ga 30$ years does the whole core become transparent.
The exact X-ray luminosity and spectrum is uncertain. Observations only probe energies down to $\sim 0.5$ keV. For low shock velocities most of the X-ray flux is below the low energy interstellar X-ray cut-off at $\sim 0.5$ keV. In [@Groningsson2008a] it is shown that the shocked lines have a FWHM $\sim 200$ . Even if there could be an inclination effect, which could increase this velocity by a factor up to $\sim 1.4$, the typical shock velocities giving rise to these lines are $\la 300$ , corresponding to shock temperatures $\la 0.1$ keV. In addition, [@Groningsson2008b] show that shocks into the densest parts of the ring are radiative up to $\sim 500$ , or $\sim 0.25$ keV. Therefore, a large fraction of the X-rays/EUV flux may well be below the observed ISM absorption threshold at $\sim 0.5$ keV.
The X-rays give rise to secondary electrons by photoionization in the same way as the gamma-rays and positrons from radioactivity [@Shull1985; @Xu1991; @Kozma1992]. These in turn will lose their energy in non-thermal excitations, ionizations and Coulomb heating of the free thermal electrons. The resulting optical/UV spectrum from the X-ray input will therefore not be very different from that of the radioactive decay, as long as the ionization of the gas is low. For a low ionization each hydrogen ionization requires $\sim 30$ eV at the relevant level of ionization [@Xu1991]. Once the ionization increases above $\sim 10^{-2}$ the efficiency of excitation and ionization decreases, while that of the heating increases. The cooling, balancing the heating, is close to the core region at these epochs mainly done by thermal, collisional excitations of mid- and far-IR fine structure lines and molecules, and possibly cool dust. This emission will therefore be difficult to observe directly, although far-IR and mm observations may help. In the envelope adiabatic cooling dominates.
Compared to the radioactive excitation some important differences arise due to the fact that the X-ray illumination is from the outside, while that of the gamma-rays and positrons is from the central regions. The positrons from the [${}^{44}$]{}Ti decay, which dominate the radioactive energy input at these epochs, will be deposited locally, mainly in the Fe-rich gas, unless the magnetic field in the ejecta is extremely low [see discussion in @Jerkstrand2011]. The X-rays will on the other hand penetrate to different depths, depending on their energy.
The photoelectric energy deposition is sensitive to the density and abundance distributions in the ejecta. To calculate this we use the ejecta model, 14E1 from [@Shigeyama1990], mixed as in [@Blinnikov2000]. This is a spherically symmetric model, where the chemical mixing has been modified to reproduce the light curve. In reality, Rayleigh-Taylor instabilities will mix the different nuclear burning zones in velocity, resulting in both large abundance and density inhomogenites [e.g., @Hammer2010]. Nevertheless, this model should show the main qualitative features of the X-ray deposition.
Using the same code as was used in L11 for calculating the total energy deposition of the X-rays, we can calculate also the distribution of the energy deposition in radius. We assume here that the X-ray sources are located in the plane of the equatorial ring at $6.1{\times 10^{17}}$ cm and the ejecta extend to the reverse shock at $ R_{\rm s} = 4.9\times10^{17}$ cm. In addition, we assume spherical geometry for the ejecta, which is certainly an oversimplification.
In Fig. \[fig\_endep\] we show the fractional energy deposition per cm integrated over the polar angle, $df/dr$, for energies between 0.1 – 5 keV at 20 and 30 years after explosion. The total energy deposition in a shell with radius $r$ and thickness $dr$ is then $dL(E) = L_{\rm X-ray}(E) \ df(E)/dr \ dr$. As we show below, this is, however, only a rough approximation and the real deposition function is two dimensional or even three dimensional.
We here see that for the hard X-rays with $\ga 1$ keV most of the energy is deposited at the outer boundary of the core, while the softer X-rays are deposited in the hydrogen envelope. At 30 years the soft X-rays are deposited in a more narrow velocity interval than at 20 years, caused by the increasing ejecta density close to the reverse shock (Eq. \[eq\_hiden\]). The harder X-rays, however, are deposited deeper in the core, where the flatter density distribution does not have the same effect. The fact that the core is very heterogeneous in abundance, with a large fraction of the volume occupied by the iron bubble [@Li1993], having a large X-ray opacity, means that most of the hard X-rays will be absorbed efficiently here.
Because the interaction of the ejecta and ring is mainly taking place in a thin region around the equatorial ring, the X-ray illumination of the ejecta will be highly non-uniform in the polar direction. Most of the energy deposition will occur close to the equatorial plane, but especially at higher energies a large fraction of the ejecta will be transparent and the deposition more distributed also above the plane. To calculate the deposition in 3-D we assume that the X-rays are injected in a thin region in the equatorial plane, and azimuthally uniform. This is a reasonable approximation for observations obtained after $\sim 2004 $, but for earlier epochs the distribution of the individual hotspots in the azimuthal direction should be taken into account. In Fig. \[fig\_endep\_2d\] we show the deposition at different energies in the poloidal plane.
Again, we note the larger penetration of the hard X-rays. More interesting is, however, that the X-rays are deposited spatially fairly locally. At low energies this is close to the ring and also close to the equatorial plane, while the higher energies are deposited in a more extended region closer to the core. We also note the higher concentration of the energy deposition to the ring at 30 years, compared to 20 years.
As mentioned in L11, the restricted radial range of the energy deposition may affect the morphology of the HST images, which show a ringlike structure in the ejecta. This may be explained as a result of the limited penetration of the X-rays from the shock, and therefore reflects the energy deposition in the ejecta rather than the true density distribution, as was seen in Fig. \[fig\_endep\]. This is discussed in more detail in L13.
Figure \[fig\_endep\] also shows that if most of the X-ray luminosity is below $\sim 0.5$ keV, as may well be the case, the outer parts of the ejecta may absorb a larger fraction of the energy, while the core only receives a minor fraction of the X-ray input. A further brightening of the outer parts can therefore be expected. As has been mentioned earlier, recent observations by [@Park2011], however, show a flattening of the soft (0.5 – 2 keV) X-ray flux (see however [@Maggi2012]), while the hard (3 – 10 keV) flux continues to increase.
In the high velocity hydrogen envelope the density is low, and the recombination timescale long. For a density profile similar to that in Eq. (\[eq\_hiden\]) the density at the reverse shock is $70 - 200 \ \ccm$ at an age of 20 years. In the absence of X-ray input we find using the code in [@2002NewFransson] that the temperature is here expected to be $\sim
10-20$ K . Extrapolating the Case B recombination rate from [@Martin1988] gives a recombination rate $\sim 2
{\times 10^{-11}} \ \rm cm^{3} s^{-1}$ at 20 K and a recombination time $$t_{\rm rec} = 22 \ k^{-1} x_e^{-1} \left({r \over R_s}\right)^{8.6} \left({t \over 20 \ {\rm yrs}}\right)^{-5.6} \ \rm years,
\label{eq_rect}$$ where $x_e$ is the ionization fraction. Equation (\[eq\_rect\]) shows that the recombination time decreases fast both with time and with decreasing radius. As long as $x_e \ll 1$ it will, however, be long at least close to the shock. As the ionization increases recombination becomes more and more important.
From the same calculation as above, again including only the radioactive input, we find that the degree of ionization in the envelope is nearly constant with time at $x_e \approx 7{\times 10^{-3}}$, as the ionization is frozen-in. Because of the X-ray illumination the ionization may, however, increase considerably. Because of this and the large density gradient, the recombination timescale may become short only a short distance inside the reverse shock. A stationary situation may then be at hand where absorption and emission balance.
For the regions where the recombination timescale is long, the ionization of the envelope is determined by the number density of H atoms in the part of the envelope where the X-rays are deposited and the time integrated ionizing luminosity, rather than the instantaneous luminosity. Further in, once the ionization has increased enough there may be a balance between ionizations and recombination.
Integrating the X-ray luminosity from [@Park2011] over time gives a total X-ray energy $E_{\rm X-ray} \approx 1.6 \times 10^{44}$ ergs in the 0.5-2 keV band. Two thirds of this is emitted in the period 19 – 22 years after explosion. On top of this there is a minor contribution from hard X-rays and a possibly dominant contribution below 0.5 keV. Roughly half of the X-ray energy will be emitted inwards to the ejecta. As shown above, most of the X-rays and EUV photons below $\sim 0.3$ keV are absorbed by the envelope. The luminosity below this energy is therefore most important for the ionization of the envelope, but also the most uncertain observationally.
To take into account the fact that we do not know the level of the unobserved soft X-ray flux, we assume that the total time integrated luminosity emitted in a narrow energy interval in the EUV is $E_{\rm EUV} = f \times 10^{44}$ ergs, where $f$ is a scaling factor reflecting the uncertainty in the EUV flux. To get an approximate estimate of the ionization in the ejecta we further assume that recombinations are slow and that each volume is absorbing a fraction of $E_{\rm EUV}$ given in Fig. \[fig\_endep\_2d\]. We also ignore the decreasing efficiency of ionization as the ionization increases. Finally, for simplicity we ignore the expansion, which is justified by the fact that most of the X-rays are emitted during a period short compared to the age of the supernova. These assumptions affect the quantitative results, but hardly the qualitative. With these assumptions the ionization of a given volume depends on the deposited energy and is inversely proportional to the density.
In Fig. \[fig\_ion\_2d\] we show the resulting distribution for two values of the efficiency of ionization, $f=1$ and $f=0.1$, which may bracket the range. Not surprising, the distribution reflects the energy input in Fig. \[fig\_endep\_2d\]. More interesting is that close to the shock the ionization is expected to be fairly high, in particular for the case where most of the ionizing flux is below $\sim 0.3$ keV. Comparing the 0.1 keV and 0.35 keV cases, we note the higher, but more concentrated ionization in the former case for the same total energy.
If the recombination time close to the reverse shock is long most of the emission in the H I lines comes from direct excitation of these. The fraction of the deposited energy going into excitation of levels with $n > 2$ is $\sim 12 \%$ for $x_e = 10^{-2}$ [@Xu1991]. Of this $\sim 10 \%$ is emitted as $\Ha$ emission. Assuming that we underestimate the UVES luminosity by a factor of $\sim 2$, as was found from the 2004 STIS observation (Sect. \[sec\_stis\]), our observation at $\sim$ 8000 days (Fig. \[fig\_core\_rev\]) result in $L({\rm H}\alpha) \approx 3.2{\times 10^{34}} \ \ergs$. To explain this with direct excitation only, an X-ray luminosity of $\sim 10^2 L({\rm H}\alpha) \approx 3.2{\times 10^{36}} \ \ergs$ is therefore needed. In the regions where the recombination time may become short compared to the age also the non-thermal ionizations and subsequent recombinations will contribute to the $\Ha$ luminosity, increasing this by a factor of $\sim 3$, compared to the direct excitations. The observed 0.5 – 2 keV X-ray luminosity at 8000 days was $\sim 3.6 \times10^{ 36} \ergs$ [@Park2011]. An additional $\sim 14 \%$ was in the hard 3 - 10 keV range. There does therefore seem to be enough X-ray luminosity to explain the observed $\Ha$ luminosity.
The conclusion of this discussion is that the ionization in both the envelope as a whole and the core are likely to remain low for at least the next ten years. A dramatic change of the spectrum is therefore not expected, although the line ratios may slowly change.
The estimate by [@Smith2005] of the ionization assumes that most of the X-rays are absorbed close to the reverse shock. The H I density in front of the reverse shock is at 20 years $\sim 90 \ \ccm$. The mean free path close to the Lyman limit is then $\sim 1.6{\times 10^{15}} (E/13.6 \ {\rm eV})^3 (n/100 \ \ccm)^{-1}$ cm, so most of the ionizing photons are indeed absorbed close to the reverese shock as long as the photon energies are not too high. As Figs. \[fig\_endep\] and \[fig\_endep\_2d\] show, this is in general the case for photon energies $\la 0.1$ keV, but not for higher energies. The fact that we still observe both H$\alpha$ and Ly$\alpha$ shows that the X-rays are not yet able to pre-ionize the gas completely. As we see from Fig. \[fig\_ion\_2d\], the ionization decreases substantially above the ring plane, and at least a fraction of the H$\alpha$ may originate here.
Except for the Mg II [$\lambda \lambda$ ]{}9218, 9244 lines, there are no major changes in the relative fluxes of the lines from the core (Fig. \[fig\_ejecta\_spectra\]). The Mg II lines, however, increase substantially compared to e.g., the \[Ca II\] lines. If these lines are pumped by Ly$\alpha$ there are several factors which influence the outgoing flux. The most straightforward is the increasing Ly$\alpha$ flux, which should be reflected directly in the Mg II lines. However, also the fractional population of Mg II, which determines the optical depth of the Mg II lines, and therefore the pumping efficiency, may increase because of the ionizing effects of the X-rays. Finally, the fraction of the envelope which is in velocity resonance with the pumping transition may change with time.
Dust
----
Strong evidence for early dust formation in the ejecta was found from both blueshifts in the line profiles [@Lucy1989] and a thermal IR excess [@Wooden1993]. The last far-IR observations before the ring interaction are from day 1731, where [@Bouchet1996] find a dust temperature of $\sim$155 K. There is also from the line profiles of Mg I\] [$\lambda$ ]{}4571 some evidence for dust from the HST observations at 8 years with an effective optical depth close to unity [@Jerkstrand2011]. This is, however, likely to be in the form of very dense clumps with high optical depth. The effective optical depth is therefore only a measure of the covering factor of the dust clumps.
More recently [@Matsuura2011] have from Herschel far-IR observations found evidence for a large dust mass, estimated to $0.4 - 0.7$ [$M_\odot$]{}, but could be up to $2.4$ [$M_\odot$]{}. The mass estimate is, however, highly dependent on the dust temperature and the mass could be several orders of magnitude lower if the temperature is higher than adopted by [@Matsuura2011], as discussed from APEX observations by [@Lakievic2012].
As was discussed in Sect. \[sec\_stis\], the 1999 $\Ha$ line profile may indicate a $\sim 15 \%$ deficit on the red side below $\sim 2000 \ \kms$ (Fig. \[fig\_ha\_symm\]). This only represents weak evidence for dust in the ejecta. The 2D spectra do, however, show a clear asymmetry with most emission on the blue side. While the observed asymmetrical emission in $\Ha$ is a possible explanation the prevalence of blue emission is indicative of dust obscuration. Because most of $\Ha$ arises outside the metal core, where the dust presumably is located, it is, however, less affected by dust in the metal core than e.g, the \[O I\] emission. There are also some indications of asymmetries from the \[Ca II\] line profile (Fig. \[fig6b\]), consistent with that in $\Ha$. As already mentioned, this line, however, originates in the same region as $\Ha$.
The most direct evidence from the line profiles at early epochs came from the evolution of the \[O I\] [$\lambda \lambda$ ]{}6300, 6364 lines [@Lucy1989]. Lucy et al. found a blueshift of the peak by $\sim 600$ at the time of the peak velocity shift, $\sim 530$ days. Figure \[fig\_stis\_profiles\] represents the last observation of this line before the spectrum becomes affected by the X-rays. We here note that the peak of the \[O I\] [$\lambda$ ]{}6300 line also in this late observation is blueshifted by $\sim 1000$ . The resolution of this spectrum is limited, but the blueshift is consistent with that determined by [@Lucy1989].
L13 argue that the ’hole’ seen in the optical HST images is the result of the external X-ray illumination rather than dust obscuration. We can, however, not exclude that this region contains some dust. After all, this metal rich region is the most likely dust forming region. The VLT/SINFONI IFU spectra from 2005 and 2010 also show a smooth decrease of the red wing of the \[Si I\]/\[Fe II\] [$\lambda$ ]{}1.644 $\mu$m line at velocities $\la 3000$ (L13). This is roughly what is expected from dust internal to the line emitting region.
Summarizing this discussion, we find indications of dust from the 1999 \[O I\] [$\lambda$ ]{}6300 line (and also the \[Si I\]/\[Fe II\]) profile, but also caution that the asymmetries in the core may also give this effect. From the high resolution observation of the H$\alpha$ line there is, however, little indication of dust, although this line should be less affected by this.
Summary and Conclusions {#sec_summary}
=======================
We have here discussed the spectral evolution of the ejecta and reverse shock emission from 2000 to 2012 (days $\sim 4400 - 9100$). Together with the HST imaging this represents a unique data set. Our main conclusions are summarised below:
- Both the H$\alpha$, \[Ca II\] and Mg II lines from the inner ejecta have increased by a factor of 4-6 from 2000 to 2012. This confirms the broad band increase found from the HST observations by L11. There is a flattening of the light curve later than $\sim 7000$ days, possibly correlated with a similar flattening of the X-ray flux and the flux from the narrow lines from the shocked ring.
- The reverse shock flux behaves in a similar way. The leveling off of the flux is probably not a result of an increasing ionization of the pre-shock hydrogen, but may be a result of an arrival of protrusions of the inner, dense core region at the reverse shock, as seen from the HST imaging.
- The reverse shock emission extends to velocities $\ga 11,000$ . This is larger than can be contained within the equatorial ring and indicates an anisotropic expansion of the ejecta above and below the ring plane.
- The Balmer and metal lines have an inner ejecta component with a width of $\sim 4500$ ,
- From the last radioactively dominated STIS spectrum in 1999 (day 4381 – 4387) we find similar velocities for the $\Ha$, Na I and \[Ca II\] lines, consistent with an origin in the hydrogen rich gas. The \[O I\] has a lower velocity, mainly coming from the core.
- There is from $\Ha$ and \[O I\] [$\lambda$ ]{}6300 evidence of a blueshift similar to what was seen as a dust indicator at early epochs.
- We identify a line feature at $\sim$ 9220 Å with an Mg II line pumped by Ly$\alpha$ fluorescence. The Mg II emission is mainly from primordial magnesium in the H and He rich ejecta. This line has been increasing faster than the other lines from the inner ejecta.
- The deposition of the X-rays from the shock region affects mainly the hydrogen envelope and outer core region of the ejecta. Only X-rays with energy $\ga 5$ keV reach the inner core region.
- The changes in the morphology of the ejecta during the last decade is probably mainly a result of the increasing ionization and heating of the ejecta by the X-rays. The steep density gradient in the hydrogen envelope may lead to a more concentrated ionization to the reverse shock with time.
- The excitation by the X-rays is similar to the gamma-ray and positron deposition as long as the ionization is low. Most of the heating is balanced by far-IR fine structure and molecular lines, while the optical and near-IR lines are powered by non-thermal excitation and ionization.
Future observations are needed to confirm the evolution and follow the increasing ionization of the ejecta by the external X-rays.
This work is supported by the Swedish Research Council and the Swedish National Space Board. Based on ESO observational programs 66.D-0589(A), 70.D-0379(A), 074.D-0761(A), 078.D-0521(A,B), 080.D-0727(A,B), 082.D-0273(A,B), 086.D-0713(A), 088.D-0638(A,C). Support for GO program numbers 02563, 03853,04445, 05480, 06020, 07434, 08243, 08648, 09114, 09428,10263, 11181, 11973, 12241 was provided by NASA through grants from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.
[*Facilities:*]{} , ,
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We use PYTHIA event generator to simulate the process of muon pair production in antiproton scattering upon the proton target at the energy of the antiproton equal to $14GeV$, which may be one of the energies for the future PANDA (GSI, Darmstadt) experiment operation.'
author:
- |
A.N. Skachkova, N.B. Skachkov,\
Joint Institute for Nuclear Research, Dubna, 141980, Russia\
E-mail:Anna.Skachkova@cern.ch, skachkov@jinr.ru
title: 'MUON PAIR PRODUCTION IN PROTON-ANTIPROTON INTERACTIONS AT INTERMEDIATE ENERGIES.'
---
Introduction. {#intro}
=============
Measurements of lepton pair production in hadron-hadron interactions is of a big interest from the point of view of study of the quark-parton structure of hadrons [@Matv] and [@Drell].
The best example is the discovery of J (also called as J/Psi) charmed meson confirmed later by e+e- experiment tool to get out the information about the parton distribution functions (PDF) in hadrons as it was already shown in a number of high energy experiments [@DYexper] and theoretical papers, devoted to the data analysis in the framework of QCD. In this connection it is worth to mention that up to now there is no good theoretical understanding of the physics of this process. The best example of this situation is the wide use of such a “theoretical method” of elimination of quite a noticeable discrepancy among the lepton pair production data and the theory predictions (quark-parton model and its QCD extensions) as a simple multiplication of the theory predictions by the so called phenomenological “K-factors” which values vary in the interval from 1.5 to 2.
Intermediate energy experiments may play an important role as they allow to study the range where the perturbative methods of QCD (pQCD) come into the interplay with a rich physics of bound states. The physics of hadron resonances formation and decays is strongly connected with the confinement problem , i.e. with the QCD dynamics at large distances. Its high precision and detail experimental study would allow to make a serious discrimination between a large variety of non-perturbative approaches and models that are already proposed as well as among those which are under development.
Therefore to define a boundary among perturbative and non-perturbative approaches it may be instructive to make some estimations on kinematical distributions, connected with the $\mu^{+}\mu^{-}$- or $e^{+}e^{-}$- pairs production in the range of future PANDA experiment energies basing upon the well known PYTHIA [@Sjost] event generator, which is widely and successfully used in high energy experiments. Such an experience may allow to set the low energy limits for pQCD application as well as it may be also useful for comparison with the predictions of the present (and future) non-perturbative approaches and finding out their difference.
It is worth to mention also that in any case the dominant mechanism of the lepton-anti-lepton pair production would be the same in the most approaches because in this special process $ p\bar{p} \rightarrow l\bar{l} + X $ the dominant contribution in any model would come from the same quark pair annihilation amplitude ${q\bar{q} \to l\bar{l}}$. PYTHIA includes this amplitude as well as proper account of relativistic kinematics
To this reason, in present Note we shall simulate in framework of PYTHIA6 the scattering of the anti-protons with the energy of $14 GeV$ on the proton target, taken as to be at the rest frame Thus the present simulation would not include the effects connected with the real detector conditions which would be the subject of the following papers. Thus, the present simulation has the importance as its own because it gives in some sense the “unbiased distributions” of particles as well as the corresponding quark - parton distributions and allows, in principle, to estimate the size of the corrections needed to reconstruct the original input parton distributions. Let us mention that in what follows we choose MRST4 [@MRST] parameterization of structure functions , which is one from the list of proton parton-distributions set of PYTHIA6.
It should be noted also that the main part of this Note (Sections 1. and 2.) can be easily applied to a case of electron-positron pair production. The main difference of $e^{+}e^{-}$ case with that one of muon pair production is that for $e^{+}e^{-}$ case there would not a big need of discussion of appearance of fake electrons in the same “signal” annihilation event due to negligibly small contribution of fake electrons from pions decays as comparing with their muon decay channel.
MUON DISTRIBUTIONS IN $P\bar P$ COLLISIONS.
===========================================
Here we present some distributions of muon physical variables obtained by use of PYTHIA event generator. Its parameters were set to those values that allow fast simulation for the antiproton beam with the energy $E_{beam}$=14 GeV, which corresponds to the center of mass frame energy of $p\bar p$ collision $E_{cm}=5.3 GeV$.
The muons produced in the corresponding hard QCD $2\rightarrow 2$ subprocess ${q\bar{q} \to \gamma^{*} \to \mu^{+}\mu^{-}}$ would be called in what follows as the “signal” ones, while those which will appear in event due to meson decays would be called as “decay” muons.
The simulation was done for a case when both initial state radiation (ISR) and final one (FSR) were switched on (i.e. with the following values of PYTHIA parameters MSTP(61)= MSTP(71)=1). The number of generated events was taken to be 100 000.
The distributions of the signal muons energy $E^{\mu^{+/-}}$ values as well as of the modulus of the transverse momentum $PT^{\mu^{+/-}}$ and of the polar angle $\theta^{\mu^{+/-}}$ (measured from the beam direction) versus the number of generated events are shown (from top to bottom, respectively) in Figure 1 of Section 6, which contains all the Figures. The left column in Figure 1 is for $\mu^{-}$ distributions and the right one for $\mu^{+}$). There is no difference seen between $\mu^{+}$ and the analogous e $\mu^{-}$ distributions in a case when the influence of magnetic field is not considered).
One can see from the top raw of Figure 1 that the energy of muons may vary in the interval $ 0 < E^{\mu} < 10GeV$ with a mean value $<E^{\mu}> = 2.6 $ GeV and a peak at a rather small value $E^{\mu}_{peak}$ =0.5 GeV.
The $PT^{\mu}$ spectrum , see middle raw of Figure 1, has an analogous peak $PT^{\mu}_{peak} = 0.5$ GeV/c. After this point both $E^{\mu}$ and $PT^{\mu}$ spectra falls rather steeply but the spectrum of $PT^{\mu}$ is confined in a more narrow interval $0 < PT^{\mu} < 2$ GeV/c .
The spectrum of events number (Nevent) versus the polar angle ${\theta^{\mu}}$ ( see bottom raw) has a peak at ${\theta^{\mu}} =10^{o} $ and the mean value $ <\theta^{\mu}> =27.5^{o}$.
From these plots we see that while the most of signal muons are scattered into the forward direction $\theta^{\mu}<90^{o}$ still there is a small number of them that fly in the backward hemisphere ($\theta^{\mu} > 90^{o} $). We shall discuss this point in more details in what follows.
From the view point of the background (the main source for it are the muons from charged pions and other hadrons decays) estimation it is usefull to have the set of plots analogouse to the previouse one, but done separatley for the signal muons having the largest energy ( “fast” muons) in the muon pair produced in an event , and, correspondently, for those having the smaller energy (“slow” muon). This set of plots for the signal muons are given in Figure 2 of the Appendix, where the left column is for the “slow” muon distributions and the right one is for “fast” muons.
We see from Figure 2 that the energy spectrum (top raw) of the fast signal muons grows steeply from the point $E^{\mu}_{fast}=0.5 GeV $ (practically $99\%$ of fast muons have $E^{\mu}_{fast} > 1 GeV $) to the peak at the point $E^{\mu}_{fast}$ = $2.7 GeV$ and then vanishes at $ E^{\mu}_{fast}$ = $10 GeV $.
In contrast to this picture the spectrum of less energetic signal munons (left raw) has a peak at the point $E^{\mu}_{slow}$=0.5 GeV (where the spectrum of fast muons only starts) and it practicaly vanishes at the point, which corresponds to the mean value of the most energetic signal muons, i.e. to $ < E^{\mu}_{fast} >$ = 3.9 GeV. Thus, one may say that the spectrum of slow muons in a pair is very different from that of the fast ones.
The difference between the $PT^{\mu}$ spectra (see middle raw of Figure 2) of the fast and slow muons seams not to be so large and it results only in about 340 MeV shift to the left of the peak position as well in the corresponding shift of the end point of slow muons spectrum. Both of these spectra demonstrait that the most of slow as well as of fast muons have $PT^{\mu} > 0.2$ GeV.
This similarity of $PT^{\mu}$ spectra of fast and slow muons results (due to a large difference of their energy spectra, i.e. of $P^{\mu}_{z}$ compronent) in a large difference of their polar angle ${\theta^{\mu}}$ distributions (see bottom raw in Figure 2): ${\theta^{\mu}_{slow}}$ spectrum is shifted to the right as comparinring to that one for ${\theta^{\mu}_{fast}}$ and their mean value $ < {\theta^{\mu}_{slow}} > $ = $38.2^{o}$ is a more than two times higher than the analogouse mean value for the fast one: $ < {\theta^{\mu}_{fast}} > = 16.5^{o}$ .
Another and the most important difference that is seen from these angular distributions is that all fast muons fly in forward direction ${\theta^{\mu}_{fast}} < 90^{o}$ and their spectrum practically finishes at ${\theta^{\mu}_{fast}} = 60^{o}$, while about $17\%$ of slow muons have ${\theta^{\mu}_{slow}} > 60^{o}$ and about of $5\%$ of them may scatter into the back hemisphere.
SRUCTURE FUNCTIONS:\
u- AND d- QUARK DISTRIBUTIONS.
==================================
Up to now we disscussed the results obtained without any other kinematical cuts than those implemented onto internal PYTHIA varaibles and needed to run this generator at as low as possible values of beam energy. In our generation we have restricted the value of the invariant mass of $\mu^{+}\mu^{-}$ signal pair, produced in event. Namely we have taken the last parameter $ M_{inv}^{\mu^{+}\mu^{-}}= {\hat{s}_{\;min}}$ to be restricted by the inequality $ M_{inv}^{\mu^{+}\mu^{-}} > 1$ GeV.
The distributions of Bjorken x-variables are shown in Figure 3 for up- and antiup- quarks (top raw) and for down- and antidown- quarks (bottom raw) correspondingly. In what follows we shall refer to these distributions as to the “unbiased” ones as the influence of different cuts onto muons energy $E^{\mu}_{cut}$ as well as the angle cuts, connected with possible different geometry size of muon muon system would be studied latter. As it can be easialy seen from this plots the obtained distributions do not start from the point x=0, what take place for the used parton distribution parametrizations used in PYTHIA. Such a differnce at low values of x appears due to the mention above cut on the value of muon pair invariant mass.
The number of entries at these plots reflects the valence quark flavour structure of the proton (the analoguse distributions and the number of entries for the antiquarks show the absolute similarity of quark and antiquarks distributions).
FAKE MUONS FROM MESON DECAYS.
==============================
All what was said above may be to a good approximation applied also to a case of electron-positron pair production in the final state. The most difference of $e^{+}e^{-}$ case with that one of ${\mu^{+}\mu^{-}}$ pair production appears, as it was allready mentioned in the Introduction, when one turns to the problem of background or fake leptons in the same signal lepton pair production process.
Really, the events with a pair of signal muons should contain also some hadrons in the final state. The pions, produced directly or in the decays cascades of other hadrons, may decay in the detector volume and thus serve as a source of background muons that may fake the signal ones produced in the annihilation subprocess.
Figure 4 includes in the left column (as in the Figures 1 and 2) the distributions (from top to bottom) of the number of events versus the energy $E_{PI}$= $E_{\pi}$, the transverse momentum $ PT_{PI}$ = ${PT_{\pi}}$ and versus the polar angle $TETA_{PI}$= $\theta_{\pi}$ of produced pions.
The right hand column contains only one plot with the distribution of the total number NPI of charged ${\pi}$-mesons in the signal events. One may see that there is a big number of events ( about$30\%$ ) which do not contain charged pions at all in the final state (they mostly have nucleon pairs in the final state).
This result is very important. Despite the fact that PYTHIA provides a good (at least one of the bests if not the best one) but still a model approximation to the real hadronization effects, it indicates (in the absence of a complete physical and theoretical understanding of parton to hadrons fragmentation processes) that it may happen so that about of one third of events with signal muon pairs may appear practically without additional fake ( or internal background ) decay muons content. That means that muon channel may be quite competitive to the $e^{+}e^{-}$ one. In what follows we shall add more arguments in a favor of this suggestion.
From the same right hand plot we also see that about of $25\%$ of events have only one charged pion in the final state (appearing mostly in ${\Delta}$ resonance decays), and a bit less than $25\%$ of events have two charged pions. It demonstrates also that about of $5\%$ of events have 3 charged pions and there is only about of $1.5\%$ of events with 4 final state charged pions.
Figure 5 includes in the left column, respectively from top to bottom, the energy $E^{\mu}_{dec}$, $PT^{\mu}_{dec}$ and polar angle $\theta^{\mu}_{dec}$ distributions for muons appearing from the discussed above charged ${\pi}$ -meson decays. These distributions follow the analogue spectra of parent pions, presented in Figure 4.
By comparing these plots with those from signal muon pairs shown in Figure 1, one may conclude that the energy mean value of the signal muons $< E^{mu} > $ =2.6 GeV does corresponds to the point in the energy distribution of fake decay muons where the contribution of the decay muons is already low. The same may be said about the PT- distribution of decay muons. The mean value of the PT- distribution of signal muons $PT^{mu}_{signal} = 0.7$ GeV (see Figure 1 ) does corresponds to the point where the spectrum of $PT^{mu}_{decay}$ practically vanishes.
The E- and PT- spectra of decay muons are a bit softer than those of “slow” signal ones. They are shifted by 300 MeV (i.e. by $30\%)$ to the left (see their mean values) as comparing with those of signal slow muons. Analogously the fake decay muons polar angle spectrum is shifted also to the left by $30\%$, i.e. by $14^{o}$. Still, due to a big similarity of decay and “slow”signal muons the former may appear as a serious background.
Therefore a reasonable cut upon the muons energy $E^{mu}$ (as well as on the $PT^{mu}$) may lead to an essential reduction of the decay mesons contribution and allow to keep the main part of signal events. Thus a cut $PT^{mu}> 0.2$ GeV may allow to get rid of a half of decay mesons at the coast of lost of $8\%$ of signal events . The analogous strict cut $PT^{mu} < 0.4$GeV leaves more than $75\%$ of signal muons and less than $15\%$ of decay mesons .
Another way to discriminate the signal “slow” muons from the decay ones is to use the information about the position of the muon production vertex.
The right hand side raw in Figure 5 contains (from top to bottom) the x-, y- and z-(i.e. along the beam direction) components of muon production vertex position, as given by PYTHIA simulation, i.e. in a space free of experimental setup around the interaction point. The distances at these distribution plots are given in millimeters. Thus one may see that the tails of the z-distribution may expand up to 100 meters, while those along x- and y- axis reach 40 meters distance. From this right hand plots it is clearly seen that in a case of mentioned above $66\%$ of signal events that include the decay muons (i.e. addition to the signal muon pair, produced in annihilation subprocess) the size of the detector (i.e., the “decay volume”) may strongly reduce the number of charged ${\pi}$-mesons which potentially may produce muons in the decays because the most of parent pions would interact with the detector components. Therefore one may expect that in the real experimental conditions the situation with the contribution of the additional decay muons may become much easy. The detailed GEANT simulation, which is in our nearest plans, should allow to get more definite predictions about the decay mesons contribution to the background.
Conclusion.
===========
The energy, transverse momenta and polar angle (with respect to the beam axis) distributions of muons that may be produced in pairs in $ p\bar{p} \rightarrow l\bar{l} + X $ process , i.e. the distributions of signal muons, are presented for a case of the proton target rest frame. These distributions were obtained by use of PYTHIA generator which parameters were adjusted to perform effective generation of physical events in a case of $\bar{p}$ beam energy equal to 14 GeV.
It is shown that those of muons that are the most energetic in a moun pair, i.e. fast muons, predominantly fly in forward direction (with the mean value $ < {\theta^{\mu}_{fast}} > = 16.5^{o}$) , while those which are less energetic in a pair , i.e. slow muons, fly at larger angles (their mean value is $ < {\theta^{\mu}_{slow}} > $ = $38.2^{o}$). It is found that some of slow signal muons may fly in backward hemisphere also. Thus, a good angle coverage by muon system would be very usefull.
The analogous distributions for muons that appear from hadrons decays (mostly from pions), i.e. for decay muons, have shown that decay muons may fake the signal slow muons in the same events and thus, in principle, may be considered to appear as a serious bakground. Nevertheless, the distributions of the number of generated events versus the number of charged pions in lepton pair production event has shown that, fortunatly, more than $30\%$ of signal events do not have charged pions in the final state. On a top of this it seams to be clear from the obtained distribution of the space position of fake decay muon production vertex that a large amount of these muons may be potentialy rejected with the account of the real detector decay volume. We plan as our next step to perform such type of complete GEANT detector simulation with the analysed here events, generated by use of PYHTIA.
The auhtors are very gratefull to G.D.Alexeev for suggestion of this topic for study, the interest to this work and multiple stimulating discussions of the questions concerned.
[50]{}
V.A. Matveev, R.M. Muradian, A.N. Tavkhelidze, JINR P2-4543, JINR, Dubna, 1969; SLAC-TRANS-0098, JINR R2-4543, Jun 1969; 27p.
S.D. Drell, T.M. Yan, SLAC-PUB-0755, Jun 1970, 12p.; Phys.Rev.Lett. [**25**]{}(1970)316-320, 1970.
CERN UA1 Collaboration, C. Albajar et al., Phys. Lett., [ **B209**]{} (1988) 397; FNAL E772 Collaboration, P.L. Gaughey at al. Phys. Rev. [ **D50**]{} (1994) 3038;
T.Sjostrand, Computer Phys. Commun.[**39**]{} (1986) 347, T.Sjostrand and M.Bengtsson, Computer Phys. Commun.[**43**]{} (1987 )367.
A.D. Martin [*et al.*]{}, Eur. Phys. J. [**C4**]{} (1998) 463.
R. Brun and F. Rademakers, ROOT - An Object Oriented Data Analysis Framework, Proceedings AIHENP’96 Workshop, Lausanne, Sep. 1996, Nucl. Inst. & Meth. in Phys. Res. [ **A389**]{} (1997) 81. See also http://root.cern.ch/.
Figures.
=========
-5mm ![ Signal muons energy $E^{\mu^{+/-}}$ (top raw), the modulus of the transverse momentum $PT^{\mu^{+/-}}$ (middle raw) and the polar angle $\theta^{\mu^{+/-}}$ (bottom raw) distributions. Left column is for $\mu^{-}$ and right one for $\mu^{+}$.[]{data-label="fig:1"}](fig1.eps "fig:"){width="18cm" height="20cm"}
-5mm ![ Signal muons distributions for muons with the largest energy (“fast” muons) in the muon pair $E^{\mu}_{fast}$ (left column) and the smaller energy (“slow” muons) $E^{\mu}_{slow}$ (right column). Top raw includes their energies, in middle raw are PT and in bottom raw are $\theta^{\mu}$.[]{data-label="fig:2"}](fig2.eps "fig:"){width="18cm" height="20cm"}
-5mm ![ Top raw shows the distributions of valence up -quarks and anti-quarks, while bottom raw includes the analogous distributions of down-quarks and anti-quarks.[]{data-label="fig:3"}](fig3.eps "fig:"){width="18cm" height="20cm"}
-5mm ![ The left column includes the distributions (from top to bottom, respectively) of number of events versus the energy $E_{PI}$=$E_{\pi}$, the transverse momentum $PT_{PI}$= $PT_{\pi}$ and versus the polar angle $TETA_{PI}$= ${\theta_{\pi}}$ of produced pions. The right hand plot shows the distribution of the total number (NPI) of charged ${\pi}$-mesons in the signal events.[]{data-label="fig:4"}](fig4.eps "fig:"){width="18cm" height="20cm"}
-5mm ![ The left column includes the distributions (from top to bottom, respectively) of number of events versus the energy $E_{PI}$=$E_{\pi}$, the transverse momentum $PT_{PI}$= $PT_{\pi}$ and versus the polar angle $TETA_{PI}$= ${\theta_{\pi}}$ of produced pions. The right column (from top to bottom) shows the distributions of x-, y- and z- components of decay muon vertex position.[]{data-label="fig:5"}](fig5.eps "fig:"){width="18cm" height="20cm"}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Distant starlight passing through the Earth’s atmosphere is refracted by an angle of just over one degree near the surface. This focuses light onto a focal line starting at an inner (and chromatic) boundary out to infinity, offering an opportunity for pronounced lensing. It is shown here that the focal line commences at ${\sim}85$% of the Earth-Moon separation, and thus placing an orbiting detector between here and one Hill radius could exploit this refractive lens. Analytic estimates are derived for a source directly behind the Earth (i.e. on-axis) showing that starlight is lensed into a thin circular ring of thickness $W H_{\Delta}/R$, yielding an amplification of $8 H_{\Delta}/W$, where $H_{\Delta}$ is the Earth’s refractive scale height, $R$ is its geopotential radius, and $W$ is the detector diameter. These estimates are verified through numerical ray-tracing experiments from optical to light with standard atmospheric models. The numerical experiments are extended to include extinction from both a clear atmosphere and one with clouds. It is found that a detector at one Hill radius is least affected by extinction since lensed rays travel no deeper than 13.7km, within the statosphere and above most clouds. Including extinction, a 1metre Hill radius “terrascope” is calculated to produce an amplification of ${\sim} 45,000$ for a lensing timescale of ${\sim} 20$hours. In practice, the amplification is likely halved in order to avoid daylight scattering i.e. $22,500$ ($\Delta$mag=10.9) for $W=$, or equivalent to a optical/infrared telescope.'
author:
- David Kipping
title: 'THE “TERRASCOPE”: ON THE POSSIBILITY OF USING THE EARTH AS AN ATMOSPHERIC LENS'
---
Introduction {#sec:intro}
============
Modeling Atmospheric Refraction {#sec:refraction}
===============================
Ray Tracing Simulations {#sec:raytracing}
=======================
Calculation Results {#sec:results}
===================
Discussion {#sec:discussion}
==========
DMK is supported by the Alfred P. Sloan Foundation. Thanks to members of the Cool Worlds Lab and the NASA Goddard Institute for Space Science group for useful discussions in preparing this manuscript. Thanks to Jules Halpern, Duncan Forgan, Caleb Scharf and Claudio Maccone for reviewing early drafts and discussions of this work. Special thanks to Tiffany Jansen for her assistance with coding questions. Finally, thank-you to the anonymous reviewer for their constructive feedback.
[99]{} Birch, K. P. & Downs, M. J., 1994, Metrologia, 31, 315. Cassini, J. D., 1740, “Tables astronomiques”, p. 34 Conrad, C., Gordon, R. F., Bean. A. L., 1969, “Earth Eclipses the Sun - Apollo 12”, NASA JPL https://moon.nasa.gov/resources/199/earth-eclipses-the-sun-apollo-12/ Eddington A. S., 1919, Obs, 42, 119. Edlén, B., 1994, Metrologia, 2, 71. Einstein, A., 1916, Annalen der Physik, 49, 769. Elvis, M., 2016, “The Crisis in Space Astrophysics and Planetary Science: How Commercial Space and Program Design Principles will let us Escape”, Frontier Research in Astrophysics II (arXix e-prints:1609.09428). Heidmann, J. & Maccone, C., 1994, Acta Astron., 32, 409. Hubbard, W. B., Nicholson, P. D., Lellough, E., et al., 1978, Icarus, 72, 635. Kneizys, F. X., Shettle, E. P., Abreu, L. W., Chetwynd, J. H., Anderson, G. P., Gallery, W. O., Selby, J. E. A., Clough, S. A., 1988, “User’s guide to LOWTRAN7”, Air Force Geophysics Lab, Tech. Rep. Kraus, J. D., 1986, Radio Astronomy, Cygnus-Quasar Books, Powell, Ohio, p. 6. Monnier, J. D., 2003, Reports on Progress in Physics, 66, 789. National Geophysical Data Center: U.S. standard atmosphere (1976), 1992, Planet. Space Sci., 40, 553. Rasmussen, C. E. & Williams, C., 2006, “Gaussian Processes for Machine Learning”, MIT Press, Cambridge Michael, S. L., 1999, “Statistical Interpolation of Spatial Data: Some Theory for Kriging”, Springer, New York Turyshev, S. G. & Andersson, B.-G., 2003, MNRAS, 341, 577. van Belle, G. T., Meinel, A. B. & Meinel, M. P., 2004, in SPIE Conf. Ser. 5489, ed. J. M. Oschmann, Jr., 563. von Eshleman R., 1979, Sci, 205, 1133. Wang, Y., 1998, Proc. SPIE, 3356, 665 Wylie, D. P., Menzel, P. W., Woolf, H., M. & Strabala, K. I., 1994, Journal of Climate, 31, 1972. Wylie, D. P. & Menzel, P. W., K. I., 1994, Journal of Climate, 12, 170.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
and Z. Paragi\
Joint Institute for VLBI ERIC, Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands\
E-mail:
bibliography:
- '/Users/hawky/Documents/Reference/bibliography.bib'
title: Localizations of Fast Radio Bursts on milliarcsecond scales
---
Introduction
============
Fast Radio Bursts (FRBs) are extragalactic transient sources of unknown physical origin [@lorimer2007; @thornton2013; @petroff2016]. These events exhibit millisecond-duration radio flashes that are highly dispersed, with dispersion measures (DMs) consistent with extragalactic origins. The first FRB was discovered in 2007 after a re-analysis of data from the Parkes Radio Telescope Galactic pulsar survey. A single signal with a peak brightness of $\sim30$ Jy lasting $\sim 5$ ms was detected, and the inferred DM suggested a distance to the source of the order of 1 Gpc [@lorimer2007]. This value for the distance would imply an extremely luminous event that could not be understood by any known system.
The field did not boost until more of such events were clearly found [@thornton2013]. Nowadays, tens of FRBs have been reported from several single-dish telescopes [@petroff2016; @keane2018]. However, the origin of FRBs remains unknown, although a large number of possible scenarios has been proposed so far (see e.g. [@katz2016; @katz2018; @pen2018]).
Although the estimated burst energies are comparable to the spin-down energy of known pulsars such as the Crab, the observed burst luminosities are $\sim 10^{10\text{--}15}$ times larger than the typical ones of individual pulses from pulsars. Candidate classes of objects that liberate this much (and even much higher) total energies are gamma-ray bursts (GRBs) and supernova (SN) explosions. However, in these cases the derived rate of events ($10^{3\text{--}4}\ \mathrm{sky^{-1}\ d^{-1}}$) [@keane2015; @law2015] does not match the ones in GRBs or SNe. In addition, multiwavelength searches have not revealed robust afterglow detections to date (although candidate events have been reported [@keane2016; @mahony2018], these were not confirmed [@giroletti2016]).
The interest on FRBs not only resides on their intriguing nature, but on their direct and indirect implications on several astrophysical fields. If FRBs are detected in great numbers at cosmological distances, they could shed light on the distribution of baryonic matter in the Universe, and the nature of dark energy [@macquart2015].
Fast transients with the European VLBI Network (EVN)
----------------------------------------------------
Very long baseline interferometry (VLBI) observations are ideal to precisely localize transient sources on milliarcsecond (mas) scales and to determine their morphological and temporal evolution. The accurate astrometry obtained from these observations is critical for extragalactic sources, where associations require the identification (and distinction) of local sources within the host galaxies. Simultaneously, the reached (mas) resolution allows the study of morphological changes in the transient events (e.g. the evolution of afterglows) on timescales as short as weeks or years.
The European VLBI Network (EVN)[^1] is a network of antennas spread mainly within Europe, Asia, and Africa. The available baselines (up to 10000 km) and the observing frequencies (primarily between 1.4 and 22 GHz) imply a resolution of the order of mas. The data, recorded locally at each station, are sent to The Joint Institute for VLBI ERIC (JIVE) in The Netherlands for correlation with the EVN Software Correlator (SFXC) [@keimpema2015].
One of the operating modes of the EVN is the real-time correlation (e-EVN), where data are directly sent in real-time and electronically to JIVE. Runs of 24 h are conducted on average once per month. This sampling makes the e-EVN an ideal tool for monitoring programs and transient studies. Highlights of synchrotron transients science, and a brief description of the years of work put into detecting and localizing fast transients with the e-EVN are described by [@paragi2016].
The only known repeating FRB
============================
The Arecibo Telescope discovered its first FRB in 2012, FRB 121102 [@spitler2014]. Multiple bursts were soon discovered, making this source the first and only repeating FRB known so far[^2] [@spitler2016; @scholz2016]. The measured DM of $\sim 560\ \mathrm{pc\ cm^{-3}}$ implied a column density around a factor of three larger than the expected one from the contribution of our Galaxy in that direction (which is close to the Galactic anticenter). This made the source of a likely extragalactic origin.
Despite multiple bursts have been detected, no periodicities have been found so far. However, there are active periods when a higher number of bursts is observed. Most of the bursts have been seen at $\sim 1.4$ GHz, but there have been detections at 4–8 GHz as well [@gajjar2018; @spitler2018]. The bursts are typically narrow-band, with characteristic widths of $\sim 500\ \mathrm{MHz}$ [@law2017]. Complex time/frequency structures and the existence of sub-bursts as narrow as $\sim 30\ \mathrm{\upmu s}$ have also been revealed [@hessels2018]. It remains unclear if these structures are intrinsic [@hessels2018], or they appear due to propagation effects in the local medium [@cordes2017].
Although similar structures have been reported in other FRBs, no repetitions have been observed in any other FRB to date (2018), even after exhaustive searches [@shannon2018]. This fact has raised questions about FRB 121102 being representative or not of the whole population of FRBs [@palaniswamy2018].
The precise localization of FRB 121102
======================================
The existence of multiple bursts in FRB 121102 allowed the scheduling of interferometric observations on a FRB for the first time. The Arecibo detections provided an a-priori localization with an uncertainty of a few arcminutes, enough to fit within the primary beam of typical dishes.
Two teams initiated interferometric observations of FRB 121102 independently: an US-based team with the Karl G. Jansky Very Large Array (VLA), starting in November 2015, and a European collaboration using the EVN (including the Arecibo Telescope), in February 2016. Five epochs of e-EVN observations during the spring showed no bursts. In the summer of that year, high activity was reported from Arecibo single-dish observations. The subsequent VLA (and later EVN) follow-up observations resulted in the detection of several bursts, and their association at the sub-arcsecond accuracy level with a faint (180-$\mathrm{\upmu Jy}$ at 3 GHz) persistent radio source that was compact even on EVN scales [@chatterjee2017] (see Figure \[fig:vla\]). These bursts appeared $\sim 10^{4\text{--}5}$ times brighter than the persistent emission. There was only low-level ($\sim 10\%$) variability observed in the persistent radio source and it did not correlate with the bursts, therefore there was no sign of relativistic outflows. The spectrum of this source also remains stable along the time, with a relatively flat emission between 1 and 10 GHz and a cutoff of unclear origin at higher frequencies [@chatterjee2017].
![Radio and optical images of the field around FRB 121102. The main figure shows the field as seen by the VLA at 3 GHz, with a resolution of 2 arcsec and a rms noise level of $2\ \mathrm{\upmu Jy\ beam^{-1}}$. The white square is centered at the position where the bursts are located, and the white circles represent previous Arecibo burst detections and their uncertainty regions. The secondary figure represents a zoom on the white square showing the optical image as seen by Gemini at [*r*]{} band. The two lines point to the position of FRB 121102. Persistent radio and optical counterparts are detected coincident with such position. See [@chatterjee2017].[]{data-label="fig:vla"}](images/nature-continuum-image.pdf){width="\textwidth"}
We detected bursts from FRB 121102 in one observation with the EVN in September 2016. Four bursts were detected, with one of them around ten times brighter than the other ones ($\sim 4\ \mathrm{Jy}$ to be compared to $\sim 0.2$–$0.5\ \mathrm{Jy}$). These data allowed us to study both the persistent source and the burst location on milliarcsecond scales [@marcote2017].
Whereas the bursts were detected in a 1.7-GHz observation, the persistent source was imaged in both 1.7 and 5.0-GHz observations. The data at the highest frequency allowed us to put strong constraints on the source size ($< 0.7\ \mathrm{pc}$). Simultaneously, we provided accurate localizations for the four detected bursts. Whereas the strongest burst produced the most accurate measurement, we determined a weighted average position from all of them. This average position is coincident with the persistent source within a projected separation of $< 40\ \mathrm{pc}$ at 95% confidence level [@marcote2017] (see Figure \[fig:evn\]). The obtained separation constraint suggests a direct physical connection between the bursts and the persistent source. The robustness of the burst astrometry was tested by a detailed analysis of single-pulse imaging from data of the known pulsar B0525+21, which was observed within these EVN observations [@marcote2017].
![Radio images of the field around FRB 121102 as seen by the EVN at 1.7 GHz with a resolution of $21\times 2\ \mathrm{mas^2}$ (represented by the white contours, that start at the $2$-$\sigma$ noise level of $10\ \mathrm{\upmu Jy\ beam^{-1}}$ and increase by factors of $2^{1/2}$), and at 5.0 GHz with a resolution of $4 \times 1\ \mathrm{mas^2}$ and a rms of $14\ \mathrm{\upmu Jy\ beam^{-1}}$ (represented by the colorscale). The synthesized beams at each frequency are represented by the gray ellipses at the right and left bottom corners, respectively. The crosses represent the positions of the single bursts and their lengths represent the 1-$\sigma$ statistical uncertainty in each direction. The red crosses represent the strongest burst detected by the EVN, which is expected to exhibit the most accurate astrometry, and the gray ones represent the other three bursts detected during the same observation. The black cross represents the weighted average position of the bursts, used for further discussions. See [@marcote2017].[]{data-label="fig:evn"}](images/evn-loc.pdf){width="\textwidth"}
The data recorded by Arecibo during the EVN observations also allowed us to study the narrow-band frequency structure of the bursts. A fine-scale frequency structure of $\sim 58\ \mathrm{kHz}$ was measured, with is consistent with Galactic scintillation when the light enters the Galaxy [@hessels2018].
![Optical image of the field around FRB 121102 as seen by the [*HST*]{}/WFC3 at [*J*]{} band. FRB 121102 (represented by a yellow cross) is coincident with a star-forming region (highlighted by the red circle) that dominates the optical emission of the host, dwarf, galaxy. The purple ellipse highlights the mean extension of the galaxy. The strong object at the right bottom is a field star. Adapted from [@bassa2017].[]{data-label="fig:hst"}](images/hst.pdf){width="\textwidth"}
Optical observations with the Gemini [@tendulkar2017] and [*HST*]{} [@bassa2017] were conducted to unveil the host of FRB 121102. These data unveiled a low-metallicity dwarf ($< 10\ \mathrm{kpc}$ in diameter) galaxy located at a redshift of $\sim 0.193$ (luminosity distance of 972 Mpc). The bursts, and the associated radio persistent source, were located inside a star-forming region with a radius of 1.2 kpc that dominates the optical emission of the whole galaxy (see Figure \[fig:hst\]). The properties of this galaxy resemble the ones where hydrogen-poor superluminous supernovae (SLSN) and long-duration gamma-ray bursts (LGRBs) are typically located. This suggests a possible connection between FRBs and these sources.
At other wavelengths, deep optical [@hardy2017; @magic2018], X-ray and GeV gamma-ray [@scholz2017], and TeV gamma-ray [@magic2018] burst searches have been conducted, with null results so far. It thus remains unclear if FRBs exhibit pulsed emission outside the radio wavelengths.
Detailed observations of the bursts from FRB 121102 at 5 GHz revealed the presence of an unexpectedly high Faraday rotation measure (RM) of $\sim 1.4 \times 10^5\ \mathrm{rad\ m^{-2}}$, and a 100% linear polarization in the bursts [@michilli2018]. Such large RM values have only been reported so far in the vicinity of Sagittarius A$^{\ast}$, i.e. in the surroundings of massive black holes.
Scenarios explaining FRB 121102
===============================
Any model expected to explain FRB 121102 needs to account for the existence of bursts and the presence of the compact and persistent radio source. Most probable scenarios assume a young (few decades old) and energetic magnetar (or highly magnetized neutron star) origin [@connor2016; @cordes2016]. These scenarios would naturally explain the observed bursts, which could likely be originated by coherent curvature radiation [@ghisellini2017].
The persistent source would then be explained by the presence of a superluminous supernova (SLSN) [@murase2016; @piro2016; @kashiyama2017; @margalit2018]. Although this would explain the co-localization of the bursts within the persistent source, and matches the environment where FRB 121102 is located (low-metallicity star-forming dwarf galaxy), the properties of the persistent source still challenge this explanation (such as its luminosity or lack of long-term variability).
Other scenarios consider the presence of a young magnetar which is interacting with the jet or surroundings of a massive black hole [@pen2015; @cordes2016; @zhang2017; @zhang2018]. These scenarios would naturally explain the existence of bursts (from the magnetar), the properties of the persistent source, and the high RM. Some theoretical models considering that bursts are originated due to strong plasma turbulence within the jet of a massive black hole [@vieyro2017] or as synchrotron masers [@ghisellini2017] have also been proposed, but these ones present a reduced observational support. In addition, there are a large number of speculative scenarios [@pen2018].
Conclusions and future prospects
================================
The FRB field has progressed a lot in the past few years. Several tens of these objects have been found, but the existence of only one repeater[^3] and only one precise localization hampers our understanding of these sources. It remains unclear if repeating and non-repeating FRBs belong to two different types of systems, or if the repetition of bursts is only subject to particular conditions in similar systems.
The available data do not support the presence of afterglows after the creation of bursts, having strong implications on future FRB searches. New FRB localizations, which are fundamental to unveil the nature of these objects, would require interferometric observations at the arrival burst times. Only detections of bursts during high resolution (interferometric) observations would allow us to associate the bursts to persistent sources, as opposed to other fields such as gravitational wave events (where the presence of afterglows allows us to locate the event from later observations).
The precise localization of FRB 121102 has led to the first detailed studies of a FRB. We have unveiled the environment of this source to exhibit extreme conditions and to be associated with a persistent and compact radio source embedded in a low-metallicity star-forming region of a dwarf galaxy. Genuine localizations of more FRBs are mandatory to unveil their hosts and trace their properties that are common to the whole population.
Upcoming facilities such as WSRT/Apertif, ASKAP, CHIME, or UTMOST have dedicated searches for new FRBs. These searches will lead to new associations that will allow us to understand the uniqueness of FRB 121102 among the whole population. However, we note that the resolution reached by VLBI observations would be in any case mandatory to study in detail the local environment to the FRBs in their host galaxies. The same methodology we have developed for the detection and localization of fast transients in general [@paragi2016], and the repeater FRB 121102 in particular [@marcote2017], could be used for future localizations of repeating FRBs.
Direct VLBI imaging of single pulses is also a (remote) possibility. However, blind commensal searches for FRBs in single dish data obtained during Very Long Baseline Array (VLBA) experiments have not been successful to date yet [@burkespolaor2016]. While the EVN has much more sensitive individual dishes to look for single pulses, these have smaller field of views as well. Therefore it is not clear how productive single-pulse FRB searches could be with the EVN. From the current FRB rates we estimate that if the EVN Archive contained full field of view raw data for all experiments since its existence ($\sim 1999$–$2000$), we could have identified and localized a few more FRBs at least from archival EVN observations.
[^1]: See the EVN website: <http://www.evlbi.org>.
[^2]: A second repeater has just been discovered by CHIME [@chime2019].
[^3]: A second repeater has just been discovered by CHIME, with no clear counterparts found yet [@chime2019].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The paper is devoted to quantization of extensive games with the use of both the Marinatto-Weber and the Eisert-Wilkens-Lewenstein concept of quantum game. We revise the current conception of quantum ultimatum game and we show why the proposal is unacceptable. To support our comment, we present the new idea of the quantum ultimatum game. Our scheme also makes a point of departure for a protocol to quantize extensive games.'
author:
- Piotr Frackiewicz
- |
<span style="font-variant:small-caps;">Piotr Frackiewicz</span>\
Institute of Mathematics, Polish Academy of Sciences\
00-956 Warsaw, Poland
title: |
Quantum information approach\
to the ultimatum game
---
\[section\] \[section\] \[proposition\][Example]{}
Introduction
============
During the last twelve years of research into quantum games the theory has been already extended beyond $2\times2$ games. Since majority of noncooperative conflict problems are described by games in extensive form, it is interesting to place extensive games in the quantum domain. Although there is still no commonly accepted idea of how to play quantum extensive games, we have proved in [@fracor3] that it is possible to use the framework [@eisert2] of strategic quantum game to get some insight into quantum extensive games. Namely, we have shown that a Hilbert space $\mathscr{H} = \mathds{C}^2 \otimes \mathds{C}^2 \otimes
\mathds{C}^2$, a unit vector $|\psi_{\mathrm{in}}\rangle \in
\mathscr{H}$, the collection of subsets $\{\mathcal{U}_{j}\}_{j=1,2,3}$ of $\mathsf{SU}(2)$, and appropriately defined functionals $E_{1}$ and $E_{2}$ express the normal representation of a two stage sequential game. Moreover, it allows to get a result inaccessible in the game played classically. In this paper, the above-mentioned quantum computing description will be used to the two proposed variants of the ultimatum game [@guth]. It is a game in which two players take part. The first player proposes one of two proposals how to divide a fixed amount of good. Then the second player either accepts or rejects the proposal. In the first case, each player receives the part of goods according to player 1’s proposal. In the second case, the players receive nothing. A game-theoretic analysis shows that player 1 is in a better position. Since player 2’s rational move is to accept each proposal, player 1’s rational move is to make the best proposal for her. As we will show in this article, the Eisert-Wilkens-Lewenstein (EWL) approach [@eisert] as well as Marinatto-Weber (MW) approach [@marinatto] can change the scenario of the ultimatum game significantly improving the strategic position of player 2. Our paper also provides an argument indicating that the previous idea [@mendes] of quantum ultimatum game is not sufficient to describe the game in the quantum domain. We will explain that, in fact, the formerly proposed protocol does not quantize the ultimatum game but another $2\times2$ game. The last part of the paper is devoted to a form of a game tree where we provide the procedure how to determine the game tree when the game is played according to the MW approach.
Preliminaries to game theory
============================
Definitions in the preliminaries are based on [@osborne]. This section starts with a definition of a finite extensive game.
Let the following components be given.
- A finite set $N = \{1,2,\dots,n\}$ of players.
- A set $H$ of finite sequences that satisfies the following two properties:
1. the empty sequence $\emptyset$ is a member of $H$;
2. if $(a_k)_{k = 1,2,\dots, K} \in H$ and $K>1$ then $(a_k)_{k =
1,2,\dots, K-1} \in H$.
Each member of $H$ is a history and each component of a history is an action taken by a player. A history $(a_{1}, a_{2},\dots,
a_{K}) \in H$ is terminal if there is no $a_{K+1}$ such that $(a_{1}, a_{2},\dots, a_{K}, a_{K+1}) \in H$. The set of actions available after the nonterminal history $h$ is denoted $A(h) = \{a
\colon (h,a) \in H\}$ and the set of terminal histories is denoted $Z$.
- The player function $P \colon H \setminus Z
\rightarrow N \cup \{c\}$ that points to a player who takes an action after the history $h$. If $P(h) = c$ then chance (the chance-mover) determines the action taken after the history $h$.
- A function $f$ that associates with each history $h$ for which $P(h) = c$ an independent probability distribution $f(\cdot|h)$ on $A(h)$.
- For each player $i\in N$ a partition $\mathcal{I}_{i}$ of $\{h \in H \setminus Z: P(h) = i\}$ with the property that for each $I_{i} \in \mathcal{I}_{i}$ and for each $h$, $h'$ $\in I_{i}$ an equality $A(h) = A(h')$ is fulfilled. Every information set $I_{i}$ of the partition corresponds to the state of player’s knowledge. When the player makes move after certain history $h$ belonging to $I_{i}$, she knows that the course of events of the game takes the form of one of histories being part of this information set. She does not know, however, if it is the history $h$ or the other history from $I_{i}$.
- For each player $i \in N$ a utility function $u_{i}\colon Z \to
\mathds{R}$ which assigns a number (payoff) to each of the terminal histories.
A six-tuple $\left(N, H, P, f, \{\mathcal{I}_{i}\},
\{u_{i}\}\right)$ is called a finite extensive game. \[edefinition\]
Our deliberations focus on games with perfect recall (although Def. \[edefinition\] defines extensive games with imperfect recall as well) - this means games in which at each stage every player remembers all the information about a course of the game that she knew earlier (see [@myerson] and [@osborne] to learn about formal description of this feature).
The notions: action and strategy mean the same in static games, because the players choose their actions once and simultaneously. In the majority of extensive games a player can make her decision about an action depending on all the actions taken previously by herself and also by all the other players. In other words, players can make some plans of actions at their disposal such that these plans point out to a specific action depending on the course of a game. Such a plan is defined as a strategy in an extensive game.
A pure strategy $s_{i}$ of a player $i$ in a game $(N, H, P,
f_{c}, \{\mathcal{I}_{i}\}, \{u_{i}\})$ is a function that assigns an action in $A(I_{i})$ to each information set $I_{i} \in
\mathcal{I}$. \[strategy\]
Like in the theory of strategic games, [*a mixed strategy*]{} $t_{i}$ of a player $i$ in an extensive game is a probability distribution over the set of player $i$’s pure strategies. Therefore, pure strategies are of course special cases of mixed strategies and from this place whenever we shall write [*strategy*]{} without specifying that it is either pure or mixed, this term will cover both cases. Let us define an [*outcome $O(s)$*]{} of a strategy profile $s = (s_{1}, s_{2},\dots, s_{n})$ in an extensive game without chance moves to be a terminal history that results when each player $i \in N$ follows the plan of $s_{i}$. More formally, $O(s)$ is the history $(a_{1}, a_{2},\dots, a_{K})
\in Z$ such that for $0 \leq k < K$ we have $s_{P(a_{1},
a_{2},\dots, a_{k})}(a_{1}, a_{2},\dots, a_{k}) = a_{k+1}$. If $s$ implies a history that contains chance moves, the outcome $O(s)$ is an appropriate probability distribution over histories generated by $s$.
Let an extensive game $\mathrm{\Gamma} = \left(N, H, P,
\{\mathcal{I}_{i}\}, \{u_{i}\}\right)$ be given. The normal representation of $\mathrm{\Gamma}$ is a strategic game $\left( N,
\{S_{i}\}, \{u_{i}'\} \right)$ in which for each player $i \in N$:
- $S_{i}$ is the set of pure strategies of a player $i$ in $\mathrm{\Gamma}$;
- $u_{i}'\colon \prod_{i \in N}S_{i} \to
\mathds{R}$ defined as $u_{i}'(s)\mathrel{\mathop:}=u_{i}(O(s))$ for every $s \in \prod_{i \in N}S_{i}$ and $i \in N$.
One of the most important notions in game theory is a notion of an equilibrium introduced by John Nash in [@nash]. A Nash equilibrium is a profile of strategies where the strategy of each player is optimal if the choice of its opponents is fixed. In other words, in the equilibrium none of the players has any reason to unilaterally deviate from an equilibrium strategy. A precise formulation is as follows:
\[nashequilibrium\] Let $(N, S_{i}, \{u_{i}\}_{i \in N})$ be a strategic game. A strategy profile $(t^*_{1}, t^*_{2},\dots,t^*_{n})$ is a Nash equilibrium (NE) if for each player $i \in N$ and for all $s_{i}
\in S_{i}$: $$\begin{aligned}
u_{i}(t^*_{i}, t^*_{-i}) \geq u_{i}(s_{i}, t^*_{-i}) ~~
\mbox{where} ~~ t^*_{-i} =
(t^*_{1},\dots,t^*_{i-1},t^*_{i+1},\dots,t^*_{n}).\label{nashequation}\end{aligned}$$
A Nash equilibrium in an extensive game with perfect recall is a Nash equilibrium of its normal representation, hence Def. \[nashequilibrium\] applies to strategic games as well as extensive ones.
The ultimatum game {#section:ultimatum}
==================
The ultimatum game is a problem in which two players face a division of some amount $\geneuro$ of money. The first player makes the second one a proposal of how to divide $\geneuro$ between them. Then the second player has to decide either accept or reject that proposal. The acceptance means each player receives a part of $\geneuro$ according to the first player’s proposal. If the second player rejects, each player receives nothing. Let us consider the a variant of the ultimatum game in which player 1 has two proposals to share $\geneuro$: a fair division $u_{\mathrm{f}}=(\geneuro/2, \geneuro/2)$ and unfair one $u_{\mathrm{u}} = (\delta\geneuro, (1-\delta)\geneuro)$, where the $\delta$ is a fixed factor such that $1/2 < \delta < 1$. This problem is an extensive game with perfect information that takes the form: $$\begin{aligned}
\label{Gamma1ultimatum}
\Gamma_{1} = \left(\{1,2\}, H, P, \{\mathcal{I}_{i}\}, u\right)\end{aligned}$$ with components defined as follows:
- $H = \{\emptyset, c_{0}, c_{1}, (c_{0}, d_{0}), (c_{0},
d_{1}), (c_{1}, e_{0}), (c_{1}, e_{1})\}$;
- $P(\emptyset) =
1$, $P(c_{0}) = P(c_{1}) = 2$;
- $\mathcal{I}_{1} =
\{\emptyset\}$, $\mathcal{I}_{2} = \{\{(c_{0})\},\{(c_{1})\}\}$;
- $u(c_{0}, d_{0}) = (\geneuro/2, \geneuro/2)$, $u(c_{1},
e_{0}) = (\delta\geneuro, (1-\delta)\geneuro)$,\
$u(c_{0},
d_{1}) = u(c_{1}, e_{1}) = (0,0)$.
The extensive and the normal representation of $\Gamma_{1}$ is shown in Figure \[figure3\]. Equilibrium
![A two proposal ultimatum game $\Gamma_{1}$: an extensive form a) and a normal form b).[]{data-label="figure3"}](classical_ultimatum.eps)
analysis of the normal representation gives us three pure Nash equilibria: $(c_{0}, d_{0}e_{1})$, $(c_{1}, d_{0}e_{0})$ and $(c_{1}, d_{1}e_{0})$. There are also mixed equilibria: a profile where player 1 chooses $c_{0}$ and player 2 chooses $d_{0}e_{0}$ with probability $p \leq 1/(2\delta)$ and $d_{0}e_{1}$ with probability $1-p$, and a profile where player 1 decides to play $c_{1}$ and player 2 chooses any probability distribution over strategies $d_{0}e_{0}$ and $d_{1}e_{0}$. However, we can put these ones aside since both mixed equilibria do not contribute to the utility outcomes of $\Gamma_{1}$. They generate the same utility outcomes as the pure ones: $(\geneuro/2, \geneuro/2)$ and $(\delta\geneuro, (1-\delta)\geneuro)$, respectively. The key feature that make the game $\Gamma_{1}$ so curious is that only equilibrium profile $(c_{1}, d_{0}e_{0})$ with unfair outcomes $(\delta\geneuro, (1-\delta)\geneuro)$ is a reasonable scenario among all the equilibria of the ultimatum game (many experiments show that people are inclined to choose fair division $(\geneuro/2, \geneuro/2)$, however we stick to the natural assumption of game theory that players are striving to maximize their payoffs). The strategy combination $(c_{1}, d_{0}e_{0})$ is the unique equilibrium that is [*subgame perfect*]{} (the idea of subgame perfection is the well-known equilibrium refinement formulated by Selten [@selten]) i.e. it is a profile of strategies that induces a Nash equilibrium in every subgame (there are three subgames in $\Gamma_{1}$: the entire game, a game after the action $c_{0}$ and a game after the action $c_{1}$). At the same time the subgame perfection rejects equilibria that are not credible. Let us consider the profile $(c_{0}, d_{0}e_{1})$. Here, the strategy $d_{0}e_{1}$ of player 2 demands the action $e_{1}$ when player 1 chooses $c_{1}$. However, when $c_{1}$ occurs, a rational move of player 2 is $e_{0}$. Similar analysis shows that also $(c_{1}, d_{1}e_{0})$ is not subgame perfect equilibrium. Although the notion of subgame perfection is related to the extensive form of a game, we can easily determine subgame perfect equilibria in any two stage extensive game with perfect information (or even in a wider class of extensive games) through an analysis of its normal representation. In the game $\Gamma_{1}$ an action taken by player 1 determines a subgame in which only player 2 makes a move. Thus subgame perfect equilibrium in $\Gamma_{1}$ is a Nash equilibrium with a property that a strategy of player 2 is the best response to every strategy of player 1 (i.e., a strategy that weakly dominates the others).
Criticism of the previous approach to\
quantum ultimatum game
======================================
A misrepresentation of the classical ultimatum game is the source of its incorrect quantum representation in [@mendes]. The author describes the ultimatum problem as a $2\times2$ game and then applies the MW and the EWL schemes to construct the quantum game. However, as we have seen in Figure \[figure3\]b, $2\times4$ is a minimal dimension allowing to represent the ultimatum game in normal form. A hypothetical case of the ultimatum game in which player 2 has only two strategies after an action taken by player 1 implies that player 2 is deprived of capability to make her move conditioned on the action of the first player. That is tantamount to an event where the players take their actions at the same time or one of the players chooses her action as the second but she does not have any information about an action taken by her opponent. It does not correspond to a description of the ultimatum game where the second player knows a proposal of her opponent and depending on the move of the first player she makes her action. Although the player 2 has only two actions: [*accept*]{} or [*reject*]{} in the two-proposal ultimatum game, in fact she has four pure strategies defined as her plans of an action at each of her information sets. Therefore, a $2\times2$ strategic game cannot depict the ultimatum game. Consequently, the MW and the EWL approach used for quantization of a $2 \times 2$ game cannot produce a quantum version of this game. Neither of these quantum realizations contains the classical ultimatum game.
The quantum ultimatum game obtained by quantization of the normal representation of the classical game
======================================================================================================
First, let us remind the protocol for playing quantum games defined in [@fracor3]. It is a six-tuple: $$\begin{aligned}
\Gamma^{\mathrm{QI}} = \left(\mathscr{H}, N,
|\psi_{\mathrm{in}}\rangle, \xi, \{\mathcal{U}_{j}\},
\{E_{i}\}\right) \label{sixtuple}\end{aligned}$$ where the components are defined as follows:
- $\mathscr{H}$ is a complex Hilbert space $\bigotimes_{j=1}^m
\mathds{C}^2$ with an orthonormal basis $\mathcal{B}$.
- $N$ is a set of players with the property that $|N| \leq m$.
- $|\psi_{\mathrm{in}}\rangle$ is the initial state of a system of $m$ qubits $|\varphi_{1}\rangle,
|\varphi_{2}\rangle,\dots,|\varphi_{m}\rangle$.
- $\xi\colon
\{1,2,\dots,m\} \to N$ is a surjective mapping. A value $\xi(j)$ indicates a player who carries out a unitary operation on a qubit $|\varphi_{j}\rangle$.
- For each $j \in \{1,2,\dots,m\}$ the set $\mathcal{U}_{j}$ is a subset of unitary operators from $\mathsf{SU}(2)$ that are available for a qubit $j$. A (pure) strategy of a player $i$ is a map $\tau_{i}$ that assigns a unitary operation $U_{j} \in \mathcal{U}_{j}$ to a qubit $|\varphi_{j}\rangle$ for every $j \in \xi^{-1}(i)$. The final state $|\psi_{\mathrm{fin}}\rangle$ when the players have performed their strategies on corresponding qubits is defined as: $$\begin{aligned}
|\psi_{\mathrm{fin}}\rangle\mathrel{\mathop:}=
(\tau_{1},\tau_{2},\dots,\tau_{n})|\psi_{\mathrm{in}}\rangle =
\bigotimes_{i\in N}\bigotimes_{j\in \xi^{-1}(i)}U_{j}
|\psi_{\mathrm{in}}\rangle. \label{finalstate}\end{aligned}$$
- For each $i\in N$ the map $E_{i}$ is a utility (payoff) functional that specifies a utility for the player $i$. The functional $E_{i}$ is defined by the formula: $$\begin{aligned}
\label{eformula}
E_{i} = \sum_{|b\rangle \in \mathcal{B}}v_i(b)|\langle b
|\psi_{\mathrm{fin}}\rangle|^2, ~~ \mbox{where} ~~ v_i(b) \in
\mathds{R}.\end{aligned}$$
The above scheme is adapted for extensive games with two available actions at each information set so that we could use only qubits for convenience. Any game richer in actions can be transferred to quantum domain by using quantum objects of higher dimensionality.
The idea framed in [@fracor3] bases on identifying unitary actions taken on a qubit with actions taken in an information set of classical game. Therefore, three qubits are required to express the ultimatum game in quantum information language. Since the first player has one information set and the second player has two ones, player 1 performs a unitary operation on only one qubit and player 2 operates on the rest. Like in [@mendes] we examine the two approaches: the MW approach and the EWL approach to quantizing $\Gamma_{1}$.
The MW approach {#subsection_MW}
---------------
Let us consider the following six-tuple: $$\begin{aligned}
\Gamma^{\mathrm{MW}}_{1} = \left(\mathscr{H}_{c}, \{1,2\},
|\psi_{\mathrm{in}}\rangle, \xi, \{\{\sigma_{0},
\sigma_{1}\}_{i}\}, \{E_{i}\} \right), \label{mw2}\end{aligned}$$ where:
- $\mathscr{H}_{c}$ is a Hilbert space $\bigotimes^3_{j=1}\mathds{C}^2$ with the computational basis states $|x_{1},x_{2},x_{3}\rangle$, $x_{j}=0,1$;
- the initial state $|\psi_{\mathrm{in}}\rangle$ is a general pure state of three qubits: $$\begin{aligned}
|\psi_{\mathrm{in}}\rangle = \sum_{x \in \{0,1\}^3} \lambda_{x}|x
\rangle, ~~\mbox{where}~~ \lambda_{x} \in \mathds{C}
~~\mbox{and}~~ \sum_{x \in \{0,1\}^3}|\lambda_{x}|^2 = 1;
\label{generalinitialstate}\end{aligned}$$
- the map $\xi$ on $\{1,2,3\}$ given by the formula: $\xi(j) =
\left\{\begin{array}{lll}
1, & \mbox{if} & j=1;\\
2, & \mbox{if} & j \in \{2,3\}.
\end{array} \right.$
- $\sigma_{0} = \left(\begin{array}{cc} 1 &
0 \\
0 & 1
\end{array}\right)$ and $\sigma_{1} = \left(\begin{array}{cc} 0 &
1 \\
1 & 0
\end{array}\right)$;
- the payoffs functionals $E_{i}$, $i=1,2$, are of the form: $$\begin{aligned}
\begin{split}
&E_{1} =
\frac{1}{2}\geneuro\sum_{x_{3}}|\langle00,x_{3}|\psi_{\mathrm{fin}}\rangle|^2
+ \delta\geneuro\sum_{x_{2}}|\langle
1,x_{2},0|\psi_{\mathrm{fin}}\rangle|^2;\\ &E_{2} =
\frac{1}{2}\geneuro
\sum_{x_{3}}|\langle00,x_{3}|\psi_{\mathrm{fin}}\rangle|^2 +
(1-\delta)\geneuro\sum_{x_{2}}|\langle
1,x_{2},0|\psi_{\mathrm{fin}}\rangle|^2. \end{split}
\label{xoperators}\end{aligned}$$
By definition of $\xi$ in $\Gamma_{1}$, player 1 acts on the first qubit and treats the operators $\sigma^1_{0}$ and $\sigma^1_{1}$ as her strategies. Player 2 acts on the second and the third qubit, hence her pure strategies are $\sigma^2_{0} \otimes
\sigma^3_{0}$, $\sigma^2_{0} \otimes \sigma^3_{1}$, $\sigma^2_{1}
\otimes \sigma^3_{0}$ and $\sigma^2_{1} \otimes \sigma^3_{1}$ (the upper index denotes a qubit on which an operation is made). Let us determine for each profile $\left(\sigma^1_{\kappa_{1}},
\left(\sigma^2_{\kappa_{2}}, \sigma^3_{\kappa_{3}}\right)\right)$, where $\kappa_{1}, \kappa_{2}, \kappa_{3} \in \{0,1\}$, the corresponding expected utility $E_{i}$ by using formulae (\[finalstate\])-(\[eformula\]) and the specification of (\[mw2\]). We illustrate it using as an example $E_{i}\left(\sigma^1_{0}, \left(\sigma^2_{1},
\sigma^3_{0}\right)\right)$ for $i=1,2$. The initial state after the players choose the profile $\left(\sigma^1_{0},\left(\sigma^2_{1},
\sigma^3_{0}\right)\right)$ takes the form $|\psi_{\mathrm{fin}}\rangle=\sigma^1_{0}\otimes\sigma^2_{1}\otimes\sigma^3_{0}|\psi_{\mathrm{in}}\rangle.$ Thus, we have: $$\begin{aligned}
\label{finalstatejeden}
|\psi_{\mathrm{fin}}\rangle = \sum_{x_{1},x_{2},x_{3} \in \{0,1\}}
\lambda_{x_{1},x_{2},x_{3}}|x_{1},\overline{x}_{2}, x_{3}\rangle,\end{aligned}$$ where $\overline{x}_{2}$ is the negation of $x_{2}$. Putting the final state (\[finalstatejeden\]) into the first of Eq. (\[xoperators\]) we obtain: $$\begin{aligned}
E_{1}\left(\sigma^1_{0},\left(\sigma^2_{1},
\sigma^3_{0}\right)\right) =
\frac{1}{2}\geneuro\left(|\lambda_{010}|^2 +
|\lambda_{011}|^2\right) +
\delta\geneuro\left(|\lambda_{100}|^2+|\lambda_{110}|^2 \right).\end{aligned}$$ Obviously, we have $(1-\delta)\geneuro$ instead of $\delta\geneuro$ in the expected utility $E_{2}$. Therefore, the payoff vector $(E_{1}, E_{2})$ is $u_{\mathrm{f}}\left(|\lambda_{010}|^2 + |\lambda_{011}|^2\right)
+ u_{\mathrm{u}}\left(|\lambda_{100}|^2 +
|\lambda_{110}|^2\right)$ in that case. Payoff vectors $(E_{1},
E_{2})$ for all possible profiles $\left(\sigma^1_{\kappa_{1}},
\left(\sigma^2_{\kappa_{2}}, \sigma^3_{\kappa_{3}}\right)\right)$ are placed in the matrix representation in Figure \[figure4\]
![The MW approach to the normal representation of $\Gamma_{1}$.[]{data-label="figure4"}](mw_ultimatum.eps)
(for convenience we convert binary indices $(x_{1}, x_{2},
x_{3})_{2}$ of $\lambda_{x_{1},x_{2}, x_{3}}$ to the decimal numeral system).
Let us examine the game in Figure \[figure4\] to answer to what degree passing to the quantum domain may influence the result of the game. Notice first that (\[mw2\]) is indeed the quantum game in the spirit of the MW approach - the normal representation of $\Gamma_{1}$ can be obtained from $\Gamma^{\mathrm{MW}}_{1}$ by putting $|\lambda_{0}|^2 = 1$ and $|\lambda_{x}|^2 = 0$ for $x=1,2,\dots ,7$, i.e., if we put $|\psi_{\mathrm{in}}\rangle =
|000\rangle$. More generally: $\Gamma^{\mathrm{MW}}_{1}$ coincides to a game isomorphic to the normal representation of $\Gamma_{1}$ if we put as $|\psi_{\mathrm{in}}\rangle =
|x_{1},x_{2},x_{3}\rangle$ any basis state. Then $\Gamma^{\mathrm{MW}}_{1}$ is equal to $\Gamma_{1}$ up to the order of players’ strategies. The game $\Gamma_{1}$ favors player 1 as we have learnt in Section \[section:ultimatum\] Thus, an interesting problem is to look for another form of the initial state (\[generalinitialstate\]) that imply fairer solution unavailable in the game $\Gamma_{1}$. Let us study first: $$\begin{aligned}
|\psi_{\mathrm{in1}}\rangle = \frac{1}{2}\left(|000\rangle +
|001\rangle + |100\rangle + |110\rangle \right).
\label{initialstate1}\end{aligned}$$ Through the substitution $|\lambda_{0}|^2 = |\lambda_{1}|^2 =
|\lambda_{4}|^2 = |\lambda_{6}|^2 = 1/4$ (the other squares of the moduli equal 0) to entries of the matrix representation in Figure \[figure4\] we obtain a game where the only reasonable equilibrium profile is $\sigma^1_{0} \otimes \sigma^2_{0} \otimes
\sigma^3_{0}$ with corresponding expected utility vector $E =
(E_1, E_{2})$ equal $\left(u_{\mathrm{f}} +
u_{\mathrm{u}}\right)/2$. The other pure equilibria: $\sigma^1_{1}
\otimes \sigma^2_{0} \otimes \sigma^3_{1}$ and $\sigma^1_{1}
\otimes \sigma^2_{1} \otimes \sigma^3_{1}$ - both generating the utility outcome $\left(u_{\mathrm{f}} + u_{\mathrm{u}}\right)/4$ are obviously worse for both players so they won’t be chosen. Moreover, $\sigma^1_{0} \otimes \sigma^2_{0} \otimes \sigma^3_{0}$ is an imitation of a subgame perfect equilibrium - the strategy of the second player $\sigma^2_{0}\otimes\sigma^3_{0}$ is the best response to any strategy of the first player. To sum up, the initial state (\[initialstate1\]) is beneficial to player 2 compared with the classical case. It turns out that the answer to the question: is there any $|\psi_{\mathrm{in}}\rangle$ allowing to obtain a fair division of $\geneuro$, is also positive. Let us consider any state of the form: $$\begin{aligned}
|\psi_{\mathrm{in2}}\rangle = \sqrt{\frac{1}{2\delta'}}|000\rangle
+ \sqrt{1-\frac{1}{2\delta'}}|001\rangle, ~~ \mbox{where}
~~\frac{1}{2} < \delta < \delta' < 1. \label{initialstate2}\end{aligned}$$ Once again the profile $\sigma^1_{0} \otimes \sigma^2_{0} \otimes
\sigma^3_{0}$ constitutes a Nash equilibrium and the strategy of the second player $\sigma^2_{0} \otimes \sigma^3_{0}$ weakly dominates her other strategies as a result of putting $|\lambda_{0}|^2 = 1/2\delta'$ and $|\lambda_{1}|^2 =
1-1/2\delta'$ in the game in Figure \[figure4\]. Since there are no other profiles with that property, $\sigma^1_{0} \otimes
\sigma^2_{0} \otimes \sigma^3_{0}$ is the most reasonable scenario that implies $E_{1,2}(\sigma^1_{0} \otimes \sigma^2_{0} \otimes
\sigma^3_{0}) =\geneuro/2$. The superposition of the third qubit (the second qubit of player 2) is essential to obtain fair division result since it is impossible to achieve $\delta\geneuro$ by player 1 then. Therefore, the payoff $\geneuro/2$ becomes the most attractive for her now.
The conclusions we can draw from the analysis of the MW approach to the ultimatum game are as follows. First, the game $\Gamma^{\mathrm{MW}}_{1}$ that begins with $|\psi_{\mathrm{in}1}\rangle$ discloses a game tree different from the one in Figure \[figure3\]a). If there is a protocol for quantizing the extensive game $\Gamma_{1}$ directly without using its normal representation as in our case, then the output game tree must be different from the game tree of $\Gamma_{1}$ in general. It follows form the fact that the game tree in Figure \[figure3\]a with any four utility outcomes assigned to its terminal histories implies the normal representation specified by only these four payoff outcomes. However, the game $\Gamma^{\mathrm{MW}}_{1}$ where the initial state take the form of (\[initialstate1\]) has five different outcomes. Notice, that is not irrelevant issue bearing in mind the fact that the bimatrix of a strategic game played classically as well as played by the MW protocol always have the same dimension.
The case where game begins with the state (\[initialstate2\]) is applied shows that even a separable initial state can influence significantly a result of $\Gamma_{1}$. It is not strange property. Any superposition of a player’s qubit causes some limitation on players’ influence on their qubits as we have seen in the case (\[initialstate1\]). In particular, if each qubit of the initial state is in the state $|+\rangle = \left(|0\rangle +
|1\rangle\right)/\sqrt{2}$, no player can affect amplitudes of her qubit applying only $\sigma_{0}$ and $\sigma_{1}$ (measurement outcomes 0 and 1 on qubit occur with the same probability). Then the result of the game only depends on the initial state $|\psi_{\mathrm{in}}\rangle = |+\rangle|+\rangle|+\rangle$.
The EWL approach
----------------
As we have seen, the two-element set of unitary operators is too simple in some cases. The two-parameter unitary operations used in the EWL protocol allow to avoid player’s powerlessness when she acts on $|+\rangle$, and generally each player can essentially affect amplitudes of the initial state. Thus, it is interesting to find a result of the ultimatum game played according to the EWL approach. Let the following six-tuple be given: $$\begin{aligned}
\label{gammaewl}
\Gamma^{\mathrm{EWL}}_{1} = \left(\mathscr{H}_{e}, \{1,2\},
|\psi_{000}\rangle, \xi, \{\{U(\theta,\beta)\}_{i}\},
\{E_{i}\}\right),\end{aligned}$$ where:
- $\mathscr{H}_{e}$ is a Hilbert space $\bigotimes_{j=1}^3
\mathds{C}^2$ with the basis $\{|\psi_{x_{1}, x_{2},
x_{3}}\rangle\colon x_{j} = 0,1\}$ of entangled states defined as follows: $$\begin{aligned}
|\psi_{x_{1},x_{2},x_{3}}\rangle = \frac{|x_{1},x_{2},x_{3}\rangle
+
i|\overline{x}_{1},\overline{x}_{2},\overline{x}_{3}\rangle}{\sqrt{2}};
\label{basestate}\end{aligned}$$
- the mapping $\xi$ is the same as in six-tuple (\[mw2\]);
- the player’s actions $\{U(\theta,\beta)\colon \theta \in [0,\pi],
\beta \in [0,\pi/2]\}$, studied, for example, in the paper [@flitney], form an alternative to two-parameter unitary operations used in [@eisert]. They are of the form: $$\begin{aligned}
U(\theta, \beta) = \left(\begin{array}{cc} \cos(\theta/2) &
ie^{i\beta}\sin(\theta/2) \\
ie^{-i\beta}\sin(\theta/2) & \cos(\theta/2)
\end{array}\right); \label{twoparameter}\end{aligned}$$
- $E_{i}$ for $i=1,2$ are the payoff functionals (\[xoperators\]) defined for the basis (\[basestate\]): $$\begin{aligned}
\begin{split}
&E_{1} = \frac{1}{2}\geneuro \sum_{x_{3}} |\langle
\psi_{00{,}x_{3}}|\psi_{\mathrm{fin}}\rangle|^2 + \delta\geneuro
\sum_{x_{2}} |\langle\psi_{1{,}x_{2}{,}0}|\psi_{\mathrm{fin}}\rangle|^2;\\
&E_{2} = \frac{1}{2}\geneuro \sum_{x_{3}} |\langle
\psi_{00{,}x_{3}}|\psi_{\mathrm{fin}}\rangle|^2 +
(1-\delta)\geneuro \sum_{x_{2}}
|\langle\psi_{1{,}x_{2}{,}0}|\psi_{\mathrm{fin}}\rangle|^2.
\label{x2operators}
\end{split}\end{aligned}$$
Each strategy $U_{1}$ of player 1 is simply $U(\theta_{1},
\beta_{1})$. The strategies of the second player are chosen in a manner similar to $\Gamma^{\mathrm{MW}}_{1}$ - they are tensor products $U_{2} \otimes U_{3} = U(\theta_{2}, \beta_{2}) \otimes
U(\theta_{3}, \beta_{3})$. The final state $|\psi_{\mathrm{fin}}\rangle$ corresponding to a profile $\tau =
((\theta_{1}, \beta_{1}), (\theta_{2}, \beta_{2}, \theta_{3},
\beta_{3}))$ is as follows: $$\begin{aligned}
|\psi_{\mathrm{fin}}\rangle = U_{1}\otimes U_{2} \otimes
U_{3}|\psi_{000}\rangle = \frac{1}{\sqrt{2}}\sum_{x \in
\{0,1\}^3}\upsilon_{x}|x\rangle, \label{rhogamma1}\end{aligned}$$ where $$\begin{aligned}
\upsilon_{x_{1},x_{2},x_{3}} &= i^{\sum x_{j}}e^{-i\sum
x_{j}\beta_{j}}\prod_{j}\cos\left(\frac{x_{j}\pi -
\theta_{j}}{2}\right)\notag\\&\quad+(-i)^{\sum x_{j}}e^{i\sum
\overline{x_{j}}\beta_{j}}\prod_{j}\cos\left(\frac{\overline{x_{j}}\pi
- \theta_{j}}{2}\right),\end{aligned}$$ and $j=1,2,3$, $x_{j} = 0,1$, and $\overline{x}_{j}$ is negation of $x_{j}$. Putting (\[x2operators\]) and (\[rhogamma1\]) into formula (\[eformula\]) we obtain the following expected payoff vector $(E_{1}(\tau), E_{2}(\tau))$: $$\begin{aligned}
\label{payoffewl}
&(E_{1}(\tau), E_{2}(\tau))= \notag\\
&\qquad
u_{\mathrm{f}}\Biggl[\cos^2\frac{\theta_{1}}{2}\cos^2\frac{\theta_{2}}{2}\left(\cos^2\frac{\theta_{3}}{2}
+ \sin^2\frac{\theta_{3}}{2}\cos^2\beta_{3}\right)\notag\\
&\qquad+\sin^2\frac{\theta_{1}}{2}\sin^2\frac{\theta_{2}}{2}\left(\sin^2\frac{\theta_{3}}{2}\sin^2(\beta_{1}
+ \beta_{2} + \beta_{3}) +
\cos^2\frac{\theta_{3}}{2}\sin^2(\beta_{1} + \beta_{2})
\right)\Biggr]\notag\\
&\qquad+u_{\mathrm{u}}\Biggl[\sin^2\frac{\theta_{1}}{2}\cos^2\frac{\theta_{3}}{2}\left(\cos^2\frac{\theta_{2}}{2}\cos^2\beta_{1}
+ \sin^2\frac{\theta_{2}}{2}\cos^2(\beta_{1} + \beta_{2})\right)\notag\\
&\qquad+\cos^2\frac{\theta_{1}}{2}\sin^2\frac{\theta_{3}}{2}\left(\sin^2\frac{\theta_{2}}{2}\sin^2(\beta_{2}
+ \beta_{3}) +
\cos^2\frac{\theta_{2}}{2}\sin^2\beta_{3}\right)\Biggr].\end{aligned}$$ Let us check first that $\Gamma^{\mathrm{EWL}}_{1}$ generalizes the classical ultimatum game $\Gamma_{1}$. Pure strategies of the first player are represented by $U(0,0)$ and $U(\pi,0)$. Similarly, the set of strategies of the second player in $\Gamma_{1}$ is represented by a set $\{U(\theta_{2},0)\otimes
U(\theta_{3},0)\colon \theta_{2}, \theta_{3} \in \{0,\pi\}\}$ since the set of profiles $$\begin{aligned}
\{\left((\theta_{1},0),(\theta_{2},0,\theta_{3},0)\right)\colon
\theta_{1},\theta_{2},\theta_{3}\in \{0,\pi\}\} \end{aligned}$$ in (\[gammaewl\]) and the set of profiles $$\begin{aligned}
\{(c_{k_{1}},d_{k_{2}}e_{k_{3}}) \colon k_{1},k_{2},k_{3} \in
\{0,1\}\} \end{aligned}$$ in (\[Gamma1ultimatum\]) generate the same payoffs. Equivalents of behavioral strategies of $\Gamma_{1}$ (i.e., independent probability distributions $p$, $q$ and $r$ over the actions $c_{k_{1}}$, $d_{k_{2}}$ and $e_{k_{3}}$, respectively, specified by players at their own information sets) can be found among unitary strategies as well. If we restrict unitary actions to $U(\theta, 0)$, i.e., to profiles of the form $((\theta_{1},0), (\theta_{2},0,\theta_{3},0))$, $\theta_{j}
\in [0,\pi]$, the right-hand side of Eq. (\[payoffewl\]) takes the form: $$\begin{aligned}
u_{\mathrm{f}}\cos^2\frac{\theta_{1}}{2}\cos^2\frac{\theta_{2}}{2}
+
u_{\mathrm{u}}\sin^2\frac{\theta_{1}}{2}\cos^2\frac{\theta_{3}}{2}.\end{aligned}$$ By substituting $p$ for $\cos^2(\theta_{1}/2)$, $q$ for $\cos^2(\theta_{2}/2)$, and $r$ for $\cos^2(\theta_{3}/2)$ we get the expected payoffs corresponding to any behavioral strategy profile $((p,1-p),((q,1-q), (r,1-r)))$ in $\Gamma_{1}$.
Let us examine an impact of the unitary strategies on a result of the EWL approach to $\Gamma_{1}$. In particular we ask the question if the unfair division $u_{\mathrm{u}}$ or the fair division $u_{\mathrm{f}}$ in $\Gamma^{\mathrm{EWL}}_{1}$ is more probable. Notice, that the profile $((\theta_{1}, \beta_{1}),
(\theta_{2}, \beta_{2}, \theta_{3},\beta_{3})) = ((\pi, 0), (0, 0,
0, 0))$ (corresponding to subgame perfect equilibrium $(c_{1},
d_{0}e_{0})$ in $\Gamma_{1}$) is not Nash equilibrium in $\Gamma^{\mathrm{EWL}}_{1}$. The second player can gain by choosing, for example, $(\theta_{2}, \beta_{2},
\theta_{3},\beta_{3}) = (\pi,\pi/2, \pi, 0)$ instead of $(0, 0, 0,
0)$. Then she obtains the fair devision payoff. Moreover, for any other strategy of the first player $(\theta_{1}, \beta_{1})$, player 2 can select, for instance, $(0, 0, 0, 0)$ to obtain a payoff being a mixture of $u_{\mathrm{f}}$ and $u_{\mathrm{u}}$. This proves that the unfair division $u_{\mathrm{u}}$ cannot be a result in (\[gammaewl\]). The fair division $u_{\mathrm{f}}$ in turn can be achieved through continuum of Nash equilibria. Let us denote by $\mathrm{NE}(\Gamma^{\mathrm{EWL}}_{1})$ the set of all Nash equilibria of $\Gamma^{\mathrm{EWL}}_{1}$. An examination of (\[payoffewl\]) shows that: $$\begin{aligned}
\left\{((\pi,\beta_{1}),(\pi,\beta_{2},\pi, \beta_{3})) \colon
\beta_{2}+ \beta_{3} \leq \frac{\pi}{4}\,, \sum^3_{j = 1}\beta_{j}
= \frac{\pi}{2}\right\} \subset
\mathrm{NE}(\Gamma^{\mathrm{EWL}}_{1}) \label{subset1}\end{aligned}$$ as well as $$\begin{aligned}
\left\{((0,\beta_{1}),(0,\beta_{2},\pi,0)) \colon \beta_{1},
\beta_{2} \in \left[0,\frac{\pi}{2}\right]\right\} \subset
\mathrm{NE}(\Gamma^{\mathrm{EWL}}_{1}). \label{subset2}\end{aligned}$$ Moreover, all strategy profiles of these sets generate the payoff vector $u_{\mathrm{f}}$ for any division factor $1/2<\delta<1$. To prove inclusion (\[subset1\]) let us consider any strategy $(\theta'_{1},\beta'_{1})$ of player 1 given that player 2’s strategy from (\[subset1\]) is fixed. Then for $\beta_{2}+\beta_{3}\leq \pi/4$ we have $$\begin{gathered}
\label{multiequation}
E_{1}((\theta'_{1},\beta'_{1}),(\pi,\beta_{2},\pi,\beta_{3}))\\
=
\left[\frac{1}{2}\geneuro\sin^2\frac{\theta'_{1}}{2}\sin^2(\beta'_{1}+\beta_{2}+\beta_{3})+
\delta\geneuro\cos^2\frac{\theta'_{1}}{2}\sin^2(\beta_{2}+\beta_{3})\right].\end{gathered}$$ Since $\beta_{2}+\beta_{3}\leq \pi/4$, the maximum value of (\[multiequation\]) is achieved if the second element of the sum is 0. It implies that the best response of player 1 is $\theta'_{1} = \pi$ and $\beta_{1}'= \pi/2 - \beta_{2}-\beta_{3}$. The second player cannot gain by deviating as well because she always obtains no more than $\geneuro/2$ in $\Gamma^{\mathrm{EWL}}_{1}$. Therefore, each profile of set (\[subset1\]) indeed constitutes Nash equilibrium. Inclusion (\[subset2\]) can be proved in similar way. Notice that there are also Nash equilibria different from (\[subset1\]) and (\[subset2\]) that generate the payoff outcome $\geneuro/2$ for both players. For example, a strategy profile $((\pi, \pi/4),(\pi,
\pi/4, \pi/2,0))$.
Intuitively, a huge number of fair solutions in $\Gamma^{\mathrm{EWL}}_{1}$ being NE together with a lack of an equilibrium outcome $u_{\mathrm{u}}$ favors the second player in comparison to the classical game $\Gamma_{1}$ . However, it does not assure the second player the fair payoff $\geneuro/2$ yet. Since the players choose their strategies simultaneously, they cannot coordinate them. If the first player unilaterally deviates from a strategy dictated by (\[subset2\]) and she plays a strategy being a part of (\[subset1\]) then both players receive nothing as we have $E_{1,2}((\pi,\beta_{1}), (0, \beta_{2}, \pi,
0)) = 0$ for all $\beta_{1}, \beta_{2} \in \left[0,\pi/2\right]$. On the other hand, it turns out that the statement that each of these equilibria is equally likely to occur is not true. Let us investigate which equilibria in $\Gamma_{1}$ are preserved in $\Gamma^{\mathrm{EWL}}_{1}$ bearing in mind that the unitary strategies $U(\theta,0)$ are quantum counterparts to classical moves in $\Gamma_{1}$. As we have seen there is no equilibrium profile in $\Gamma^{\mathrm{EWL}}_{1}$ that allows the first player to gain $\delta\geneuro$. Therefore, in particular, the unfair division equilibrium $(c_{1},d_{0}e_{0})$ of $\Gamma_{1}$ cannot be generated by any unitary operations $U(\theta,0)$. However, each fair division equilibrium (pure or mixed) of (\[Gamma1ultimatum\]) can be reconstructed in (\[gammaewl\]). The profile $((0,0),(0,0,\pi,0))$ corresponding to the equilibrium $(c_{0},d_{0}e_{1})$ in $\Gamma_{1}$ is Nash equilibrium of $\Gamma^{\mathrm{EWL}}_{1}$ since it is element of the set (\[subset2\]). Next, the mixed equilibria mentioned in Section \[section:ultimatum\] can be implemented in $\Gamma^{\mathrm{EWL}}_{1}$ as follows: they are the profiles where the first player chooses $U(0,0)$ and the second player chooses either $U(0,0)\otimes U(0,0)$ with probability $p \in [0,
1/2\delta]$ and $U(0,0)\otimes U(\pi,0)$ with probability $1-p$, or in a language of behavioral strategies she just takes an operator from $\bigl\{U(0,0) \otimes U(\theta,0)\colon \theta \in
\bigl[2\arccos\bigl(1/\sqrt{2\delta}\bigr), \pi/2\bigr]\bigr\}$. According to the concept of Schelling Point [@schelling] players tend to select a solution that is the most natural as well as the most distinctive among all possible choices. Therefore, if we assume that the players prefer the fair division, they choose a profile that is an equilibrium of both $\Gamma_{1}$ and $\Gamma^{\mathrm{EWL}}_{1}$ among all equal equilibria of $\Gamma^{\mathrm{EWL}}_{1}$. Since all these shared equilibria generate the same outcome, the pure equilibrium is the most natural and it ought to be chosen as the Schelling Point.
Extensive form of the quantum ultimatum game
============================================
In subsection \[subsection\_MW\] we made observation that an extensive game and its quantum realization differ not only in utilities but also in game trees. Now, we are going to give the answer to the question how would a game tree of such quantum realization look like? Let us reconsider an extensive game form given by the game tree on Figure \[figure3\]a, where the components $H$, $P$ and $\mathcal{I}_{i}$ are derived from $\Gamma_{1}$, and the outcomes $O_{00}, O_{01}, O_{10}$ and $O_{11}$, are assigned to the terminal histories $(c_{0}, d_{0}),
(c_{0}, d_{1}), (c_{1}, e_{0})$ and $(c_{1}, e_{1})$, respectively, instead of particular payoff values. Let us denote this problem as: $$\begin{aligned}
\Gamma_{2} = \left(\{1,2\}, H, P, \mathcal{I}_{i}, O\right).
\label{gamma3}\end{aligned}$$ Then the tuple $\Gamma^{\mathrm{MW}}_{2}$ associated with $\Gamma_{2}$ is derived from $\Gamma^{\mathrm{MW}}_{1}$ and only the payoff functionals $E_{i}$ undergo appropriate modifications. Let us write $\Gamma^{\mathrm{MW}}_{2}$ in the language of density matrices, for convenience. That is: $$\begin{aligned}
\Gamma^{\mathrm{MW}}_{2} = \left(\mathscr{H}_{c}, \{1,2\},
\rho_{\mathrm{in}}, \xi, \{\sigma_{0}, \sigma_{1}\}_{i}, X\right),
\label{gameMW3}\end{aligned}$$ where
- $\rho_{\mathrm{in}}$ is a density matrix of the initial state (\[generalinitialstate\]);
- the outcome operator $X$ is a sum of $X^{0} + X^{1}$ defined as: $$\begin{aligned}
\begin{split}
&X^0 = O_{00}|00\rangle \langle 00| \otimes \mathds{1} +
O_{01}|01\rangle
\langle 01| \otimes \mathds{1};\\
&X^1 = O_{10}|1\rangle \langle 1| \otimes \mathds{1} \otimes
|0\rangle \langle 0| + O_{11}|1\rangle \langle 1| \otimes
\mathds{1} \otimes |1\rangle \langle 1|. \end{split}\end{aligned}$$
In this case, the density matrix $\rho_{\mathrm{fin}}$ of the final state $|\psi_{\mathrm{fin}}\rangle$ takes a form $$\begin{aligned}
\label{finalstateultimatum}
\rho_{\mathrm{fin}} = \sigma^1_{\kappa_{1}} \otimes
\sigma^2_{\kappa_{2}}\otimes \sigma^3_{\kappa_{3}}
\rho_{\mathrm{in}} \sigma^1_{\kappa_{1}} \otimes
\sigma^2_{\kappa_{2}}\otimes \sigma^3_{\kappa_{3}}.\end{aligned}$$ The outcome functionals (\[eformula\]) are then equivalent to the following one: $$\begin{aligned}
\label{eformulaultimatum}
E\left( \sigma^1_{\kappa_{1}},\left(\sigma^2_{\kappa_{2}},
\sigma^3_{\kappa_{3}}\right) \right) =
\mathrm{tr}\left(X\rho_{\mathrm{fin}}\right).\end{aligned}$$ In order to give a extensive form to determine the final state $\rho_{\mathrm{fin}}$ in $\Gamma^{\mathrm{MW}}_{2}$ let us modify the way (\[finalstateultimatum\]) of calculating the final state $\rho_{\mathrm{fin}}$. To begin with, player 1 acts on the first qubit. Next, player 2 carries out a measurement on that qubit in the computational basis to find out what is a current state of the game. Then she performs an operation on either the second or the third qubit of the post-measurement state depending on whether the measurement outcome 0 or 1 has occurred. The operation of the second player ultimately defines the final state that is inserted to the formula (\[eformulaultimatum\]). The procedure can be formalized as follows:\
-------- --------------------------------------------------------------------------------------------------------------------------------------------------------- -- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[**Sequential procedure**]{}
[1.]{} [$\sigma^1_{\kappa_{1}}\rho_{\mathrm{in}}\sigma^1_{\kappa_{1}} = \rho_{\kappa_{1}}$]{} [the player 1 performs an operation $\sigma^1_{\kappa_{1}}$ on her qubit of the initial state $\rho_{\mathrm{in}}$]{}
[2.]{} [$\displaystyle\frac{M_{\iota}\rho_{\kappa_{1}} M_{\iota}}{\mathrm{tr}(M_{\iota}\rho_{\kappa_{1}})} = \rho_{\kappa_{1},\iota}$, $p_{\kappa_{1},\iota} = [the player 2 prepares the measurement $\{M_{0},
\mathrm{tr}(M_{\iota}\rho_{\kappa_{1},\iota}$)]{} M_{1}\}$ defined by $M_{\iota} = |\iota \rangle \langle \iota|
\otimes I \otimes I, \, \iota =0,1$ on the first qubit of the state $\sigma_{\kappa_{1}}\rho_{\mathrm{in}}\sigma_{\kappa_{1}}$ (the probability of obtaining result $\iota$ is denoted by $p_{\kappa_{1},\iota}$)]{}
[3.]{} [$\sum_{\iota} p_{\kappa_{1},\iota}\sigma^{2+\iota}_{\kappa_{2+\iota}} \rho_{\kappa_{1},\iota} \sigma^{2+\iota}_{\kappa_{2+\iota}} = [if a measurement outcome $\iota$ occurs, the player 2 performs an operaton $\sigma_{\kappa_{\iota+2}}$ on $\iota + 2$ qubit of the post-measurement state]{}
\rho'_{\mathrm{fin}}$]{}
-------- --------------------------------------------------------------------------------------------------------------------------------------------------------- -- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
It turns out that for any strategy profile $\left(\sigma^1_{\kappa_{1}}, \left(\sigma^2_{\kappa_{2}},
\sigma^3_{\kappa_{3}}\right)\right)$ the final state $\rho_{\mathrm{fin}}$ defined both by the formula (\[finalstateultimatum\]) and by the sequential procedure determine the same outcome of the game $\Gamma^{\mathrm{MW}}_{2}$.\
Let density operator $\rho_{\mathrm{in}}$ of a state (\[generalinitialstate\]) be given. Then the state $\rho'_{\mathrm{fin}}$ after the third step of procedure can be expressed as: $$\begin{aligned}
\rho'_{\mathrm{fin}} &=
\sigma^2_{\kappa_{2}}M_{0}\rho_{\kappa_{1}}
M_{0}\sigma^2_{\kappa_{2}} +
\sigma^3_{\kappa_{3}}M_{1}\rho_{\kappa_{1}}
M_{1}\sigma^3_{\kappa_{3}}\notag\\ &=
M_{0}\sigma^2_{\kappa_{2}}\rho_{\kappa_{1}}\sigma^2_{\kappa_{2}}M_{0}
+
M_{1}\sigma^3_{\kappa_{3}}\rho_{\kappa_{1}}\sigma^3_{\kappa_{3}}M_{1}.\end{aligned}$$ Since $X^{\kappa}M_{\iota} = \delta_{\kappa \iota}X^{\kappa}$, where $\delta_{\kappa \iota}$ is the Kronecker’s delta, we obtain: $$\begin{aligned}
\mathrm{tr}(X\rho'_{\mathrm{fin}}) =
\mathrm{tr}(X^0\sigma^2_{\kappa_{2}}\rho_{\kappa_{1}}\sigma^2_{\kappa_{2}}
+ X^1\sigma^3_{\kappa_{3}}\rho_{\kappa_{1}}\sigma^3_{\kappa_{3}}).
\label{equation1}\end{aligned}$$ Notice that operation $\sigma_{1}$ on the second (third) qubit of any state (\[generalinitialstate\]) does not influence the measurement of outcomes $O_{10}$ and $O_{11}$ ($O_{00}$ and $O_{01}$), because of the form of $X^{1}$ ($X^{0}$), which means that: $$\begin{aligned}
\mathrm{tr}(X^{\iota}\sigma^{2+\iota}_{\kappa_{2+\iota}}\rho_{\kappa_{1}}\sigma^{2+\iota}_{\kappa_{2+\iota}})
= \mathrm{tr}(X^{\iota} \sigma^2_{\kappa_{2}} \otimes
\sigma^3_{\kappa_{3}}\rho_{\kappa_{1}}\sigma^2_{\kappa_{2}}
\otimes \sigma^3_{\kappa_{3}})\quad \mbox{for} \quad \iota = 0,1.
\label{notinfluence}\end{aligned}$$ Inserting (\[notinfluence\]) into the formula (\[equation1\]) we get: $$\begin{aligned}
\mathrm{tr}\left(X\rho'_{\mathrm{fin}}\right) =
\mathrm{tr}\Biggl(\left(X^{0} +
X^{1}\right)\Biggl(\bigotimes^3_{j=1}
\sigma^j_{\kappa_{j}}\rho_{\mathrm{in}}\bigotimes^3_{j=1}
\sigma^j_{\kappa_{j}}\Biggr)\Biggr). \label{formulakoniec}\end{aligned}$$ The right-hand side of (\[formulakoniec\]) is equal the expected outcome given by formula (\[eformulaultimatum\]). Thus, the two ways of determining the final state are outcome-equivalent.
We claim that performing quantum measurement is a more natural manner to play quantum games than observation of player’s actions taken previously - the way suggested by games played classically. Since the result of a quantum game is determined by the measurement outcome of the final state instead of actions taken by players, each stage of the quantum game also ought to be set via a quantum measurement of a current state. Moreover, when we suppose the second player’s move dependence on actions of the first player in $\Gamma_{2}$ then it implies the same game tree as in Figure. \[figure3\]a). This way, however, stands in contradiction to the results in subsection \[subsection\_MW\] that tell us that the game trees must be different. Of course, if the initial state is $|000\rangle \langle 000|$ (i.e., when game given by (\[gameMW3\]) boils down to a game (\[gamma3\])), observation of the course of the game played classically and with the use of quantum measurement coincide.
Let us study what a game tree corresponding to the game $\Gamma^{\mathrm{MW}}_{2}$ is yielded by the above-mentioned procedure. According to the first step, the initial history $\emptyset$ is followed by two actions of the first player. Next, the measurement on the first qubit is made. The two possible measurement outcomes $\iota = 0,1$ can be identified with two actions (following each player 1’s move) of a chance mover that are taken with probability $p_{\kappa_{1},\iota}$. Finally the player 2 acts on $\iota +2$ qubit of the state $\rho_{\iota}$ after each history associated with the outcome $\iota$. Therefore, all histories followed by given outcome $\iota$ constitutes an information set of player 2. Such description in a form of a game tree is illustrated in Figure \[figure5\].
![The extensive game associated with the quantum realization $\Gamma^{\mathrm{MW}}_{2}$.[]{data-label="figure5"}](quantum_ultimatum.eps)
The outcomes $O'_{0.\kappa_{1}{,}\kappa_{2}}$ and $O'_{1.\kappa_{1}{,}\kappa_{3}}$ are determined by the following equations: $$\begin{aligned}
O'_{0.\kappa_{1}{,}\kappa_{2}} = \mathrm{tr}(X \sigma_{\kappa_{2}}
\rho_{\kappa_{1},0} \sigma_{\kappa_{2}}),\quad
O'_{1.\kappa_{1}{,}\kappa_{3}} = \mathrm{tr}(X \sigma_{\kappa_{3}}
\rho_{\kappa_{1},1} \sigma_{\kappa_{3}}). \label{ostatnierownania}\end{aligned}$$ We have proved that the two approaches: (\[finalstateultimatum\]) and the sequential one to calculate the final state are outcome-equivalent. Therefore, it should be expected that extensive forms of $\Gamma_{2}$ and $\Gamma^{\mathrm{MW}}_{2}$ coincide when the initial state is a basis state. In fact, given $\rho_{\mathrm{in}} = |000\rangle
\langle 000|$ the probabilities $p_{\kappa_{1},\iota}$ are expressed by the formula $p_{\kappa_{1},\iota} =
\delta_{\kappa_{1},\iota}$, where $\kappa_{1}, \iota \in \{0,1\}$. Then, the available outcomes given by Eq. (\[ostatnierownania\]) are as follows: $O'_{0.00} = O_{00}$, $O'_{0.01} = O_{01}$, $O'_{1.10} = O_{10}$ and $O'_{1.11} = O_{11}$. By identifying $\sigma^1_{\kappa_{1}} \mathrel{\mathop:}= c_{\kappa_{1}}$, $\sigma^2_{\kappa_{2}} \mathrel{\mathop:}= d_{\kappa_{2}}$, $\sigma^3_{\kappa_{3}} \mathrel{\mathop:}= e_{\kappa_{3}}$ the extensive game in Figure \[figure5\] represents game $\Gamma_{2}$.
Conclusion
==========
We have shown that our proposal extends the ultimatum game in the quantum area. Although proposed scheme is suitable only for a normal representation of the ultimatum game in which some features of corresponding game in extensive form are lost, it passes on valuable information about how passing to the quantum domain influences a course of extensive games. The dominant position of player 1, when the ultimatum game is played classically, can be weakened in the case of playing the game via both the MW approach and the EWL approach. Another thing worth noting is that the the quantization significantly extends the game tree compared with classical case. It makes the normal representation to be more convenient way to analyze the game than the way of extensive form.
[99]{}
Eisert, J., Wilkens, M., Lewenstein, M., [*Phys. Rev. Lett.*]{} [**83**]{} (1999), 3077. Eisert, J., Wilkens, M., [*J. Mod. Opt.*]{} [**47**]{} (2000), 2543. Flitney, A. P., Hollenberg, L. C. L., [*Phys. Lett. A*]{} [**363**]{} (2007), 381. Frackiewicz, P., arXiv:1107.3245v2 (2011). Güth, W., Schmittberger, R., Schwarze, B., [*J. Econ. Behav Organ.*]{} [**3**]{} (1982), 367. Harsanyi, J. C., Selten, R., [*A General Theory of Equilibrium Selection in Games*]{}, MIT Press, Cambridge, MA. 1988. Marinatto, L., Weber, T., [*Phys. Lett. A*]{} [**272**]{} (2000), 291. Mendes, R. V., [*Quantum Inf. Process.*]{} [**4**]{} (2005), 1. Myerson, R. B., [*Game Theory: Analysis of Conflict*]{}, Harvard University Press 1991. Nash, J. F., [*Ann. Math.*]{}, [**54**]{} (1951), 289. Osborne, M. J., Rubinstein, A., [*A Course in Game Theory*]{}, MIT Press, Cambridge, MA. 1994. Schelling, T. C., [*The Strategy of Conflict*]{}, Harvard University Press, Cambridge, MA. 1960. Selten, R., [*Zeitschrift für die gesamte Staatswissenschaft*]{} [**121**]{} (1965), 301.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
J.-F. POMMARET\
CERMICS, Ecole des Ponts Paris-Tech, France\
(http://cermics.enpc.fr/$\sim$pommaret/home.html)
date:
title: |
DIFERENTIAL GALOIS THEORY\
AND MECHANICS
---
[**ABSTRACT**]{}\
The [*classical Galois theory*]{} deals with certain finite algebraic extensions and establishes a bijective order reversing correspondence between the intermediate fields and the subgroups of a group of permutations called the [*Galois group*]{} of the extension. It has been the dream of many mathematicians at the end of the nineteenth century to generalize these results to systems of algebraic partial differential (PD) equations and the corresponding finitely generated differential extensions, in order to be able to add the word [*differential*]{} in front of any classical statement. The achievement of the Picard-Vessiot theory by E. Kolchin between 1950 and 1970 is now well known.\
The purpose of this paper is to sketch the general theory for such differential extensions and algebraic pseudogroups by means of new methods mixing [*differential algebra*]{}, [*differential geometry*]{} and [*algebraic geometry*]{}. As already discovered by E. Vessiot in 1904 through the use of [*automorphic systems*]{}, a concept never acknowledged, the main point is to notice that the Galois theory (old and new) is mainly a study of [*principal homogeneous spaces*]{} (PHS) for algebraic groups or pseudogroups. Hence, all the formal theory of PD equations developped by D.C. Spencer around 1970 must be used together with modern algebraic geometry, in particular tensor products of rings and fields.\
However, the combination of these new tools is not sufficient and we have to create the analogue for Lie pseudogroups of the so-called [*invariant derivations*]{} introduced by A. Bialynicki-Birula after 1960 in the study of algebraic groups and [*fields with derivations*]{}. We shall finally prove the usefulness of the resulting [*differential Galois theory*]{} through striking applications to mechanics, revisiting [*shell theory*]{}, [*chain theory*]{}, the [*Frenet-Serret formulas*]{} and the integration of [*Hamilton-Jacobi equations*]{}.\
[**KEY WORDS**]{} Classical Galois theory, Differential Galois theory, Differential algebra, Differential extensions, Tensor products of rings, Automorphic systems, Algebraic groups, Algebraic pseudogroups, Principal homogeneous spaces, Shell theory, Chain theory, Frenet-Serret formulas, Hamilton-Jacobi equations.\
[**FOREWORD**]{}\
The [*classical Galois theory*]{} deals with certain finite algebraic extensions and establishes a bijective order reversing correspondence between the intermediate fields and the subgroups of a group of permutations called the [*Galois group*]{} of the extension.\
It has been the dream of many mathematicians at the end of the nineteenth century to generalize these results to systems of linear or algebraic ordinary differential (OD) or partial differential (PD) equations and the corresponding finitely generated differential extensions, in order to be able to add the word [*differential*]{} in front of any classical statement. Among the tentatives, we may quote the [*Picard-Vessiot theory*]{} dealing with differential extensions having finite transcendence degree, where the Galois group is an [*algebraic group*]{} that can be considered as a linear algebraic group of matrices. We may also quote the [*Drach-Vessiot theory*]{} dealing with differential extensions having an infinite transcendence degree but a finite differential transcendence degree where the Galois group is an [*algebraic Lie pseudogroup*]{}. The achievement of the Picard-Vessiot theory by E. Kolchin and coworkers between $1950$ and $1970$ is now well known.\
The purpose of this chapter is to sketch the general theory for arbitrary partial differential extensions and algebraic Lie pseudogroups by means of new methods mixing [*differential algebra*]{}, [*differential geometry*]{} and [*algebraic geometry*]{}. As already discovered by Vessiot in $1904$ through the use of [*automorphic systems*]{}, a concept still neither known nor acknowledged, the main point is to notice that the Galois theory (old and new) is mainly a study of [*principal homogeneous spaces*]{} (PHS) for algebraic groups or pseudogroups. Hence all the modern formal theory of OD or PD equations (D.C. Spencer and coworkers around 1970) must be used together with the modern algebraic geometry missing in the work of Kolchin, in particular tensor products of rings and fields.\
However, as will be shown by means of explicit counterexamples, the combination of these new tools is not sufficient and we have to create the analogue for Lie pseudogroups of the so-called [*invariant derivations*]{} introduced by A. Bialynicki-Birula after 1960 in the study of algebraic groups and [*fields with derivations*]{}.\
After recalling the mathematical foundations of the resulting [*differential Galois theory*]{}, our main purpose will be to prove its usefulness through striking applications to mechanics, revisiting in particular [*shell theory*]{}, [*chain theory*]{}, the [*Frenet-Serret formulas*]{} and the integration of [*Hamilton-Jacobi equations*]{}.\
[**1) INTRODUCTION**]{}\
Evariste Galois died on may 311832, at the age of $21$ in a duel. Though he introduced the word “ [*group*]{} ” in mathematics for the firs time in $1830$, his work has only been known fifteen years later. Then group theory attracted more and more people, slowly passing from the field of pure algebra to the field of differential algebra, with applications ranging from the domain of pure geometry to the domain of differential geometry.\
A major step ahead has been achieved by Sophus Lie in 1880 with the introduction of [*Lie groups of transformations*]{} and, in $1890$, with the understanding that these groups of transformations were in fact only examples of a wider class, now called [*Lie pseudogroups of transformations*]{}, that is groups of transformations solutions of a system of OD or PD equations, in general non-linear and of rather high order. Let us illustrate this point of view with a few examples that will be used later on in a quite different setting. The group $y=ax+b$ of affine transformations of the real line can be considered after differentiating twice as the set of solutions of the second order OD equation $y_{xx}=0$ with standard notations, on the invertibility condition $y_x=a\neq 0$. However, it is not so evident that the group of projective transformations $y=(ax+b)/(cx+d)$ is made by the invertible solutions $y_x\neq 0$ of the third order schwarzian OD equation $(y_{xxx}/y_x) - \frac{3}{2}(y_{xx}/y_x)^2=0$ and it is a pure chance that an explicit integration can be exhibited by any student after some work. Hence the concet of [*parameter*]{} which is crucial in the first approach has no longer any meaning in the second approach. Going on this way in ${\mathbb{R}}^2$, volume preserving transformations are defined by the jacobian condition $y^1_1y^2_2-y^1_2y^2_2=1$ while complex transformations are defined by the linear Cauchy-Riemann PD equations $y^1_1-y^2_2=0, y^1_2 + y^2_1=0$ and one may introduce similarly contact transformations of ${\mathbb{R}}^3$ or symplectic transformations of ${\mathbb{R}}^4$ and so on, with no longer any parameter involved.\
With more details, the idea is to consider a group $G$ as a manifold of dimension $p$ with local coordinates $a=(a^1, ... ,a^p)$ indexed by greek indices $\rho, \sigma, \tau$, with a [*composition law*]{} $G\times G \rightarrow G:(a,b)\rightarrow ab$, an [*invers law*]{} $GÊ\rightarrow G: a \rightarrow a^{-1}$ and an identity $e \in G$ such that $a(bc)=(ab)c, ae=ea=a, aa^{-1}=a^{-1}a=e, \forall a,b,c \in G$. The group $G$ is said to [*act*]{} on a manifold $X$ with local coordinates $x=(x^1, ..., x^n)$ if there is an [*action map*]{} $X\times G \rightarrow X$ or rather its [*graph*]{} $X\times G \rightarrow X \times X:(x,a) \rightarrow (x,y=ax=f(x,a))$. The point $x$ is called the [*source*]{} of the transformation while the point $y$ is called the [*target*]{}. The action is said to be [*free*]{} if its graph is injective and [*transitive*]{} if its graph is surjective. Moreover, $X$ is said to be a [*principal homogeneous space*]{} (PHS) for $G$ if the graph is an isomorphism. The set $Gx=\{ax \mid a\in G \}$ is called the [*orbit*]{} of $x$ under $G$ and the action is said to be [*effective*]{} if $ax=x, \forall x \in X \Rightarrow a=e$.\
Let $T$ be the tangent vector bundle of vector fields on $X$, $T^*$ be the cotangent vector bundle of 1-forms on $X$ and ${\wedge}^sT^*$ be the vector bundle of s-forms on $X$ with usual bases $\{dx^I=dx^{i_1}\wedge ... \wedge dx^{i_s}\}$ where we have set $I=(i_1< ... <i_s)$. Also, let $S_qT^*$ be the vector bundle of symmetric q-covariant tensors. Moreover, if $\xi,\eta\in T$ are two vector fields on $X$, we may define their [*bracket*]{} $[\xi,\eta]\in T$ by the local formula $([\xi,\eta])^i(x)={\xi}^r(x){\partial}_r{\eta}^i(x)-{\eta}^s(x){\partial}_s{\xi}^i(x)$ leading to the [*Jacobi identity*]{} $[\xi,[\eta,\zeta]]+[\eta,[\zeta,\xi]]+[\zeta,[\xi,\eta]]=0, \forall \xi,\eta,\zeta \in T$. We have also the useful formula $[T(f)(\xi),T(f)(\eta)]=T(f)([\xi,\eta])$ where $T(f):T(X)\rightarrow T(Y)$ is the tangent mapping of a map $f:X\rightarrow Y$. Finally, we may introduce the [*exterior derivative*]{} $d:{\wedge}^rT^*\rightarrow {\wedge}^{r+1}T^*:\omega={\omega}_Idx^I \rightarrow d\omega={\partial}_i{\omega}_Idx^i\wedge dx^I$ with $I=\{i_1< ... <i_r\}$ and we have $d^2=d\circ d\equiv 0$ in the [*Poincaré sequence*]{}:\
$${\wedge}^0T^* \stackrel{d}{\longrightarrow} {\wedge}^1T^* \stackrel{d}{\longrightarrow} {\wedge}^2T^* \stackrel{d}{\longrightarrow} ... \stackrel{d}{\longrightarrow} {\wedge}^nT^* \longrightarrow 0$$ Defining the [*algebraic bracket*]{} $j_q([\xi,\eta])=\{j_{q+1}(\xi),j_{q+1}(\eta)\}$ with a slight abuse of language and recalling the [*Spencer operator*]{} $D:J_{q+1}(E)\rightarrow T^*\otimes J_q(E): f_{q+1} \rightarrow j_1(f_q)-f_{q+1}$ with $(Df_{q+1})^k_{\mu ,i}=
{\partial}_if^k_{\mu} - f^k_{\mu +1_i}$ for any vector bundle $E$, we may set $E=T$ and define a [*differential bracket*]{} of Lie algebra on $J_q(T)$ by the formula:\
$$[{\xi}_q, {\eta}_q]= \{ {\xi}_{q+1},{\eta}_{q+1}\} + i(\xi)D{\eta}_{q+1} - i(\eta)D{\xi}_{q+1}, \,\,\, \forall {\xi}_q,{\eta}_q\in J_q(T)$$ which does not depend on the lifts and where $i()$ is the standard [*interior multiplication*]{} of a $1$-form by a vector \[19,22,23\].\
We now recall two results of Lie that will be of constant use in this chapter:\
$\bullet$ [*First fundamental theorem*]{}:\
The orbits $x=f(x_0, a)$ satisfy $\partial x^i/\partial a^{\sigma}={\theta}^i_{\rho}(x){\omega}^{\rho}_{\sigma} (a)$ with $det(\omega)\neq 0$. The vector fields ${\theta}_{\rho}={\theta}^i_{\rho}(x){\partial}_i$ are called [*infinitesimal generators*]{} of the action and are linearly independent over the constants when the action is effective.\
$\bullet$ [*Second fundamental theorem*]{}:\
If $\{{\theta}_1, ... ,{\theta}_p\}$ is a basis of the infinitesimal generators of the effective action of a Lie group $G$ on $X$, then $[{\theta}_{\rho},{\theta}_{\sigma}]=c^{\tau}_{\rho\sigma}{\theta}_{\tau}$ where the $c^{\tau}_{\rho\sigma}$ are the [*structures constants*]{} of the [*Lie algebra*]{} ${\cal{G}}=T_e(G)$.\
Coming back to the work of Lie and Vessiot on what is now called [*Lie pseudogroup*]{} while [*Lie groups*]{} correspond to actions of groups, we denote by $aut(X)$ the pseudogroup of all local diffeomorphisms of $X$ and define the sub-fibered manifold ${\Pi}_q(X,X)\subset J_q(X\times X)$ by the condition $det(y^k_i)\neq 0$.\
[**DEFINITION 1.1**]{}: We may say that $\Gamma \subset aut(X)$ is a Lie pseudogroup of transformations defined by a system of OD or PD equations ${\cal{R}}_q\subset {\Pi}_q(X,X)$ with $n$ independent variables $x$ and the same number of unknowns $y$ if, whenever $y=f(x)$ and $z=h(y)$ are two local invertible transformations solutions of this system that can be composed in $aut(X)$, then $z=h\circ f (x)$ and $x=f^{-1}(y)=g(y)$ are again solutions.\
As seen on the previous examples, such a definition is totally useless in actual practice unless one can provide an explicit form for the generic solutions of the defining system of ODE or PDE which may be highly nonlinear. Let us now introduce [*two*]{} manifolds, namely the [*source manifold*]{} $X$ with $dim(X)=n$ and the [*target manifold*]{} $Y$ with $dim(Y)=m$ and local coordinates $y=(y^1, ... ,y^m)$. Hence, considering a target transformation $\bar{y}=g(y)\in \Gamma$, a map $f:X\rightarrow Y:x\rightarrow y=f(x)$ and using the chain rule for derivatives, we obtain successively ${\bar{y}}_x=\frac{\partial g}{\partial y} y_x, {\bar{y}}_{xx}=\frac{\partial g}{\partial y} y_{xx}+ \frac{{\partial}^2g}{\partial y \partial y}y_xy_x$ in a symbolic way, and so on. Accordingly, we may look for [*differential invariants*]{}, namely functions $\Phi (y,y_x,y_{xx}, ...)$ preserved by the action of $\Gamma$ on the target, that is such that $\Phi (\bar{y},{\bar{y}}_x, {\bar{y}}_{xx},...)=\Phi (y,y_x,y_{xx}, ...)$. We may also look for infinitesimal transformations of the target $y \rightarrow \bar{y}=y + t {\eta}(y) + ...$ where $t$ is a small constant parameter and extend it to ${\bar{y}}_x=y_x+t \frac{\partial \eta}{\partial y}y_x, {\bar{y}}_{xx}=y_{xx}+t (\frac{\partial \eta}{\partial y}y_{xx}+\frac{{\partial}^2\eta}{\partial y\partial y}y_xy_x),... $. If $\mu=({\mu}_1,...,{\mu}_n)$ is a multi-index with length $\mid \mu \mid={\mu}_1 + ... + {\mu}_n$ and $\mu + 1_i=({\mu}_1, ..., {\mu}_{i-1},{\mu}_i+1, {\mu}_{i+1},...,{\mu}_n)$, we may set $y_q=\{y^k_{\mu} \mid 0\leq k\leq m,0\leq \mid \mu \mid \leq q,y_0=y\}$ and introduce the following [*formal derivative*]{} on functions of $(x,y_q)$ in order to get functions of $(x,y_{q+1})$:\
$$\frac{d}{dx^i}=\frac{\partial }{\partial x^i} + y^k_{\mu +1_i} \frac{\partial }{\partial y^k_{\mu}}$$ We may thus define the $q$-prolongation ${\rho}_q(\eta)$ of the target infinitesimal transformation $\eta={\eta}^k(y)\frac{\partial}{\partial y^k}\in T(Y)$ by the formula ${\rho}_q(\eta)=d_{\mu}{\eta}^k\frac{\partial}{\partial y^k_{\mu}}$. Now, if $\Theta\subset T$ denotes the set of infinitesimal transformations of $\Gamma \subset aut(X)$, one can prove that $[R_q,R_q] \subset R_q$ when $R_q=id_q^{-1}V({\cal{R}}_q)$ if we set $id_q=j_q(id)$ for the identity transformation $y=x$. It follows that such a condition can be checked by means of computer algebra, contrary to the condition $[\Theta,\Theta]\subset \Theta$. In the formula for ${\rho}_q(\eta)$, we may replace the derivatives of $\eta$ with respect to $y$ by a section ${\eta}_q\in R_q(Y)$ and denote by $\sharp({\eta})_q$ the vertical vector in $V({\cal{R}}_q)$ obtained. One can then prove that we have the important formula:\
$$[\sharp ({\eta}_q),\sharp ({\zeta}_q)]=\sharp ([{\eta}_q,{\zeta}_q]$$ Applying the Frobénius theorem, on the resulting distribution of vertical vector fields on ${\cal{R}}_q$, we may obtain a fundamental set $\{ {\Phi}^{\tau}(y_q)$ of funtionnaly invariant [*differential invariants*]{}.
Setting now $Y=X$ and $y=x$, we may summarize the previous results as follows:\
$\bullet$ [*First fundamental result of Vessiot*]{}:\
Any source transformation commutes with any target transformation and exchanges therefore among them te differential invariants of a fundamental set. Patching coordinates, we may therefore obtain a [*natural bundle*]{} ${\cal{F}}$ over $X$ of order $q$, also called [*bundle of geometric objects*]{} of order $q$, both with a section $\omega$ in such a way that $\Gamma=\{f\in aut(X)\mid {j_q(f)}^{-1}(\omega)=\omega \}$.\
[**EXAMPLE 1.2**]{}: With $n=2, m=2$, let us introduce the manifolds $X$ with local coordinate $(x^1,x^2)$ and let $Y$ be a copy of $X$ with local coordinates $(y^1,y^2)$. We may consider the [*algebraic Lie pseudogroup*]{} $\Gamma \subset aut(X)$ of (local, invertible) transformations of $X$ preserving the $1$-form $\alpha= x^2dx^1$ and thus also the $2$-form $\beta=dx^1\wedge dx^2$, that is to say made up by transformations $y=f(x)$ solutions of the Pfaffian system $ y^2dy^1=x^2dx^1$ and thus $dy^1\wedge dy^2=dx^1\wedge dx^2$. Equivalently, we have to look for the invertible solutions of the algebraic first order involutive system ${\cal{R}}_1\subset {\Pi}_1(X,X)$ defined by the first order involutive system of algebraic PD equations in Lie form:\
$${\Phi}^1\equiv y^2y^1_1=x^2, \hspace{5mm} {\Phi}^2\equiv y^2y^1_2= 0 \hspace{5mm}\Rightarrow \hspace{5mm}{\Phi}^3\equiv \frac{\partial (y^1,y^2)}{\partial (x^1,x^2)}=y^1_1y^2_2-y^1_2y^2_1=1$$ We let the reader check that the corresponding natural bundle over $X$ is ${\cal{F}}=T^*{\times }_X{\wedge}^2T^*$ with section $\omega = (\alpha,\beta)$ and we notice that $d_1{\Phi}^1-d_2{\Phi}^1+{\Phi}^3=0$, that is $d\alpha + \beta=0$. By chance one can obtain the generic solution $y^1=f(x^1), \hspace{2mm} y^2=x^2/(\partial f(x^1)/\partial x^1)$ where $f(x^1)$ is an arbitrary (invertible) function of one variable. The linearized system over the target $Y$ is:\
$$y^2\frac{\partial {\eta}^1}{\partial y^1}+{\eta}^2=0, \frac{\partial {\eta}^1}{\partial y^2}=0, \frac{\partial {\eta}^1}{\partial y^1}+ \frac{\partial {\eta}^2}{\partial y^2}=0$$ Now, with $n=1,m=2$, introducing a manifold $X$ of dimension $n=1$ and a [*different*]{} manifold $Y$ of dimension $2$ with a map $y=f(x)$ while considering the corresponding transformations of the jets $(y^1,y^2,y^1_x,y^2_x, ...)$, we obtain the distribution generated by:\
$$\{ {\theta}_1= \frac{\partial}{\partial y^1}, \,\, {\theta}_2= y^2\frac{\partial}{\partial y^2}-y^1_x\frac{\partial}{\partial y^1_x}+y^2_x\frac{\partial}{\partial y^2_x}, \,\, {\theta}_3= y^1_x\frac{\partial}{\partial y^2_x} \}$$ which has the only generating differential invariant $\Phi\equiv {\bar{y}}^2 {\bar{y}}^1_x=y^2y^1_x$ because its generic rank is $3$ and it is easy to check the commutation relations $[{\theta}_1,{\theta}_2]=0, [{\theta}_1,{\theta}_3]=0, [{\theta}_2,{\theta}_3]=-2{\theta}_3$. The corresponding natural bundle with local coordinates $(x,u)$ is $T^*=T^*(X)$ because it has the transition rules $(\bar{x}=\varphi(x), \bar{u}=u/\frac{\partial (\varphi(x)}{\partial x})$.\
The next proposition is important but its proof is out of the scope of this book \[22\]:\
[**PROPOSITION 1.3**]{}: For any function $\Phi \in {\Pi}_q(X,X)$ we have:\
$$\sharp({\eta}_{q+1}) d_i\Phi=d_i(\sharp({\eta}_q).\Phi) - y^k_i\sharp(D{\eta}_{q+1}(\frac{\partial}{\partial y^k})).\Phi$$
If $\Phi$ is a differential invariant at order $q$ and ${\eta}_{q+1}\in R_{q+1}$, then $D{\eta}_{q+1}\in T^*\otimes R_q$ over the target and the right member vanishes, that is $d_i\Phi$ is a differential invariant at order $q+1$. Consider now a maximal number of formal linear combinations of the $d_i{\Phi}^{\tau}$ that do not contain jets of strict order $q$. We can always suppose that they begin with a leading term equal to $1$ and we may apply $\sharp(R_{q+1})$ in order to find a contradiction unless the other coefficients of the combinations are killed by $\sharp(R_q)$ and are thus only functions of the $\Phi$, a result leading to identities of the symbolic form:\
$$I(j_1(\Phi))\equiv A(\Phi) d\Phi +B(\Phi)=0$$ Taking now the reciprocal images with respect to the corresponding sections when $Y$ is a copy of $X$, we obtain the vector bundles:\
$$T=id^{-1}(V(X\times Y), \,\,\, R_q=id_q)^{-1}(V({\cal{R}}_q), \,\,\, F={\omega}^{-1}(V({\cal{F}}))$$ and an operator:\
$${\cal{D}}: T \Rightarrow F: \xi \rightarrow {\cal{L}}(\xi)\omega= \frac{d}{dt}j_q(exp(t\xi))^{-1}(\omega)\mid_{t=0}=\Omega$$ where ${\cal{L}}(\xi)$ is the [*Lie derivative*]{} of a geometric object with respect to a vector field $\xi$. We notice that ${\cal{D}}$ is a [*Lie operator*]{} in the sense that ${\cal{D}}\xi=0, {\cal{D}}\eta=0 \Rightarrow {\cal{D}}[\xi,\eta]=0$.\
Finally, introducing the vector bundle $J_q(T)$ of $q$-jets of $T$ with sections ${\xi}_q$ over $\xi$ transforming like $j_q(\xi)$, we have $F=J_q(T)/R_q$ and we may introduce its [*Medolaghi form*]{} \[19,22\]:\
$${\Omega}^{\tau}\equiv - L^{\tau\mu}_k(\omega (x)){\xi}^k_{\mu} + {\xi}^r{\partial}_r{\omega}^{\tau}(x)=0$$ with $0< \mid \mu \mid \leq q$, a result showing that the coefficients of ${\cal{D}}$ only depend on $\omega$ but the last, exacily like the standard Lie derivative of tensors. In particular, if $\Gamma$ contains the translations $\xi=cst$, then $\omega$ [*must*]{} be locally constant.\
$\bullet$ [*Second fundamental result of Vessiot*]{}:\
The map ${\pi}^{q+1}_q:R_{q+1} \rightarrow R_q$ is surjective for a [*general section*]{} $\omega$ of ${\cal{F}}$ if and only if it is indeed surjective for the [*special section*]{} and if the [*Vessiot structure equations*]{}:\
$$I(j_1(\omega))=c(\omega)$$ are satisfied, where the section $\omega \rightarrow c(\omega)$ is invariant under any diffeomorphism and only depends on a certain number of constants called [*structure constants*]{} satisfying purely algebraic equations $J(c)=0 $ called [*Jacobi conditions*]{} by analogy with the case of Lie groups, even though there is no Lie algebra structure behind.\
[**EXAMPLE 1.4**]{}: In the preceding example we have $d\alpha=-\beta$ when $\alpha=x^2dx^1,\beta=dx^1\wedge dx^2$ and thus one structure constant $c=-1$ only. However, we may choose $\alpha=dx^1, \beta=dx^1\wedge dx^2$ a choice leading to $c=0$ and to the new (non-isomorphic) pseudogroup $y^1=x^1+a, y^2=x^Ž+g(x^1)$. Moreover, we have proved in \[26\] that, in the case of a Riemann structure, that is when ${\cal{F}}=S_2T^*$ and $\omega$ is such that $det(\omega)\neq 0$, there are indeed [*two*]{} structure constants but $J(c)$ is linear in such a way that they must be equal and there is finally the [*only*]{} constant of the constant Riemann curvature condition. We do not believe that such a result or even situation is known.\
We have explained the deep contribution of Vessiot to the formal theory of Lie pseudogroups, made as early as in 1903 \[19,22,26\] but still neither known nor acknowledged today, and we are in position to sketch the other important contribution of Vessiot to the differential Galois theory, made as early as in 1904 \[29\] but still neither known nor acknowledged today.\
First of all, a group of permutation can be represented as a group of matrices with entries equal to $0$ or $1$, having a single $1$ in each row or column. For example, the group ${\mathfrak{S}}_2$ of permutations in $2$ variables is $\{(12)\rightarrow (12), (12) \rightarrow (21)\}$, the first matrix is the identity $2\times 2$ matrix while the second is the anti-diagonal $2\times 2$ matrix. Then it is well known that any algebraic group can be realized by a linear algebraic group of matrices. Also, an algebraic pseudogroup must be defined by a system of algebraic OD or PD equations, the above permutation group being defined by the algebraic equations $( y^1 +y^2=x^1 + x^2, y^1y^2=x^1x^2)$. However, the pseudogroup defined by the condition that the Jacobian matrix should be of the form:\
$$\left ( \begin{array}{ccc}
1 & 0 & 0 \\
A & 1 & 0 \\
0 & 0 & e^A
\end{array} \right )$$ is [*not*]{} an algebraic pseudogroup because one of the defining OD equations is easily seen to be $ y^3_3 - \exp(y^2_1)=0$. We have the inclusions:\
$$finite \,\,\, groups \,\,\, \subset \,\,\, algebraics \,\; groups \, \, \, \subset \,\,\, algebraic \,\,\, pseudogroups \,\,\,$$
Now, if $K\subset L$ are fields, then $L$ can be considered as a vector space over $K$ and we shall set $\mid L/K\mid=dim_K(L)$. Also,if the extension $L/K$ is not an algebraic extension, that is if $L$ contains elements which are not algebraic over $K$, that is which are not roots of a polynomial with coefficients in $K$, then the maximum number of algebraically independent such elements is well defined, only depends on $L/K$ and is called the [*transcendence degree*]{} $trd(L/K)$ of $L/K$. More generally, a [*differential integral domain*]{} $A$ is a ring without any divisor of zero and with $n$ commuting derivations ${\partial}_1,...,{\partial}_n$ such that ${\partial}_i(a+b)={\partial}_ia + {\partial}_i b, {\partial}_i(ab)=({\partial}_ia)b + a {\partial}_ib, \forall a,b\in A, \forall i=1,...,n$. Such a definition can easily be extended in order to define a [*differential field*]{} by introducing the field $K=Q(A)$ of quotients of $A$ while setting ${\partial}_i(a/b)=({\partial}_ia)b-a{\partial}_ib)/b^2, \forall a,b\in A$ with $b\neq 0$. Similarly, if $L/K$ is a [*differential extension*]{}, that is if $K\subset L$ and the derivations of $K$ are induced by those of $L$, then the maximum number of elements of $L$ which are not differentially algebraic, that is which are not solutions of a differential polynomial PD equation with coefficients in $K$, is well defined and is called the [*differential transcendence degree*]{} $diff\, trd (L/K)$. The first basic idea of Vessiot has been first to establish a kind of “[*classification*]{} ” of the [*differential Galois theory*]{}, namely:\
$\bullet$ CLASSICAL GALOIS THEORY : $\mid L/K \mid < \infty $.\
systems of algebraic equations $\leftrightarrow$ finite groups.\
$\bullet$ PICARD-VESIOT THEORY: $trd(L/K)< \infty, diff\, trd(L/K)=0$.\
Systems of algebraic OD or PD equations $\leftrightarrow$ algebraic groups.\
$\bullet$ DRACH-VESSIOT THEORY: $trd(L/K)=\infty, diff\,trd(L/K)<\infty$.\
Systems of algebraic PD equations $\leftrightarrow$ algebraic pseudogroups.\
The next crucial step is to establish a specific link between the systems and the groups and this will be the heart of this chapter. In order to sketch the underlying idea in the most elementary way, let us review for a few lines the classical Galois theory and see how one could add the word “[*differential*]{} ” in front of the concepts. If $L/K$ is an algebraic extension, we denote by $iso(L/K)$ the set (care) of isomorphisms $\varphi:L\rightarrow M$ of $L$ into another field $M$ containing $K$ and such that $\varphi(a)=a, \forall a\in K$ and $\varphi(a+b)=\varphi(a) + \varphi(b), \varphi(ab)=\varphi(a)\varphi(b), ,\forall a,b\in L$. We denote by $aut(L/K)$ the group of automorphisms of $L$ fixing $K$ and by $inv(\Gamma)$ the subfield of $L$ fixed by a group $\Gamma \subset aut(L/K)$ with $\mid \Gamma \mid$ elements. Classical Galois theory deals with [*Galois extensions*]{} and we recall the following three equivalent definitions that can be found in any textbook \[1,28\]:\
[**DEFINITION 1.5**]{}:\
1) $L/K$ is a Galois extension if $iso(L/K)=aut(L/K)=\Gamma$ with $\mid \Gamma \mid = \mid L/K\mid$.\
2 $L/K$ is a Galois extension if $inv(\Gamma)=K$ with $\Gamma = aut(L/K)$.\
3) $L/K$ is a Galois extension if $L$ is the [*splitting field*]{} of an irreducible polynomial with coefficients in $K$, that is $L$ is obtained by ajoining to $K$ all the roots of such a polynomial which can be thereore decomposed into linear factors over $L$.\
Once these definitions are assumed, the two main results of the classical Galois theory useful for applications are the following [*fundamental theorem*]{} and its corollary:\
[**THEOREM 1.6**]{}: When $L/K$ is a Galois extension, there is a bijective order reversing [*Galois correspondence*]{} between intermediate fields $K\subset K' \subset L$ and subgroups $id \subset{\Gamma}'\subset \Gamma=\Gamma(L/K)$ given by:\
$$K' \longrightarrow {\Gamma}'=aut(L/K'), \hspace{2cm} {\Gamma}' \longrightarrow K'=in({\Gamma}')$$
[**COROLLARY 1.7**]{}: Let $L/K$ be a Galois extension and $M$ be an arbitrary extension of $K$. If $L$ and $M$ are contained in a bigger field $N$ and we denote by $(L,M)$ the [*composite field*]{} of $L$ and $M$ in $N$, that is the smallest subfield of $N$ containing both $L$ and $M$, then $(L,M)/M$ is a Galois extension and there is an isomorphism $\Gamma((L,M)/M)\simeq \Gamma(L/(L\cap M))$. Moreover, $L$ and $M$ are linearly disjoint over $L\cap M$ in $(L,M)$.\
[**REMARK 1.8**]{}: The additional well known result saying that $K'/K$ is again a Galois extension if and only if ${\Gamma}'\lhd \Gamma$, that is ${\Gamma}'$ is a [*normal subgroup*]{} of $\Gamma$ will not be considered here because the study of the [*normalizer*]{} of a Lie pseudogroup $\Gamma$ in $aut(X)$ is one of the most difficult problem to be found in the formal theory of Lie pseudogroups \[19,26\].\
The first and second previous properties cannot be extended because there is no reason at all that the trnsformations of the Lie group/pseudogroup of invariance of the system do respect [*any kind*]{} of extension. As for the last definition, a system of OD or PD equations has in general an infinite number of solutions that cannot even be explicitly described or added in general. In particular, we should like to strongly react against the abstract fashion a few people are using "[*universal extension*]{}, a kind of huge reserve into which one ould put all the solutions of all systems of algebraic OD or PD equations , exactly like people use to do in in algebra with the field of complex numbers through the so-called [*fundamental theorem of algebra*]{} \[14,15,27\].\
Before providing the striking answer given by Vessiot, let us examine the auxiliary though preliminary problem of how to define a [*differential extension*]{} by relating it only to the formal theory of systems of OD or PD equations. If $K$ is a differential field as above and $(y^1,...,y^m)$ are indeterminates over $K$, we transform the polynomial ring $K\{y\}={lim}_{q\rightarrow \infty}K[y_q]$ into a differential ring by introducing as usual the [*formal derivations*]{} $d_i={\partial}_i+y^k_{\mu+1_i}\partial/\partial y^k_{\mu}$ and we shall set $K<y>= Q(K\{y\})$ as field of quotients.\
[**DEFINITION 1.9**]{}: We say that $\mathfrak{a}\subset K\{y\}$ is a [*differential ideal*]{} if it is stable by the $d_i$, that is if $d_ia\in\mathfrak{a}, \forall a \in \mathfrak{a}, \forall i=1,...,n$. We shall also introduce the [*radical*]{} ideal $rad(\mathfrak{a})=\{a\in A\mid \exists r,a^r\in \mathfrak{a}\}\supseteq \mathfrak{a}$ and say that $\mathfrak{a}$ is a [*perfect*]{} (or [*radical*]{}) differential ideal if $rad(\mathfrak{a})=\mathfrak{a}$. We say that $\mathfrak{p}\subset K\{y\}$ is a [*prime*]{} differential ideal if it is a prime ideal and a differential ideal. If $S$ is any subset of a differential ring $A$, we shall denote by $\{S\}$ the differential ideal generated by $S$ and introduce the (non-differential) ideal ${\rho}_r(S)=\{ d_{\nu}a \mid a\in S, 0 \leq \mid\mu\mid \leq r\}$ in $A$.\
[**LEMMA 1.10**]{}: If $\mathfrak{a}\subset A$ is differential ideal, then $rad(\mathfrak{a}) $ is a differential ideal containing $\mathfrak{a}$.\
[*Proof*]{}: If $d$ is one of the derivations, we have $ a^{r-1}da=\frac{1}{r}da^r \in \{a^r\}$ and thus:\
$$(r-1) a^{r-2}(da)^2 + a^{r-1}d^2a \in \{a^r\}\Rightarrow a^{r-2}(da)^3\in \{a^r\},... \Rightarrow (da)^{2r-1} \in \{a^r\}$$ Q.E.D.\
We shall say that a differential extension $L=Q(K\{y\}/\mathfrak{p})$ is a [*finitely generated*]{} differential extension of $K$ and we may define the [*evaluation epimorphism*]{} $K\{y\} \rightarrow K\{\eta \}\subset L$ with kernel $\mathfrak{p}$ where $\eta$ or $\bar{y}$ is the residual image of $y$ modulo $\mathfrak{p}$. In particular, the following Lemma will be used in the next important Theorem:\
[**LEMMA 1.11**]{}: If $\mathfrak{p}$ is a prime differential ideal of $K\{y\}$, then, for $q$ sufficiently large, there is a polynomial $P\in K[y_q]$ such that $P\notin {\mathfrak{p}}_q$ and :\
$$P{\mathfrak{p}}_{q+r} \subset rad ({\rho}_r({\mathfrak{p}}_q)) \subset {\mathfrak{p}}_{q+r}, \hspace {1cm} \forall r\geq 0$$
[**THEOREM 1.12**]{}: ([*Primality test*]{}) Let ${\mathfrak{p}}_q\subset K[y_q]$ and ${\mathfrak{p}}_{q+1}\subset K[y_{q+1}]$ be prime ideals such that ${\mathfrak{p}}_{q+1}={\rho}_1({\mathfrak{p}}_q)$ and ${\mathfrak{p}}_{q+1}\cap K[y_q]={\mathfrak{p}}_q$. If the symbol $g_q$ of the algebraic variety ${\cal{R}}_q$ defined by ${\mathfrak{p}}_q$ is $2$-acyclic and if its first prolongation $g_{q+1}$ is a vector bundle over ${\cal{R}}_q$, then $\mathfrak{p}={\rho}_{\infty}({\mathfrak{p}}_q)$ is a prime differential ideal with $\mathfrak{p} \cap K[y_{q+r}]={\rho}_r({\mathfrak{p}}_q), \forall r\geq 0 $.\
[**EXAMPLE 1.13**]{}: With $n=2$ and $K=\mathbb{Q}$, let us consider the two differential polynomials $P_1\equiv y_{22}-\frac{1}{3}(y_{11})^3, P_2\equiv y_{12}- \frac{1}{2}(y_{11})^2$ in $K\{y\}$. We have $d_2P_2-D_1P_1-y_{11}d_1P_2\equiv 0$ and the differential ideal $\mathfrak{p}=\{P_1,P_2\}\subset K\{y\}$ is prime as we have indeed:\
$$K\{y\}/ \mathfrak{p} \simeq K[y,y_1,y_2, y_{11}, y_{111},...]$$
[**COROLLARY 1.14**]{}: Every perfect differential ideal of $\{y\}$ can be expressed in a unique way as the non-redundant intersection of a finite number of prime differential ideals.\
[**COROLLARY 1.15**]{}: ([*Differential basis*]{}) If $\mathfrak{r}$ is a perfect differential ideal of $K\{y\}$, then we have $\mathfrak{r}=rad({\rho}_{\infty} ({\mathfrak{r}}_q))$ for $q$ sufficiently large.\
[**PROPOSITION 1.16**]{}: If $\zeta$ is differentially algebraic over $K<\eta>$ and $\eta$ is differentially algebraic over $K$, then $\zeta$ is differentially algebraic over $K$. Setting $\xi=\zeta - \eta$, it follows that, if $L/K$ is a differential extension and $\xi,\eta \in L$ are both differentially algebraic over $K$, then $\xi + \eta$, $\xi\eta$ and $d_i\xi$ are differentially algebraic over $K$.\
If $L=Q(K\{y\}/\mathfrak{p})$, $M=Q(K\{z\}/\mathfrak{q})$ and $N=Q(K\{y,z\}/\mathfrak{r})$ are such that $\mathfrak{p}=\mathfrak{r}\cap K\{y\}$ and $\mathfrak{q}=\mathfrak{r}\cap K\{z\}$, we have the two towers $K\subset L\subset N$ and $K\subset M\subset N$ of differential extensions and we may therefore define the new tower $K \subseteq L\cap M \subseteq <L,M> \subseteq N$. However, if only $L/K$ and $M/K$ are known and we look for such an $N$ containing both $L$ and $M$, we may use the universal property of tensor products an deduce the existence of a differential morphism $L{\otimes}_KM\rightarrow N$ by setting $d(a\otimes b)=(d_La) \otimes b+a \otimes (d_Mb)$ whenever $d_L\mid K=d_M\mid K=\partial$. The construction of an abstract [*composite*]{} differential field amounts therefore to look for a prime differential ideal in $L{\otimes}_K M$ which is a direct sum of integral domains \[20,30\].\
[**DEFINITION 1.17**]{}: A differential extension $L$ of a differential field $K$ is said to be [*differentially algebraic*]{} over $K$ if every element of $L$ is differentially algebraic over $K$. The set of such elements is an intermediate differential field $K' \subseteq L$, called the [*differential algebraic closure*]{} of $K$ in $L$. If $L/K$ is a differential extension, one can always find a maximal subset $S$ of elements of $L$ that are differentially transcendental over $K$ and such that $L$ is differentially algebraic over $K<S>$. Such a set is called a [*differential transcedence basis*]{} and the number of elements of $S$ is called the [*differential transcendence degree*]{} of $L/K$.\
[**THEOREM 1.18**]{}: The number of elements in a differential basis of $L/K$ does not depent on the generators of $L/K$ and his value is $difftrd(L/K)=\alpha$. Moreover, if $K\subset L \subset M$ are differential fields, then $difftrd(M/K)=difftrd(M/L) + difftrd(L/K)$.\
[**THEOREM 1.19**]{}: If $L/K$ is a finitely generated differential extension, then any intermediate differential field $K'$ between $K$ and $L$ is also finitely generated over $K$.\
We shall now slightly transform the third property of Definition $1.4$ into:\
4) $L/K$ is a Galois extension if $L{\otimes}_KL\simeq L \oplus ... \oplus L$ with $\mid L/K \mid$ terms.\
However, we have $L\oplus ... \oplus L =L{\otimes}_{\mathbb{Q}} (\mathbb{Q} \oplus ... \oplus \mathbb{Q} )$ with $\mid L/K \mid$ terms in the direct sum of fields equal to $\mathbb{Q}$ which is isomorphic to $\mathbb{Q}[\Gamma]$ because $\Gamma$ is a group of permutations which splits entirely over $\mathbb{Q}$ as we have already seen by exhibiting square invertible matrices with coefficients equal to $0$ or $1$ only. As a byproduct, we obtain the isomorphism:\
$$L{\otimes}_{\mathbb{Q}} L\simeq L {\otimes }_k k[\Gamma]$$ where each member is a direc sum of fields \[4,20,30\].\
Let finally $k\subset K \subset L$ be fields and consider an irreducible algebraic set $X$ or [*variety*]{} defined over $K$ by a prime ideal $\mathfrak{p}\subset K[y]$. We may denote as usual by $K[X]=K[y]/\mathfrak{p}$ the ring of polynomial functions on $X$ [*which is an integral domain*]{} and introduce its field of quotients $L=K(X)=Q(K[y]/\mathfrak{p})$. We shall say that $X$ is the [*model variety*]{} of the extension. Similarly, if $G$ is an algebraic group defined on $k$, we denote by $k[G]$ the ring of polynomial functions on $G$. Accordingly, if $X$ is a PHS for $G$, we have $X\times X\simeq X\times G$ and obtain therefore the [*fundamental isomorphism*]{} also called [*Hopf duality*]{}:\
$$Q(L{\otimes}_KL) \simeq Q(L{\otimes}_kk[\Gamma])$$ where both members are direct sums of fields. It follows that classical Galois theory is a theory of algebraic PHS and we thus obtain the key idea of Vessiot obtained as early as in 1904 in a clever paper where each chapter is studying the previous classification:\
DIFFERENTIAL GALOIS THEORY IS A THEORY OF DIFFERENTIAL ALGEBRAIC PHS FOR ALGEBRAIC PSEUDOGROUPS.\
Of course the usual definition of of a PHS saying that, if $y=f(x)$ and $\bar{y}=\bar{f}(x)$ are two solutions of the defining system of algebraic OD or PD equations, then there exists one and only one transformation $\bar{y}=g(y)\in \Gamma$ such that $\bar{f}=g\circ f$ is [*totally useless*]{} in actual practice, thoug it can be checked sometimes. Nevertheless, we may state with Vessiot:\
[**DEFINITION 1.20**]{}: A system of OD or PD equations having such an above property is called an [*automorphic system*]{} for $\Gamma$.ÊThe corresponding differential extension is called a [*differential automorphic extension*]{}.\
[**REMARK 1.21**]{}: Contrary to the situation existing in he classical Galois theory, the irreducible components of a PHS may not be themselves PHS for subgroups. We notice that the equation $y^4+1=0$ defines a PHS for the group $\{ \bar{y}=\epsilon y\mid {\epsilon}^4=1 \}$ made up by the four roots of unity $(\epsilon=1,i,-1,-i)$. However, we have the identity $y^4+1\equiv (y^2-2y+1)(y^2+2y+1)$.\
[**EXAMPLE 1.22**]{}: Coming back to example $1.2$ with $n=1,m=2$, the defining system $y^2y^1_x=\omega \in K$ is an automorphic system for the algebraic pseudogroup $\Gamma=\{{\bar{y}}^1=g(y^1), {\bar{y}}^2= y^2/(\partial g/ \partial y^1)\}$. Indeed, giving $y^1=f^1((x)$, we get $y^2=\omega / {\partial}_xf^1$ provided the derivative is non-zero. Hence we can get $x=h(x^1)$ by the implicit function theorem and set ${\bar{y}}^1= \bar{f}(h(y^1))=g(y^1)$. However, the new system $y^2y^1_x - y^1y^2_x=\omega \in K$ cannot be integrated in an explicit way and it does not seem evident to prove that it is an automorphic system for the algebraic pseudogroup preserving the $1$-form $y^2dy^1-y^1dy^2$ and the $2$-form $dy^1\wedge dy^2$. There is no reason to tansform also “$x$” which does not appear explicitly in $K$. Finally, if we set:\
$$K=\mathbb{Q}<y^2y^1_x> \subset K'= \mathbb{Q}<y^2y^1_x, y^2_x> \subset L=\mathbb{Q}<y^1,y^2>$$ we get at once the subpseudogroup ${\Gamma}'=\{{\bar{y}}^1=y^1 + a, {\bar{y}}^2=y^2\}$ preserving $K'$ with $a=cst$ and cannot obtain a Galois correspondence because it leaves invariant the intermediate differential field $K"=\mathbb{Q}<y^2y^1_x, y^2>$ which strictly contains $K'$. On the contrary, if we choose now $K'=\mathbb{Q}<y^2y^1_x, y^2_x/y^2>$, we get ${\Gamma}'=\{{\bar{y}}^1=ay^1+b, {\bar{y}}^2=(1/a)y^2$. Hence there are only two possible ways to escape from such a contradiction: one is to say that the Galois correspondence does not exist in the differential Galois theoy while the other is to say that not all intermediate differential fields can be chosen.\
[**EXAMPLE 1.23**]{}: ([*Picard-Vessiot*]{}) As another reason not to believe in the Picard-Vessiot theory of Kolchin and others, let us prove that it cannot even allow to study the simplest second order OD equation $y_{xx}=0$. For this, following Vessiot, let us copy twice this equation in order to obtain the automorphic system $P_1\equiv y^1_{xx}=0, P_2 \equiv y^2_{xx}=0$ for the action of the linear group $GL(2)$ with ${\bar{y}}^1=ay^1+by^2, {\bar{y}}^2=cy^1+dy^2$ where $a,b,c,d=cst$. With $K=\mathbb{Q}$, we have $L=Q(K\{y^1\}/\{P_1,P_2\})\simeq K(y^1,y^2, y^1_x,y^2_x)$ which is a differential field with $d_xy^k=y^k_x,d_xy^k_x=0$ providing therefore a differential automorphic extension $L/K$ with $trd(L/K)=4$. Choosing the intermediate ifferential field $K'=K(y^1_x)$, we obtain easily the subgroup ${\Gamma}'$ of $GL(2)$ defined by ${\bar{y}}^1=y^1, {\bar{y}}^2= cy^1+dy^2$ which preserves $K"=K(y^1)$ strictly containing $K'$. Finally, choosing $K'=K(y^1y^2_x - y^2y^1_x) $, we get ${\Gamma}'=SL(2)$ and thus $K"=K'$.\
Let us consider now a quite different problem brought by the above examples. Indeed, for certain groups or subgroups, we have found Lie groups with constant parameters, namely $a=cst$ in the first and $a,b,c,d=cst$ in the second. such a result has no formalmeaning because no explicit integration can be achieved in general. Hence, setting $C=cst(K)=\{a\in K\mid {\partial}_ia=0, \forall i=1,...,n\}$ for the [*subfield of constants*]{} of $K$, the question is now where to find these constant parameters. The next result will prove the confusion that has been done on the concept of constants. When the PHS $X$ for a Lie group $G$ is $G$ tself, it has been first discovered by A. Bialynicki-Birula in $1960$ and presented in two very difficult papers \[3,4\].\
[**THEOREM 1.24**]{}: ([*PHS revisited*]{}) The group parameters are constant on $X \times X$.\
[*Proof*]{}: Let $X$ be PHS for $G$ and consider the [*graph isomorphism*]{} $X\times G \simeq X \times X$ under the action of $G$ on $X$ already considered. We have $dim(X)=dim(G) \Rightarrow n=p$ and thus:\
$$det \left ( \begin{array}{cc}
1 & 0 \\
\frac{\partial f}{\partial x} & \frac{\partial f}{\partial a}
\end{array} \right)
= det(\frac{\partial f}{\partial a}) \neq 0$$ Using this isomorphism, we may exhibit $p$ functions $a=\varphi (x,y)$ when $dim(G)=p$ in such a way to have the $dim(X)=n$ identities $y\equiv f(x,\varphi(x,y))$. Let $\delta={\xi}^i(x)\frac{\partial}{\partial x^i}\in \Delta$ be a transformation commuting with all the infinitesimal generators $\theta \in \Theta$ of the action and extend $\delta$ to $X\times X$ or rather $X\times Y$ where $Y$ is a copy of $X$ by setting anew $ \delta={\xi}^i(x)\frac{\partial}{\partial x^i} + {\xi}^k(y)\frac{\partial}{\partial y^k}$. Applying to the above identity, we obtain the formula:\
$${\xi}^k(y)={\xi}^i(x)\frac{\partial f^k}{\partial x^i}(x,a) + (\delta.a^{\tau})\frac{\partial f^k}{\partial a^{\tau}}(x,a)$$ whenever $a=\varphi(x,y)$. Then $\delta$ commutes with the action, that is $[\delta,\theta]=0$ and thus $\xi (y) = \xi(x)\frac{\partial f}{\partial x}$. But we have $n=p$ and thus $ det(\frac{\partial f}{\partial a})\neq 0 \Rightarrow \delta . a=0 $. Finally, as $[\Delta, \Theta]=0$ and $\Phi$ is an ivariant of thaction, we get $\delta ( \theta \Phi) - \theta (\delta \Phi)=0 \Rightarrow \theta (\delta \Phi)=0$ and thus $\delta \Phi$ [*must*]{} also be invariant by $\Theta$.\
Q.E.D.\
We discover that the group parameters must be “[*constants*]{} ” but not necessarily killed by the ${\partial}_i$ as in the standard sense of the word.\
[**COROLLARY 1.25**]{}: $k[G]\subset cst(Q(L {\otimes}_K L) \Rightarrow Q(L {\otimes}_K L)\simeq Q(L{\otimes}_k cst(Q(L {\otimes}_K L)$. where the first $L$ in the right member is identified with $L\otimes 1$.\
[**EXAMPLE 1.26**]{}: Let us consider the affine group of the real line and set $z=b^1y+b, y=a^1x + a^2\Rightarrow z=(b^1a^1)x + (b^1a^2+b^2)$ in order to get the group law $b=(b^1,b^2), a=(a^1,a^2) \Rightarrow \bar{a}=ba \Rightarrow ({\bar{a}}^1=b^1a^1,{\bar{a}}^2=b^1a^2+b^2)$. With $K=\mathbb{Q}\subset K'=\mathbb{Q}(a^2/a^1)\subset \mathbb{Q}(a^1,a^2)=L$ we obtain $G'$ by setting $b^2=0$ and get again $K'$. However, with the new $K'=\mathbb(Q)(a^1a^2)$, we get for the new $G'$ the conditions $(b^1)^2=1, b^2=0$ and the invariant field $K"=\mathbb{Q}('a^1)^2,a^1a^2, (a^2)^2)$ which is strictly bigger than $K'$. In this case, we may use the reciprocal commuting left and right invariant disributions:\
$$\Theta=\{ {\theta}_1=a^1\frac{\partial}{\partial a^1} +a^2 \frac{\partial}{\partial a^2},\,\,{\theta}_2=\frac{\partial}{\partial a^2}\}$$
$$\Delta=\{{\delta}_1=a^1\frac{\partial}{\partial a^1}, \,\, {\delta}_2=a^1\frac{\partial}{\partial a^2} \}$$ and we check indeed that ${\delta}^2(a^1a^2)=(a^1)^2 \notin K'$.\
[**EXAMPLE 1.27**]{}: ([*Picard-Vessiot revisited*]{}) Prolonging the action of $GL(2)$ up to the first order jets, we get:\
$$\left( \begin{array}{cc}
a & b \\
c & d
\end{array} \right)
=
\left( \begin{array}{cc}
{\bar{y}}^1 & {\bar{y}}^1_x\\ {\bar{y}}^2 & {\bar{y}}^2_x \end{array} \right)
\left( \begin{array}{cc}
y^1 & y^1_x \\
y^2 & y^2_x
\end{array} \right)^{-1}$$ that we can write $A=\bar{M}M^{-1}$. Now, the two reciprocal distributions are:\
[ $$\Theta= \{ {\theta}_1=y^1 \frac{\partial}{\partial y^1 } + y^1_x \frac{\partial}{\partial y^1_x },
{\theta}_2 = y^1 \frac{\partial}{\partial y^2 } + y^1_x \frac{\partial}{\partial y^2_x },
{\theta}_3 = y^2 \frac{\partial}{\partial y^1 } + y^2_x \frac{\partial}{\partial y^1_x },
{\theta}_4 = y^2 \frac{\partial}{\partial y^2 } + y^2_x \frac{\partial}{\partial y^2_x } \}$$ $$\Delta=\{ {\delta}_1=y^1 \frac{\partial}{\partial y^1 } + y^2\frac{\partial}{\partial y^2 },
{\delta}_2 = y^1 \frac{\partial}{\partial y^1_x } + y^2 \frac{\partial}{\partial y^2_x },
{\delta}_3= y^1_x \frac{\partial}{\partial y^1 } + y^2_x \frac{\partial}{\partial y^2 },
{\delta}_4= y^1_x \frac{\partial}{\partial y^1_x } + y^2_x \frac{\partial}{\partial y^2_x } \}$$ ]{} We check that $\bar{\delta }M=A\delta M$ and thus $\delta A=0$ whenever $det(M)\neq 0$, that is the well known non-zero wronskian condition. We also check that each $\delta$ stabilizes $K"$ but [*not*]{} $K'$. We finally notice that the action is generically free because the rank of $\Theta$ is equal to $4$ whenever the [*wronskian condition*]{} $y^1y^2_x-y^2y^1_x \neq 0$ is satisfied.\
[**EXAMPLE 1.28**]{} Revisiting Examples 1.2 and 1.22, we obtain the reciprocal distribution at order $1$:\
$$\Delta= \{{\delta}_1= y^1_x \frac{\partial}{\partial y^1_x} + y^2_x \frac{\partial}{\partial y^2_x}, \,\, {\delta}_2= y^2 \frac{\partial }{\partial y^2_x} \}$$ Taking into account that we have a tensor product over $K$, we may thus use the identification ${\bar{y}}^2{\bar{y}}^1_x=y^2y^1_x$ in the formulas and check that each $\delta$ separately kills:\
$$\frac{\partial {\bar{y}}^1}{\partial y^1}= \frac{{\bar{y}}^1_x}{y^1_x}=\frac{y^2}{{\bar{y}}^2}, \,\, \frac{\partial{\bar{y}}^1}{\partial y^2}=0, \frac{\partial {\bar{y}}^2}{\partial y^1}= \frac{{\bar{y}}^1_x{\bar{y}}^2_x-y^1_xy^2_x}{y^1_x{\bar{y}}^1_x}= \frac{y^2{\bar{y}}^2_x - {\bar{y}}^2y^2_x}{y^2y^1_x}
, \,\, \frac{\partial {\bar{y}}^ 2}{\partial y^2}= \frac{y^1_x}{{\bar{y}}^1_x}=\frac{{\bar{y}}^2}{y^2}$$
Now, in order to apply the differential Galois theory to mechanics, we need to answer to the following important question:\
[**QUESTION 1.29**]{}: When a given linear or nonlinear system of algebraic OD or PD equations is given, how to check in a formal way, that is to say without introducing solutions, that it is an automorphic system for an algebraic pseudogroup and can thus be considered as a model differential algebraic variety for a differential automorphic extension ? .\
Among the systems of OD/PD equations with $m=n$, we shall distinguish the ones defining Lie pseudogroups. For this, let us introduce the open sub-bundle ${\Pi}_q={\Pi}_q(X,X) \subset J_q(X\times X)$ defined by the condition $det(y^k_i)\neq 0$ when ${\cal{E}}=X\times Y$ and $Y$ is a copie of $X$. We introduce the [*source projection*]{} ${\alpha}_q:{\Pi}_q \rightarrow X:(x,y_q) \rightarrow x$ and the [*target projection*]{} ${\beta}_q:{\Pi}_q \rightarrow Y:(x,y_q) \rightarrow y$ while identifying a map $f:X \rightarrow Y$ with its graph $f \rightarrow X\times X\times X$. In particular, we denote by $id:X\rightarrow X\times X \times X:x\rightarrow'x,x)$ is called the [*identity map*]{}.\
[**DEFINITION 1.30**]{}: A fibered manifold ${\cal{R}}_q\subset {\Pi}_q$ is called a system of [*finite Lie equations*]{} or a [*Lie groupoid*]{} of oder $q$ if we have an induced [*source projection*]{} ${:alpha}_q:{\cal{R}}_q \rightarrow X$ an induced [*target projection*]{} ${\beta}_q: {\cal{R}}_q \rightarrow X$, an induced [*identity*]{} $j_q(id)=id_q$ as a section $X \rightarrow {\cal{R}}_q$, an induced [*composition*]{} ${\gamma}_q:{\cal{R}}_q {\times}_X {\cal{R}}_q$ where the fibered product is taken with respect to the target projection on the left and to the source projection on the right, both with an induced [*inversion*]{} ${\iota}_q{\cal{R}}_q \rightarrow {\cal{R}}_q$ exchanging source and target. The set $\gamma\subset aut(X)$ of (local) solutions of ${\cal{R}}_q$ is called a [*Lie pseudogroup of order q*]{}. We shall suppose that ${\cal{R}}_q$ is [*transitive*]{}that is the projectin $({\alpha}_q, {\beta}_q):{\cal{R}}_q \rightarrow X \times X$ is surjective.\
With evident notations, we set formally in a pointwise way from left to right:\
$$((x,y,\frac{\partial y}{\partial x}, ...), (y,z, \frac{\partial z}{\partial y},...)) \rightarrow (x,z,\frac{\partial z}{\partial y}.\frac{\partial y}{\partial x}, ...)$$ $$(x,y,\frac{\partial y}{\partial x}, ...) \rightarrow (y,x,(\frac{\partial y}{\partial x})^{-1},...)$$ We set $ j_q(g \circ f )=j_q(g) \circ j_q(f)$ whenever the composition is defined and $j_q(f)^{-1}=j_q(f^{-1})$.\
[**DEFINITION 1.31**]{}: Setting $y=x+t{\xi}(x)+ ...$ and liearizing, the pull-back $R_q=id_q^{-1}(V({\cal{R}}_q))\subset J_q(T)$ defines a system of [*infinitesimal Lie equations*]{} and, as we already saw, we have $[R_q,R_q] \subset R_q \Rightarrow [\Theta, \Theta ] \subset \Theta$.
When $X$ with local coordinates $(x^1,...,x^n)$ and $Y$ wit local coordinates $(y^1,...,y^m)$ and $m$ is no longer equal to $n$, we may use the preceding results, in particular the bracket on sections of $R_q(Y)$ when $\Gamma \subset aut(Y)$ is a Lie pseudogroup of transformations of $Y$ in order to find a fundamental set of generating differential invariants ${\Phi}^{\tau}(y_q)$ at order $q$. Then we know that the $d_i{\Phi}^{\tau}$ are again differential invariants at order $q+1$, though, as we shall see in many examples of mechanics, new differential invariants may be added at higher order. Accordingly, we just need a criterion in order to stop the adjonction procedure when $q$ is large enough.\
[**REMARK 1.32**]{}: In the case of an algebraic pseudogroup, the use of the Frobénius theorem may not provide rational differential invariants. However, when $m=n$, it is possible to exhibit rational differential invariants by using [*formal translation*]{} techniques first introduced by J. Drach (See \[6,9\] and \[20\], p 467 for details).\
Using the composition of jets when $m\neq n$ exactly as we did when $m=n$, we have an action morphism with graph:\
$$J_q(X \times Y){\times}_Y {\Pi}_q(Y,Y) \stackrel{(graph)}\longrightarrow J_q(X\times Y){\times}_X J_q(X \times Y)$$ and a system ${\cal{A}}_q\subset J_q(X \times Y)$ will be said to be [*invariant*]{} by the action of the Lie groupoid ${\cal{R}}_q\subset {\Pi}_q(Y,Y)$ if we have the following restricted action morphism with induced graph:\
$${\cal{A}}_q {\times}_Y {\cal{R}}_q \longrightarrow {\cal{A}}_q {\times}_X {\cal{A}}_q$$
As we shall see, the following definition and the two corresponding criteria will become crucial for the applications to mechanics.\
[**DEFINITION 1.33**]{}: ${\cal{A}}_q$ is said to be a PHS for ${\cal{R}}_q$ if the above morphism is an isomorphism. Then ${\cal{A}}_q$ is said to be an [*automorphic system*]{} for ${\cal{R}}_q$ if ${\cal{A}}_{q+r}={\rho}_r({\cal{A}}_q)$ is a PHS for ${\cal{R}}_{q+r}={\rho}_r({\cal{R}}_q), \forall r\geq 0$.\
Of course, if ${\cal{A}}_q$ is an automorphic system for ${\cal{R}}_q$, then, [*necessarily*]{}, ${\cal{A}}_q$ is a PHS for ${\cal{R}}_q$ [*and*]{} ${\cal{A}}_{q+1}$ is a PHS for ${\cal{R}}_{q+1}$ but such a double condition may not be sufficient, a result proving the importance of the following theorem (See \[20\], p 330 for the technical proof):\
[**THEOREM 1.34**]{}: ([*First criterion for automorphic systems*]{}) If an involutive system ${\cal{A}}_q \subset J_q(X\times Y)$ is a PHS for a Lie groupoid ${\cal{R}}_q \subset {\Pi}_q(Y,Y)$ and if ${\cal{A}}_{q+1}={\rho}_1({\cal{A}}_1) \subset J_q(X\times Y)$ is a PHS for the Lie groupoid ${\cal{R}}_{q+1}={\rho}_1({\cal{R}}_q) \subset {\Pi}_{q+1}(Y,Y)$, then ${\cal{R}}_q$ is an involutive system with the same non-zero characters and ${\cal{A}}_q$ is an automorphic system for ${\cal{R}}_q$.\
Similarly, we have (See \[20\], p 339 for the technical proof):\
[**THEOREM 1.35**]{}: ([*Second criterion for automorphic systems*]{}) If ${\cal{R}}_q \subset {\Pi}_q(Y,Y)$ is an involutive sytem of finite Lie equations such that the action of ${\cal{R}}_q$ on $J_q(X\times Y)$ is generically free, then the action of ${\cal{R}}_{q+r}={\rho}_r({\cal{R}}_q)$ on $J_{q+r}(X \times Y)$ is generically free and all the differential invariants are generated by a fundamental set of order $q+1$ (care to the order).\
[**EXAMPLE 1.36**]{}: With $K=\mathbb{Q}, n=1, m=2, q=2$, let $\Gamma$ be the Lie pseudogroup determined by the Lie group $G$ with $3+2=5$ parameters, defined by $\bar{y}=Ay+B$ with $det(A)=1$. We have the involutive defining system of finite Lie equations:\
$$\frac{\partial({\bar{y}}^1,{\bar{y}}^2)}{\partial (y^1,y^2)}=1, \frac{{\partial}^2\bar{y}}{\partial y\partial y}=0$$ with linearized system ${\eta}^1_1+{\eta}^2_2=0, {\eta}^k_{rs}=0, \forall k,r,s=1,2$ having a zero symbol and thus trivially involutive. The only generating differential invariant at order $2$ is $\Phi\equiv y^1_xy^2_{xx}-y^2_xy^1_{xx}$. It follows that $d_x\Phi\equiv y^1_xy^2_{xxx}-y^2_xy^1_{xxx}$ is a differential invariant at order $3$ but we have also $\Psi\equiv y^1_{xx}y^2_{xxx}-y^2_{xx}y^1_{xxx}$. The involutive distribution $\Theta$ involved has $5$ infinitesimal generators at order $2$, namely:\
$${\theta}_1= \frac{\partial}{\partial y^1}, \,\,{\theta}_2= \frac{\partial}{\partial y^2},
\,\, {\theta}_3=y^1_x\frac{\partial}{\partial y^1_x}+y^1_{xx}\frac{\partial}{\partial y¬1_{xx}} - y^2_x\frac{\partial}{\partial y^2_x} - y^2_{xx}\frac{\partial}{\partial y^2_{xx}},$$ $${\theta}^4= y^2_x\frac{\partial}{\partial y^1_x}+y^2_{xx}\frac{\partial}{\partial y^1_{xx}},
\,\, {\theta}_5= y^1_x\frac{\partial}{\partial y^2_x}+ y^1_{xx}\frac{\partial}{\partial y^2_{xx}}$$ We notice that the symbol of ${\cal{A}}_3$ vanishes if and only if $y^1_xy^2_{xx}-y^2_xy^1_{xx}\neq 0$ and, in this case, we have thus a generically free action, in agrement with the second criterion. Finally, we notice that ${\cal{A}}_2$ is a PHS for ${\cal{R}}_2$, both fibered manifolds having a fiber dimension equal to $5$ (namely $(2+2+2)-1$ for the first and $(2+4)-1$ for the second), and that ${\cal{A}}_3$ is thus an automorphic system with the same fiber dimension at any higher order.\
It is finally important to notice that the definition of prime differential ideals was not known at all by Drach and Vessiot because it has only been introduced by Ritt after 1930 (See \[20,27\] for details) and we shall correct the old definition by saying:\
[**DEFINITION 1.37**]{}: A system of algebraic PD equations is [*irreducible*]{} if it is defined by a [*prime*]{} differential ideal.\
Contrary to the classical Galois theory, the irreducible components of a PHS for a group may not be again at all PHS for subgroups. It follows that Drach, Vessiot or even Kolchin made a confusion between [*prime*]{} differential ideals and [*maximum*]{} differential ideals in order to define the Galois group of a system. Hence, the starting point of any differential Galois theory must be an irreducible automorphic system (See the introduction of \[20\] for more details). However, despite this comment, modern works on the Picard-Vessiot (PV) theory or on the differential Galois theory are based on three [*conceptual misunderstandings*]{}:\
1) [*First misunderstanding*]{}: The Galois group of a PV extension cannot be computed in general. On the other end, the system $y^1_{xx}=0,y^2_{xx}=0$ is indeed an irreducible ([*prime*]{} because [*linear*]{}) automorphic system for $GL(2)$. Hence, the Galois group of [*any*]{} linear OD equation of order $m$ must be $GL(m)$.\
2) [*Second misunderstanding*]{}: The OD equation $y_{xx}=0$ cannot be treated by Kolchin because the corresponding automorphic extension $L/K$ defined by $y^1_{xx}=0,y^2_{xx}=0$ is such that $L$ contains differential constants other than the ones of $K$, for example $y^1_x$. Hence, the definition of “[*constants*]{} ” never took into account the modern approach of Bialynicki-Birula.\
3) [*Third misunderstanding*]{}: [*Last but not least*]{}, instead of using the concept of [*algebraic pseudogroups*]{} like Drach and Vessiot, Kolchin used the concept of [*differential algebraic group*]{} initiated by Ritt \[16\].\
[**2) SHELL THEORY REVISITED**]{}\
Using the previous notations, we study first the case $n=2,m=3$ when $G$ is the group of rigid motions ($3$ translations + $3$ rotations) acting on the space $Y={\mathbb{R}}^3$ with cartesian coordinates $(y^1,y^2,y^3)$ and the space $X={\mathbb{R}}^2$ with coordinates $(x^1,x^2)$ is the parametrizing manifold for the surface considered but we shall insist on the intrinsic aspect by using [*jet theory*]{}, both fom the differential geometric and algebraic aspects. Meanwhile, we invite the reader to try to imagine how to extend the results below to arbirary $n$ and $m$.\
The starting motivation for such a study has been the clever idea of Pierre Oscar Bonnet in $1867$ (\[2\]) to look for the following vague and difficult problem:\
\
Of course, translated into the modern language of the previous sections, it just amounts to construct the corresponding [*automorphic systems*]{}, that is to exhibit a generating set of differential invariants ${\Phi}^{\tau}$, the corresponding bundle $\cal{F}$ of geometric objects and a section $\omega$ both with its compatibility conditions (CC). Meanwhile, it will be rather striking to notice that the differential Galois theory may be applied as it works in this framework because we have an algebraic Lie group of transformations, namely $\bar{y}=Ay+B$ where $A$ is an orthogonal $3\times 3$ matrix with $det(A)=1$ (see later on when such a condition is used) and $B$ is a vector. Of course, eliminating the parameters provides at once an [*algebraic Lie pseudogroup*]{} defined by the $m(m+1)/2$ differential polynomial equations in symbolic form:\
$${\delta}_{uv} \frac{\partial {\bar{y}}^u}{\partial y^k} \frac{\partial {\bar{y}}^l}{\partial y^l}={\delta}_{kl}$$ and all the second order jets vanish. The novelty of this presentation is that not only [*all*]{} known results of shell theory will appear for the first time as an effective application of a general theory with no reference to classical geometry and moving frames but other results will be obtained which are not known up to our knowledge, in particular when $m$ and $n$ are arbitrary.\
First of all, we look for the corresponding differential invariants by examining the various prolongations ${\rho}_0(\theta), {\rho}_1(\theta), {\rho}_2(\theta), {\rho}_3(\theta), ...$ of the infinitesimal generators:\
$${\theta}_1=\frac{\partial}{\partial y^1}, {\theta}_2= \frac{\partial}{\partial y^2}, {\theta}_3=\frac{\partial}{\partial y^3},$$ $${\theta}_4=y^2\frac{\partial}{\partial y^3}-y^3\frac{\partial}{\partial y^2}, {\theta}_5=y^3\frac{\partial}{\partial y^1}- y^1\frac{\partial}{\partial y^3}, {\theta}_6=y^1\frac{\partial}{\partial y^2}-y^2\frac{\partial}{\partial y^1}$$ as we need only stop when the symbol of the automorphic system is zero, according to the criteria for automorphic systems.\
Using the identity $y^1{\theta}_4+y^2{\theta}_5+y^3{\theta}^6\equiv 0$, the rank of the vertical distribution generated by the ${\rho}_0(\theta)$ on $V(X\times Y)=E$ is maximum and equal to $3$, so that there is no differential invariant of order zero and also any differential invariant cannot depend explicitly on $y$. The fiber of $V(J_1(X\times Y))\simeq J_1(V(X\times Y))=J_1(E)$ has dimension $3+6=9$ and there is therefore $9-6=3$ (rational) differential invariants of first order killed by ${\rho}_1(\theta)$, namely we get the Lie form at order $1$:\
$$\begin{array}{rcccccl}
{\Omega}_{11} & \equiv & {\sum}_k (y^k_1)^2& = & (y^1_1)^2+(y^2_1)^2+(y^3_1)^2 & = &
{\omega}_{11}(x)\\
{\Omega}_{12} & \equiv & {\sum}_k y^k_1y^k_2& = & y^1_1y^1_2+y^2_1y^2_2+y^3_1y^3_2 & = &
{\omega}_{12}(x)\\
{\Omega}_{22} & \equiv & {\sum}_k (y^k_2)^2& = & (y^1_2)^2+(y^2_2)^2+(y^3_2)^2 & = &
{\omega}_{22}(x)
\end{array}$$ or simpy:\
$${\Omega}_{ij}\equiv {\delta}_{kl}y^k_iy^l_j = {\omega}_{ij}(x)$$ It is easy to check that $\omega=({\omega}_{ij}={\omega}_{ji}) \in S_2T^* $ and $\omega$ is called the [*first fundamental form*]{} though this name can be rather confusing. Then we know that each $d_i\Omega$ is also a differential invariant and we may set:\
$${\Gamma}_{rij}=\frac{1}{2}(d_i{\Omega}_{rj}+d_j{\Omega}_{ri}-d_r{\Omega}_{ij}) \Rightarrow {\gamma}_{rij}=\frac{1}{2}({\partial}_i{\omega}_{rj}+{\partial}_j{\omega}_{ri}-{\partial}_r{\omega}_{ij})$$ in order to get the $6$ linearly and thus functionally independent second order differential invariants:\
$${\Gamma}_{rij}\equiv {\vec{y}}_r.{\vec{y}}_{ij}={\delta}_{kl} y^k_ry^l_{ij} ={\gamma}_{rij}(x)={\gamma}_{rji}(x)$$ In order to check this result, let us introduce the $2\times 3$ matrix $(y^k_i)$ of strict first order jets and consider the $3$ different $2\times 2$ subdeterminants like $y^1_1y^2_2-y^1_2y^2_1$. Then it is easy to obtain the relation:\
$$det(\omega)\equiv {\omega}_{11}{\omega}_{22} - ({\omega}_{12})^2=
{\sum}_{det} (y^1_1y^2_2-y^1_2y^2_1)^2$$ and we shall suppose from now on that $det(\omega)\neq 0$. Introducing the action and its prolongations:\
$${\bar{y}}^u=a^u_ky^k+b^u, {\bar{y}}^u_i=a^u_ky^k_i, {\bar{y}}^u_{ij}=a^u_ky^k_{ij}, ...$$ we get:\
$${\bar{y}}^1_1{\bar{y}}^2_2-{\bar{y}}^2_1{\bar{y}}^1_2= ... + (-(a^1_1a^2_3-a^2_1a^1_3))(y^3_1y^1_2-y^1_1y^3_2)+ ...$$ and obtain therefore with $A^t=transposed(A)$ and thus $A^{-1}=(1/det(A)) cof(A)^t$ by using the matrix of cofactors:\
$$( {\vec{\bar{y}}}_1 \wedge {\vec{\bar{y}}}_2)= det(A) (A^t)^{-1} ({\vec{y}}_1\wedge {\vec{y}}_2)$$ But $AA^t=I\Rightarrow A^t=A^{-1}\Rightarrow (A^t)^{-1}=A$ and, if $det(A)=1$ ([*care*]{}), that is we deal with the connected component $G$ of the identity, then:\
$$( {\vec{\bar{y}}}_1 \wedge {\vec{\bar{y}}}_2).{\vec{\bar{y}}}_{ij}= ({\vec{y}}_1\wedge {\vec{y}}_2).{\vec{y}}_{ij}$$ Accordingly, we may introduce the unexpected $3$ [*additional*]{} differential invariants:\
$${\Sigma}_{ij}\equiv (y^2_1y^3_2-y^3_1y^2_2)y^1_{ij}+(y^3_1y^1_2-y^1_1y^3_2)y^2_{ij}+(y^1_1y^2_2-y^2_1y^1_2)y^3_{ij}={\sigma}_{ij}(x)$$ and we notice that the $3\times 3$ matrix made by the factors of the second order jets $y^k_{ij}$ with $k=1,2,3$ and each $(ij)\in ((11), (12), (22))$ in $({\Sigma}_{ij},{\Gamma}_{1ij}, {\Gamma}_{2ij})$ has determinant equal to $det(\omega)$ and is thus of maximal rank $3$. It follows that the second order symbol defined by the system $(\Gamma=\gamma, \Sigma=\sigma)$ does vanish and the complete system $(\Omega=\omega, \Gamma=\gamma,\Sigma=\sigma)$ is not only automorphic but also involutive if and only if convenient [*first order*]{} generating compatibility conditions (CC) among $(\omega, \gamma, \sigma)$ are satisfied as a necessary and sufficient condition for applying the Cartan-Kähler theorem in the analytic case. The reader may compare the present approach with the one of M. Janet in (\[13\]) and with the one of P.G.Ciarlet in (\[7,11\]), always getting in mind that, in the formal theory of non-linear systems, the difficulty is to look for the various numbers of CC through techniques of acyclicity and diagram chasing which are in general very far from just doing “crossed derivatives ” (Just consider the case $n=3, m= 6$ or the many examples provided in (\[19-22,26\])).\
[*We exhibit them for the first time within the formal framework of automorphic systems, still unknown today*]{}.\
First of all, we get at once $\hspace{5mm} {\partial}_r{\omega}_{ij}= {\gamma}_{irj}+{\gamma}_{jri} $.\
As for the other CC, they are similar to the ones of a system like $y^k_{ij}=0$ with $k=1,2,3$ and $i,j=1,2$, by setting $y^1_1=1,y^2_2=1$ and the other first order jets equal to zeo, that is to say these CC are induced by the cokernel of the Spencer monomorphism $\delta: S_3T^* \otimes E\rightarrow T^*\otimes S_2T^*\otimes E$ for each $k=1,2,...,m$, that is $m(n^2(n+1)/2-n(n+1)(n+2)/6)=mn(n^2-1)/3$ which is equal to $6$ when $n=2,m=3$. Among these $6$ CC, we have the $2\times 2=4$ CC with symbols:\
$${\partial}_2{\gamma}_{i11} - {\partial}_1{\gamma}_{i12}, \hspace{3mm} {\partial}_2{\gamma}_{i12} - {\partial}_1{\gamma}_{i22}$$ coming from the ordinary Riemann tensor and it just remains to compute the $2$ [*new*]{} CC having symbols:\
$${\partial}_2{\sigma}_{11} - {\partial}_1{\sigma}_{12}, \hspace{3mm} {\partial}_2{\sigma}_{12} - {\partial}_1{\sigma}_{22}$$ in such a way that these $6$ quantities do not contain third order jets any longer. Indeed, thanks to differential algebra and the fact that the first prolongation of a nonlinear partial differential equation of order $q$ is quasi-linear in the jets of strict order $q+1$, we are thus led to a problem of pure linear algebra while studying cokernels.\
We provide details for this [*tricky*]{} computation but we know from the differential Galois theory that these $6$ CC come from differential invariants of order $2$ and [*must*]{} therefore be expressed by means of rational functions of $(\omega, \gamma, \sigma)$, a fact not at all evident at first sight as we shall see but explaining why our definion of the second form slightly supersedes the standard one of textbooks as it [*is the only one*]{} fitting with differential algebra and differential Galois theory while using only rational differential functions in $\mathbb{Q}<y^1,y^2,y^3>=\mathbb{Q}<y>$.\
Second, we get at once:\
$$d_2{\Gamma}_{111}-d_1{\Gamma}_{112}\equiv {\delta}_{kl}( y^k_{12}y^l_{11} - y^k_{11}y^l_{12})=0 \Rightarrow {\partial}_2{\gamma}_{111} - {\partial}_2{\gamma}_{112}=0$$ $$d_2{\Gamma}_{212}-d_1{\Gamma}_{222}\equiv {\delta}_{kl} (y^k_{22}y^l_{12} - y^k_{12}y^l_{22})=0 \Rightarrow {\partial}_2{\gamma}_{212} - {\partial}_2{\gamma}_{222}=0$$ Similarly, we get:\
$${\partial}_2{\gamma}_{211}-{\partial}_1{\gamma}_{212}={\partial}_1{\gamma}_{122} - {\partial}_2{\gamma}_{112} = {\vec{y}}_{11}.{\vec{y}}_{22} - {\mid} {\vec{y}}_{12}{\mid}^2$$ and obtain therefore [*only three*]{} CC [*for the only*]{} $\gamma$ by adding [*one more*]{} CC, namely:\
$${\partial}_2{\gamma}_{211}-{\partial}_1{\gamma}_{212}+{\partial}_2{\gamma}_{112} - {\partial}_1{\gamma}_{122} =0$$ a result showing that [*the other*]{} CC [*must contain*]{} $\sigma$.\
Meanwhile, we also get (See (\[19\], p 126-129\]):\
$$\frac{1}{2} ( {\partial}_{11}{\omega}_{22} + {\partial}_{22}{\omega}_{11}-2{\partial}_{12}{\omega}_{12})=
{\partial}_1{\gamma}_{212} - {\partial}_2{\gamma}_{211}= {\mid} {\vec{y}}_{12} {\mid}^2- {\vec{y}}_{11}.{\vec{y}}_{22}$$ Using the formula ${\Sigma}_{ij}\equiv ({\vec{y}}_1\wedge {\vec{y}}_2).{\vec{y}}_{ij}={\sigma}_{ij} $ and various projections in the tangent plane to the surface like $({\vec{y}}_2\wedge ({\vec{y}}_1\wedge {\vec{y}}_2)).{\vec{y}}_1=det(\omega)$, we finally obtain:\
$$\begin{array}{rcl}
det(\omega)y^k_{11} & = & {\sigma}_{11}({\vec{y}}_1\wedge {\vec{y}}_2)^k +{\gamma}_{111}
({\vec{y}}_2\wedge ({\vec{y}}_1\wedge {\vec{y}}_2))^k-{\gamma}_{211}({\vec{y}}_1\wedge ({\vec{y}}_1\wedge {\vec{y}}_2))^k \\
det(\omega)y^k_{22} & = & {\sigma}_{22}({\vec{y}}_1\wedge {\vec{y}}_2)^k +{\gamma}_{122}
({\vec{y}}_2\wedge ({\vec{y}}_1\wedge {\vec{y}}_2))^k-{\gamma}_{222}({\vec{y}}_1\wedge ({\vec{y}}_1\wedge {\vec{y}}_2))^k \\
det(\omega)y^k_{12} & = & {\sigma}_{12}({\vec{y}}_1\wedge {\vec{y}}_2)^k +{\gamma}_{112}
({\vec{y}}_2\wedge ({\vec{y}}_1\wedge {\vec{y}}_2))^k-{\gamma}_{212}({\vec{y}}_1\wedge ({\vec{y}}_1\wedge {\vec{y}}_2))^k \\
\end{array}$$ or simply:\
$$det(\omega)\,{\vec{y}}_{ij}= {\sigma}_{ij} ({\vec{y}}_1\wedge {\vec{y}}_2) +
{\gamma}_{1ij}({\vec{y}}_2\wedge ({\vec{y}}_1\wedge {\vec{y}}_2))-{\gamma}_{2ij}({\vec{y}}_1\wedge ({\vec{y}}_1\wedge {\vec{y}}_2))$$ where we recall that:\
$$det(\omega)=({\vec{y}}_2\wedge ({\vec{y}}_1 \wedge {Ê\vec{y}}_2)).{\vec{y}}_1= ({\vec{y}}_1,{\vec{y}}_2, {\vec{y}}_1 \wedge {Ê\vec{y}}_2 )=
{\mid} {\vec{y}}_1 \wedge {Ê\vec{y}}_2 {\mid }^2 = {\mid }{\vec{y}}_1{\mid}^2{\mid}{\vec{y}}_2{\mid}^2 - {\mid}{\vec{y}}_1.{\vec{y}}_2{\mid}^2$$ It follows that we have:\
$$\begin{array}{lcl}
det(\omega)\,{\vec{y}}_{11}.{\vec{y}}_{22} & = &{\sigma}_{11}{\sigma}_{22} + {\omega}_{22}{\gamma}_{111}{\gamma}_{122} +{\omega}_{11}{\gamma}_{211}{\gamma}_{222}-{\omega}_{12}({\gamma}_{111}{\gamma}_{222} + {\gamma}_{211}{\gamma}_{122}) \\
det(\omega)\,{\mid} {\vec{y}}_{12}{\mid}^2 & = & ({\sigma}_{12})^2 + {\omega}_{22}({\gamma}_{112})^2 +{\omega}_{11}({\gamma}_{212})^2- 2{\omega}_{12}{\gamma}_{112}{\gamma}_{212}
\end{array}$$ Substracting the second equation from the first and setting $det(\sigma)={\sigma}_{11}{\sigma}_{22} - ({\sigma}_{12})^2$, we get an equation of the form:\
$$det(\omega)({\vec{y}}_{11}.{\vec{y}}_{22}-{\mid} {\vec{y}}_{12} {\mid}^2)=det (\sigma) + \omega \gamma\gamma$$ where the last expression is a rational function of $j_1(\omega)\simeq (\omega,\gamma)$ according to the well known Levi-Civita isomorphism. This is [*exactly*]{} the “ [*theorema egregium*]{} ” of K.F. Gauss in (\[10\]) and, for this reason, $det(\sigma)/det(\omega)=(j_2(\omega))$ is called the [*Gauss curvature*]{} or [*total curvature*]{} and only depend on $\omega$. It is however important to notice that our definition of $\sigma$ does not involve the square root of $det(\omega)$. For this reason [*it allows to use only rational differential functions*]{} and must therefore be preferred to the standard one existing in the literature which is using the so-called [*normal vector*]{} $\vec{n}=({\vec{y}}_1\wedge {\vec{y}}_2)/\mid {\vec{y}}_1\wedge {\vec{y}}_2\mid $.\
As another motivation for such a choice, we notice that $\omega \in S_2T^*$ is usually called [*first fundamental form*]{} while $\sigma$, which is called [*second fundamental form*]{}, is also considered as a section of $S_2T^*$ but this is not correct. Indeed, for any change $\bar{x}=\varphi (x)$ of independent variables, we have successively:\
$$\frac{\partial \vec{y}}{\partial x^1}\wedge \frac{\partial \vec{y}}{\partial x^2} = (\frac{\partial \vec{y}}{\partial {\bar{x}}^1} \wedge \frac{\partial \vec{y}}{\partial {\bar{x}}^2} ) \frac{\partial ({\bar{x}}^1,{\bar{x}}^2)}{\partial (x^1,x^2)}$$ $$\frac{{\partial}^2 \vec{y}}{\partial x^i\partial x^j} = \frac{{\partial}^2 \vec{y}}{\partial {\bar{x}}^r\partial {\bar{x}}^s} \frac{ \partial {\bar{x}}^r}{\partial x^i} \frac{\partial {\bar{x}}^s}{\partial x^j} + \frac{\partial \vec{y}}{ \partial {\bar{x}}^r} \frac{{\partial}^2 {\bar{x}}^r}{\partial x^i\partial x^j}$$ with $i,j,r,s=1,2$ and we deduce that $\sigma \in S_2T^* \otimes {\wedge}^2T^*$ is in fact a [*metric density*]{}.\
As for the last two CC for $\sigma$, an easy but tedious computation does provide:\
$$\begin{array}{rcr}
det(\omega)({\partial}_2{\sigma}_{12} - {\partial}_1{\sigma}_{22}) & = & ({\gamma}_{122}\,{\omega}_{22} - {\gamma}_{222}\,{\omega}_{12}) \,{\sigma}_{11} \\
& & + (2{\gamma}_{112}\,{\omega}_{12}-2 {\gamma}_{212}\,{\omega}_{11}+{\gamma}_{211}\,{\omega}_{12}-
{\gamma}_{111}\,{\omega}_{22}) \,{\sigma}_{22} \\
& & +(2{\gamma}_{212}\,{\omega}_{12}- 2{\gamma}_{112}\,{\omega}_{22}
+2{\gamma}_{222}\,{\omega}_{11}- 2{\gamma}_{122}\,{\omega}_{12}) \,{\sigma}_{12}
\end{array}$$ and the other CC by exchanging $1$ and $2$. These equations have been first found independently by D. Codazzi (\[8\]) and G. Mainardi (\[18\]) in a classical setting, with no reference to differential algebra. It follows that, contrary to $\omega$ which can be given arbitrarily provided that $det(\omega)\neq 0$, $\sigma$ [*cannot be given arbitrarily*]{}.\
In certain cases, a known symmetry of the surface may be taken into account in order to determine explicitly $(\omega,\gamma, \sigma)$ and we provide two particular cases.\
$\bullet$ $y^1=x^1, y^2=x^2, y^3= \frac{1}{6}((x^1)^3 + (x^2)^3)$\
$${\omega}_{11}=1 + \frac{1}{4}(x^1)^4, {\omega}_{22}= 1 + \frac{1}{4}(x^2)^4, {\omega}_{12}= \frac{1}{4}(x^1x^2)^2 \Rightarrow det(\omega)= 1 +\frac{1}{4}(x^1)^4 + \frac{1}{4}(x^2)^4$$ $${\gamma}_{111}=\frac{1}{2} (x^1)^3, {\gamma}_{112}=0, {\gamma}_{212}=0, {\gamma}_{222}=\frac{1}{2}(x^2)^3, {\gamma}_{211}=\frac{1}{2}x^1(x^2)^2, {\gamma}_{122}=\frac{1}{2}(x^1)^2x^2$$ $${\sigma}_{11}= x^1, {\sigma}_{22}=x^2, {\sigma}_{12}=0 \Rightarrow det(\sigma)=x^1x^2$$ We check easily:\
$$det(\omega)x^1x^2=x^1x^2 +\frac{1}{2}{\omega}_{22}(x^1)^5x^2+\frac{1}{2}{\omega}_{11}x^1(x^2)^5 - \frac{1}{2}{\omega}_{22}(x^1x^2)^3, {\partial}_2{\sigma}_{12}-{\partial}_1{\sigma}_{22}=0$$ and all the results must remain unchanged by the permutation $1 \leftrightarrow 2$.\
$\bullet$ Again with $n=2,m=3$, let us consider the sphere of radius $R$ centered at the origin of the cartesian frame $(Oy^1y^2y^3)$ and apply a stereographic projection of the northern half sphere (draw a picture) with north pole $N$ on the equatorial plane $(Ox^1x^2)$ with coordinates $(x^1,x^2)$ from the south pole $S$. Using well known similarity of the triangles $SNM$ for a point $M$ on the sphere and $SXO$ where $X$ is the intersection of $SM$ with the equatorial plane, we get the classical formulas:\
$$r^2=(x^1)^2+(x^2)^2, L^2=\mid {\vec{SX}}^2 \mid=R^2 + r^2, {\vec{SM}}^2=D^2=(y^1)^2+(y^2)^2+ (R+y^3)^2,$$ $$LD=2R^2 \Rightarrow \frac{y^1}{x^1}=\frac{y^2}{x^2}=1+\frac{y^3}{R}=\frac{2R^2}{L^2}, \frac{y^3}{R}=\frac{R^2-r^2}{L^2}$$ and obtain the parametrization of the sphere:\
$$\frac{y^1}{R}=\frac{2Rx^1}{R^2+(x^1)^2+(x^2)^2}, \frac{y^2}{R}=\frac{2Rx^2}{R^2+(x^1)^2+(x^2)^2}, \frac{y^3}{R}=\frac{R^2-(x^1)^2-(x^2)^2}{R^2+(x^1)^2+(x^2)^2}$$ leading to:\
$${\omega}_{11}={\omega}_{22}=4R^4/(R^2+(x^1)^2+(x^2)^2)^2={\phi}, \hspace{3mm}{\omega}_{12}={\omega}_{21}=0\Rightarrow det(\omega)={\phi}^2$$ We obtain at once:\
$${\gamma}_{111}={\gamma}_{212}= - {\gamma}_{122}=\frac{1}{2}{\partial}_1\phi, \hspace{3mm} {\gamma}_{121}={\gamma}_{222}= - {\gamma}_{211}=\frac{1}{2}{\partial}_2\phi$$ and, from the previous second order CC for $\omega$, we get: $$\frac{1}{2}{\phi}^2({\partial}_{11}\phi + {\partial}_{22}\phi)= - det(\sigma) + \frac{1}{2}\phi (({\partial}_1\phi)^2 + ({\partial}_2\phi)^2)$$ An easy computation then leads to:\
$$det(\sigma)=(16R^7/(R^2 + (x^1)^2 + (x^2)^2)^4)^2=\frac{1}{R^2}{\phi}^4$$ Finally, though we have indeed:\
$${\sigma}_{ij}=det \left( \begin{array}{ccc}
y^1_1 & y^1_2& y^1_{ij} \\
y^2_1 & y^2_2 & y^2_{ij} \\
y^3_1 & y^3_2 & y^3_{ij}
\end{array} \right)$$ a direct computation is very tedious and we prefer to use the relations ${\omega}_{11}={\omega}_{22}=\phi=4R^4/L^4$ and ${\omega}_{12}=0$ in the formula:\
$${\phi}^2{\mid} {\vec{y}}_{12}{\mid }^2=({\sigma}_{12})^2+\phi (({\gamma}_{112})^2+({\gamma}_{212})^2)$$ Substituting and using the relation $L{\partial}_iL=x^i$, we obtain:\
$$y^1_{12}= - \frac{4R^2x^2}{L^4} + \frac{16R^2(x^1)^2x^2}{L^6},
y^2_{12}= - \frac{4R^2x^1}{L^4} + \frac{16R^2x^1(x^2)^2}{L^6},
y^3_{12}=\frac{16R^3x^1x^2}{L^6}$$ and thus:\
$$(y^1_{12})^2+(y^2_{12})^2+(y^3_{12})^2=\frac{16R^4((x^1)^2+(x^2)^2)}{L^8}$$ $$({\gamma}_{112})^2+({\gamma}_{212})^2=\frac{1}{4}({\partial}_1\phi)^2+({\partial}_2\phi)^2)=\frac{64R^8((x^1)^2+(x^2)^2)}{L^{12}}$$ that is to say ${\sigma}_{12}={\sigma}_{21}=0$. A similar computation left to the reader as an exercise gives ${\sigma}_{11}={\sigma}_{22}= - \frac{1}{R}{\phi}^2$ in agrement with the value of $det(\sigma)$ already obtained. As $\sigma$ is only determined up to the sign, we could also go to the north pole and notice that all the jets with $i=1,j=1$ vanish at $x^1=0,x^2=0$ but $y^1_1=y^2_2=2, y^3_{11}= - 4/R$ in order to obtain:\
$${\sigma}_{11}(0,0)=det \left( \begin{array}{ccc}
2 & 0& 0 \\
0 & 2 & 0 \\
0 & 0& -4/R
\end{array} \right) = - 16/R$$ as a simple way to know about the right sign.\
Let us now treat the general situation for arbirary $m$ and $n$ by using the formal theory of systems of partial differential eqquations and algebraic analysis (\[19-24\]). First of all, the case $n=2,m=3$ has been fully examined in (\[19\], p 126-129) and we recall it briefly in a more modern setting.\
The initial system ${\cal{A}}_1\subset J_1(X\times Y)$ is, by construction, a PHS for the Lie groupoid ${\cal{R}}_1\subset {\Pi}_1(Y,Y)$ already defined as we have a groupoid action with:\
$$dim_X ({\cal{A}}_1)=dim_Y ({\cal{R}}_1)=6$$ However, its first prolongation ${\cal{A}}_2={\rho}_1({\cal{A}}_1)\subset J_2(X\times Y)$ over $X$ is not at all a PHS for ${\cal{R}}_2={\rho}_1({\cal{R}}_1)$ over $Y$ because we have now $dim_X ({\cal{A}}_2)=9$ but $dim_Y ({\cal{R}}_2)=6$ and it follows that ${\cal{A}}_1$ is [*not*]{} an automorphic systems. Moreover, there is an additional difficulty coming from the fact that ${\cal{A}}_1$ is not even formally integrable because, introducing the second prolongation ${\cal{A}}_3={\rho}_2({\cal{A}}_1)$, there is a [*strict inclusion*]{} ${\cal{A}}^{(1)}_2 = {\pi}^3_2({\cal{A}}_3)\subset {\cal{A}}_2$ with $dim_X({\cal{A}}^{(1)}_2)=8$ as we have exhibited an additional second order equation and ${\cal{A}}_3$ is an affine bundle over ${\cal{A}}^{(1)}_2$ modelled on $g_3$. Dealing with nonlinear systems, we have the following commutative and exact diagram of affine bundles where the top row is made by the corresponding model vector bundles with $E=V(X\times Y)$:\
$$\begin{array}{rcccccl}
0\longrightarrow & g_3 & \longrightarrow & S_3T^*\otimes E & \longrightarrow &S_2T^*\otimes S_2T^* & \\
& \vdots & & \vdots & & \vdots & \\
0 \longrightarrow & {\cal{A}}_3 & \longrightarrow & J_3(X\times Y)) & \longrightarrow & J_2(S_2T^*) & \longrightarrow 0 \\
& \downarrow & & \downarrow & & \downarrow & \\
0 \longrightarrow & {\cal{A}}_2 & \longrightarrow & J_2(X\times Y) & \longrightarrow & J_1(S_2T^*) & \longrightarrow 0
\end{array}$$ By chasing, the defect of surjectivity by $1$ of the arrow on the right of the top row is equal to the defect of surjectivity by $1$ of the arrow in the left column and $dim(g_3)= dim_X({\cal{A}}_3)-dim_X({\cal{A}}^{(1)}_2)=(30-18)-8=4$, a result not evident at all as it depends on a quite difficult [*prolongation theorem*]{} (\[19,22,23):\
[**THEOREM 2.1**]{}: If $\cal{E}$ is a fibered manifold over $X$ and ${\cal{R}}_q\subset J_q({\cal{E}})$ is a system of order $q$ on $\cal{E}$, then, if the symbol $g_{q+1}$ is a vector bundle over ${\cal{R}}_q$ and $g_q$ is involutive or at least $2$-acyclic, one has the following non-trivial prolongation formula:\
$$\begin{array}{rcl}
{\cal{R}}^{(1)}_{q+r}={\pi}^{q+r+1}_{q+r}({\cal{R}}_{q+r+1})={\pi}^{q+r+1}_{q+r}({\rho}_{r+1}({\cal{R}}_q)) & = & {\rho}_r({\cal{R}}^{(1)}_q) \\
& = & ({\cal{R}}^{(1)}_q)_{+r}\subset J_{q+r}({\cal{E}}) , \forall r\geq 0
\end{array}$$
In the present situation, the symbol $g_2$ of ${\cal{A}}_2$ is defined by the $6$ linear equations:\
$${\delta}_{kl} y^k_rv^l_{ij}=0$$ with $2$ equations of class $2$ and $4$ equations of class $1$ providing characters $(2,1)$ as we may always suppose that one determinant, say $y^1_1y^2_2-y^1_2y^2_1$ does not vanish. As the $4$ corresponding CC are easily seen to be satisfied, it follows that $g_2$ is involutive, a result leading to $dim(g_{3+r})=r+4$.\
The symbol $g^{(1)}_2$ of ${\cal{A}}^{(1)}_2$ is defined by adding the only equation:\
$${\delta}_{kl}(y^k_{11}v^l_{22}+ y^k_{22}v^l_{11}-2y^k_{12}v^k_{12})=0$$ Meanwhile, $3$ among the $7$ equations can be solved with respect to $(y^1_{22},y^2_{22},y^3_{22})$ provided that the determinant ${\sigma}_{11}$ does not vanish. This additional equation reduces the second character to zero and, for the same reason as above because the equations of class $1$ are untouched. It follows that $dim(g^{(1)}_2)=2$ and $g^{(1)}_2$ is thus also involutive with characters $(2,0)$ providing therefore $dim(g^{(1)}_{2+r})=2$. We obtain therefore ${\cal{A}}^{(1)}_{2+r}=({\cal{A}}^{(1)}_2)_{+r}$ with fiber dimension equal to:\
$$dim_X({\cal{A}}^{(1)}_{2+r})= dim(g^{(1)}_3) + ... +dim(g^{(1)}_{2+r}) +dim_X ({\cal{A}}^{(1)}_2)= 2r + 8$$ and ${\cal{A}}^{(1)}_2$ is an involutive system. It follows that:\
$$dim_X({\cal{A}}_{3+r})= dim g_{3+r} + dim_X({\cal{A}}_{2+r})=(r+4) + (2r+8)= 3r + 12$$ while\
$$dim_X(J_{3+r}(X\times Y))=3(r^2+9r+20)/2, \, dim(J_{2+r}(S_2T^*))=3(r^2+7r+12)/2$$ and thus:\
$$dim({\cal{A}}_{3+r})=dim_X(J_{3+r}(X\times Y)) - dim_X(J_{2+r}(S_2T^*))$$ It follows that $\omega$ can be given arbitrarily and we have the following commutative diagram of affine bundles:\
$$\begin{array}{rcccccccl}
0 \rightarrow & g_{3+r} & \longrightarrow & S_{3+r}T^*\otimes E & \longrightarrow &S_{2+r}T^*\otimes S_2T^* & \longrightarrow & ? &\rightarrow 0 \\
& \vdots & & \vdots & & \vdots & & & \\
0 \rightarrow & {\cal{A}}_{3+r} & \longrightarrow & J_{3+r}(X\times Y) & \longrightarrow & J_{2+r}(S_2T^*) & \longrightarrow & 0 &\\
& \downarrow & & \downarrow & & \downarrow & & & \\
0 \rightarrow & {\cal{A}}_{2+r} & \longrightarrow & J_2(X\times Y) & \longrightarrow & J_{1+r}(S_2T^*) & \longrightarrow &0 &
\end{array}$$ where the defect of surjectivity of the central upper arrow in the first row is equal to the defect of surjectivity of the left downarrow that is exactly:\
$$dim_X({\cal{A}}_{2+r}) - dim_X({\cal{A}}^{(1)}_{2+r}= (3r+9) - (2r+8)=r+1$$ coming from the fact that ${\cal{A}}_2$ is [*not*]{} formally integrable. We have thus provided a modern proof of the fact that it is always possible to embed a riemannian surface into an euclidean space when $n=2,m=3$, a beautiful and tricky result first obtained by M. Janet in 1926 (\[13\]) and by E. Cartan in 1927 (\[5\]).\
Changing slightly the notations while adding the three additional Lie equations ${\Sigma}_{ij}={\sigma}_{ij}$, we obtain a PHS ${\cal{A}}_2$ for ${\cal{R}}_2$, both systems having a zero symbol and we have the successive inclusions ${\cal{A}}_2 \subset {\pi}^3_2({\rho}_1({\cal{A}}_1)) \subset {\rho}_1({\cal{A}}_1)$ with respective fiber dimensions $ 6 < 8 < 9$. According to the fundamental theorems on automorphic systems, it follows that [*only*]{} ${\cal{A}}_2$ [*is indeed an automorphic system*]{} provided that convenient CC are satisfied, being described by the top row of the new following commutative and exact diagram of affine bundles with symbolic notations, that must be compared to the previous ones:\
$$\begin{array}{rcccccccl}
& 0 & \longrightarrow & S_3T^*\otimes E & \longrightarrow &T^*\otimes (\Omega,\Gamma,\Sigma) & \longrightarrow & CC &\rightarrow 0 \\
& \vdots & & \vdots & & \vdots & & & \\
0 \rightarrow & {\cal{A}}_3 & \longrightarrow & J_3(X\times Y)) & \longrightarrow & J_1(\Omega,\Gamma,\Sigma) & & &\\
& \downarrow & & \downarrow & & \downarrow & & & \\
0 \rightarrow & {\cal{A}}_2 & \longrightarrow & J_2(X\times Y) & \longrightarrow & (\Omega,\Gamma,\Sigma) & \longrightarrow &0 &
\end{array}$$ The number of desired CC is determined by the exactness of the top row and is:\
$$Ê \begin{array}{rcl}
nb(CC) & = & dim( T^*\otimes (\Omega,\Gamma,\Sigma)) - dim (S_3T^*\otimes E)\\
& = &2 \times dim(\Omega,\Gamma,\Sigma) - 4 \times dim (E) \\
& = & 2\times (3+6+3) - 4\times 3= 24-12= 12
\end{array}$$
Among these $12$ CC, we have $6$ CC of the form $\partial \hspace{1mm} \omega=\gamma + \gamma$, then $2$ CC of the form $\partial \hspace{1mm}\gamma - \partial \hspace{1mm} \gamma =0$, then $1$ CC of the form $\partial \hspace{1mm}\gamma - \partial \hspace{1mm} \gamma + \partial \hspace{1mm}\gamma - \partial \hspace{1mm} \gamma =0$, then $1$ CC of the form:\
$$det(\omega)({\partial}_2{\gamma}_{211} - {\partial}_1{\gamma}_{212}) - \omega\gamma\gamma = det(\sigma)$$ providing Gauss theorem and finally $2$ additional CC for $\sigma$ providing the Codazzi-Mainardi equations.\
[*It is therefore only now that we do really understand the structure of shell theory and its relation with the theory of automorphic systems*]{}.\
The generalization of the previous results to arbitrary dimensions is of course quite more difficult and could provide a lot of work for the future. For simplicity, we shall restrict our study to the local isometric embedding problem of a riemann surface of dimension $n$ in an euclidean space of dimension $m$. Though surprising it may look like at first sight, our study will highly depend on the following result involving the differential rank of differential modules and [*double duality*]{} (Compare to a modern proof of the Janet conjecture in (\[24\],\[23\], p 539):\
[**THEOREM 2.2**]{}: If $K$ is a differential field and a left differential module $M={ }_DM$ is defined over the ring $D=K[d_1, ... ,d_n]=K[d]$ of differential operators with coefficients in $K$ by a finite presentation $D^p \stackrel{{\cal{D}}}{\longrightarrow} D^m \longrightarrow M \rightarrow 0 $, then ${\cal{D}}$ is formally injective if and only if $rk_D(M)=m-p$. In particular, if $m=p$, then ${\cal{D}}$ is injective if and only if $M$ is a torsion module, that is $rk_D(M)=0$.\
[*Proof*]{}: First of all, we recall that $rk_D(M)$ is the dimension of the biggest free differential module that can be contained in $M$ and that the differential rank satisfies the additivity condition $ rk_D(M)=rk_D(M') + rk _D(M")$ for any short exact sequence $0 \rightarrow M' \rightarrow M \rightarrow M" \rightarrow 0$ (\[23,24\]). A similar property is existing in the non-linear framework for differential extensions $K \subset L \subset M$ where $diff\,rk (M/K)=diff\,rk (L/K) +diff\,rk(M/L)$ and generalizes the well known classical purely algebraic situation (\[20,22,25\]). As a second comment, this result is valid even if $D$ is non-commutative and if ${\cal{D}}$ is not formally integrable and this will be the case of the situations we shall study. Applying $hom_D(\bullet,D)$ to the presentation, we may define the “[*right*]{} ” (care) differential module $N_D$ by the long exact sequence $0 \leftarrow N_D \longleftarrow D^p \longleftarrow D^m \longleftarrow hom_D(M,D) \leftarrow 0$ of [*right*]{} differential modules where we use the bimodule structure of $D={ }_DD_D$ where the left action of $P\in D$ on $D$ is defined by $Q \rightarrow PQ$ while the right action is defined by $Q \rightarrow QP$ for any $Q\in D$. Meanwhile, the right action of $P\in D$ on $f \in hom_D(M,D)$ is defined by $ (fP)(m)=f(m)P$ but a left action cannot be defined and we check that:\
$$((fP)Q)(m)=((fP)(m))Q=(f(m)P)Q=f(m)PQ=(f(PQ))(m), \forall m\in M$$ in such a way that:\
$$(fP)(Qm)=f(Qm)P=Qf(m)P=Q(f(m)P)=Q(fP)(m) , \forall P,Q\in D$$ Of course, when $D$ is commutative, both actions coincide and $hom_D(D,D)\simeq D$ because $1\in D$ and thus $f(Q)=f(Q.1)=Q(f(1))=QP$ if we set $f(1)=P$. We may also pass from the right differential module $N_D$ to a left differential module $N={ }_DN=hom_K({\wedge}^nT^*,N_D)$ by the [*side changing procedure*]{} if we notice that $D$ is generated by $K$ and $T=\{a^id_i\}\subset D$ with $T^*=hom_K(T,K)$. In this case, the [*dual operator*]{} $hom_D({\cal{D}},D)$ becomes the [*adjoint operator*]{} $ad({\cal{D}})$ used in variational calculus.\
Finally, it can be proved that the additive property of the differential rank is such that $rk_D(hom_D(M,D))=rk_D(M)$ and $rk_D(ad({\cal{D}}))=rk_D({\cal{D}})$ if we set $rk_D({\cal{D}})=rk_D(im({\cal{D}}))$. Counting the differential ranks, we get:\
$$rk_D(N)-p+m-rk_D(M)=0$$ and thus discover that $rk_D(N)=0$ if and only if $rk_D(M)=m-p$. As $D$ is an integral domain and $N$ is a torsion module, then $hom_D(N,D)=0$ and we have obtained by biduality the short exact sequence $0 \rightarrow D^p \stackrel{{\cal{D}}}{\longrightarrow } D^m \rightarrow M \rightarrow 0$. For the reader not familiar with homological algebra, if $L=ker({\cal{D}})$ in the long exact sequence $0 \rightarrow L \rightarrow D^p \stackrel{{\cal{D}}}{\rightarrow} D^m \rightarrow M \rightarrow 0 $, then $rk_D(L)=0$ and thus $L$ is a torsion module over the differential integral domain $D$. Hence, any element $x\in L$ is such that there exists [*at least one*]{} $0\neq P\in D$ such that $Px=0$ and thus $x=0, \forall x\in L$ because $D$ and thus $D^p$ do not admit divisors of zero., that is $L=0$. We can also say that $L$ is a torsion module and obtain a contradiction with the fact that $L$ is contained in the free and thus torsion-free module $D^p$, unless $L=0$.\
Q.E.D.\
Using the notations of M. Janet in (13\]), we have the non-linear system ${\cal{A}}_1$:\
$${\Omega}_{i'j'}\equiv {\delta}_{kl}y^k_{i'}y^l_{j'}={\omega}_{i'j'}, \hspace{3mm}
{\Omega}_{i'n}\equiv {\delta}_{kl}y^k_{i'}y^l_{n}={\omega}_{i'n}, \hspace{3mm}
{\Omega}_{nn}\equiv {\delta}_{kl}y^k_{n}y^l_{n}={\omega}_{nn}$$ where $i',j'=1,...,n-1$ and $k,l=1,...,m$. We may, as before, differentiate all these equations once in order to obtain the various ${\Gamma}_{rij}={\gamma}_{rij}$ with now $i,j,r=1,...,n$, in particular the $n$ differential invariants:\
$${\Gamma}_{rnn}\equiv {\delta}_{kl}(y^k_{r}y^l_{nn} )={\gamma}_{rnn}, \hspace{4mm} r=1,...,n$$ and consider the $n(n-1)/2$ new differential invariants of order two (Look for example in ([AIRY]{}) for $m=n=3$):\
$$d_{nn}{\Omega}_{i'j'} + d_{i'j'}{\Omega}_{nn} - d_{i'n}{\Omega}_{j'n} - d_{j'n}{\Omega}_{i'n}\equiv
2{\delta}_{kl}(y^k_{i'n}y^l_{j'n}- y^k_{i'j'}y^l_{nn})$$ Accordingly, the possibility to compute the character of index $n$ of the system ${\cal{A}}^{(1)}_2\subset {\rho}_1({\cal{A}}_1)$ only depends on the matrix with $m$ rows and $n+(n(n-1)/2)=n(n+1)/2$ columns needed for solving the equations with respect to the jets $y^1_{nn},...,y^m_{nn}$:\
$$(y^1_{nn} ... y^m_{nn}) \left( \begin{array}{cccc}
y^1_1 & ... & y^1_n & y^1_{i'j'} \\
... & ... & ... & ... \\
y^m_1& ... & y^m_n & y^m_{i'j'}
\end{array} \right)$$ We find the results previously obtained for $n=2,m=3$ where the corresponding $3\times 3$ square matrix has a determinant equal to ${\sigma}_{11}$. Hence, when $m=n(n+1)/2=p$ and generic jets, the rank of the above system is equal to $m$ and [*the character*]{} $n$ [*of the symbol must vanish*]{}. It follows from the previous theorem that we cannot have CC for the $\omega$ and we obtain a new proof in a modern setting of the result first found by M. Janet in $1926$.\
Coming back to the case $n=2,m=3$, we may introduce the matrix :\
$$M(x)=({\vec{y}}_1,{\vec{y}}_2,{\vec{y}}_1\wedge{\vec{y}}_2)$$ This matrix is invertible because $det(M)=\mid {\vec{y}}_1\wedge{\vec{y}}_2 {\mid}^2 = det(\omega)\neq 0$. Also, looking at the action of the group of rigid motions, we have:\
$$\bar{y}=Ay+B \Rightarrow {\vec{y}}_i=A{\vec{y}}_i \Rightarrow \bar{M}=AM\Rightarrow A=\bar{M}M^{-1}, B=\bar{y}-Ay$$ Accordingly, if $y_2=f_2(x)\neq j_2(f)(x)$ and ${\bar{y}}_2={\bar{f}}_2(x)\neq j_2(\bar{f})(x)$ are two sections of $J_2(X\times Y)$, then $M=M(x), \bar{M}=\bar{M}(x)$ leads to $A=A(x), B=B(x)$ with:\
$$\bar{f}(x)=A(x)f(x)+B(x), {\bar{f}}_x(x)=A(x)f_x(x), {\bar{f}}_{xx}(x)=A(x)f_{xx}(x)$$ and the relations:\
$${\partial}_x\bar{f}- {\bar{f}}_x=A({\partial}_xf-f_x)+{\partial}_xAf+{\partial}_xB,\hspace{3mm}
{\partial}_x\bar{f}_x - {\bar{f}}_{xx}=A({\partial}_xf_x-f_{xx})+{\partial}_xAf_x$$ If ${\bar{f}}_2\neq j_(\bar{f})$ but $f_2=j_2(f)$ and thus $f$ is one solution of the automorphic system, then we have:\
$${\partial}_x\bar{f}- {\bar{f}}_x=({\partial}_xA{A}^{-1})\bar{f}+({\partial}_xB-{\partial}_xAA^{-1}B),\hspace{3mm}
{\partial}_x\bar{f}_x - {\bar{f}}_{xx}=({\partial}_xAA^{-1}){\bar{f}}_x$$ bringing right invariant $1$-forms which are the pull-back of the Maurer-Cartan forms by the gauging procedure $X\rightarrow G:(x)\rightarrow (A(x),B(x))$ induced by the two sections $f_2,{\bar{f}}_2\in {\cal{A}}_2$. A similar left invariant version is obtained by composing the gauging with the inversion map $G\rightarrow G:(A,B) \rightarrow (A^{-1}, - A^{-1}B)$. Sections of jet bundles provide therefore a modern version of the so-called Darboux vectors but this is out of the scope of this paper.\
[**3) CHAIN THEORY REVISITED**]{}\
The shape $y=f(x)$ of a thin [*inelastic*]{} free hanging chain ( a bike chain for example ) in a vertical plan with cartesian coordinates ($x$ horizontal,$y$ vertical) under gravity $g$ is usually called [*chainette*]{}, [*catenary*]{} or sometimes [*funicular*]{}. If $\vec{T}(x)$ is the tension in the chain and $s$ the curvilinear abcissa along the chain, we have $ds=\sqrt{1+({\partial}_xf)^2}$ and the equilibrium condition of a piece of chain with length $ds$ and mass $m$ per unit of length is:\
$$\frac{d\vec{T}}{ds} + m\vec{g}=0$$ As $\vec{T}=\lambda (1,{\partial}_xf)$, we get ${\partial}_x\lambda=0 \Rightarrow \lambda =cst$ and thus ${\partial}_{xx}f/\sqrt{1+({\partial}_xf)^2}=cst=1/a$. One possible solution is $y=a\, ch(x/a)$ and we shall only consider the particular case $y=ch(x)$ obtained when $a=1$.\
As before, the problem becomes:\
\
Changing slightly the notations with $n=1,m=2$, we may now consider the algebraic group $G$ of rigid motions of the plan $Y={\mathbb{R}}^2$ with cartesian coordinates $y=(y^1,y^2)$ while $X=\mathbb{R}$ has coordinate $(x)$. The finite action is defined by $\bar{y}=Ay+B$ with $AA^t=I\Rightarrow (det(A))^2=1$ and we may also consider, as before, the connected component $G^0\subset G$ of the identity with $det(A)=1$. The three infinitesimal generators of the group action are:\
$$\{ {\theta}_1=\frac{\partial}{\partial y^1},{\theta}_2=\frac{\partial}{\partial y^2},
{\theta}_3=y^1\frac{\partial}{\partial y^2} - y^2 \frac{\partial}{\partial y^1} \}$$ and the prolongation to the jets of order $2$ only depends on ${\theta}_1, {\theta}_2$ and:\
$${\rho}_2( {\theta}_3)=y^1\frac{\partial}{\partial y^2} - y^2 \frac{\partial}{\partial y^1}+y^1_x\frac{\partial}{\partial y^2_x} - y^2_x \frac{\partial}{\partial y^1_x} + y^1_{xx}\frac{\partial}{\partial y^2_{xx}} - y^2_{xx} \frac{\partial}{\partial y^1_{xx}}$$ The corresponding PHS ${\cal{A}}_1$ is easily seen to be ${\Omega}^1\equiv (y^1_x)^2 + (y^2_x)^2=\omega (x)$ but is [*not*]{} an automorphic system. Exactly like in the study of shell, we may therefore consider the automorphic system ${\cal{A}}_2 \subset {\rho}_1({\cal{A}}_1)$:\
$${\Omega}^1\equiv (y^1_x)^2 + (y^2_x)^2=\omega (x), \hspace{3mm} \Gamma \equiv y^1_xy^1_{xx} + y^2_xy^2_{xx}=\gamma(x),
\hspace{3mm} \Sigma \equiv y^1_xy^2_{xx}- y^2_xy^1_{xx}=\sigma (x)$$ The symbol of order $2$ vanishes if and only if $\omega \neq 0$ and we shall assume such a condition. The only CC is $\gamma = \frac{1}{2}{\partial}_x\omega $ but, contrary to shell theory, $\sigma$ is arbitrary. However, our definitions allow to avoid using square roots in the geometrical approach. Indeed, setting as usual $\frac{d\vec{y} }{ds} = \vec{t}, \frac{d\vec{t}}{ds}= \kappa \vec{n}, \frac{d\vec{n}}{ds}= - \kappa \vec{t} $, we have $ds^2=\omega dx^2$ and thus $\mid \vec{t}\,{\mid}^2= 1 \Rightarrow \vec{t}.\vec{n}=0 \Rightarrow \vec{t} \wedge \vec{n}=\vec{b}\Rightarrow \mid \vec{b}\,{\mid}^2=1$ with $\vec{b}$ fixed perpendicular to the plane considered. It follows that:\
$$\frac{d\vec{y}}{dx}\wedge \frac{d^2\vec{y}}{dx^2}= (\frac{ds}{dx}\vec{t}\,)\wedge (\frac{d}{dx}(\frac{ds}{dx}\vec{t}\,))=
\omega \vec{t}\wedge \frac{d\vec{t}}{dx} = \omega \vec{t}\wedge (\frac{ds}{dx}\frac{d\vec{t}}{ds})=\omega \frac{ds}{dx}\kappa \vec{b}$$ and thus $\sigma^2={\omega}^3 {\kappa}^2 $. It is however important to notice that $\bar{y}=Ay+B \Rightarrow {\bar{y}}_x=Ay_x, {\bar{y}}_{xx}=Ay_{xx}, ...$ and thus $\bar{\sigma}=
det(A)\sigma$, that is $\Sigma$ is only invariant by $G^0$ while the new Lie equation $\Upsilon\equiv (y^1_{xx})^2 + (y^2_{xx})^2=\upsilon(x)$ is invariant by $G$. Nevertheless, an elementary computation provides the idenity $\Omega \Upsilon=(\Gamma)^2 + (\Sigma)^2 \Rightarrow \omega\upsilon={\gamma}^2 + {\sigma}^2$ and we have the successive inclusions of differential extensions $k\subset K \subset K_0 \subset L$:\
$$\mathbb{Q} < \mathbb{Q}<\Omega, \Upsilon>\subset \mathbb{Q}<\Omega, \Sigma >\subset \mathbb{Q} < y >$$ because $\Gamma \in \mathbb{Q}<\Omega>$. Then $K_0=\mathbb{Q}<\Omega,\Sigma>$ is thus purely algebraic over $K=\mathbb{Q}<\Omega,\Upsilon>$ in agrement with the fact that $G/G^0=\{1, - 1\}$ while $L=\mathbb{Q}<y>$ is a regular extension of $K_0=\mathbb{Q}<\Omega,\Sigma>$ because $G$ is defined over $k=\mathbb{Q}$ and $k(G_0)$ is a regular extension of $k$, that is $k$ is algebraically closed in $k(G_0)$. As a byproduct, we obtain now:\
$$\upsilon = \mid\frac{d^2\vec{y}}{d^2x}{\mid}^2=\mid (\frac{d^2s}{dx^2}\vec{t}+ (\frac{ds}{dx})^2\frac{d\vec{t}}{ds}{\mid}^2=
(\frac{d^2s}{dx^2})^2 + (\frac{ds}{dx})^4{\kappa}^2=({\gamma}^2/\omega ) + {\omega}^2 {\kappa}^2$$ because $\frac{d^2s}{dx^2}= \gamma / \sqrt{\omega}$ and, multiplying by $\omega$, we find back the previous formulas. It follows that $\omega=1 \Rightarrow \gamma=0 \Rightarrow \sigma=\kappa$ in the case of an inelastic chain.\
Let us now consider two sections of ${\cal{A}}_2$, namely $f_2=j_2(f)$ with $y^1=x, y^2=ch(x)$ giving $ \omega= ch^2(x), \gamma= sh(x)ch(x), \sigma=ch(x), \upsilon= {ch}^2(x)$ and ${\bar{f}}_2\neq j_2(\bar{f})$ defined by $ {\bar{y}}^1=sh(x), {\bar{y}}^2= 1, {\bar{y}}^1_x=ch(x), {\bar{y}}^2_x= 0, {\bar{y}}^1_{xx}= sh(x), {\bar{y}}^2_{xx}=1$ in order to obtain right invariant Maurer-Cartan forms. Indeed, the gauging is defined by the formulas:\
$$\bar{y}=Ay+B\Rightarrow \bar{f}(x)=A(x)f(x)+B(x), \bar{f}_x(x)=A(x)f_x(x), \bar{f}_{xx}(x)A(x)f_{xx}$$ that is to say:\
$$\forall f_2,\bar{f}_2 \in {\cal{A}}_2 \Rightarrow \exists \, gauging \,(A(x),B(x))$$ An easy computation provides:\
$$A(x)=\left( \begin{array}{cc}
\hspace{2mm} \frac{1}{ch(x)} & \frac{sh(x)}{ch(x)}\\
- \frac{sh(x)}{ch(x)} & \frac{1}{ch(x)}
\end{array} \right ) , \hspace{5mm} B(x)= \left( \begin{array}{c}
- \frac{x}{ch(x)} \\ \hspace{3mm}\frac{x\,sh(x)}{ch(x)}
\end{array} \right)$$ where of course $A$ is an orthogonal $2\times 2$ matrix with $det(A)=1$. Equivalently, we may introduce the $2\times 2$ matrices $M(x)$ and $\bar{M}(x)$ defined by:\
$$M=\left( \begin{array}{cc}
f^1_x & f^1_{xx} \\ f^2_x & f^2_{xx}
\end{array} \right) ,\, \bar{M}=\left( \begin{array}{cc}
{\bar{f}}^1_x & {\bar{f}}^1_{xx} \\ {\bar{f}}^2_x & {\bar{f}}^2_{xx}
\end{array} \right) \Rightarrow \bar{M}(x)=A(x)M(x)$$ and the inversion formula $A=\bar{M}M^{-1}$ can be used whenever $\bar{\sigma}=det(\bar{M})=det(M)=\sigma \neq 0 \Rightarrow det(A)=1$, leading to $B=\bar{f} -Af$.The components of the Spencer operator are now $Df_2=0, \, D{\bar{f}}_2$ with:\
$$\frac{\partial \bar{f}}{\partial x} - {\bar{f}}_x= (\frac{\partial A}{\partial x}A^{-1})\bar{f}+(\frac{\partial B}{\partial x} - \frac{\partial A}{\partial x}A^{-1}B), \, \frac{\partial {\bar{f}}_x}{\partial x} - {\bar{f}}_{xx}= (\frac{\partial A}{\partial x}A^{-1}){\bar{f}}_x$$ We finally obtain:\
$$\frac{\partial A}{\partial x}A^{-1}= \left( \begin{array}{cc} 0 & \frac{1}{ch(x)} \\ - \frac{1}{ch(x)} & 0 \end{array} \right), \,
\frac{\partial B}{\partial x} - \frac{\partial A}{\partial x}A^{-1}B= \left( \begin{array}{c} - \frac{1}{ch(x)} \\
\frac{sh(x)}{ch(x)}\end{array} \right)$$ where the first matrix is skew-symmetric as expected but there is no Maurer-Cartan equation because $dim(X)=1$.\
Conversely, if these $1$-forms are known and provide $(A,B)$, then any other solution $(\bar{A},\bar{B})$ is such that:\
$$\frac{\partial \bar{A}}{\partial x}{\bar{A}}^{-1}= \frac{\partial A}{\partial x}A^{-1}, \hspace{5mm}
\frac{\partial \bar{B}}{\partial x} - \frac{\partial \bar{A}}{\partial x}{\bar{A}}^{-1}\bar{B} = \frac{\partial B}{\partial x} - \frac{\partial A}{\partial x}A^{-1}B$$ ad thus:\
$$\frac{\partial}{\partial x}(A^{-1}\bar{A})=0 \Rightarrow \bar{A}=AC, C=cst \Rightarrow \bar{B}=AD+B, D=cst$$ This is a modern version of the so-called Darboux vectors which are replaced by sections of jet bundles.\
Coming back to the fundamental theorem of the differential Galois theory (\[3,4,20,22\]), the reciprocal distribution $\Delta$ commuting with the invariant distribution $\Theta$ generated by the ${\rho}_2({\theta}_{\tau}), \, \tau=1,2,3$, is generated by the $4=dim(J^0_2(E))=dim(J_2(E)-m$:\
$${\delta}_1= y^1_x\frac{\partial}{\partial y^1_x}+ y^2_x\frac{\partial}{\partial y^2_x} ,
\hspace{4mm} {\delta}_2= y^1_{xx}\frac{\partial}{\partial y^1_x}+y^2_{xx}\frac{\partial}{\partial y^2_x},
\hspace{4mm}{\delta}_3= y^1_x\frac{\partial}{\partial y^1_{xx}}+y^2_x\frac{\partial}{\partial y^2_{xx}},$$ $${\delta}_4= y^1_{xx}\frac{\partial}{\partial y^1_{xx}}+y^2_{xx}\frac{\partial}{\partial y^2_{xx}}$$ First of all, it is easy to check that all the differential extensions already considered are stable by $\Delta$ and that the distribution $\delta \otimes \delta$ or $\delta + \bar{\delta}$ where $\bar{\delta}$ is obtained from any $\delta \in \Delta$ by adding a bar over the jet coordinates, satisfies:\
$$\bar{\delta}.\bar{M}=A\delta.M, \, \bar{M}=AM , \, \bar{y}=Ay+B \Rightarrow (\delta+\bar{\delta}).A=0, \, (\delta+\bar{\delta}).B=0$$ and thus $(A,B)$ is killed by $ (\delta + \bar{\delta}), \forall \delta \in \Delta$. Moreover, if any vector field $\delta$ commutes with ${\delta}_1,...,{\delta}_4$, then, applying any $\theta \in \Theta$ to the linear combination $\delta= a_1{\delta}_1 + ... +a_4{\delta}_4$ we discover that the $a$ are kiled by $\theta$ and are thus only functions of the differential invariants. For example, if we consider $\delta=y^1_x\frac{\partial}{y^2_x} - y^2_x \frac{\partial}{\partial y^1_x}$ we obtain $\delta = - (\Gamma/\Sigma){\delta}_1 + (\Omega / \Sigma){\delta}_2$ a result not evident at all at first sight. We obtain therefore $\delta \Gamma= \Sigma$ and thus $\delta K\not\subset K$ because $\Sigma \notin K$ but $\delta K_0\subset K_0$.
We end this example with a few comments on variational calculus with constraints presented at the end of (\[23\]). To start with, we shall even suppose that the chain $C$ is elastic while hanging between two fixed points, with initial mass $m_0(x)$ per unit lentgh $x$, final mass $m(s)$ per unit legth $s$ in such a way that $m_0dx=m(s)ds$ and elastic coefficient $E$ in such a way that, under a tension $T$ we have $T=E(ds-dx)/dx=E(ds/dx)-1)$. Using previous notations, the total lagrangian including the gravitational potential and the energy of deformation stored in the chain, we have to study the following variational problem where $y^k=f^k(x), y^k_x=(df^k_x)/dx\ \forall k=1,2$ :\
$$\delta {\int}_C (m_0gy^2 + \frac{1}{2}E(\frac{ds}{dx} - 1)^2 )dx={\int}_C (m_0g\delta y^2 + T (y^1_x\delta y^1_x+y^2_x\delta y^2_x)/(\frac{ds}{dx}))dx=0$$ Integrating by part while using the relations $\frac{d y^1}{dx}/\frac{ds}{dx}= cos(\theta),\frac{d y^2}{dx}/\frac{ds}{dx}= sin(\theta) $, we obtain successively the two Euler-Lagrange equations:\
$$\begin{array}{lcl}
\delta y^1 \rightarrow \frac{d}{dx}(Tcos(\theta))=0 & \Rightarrow & Tcos(\theta)=T_0 \\
\delta y^2 \rightarrow \frac{d}{dx}(Tsin(\theta))=m_0g & \Rightarrow & T sin(\theta)=m_0gx
\end{array}$$ Setting $T_0=am_0g$ and integrating, we finally get, up to a shift of the axes:\
$$y^1= a \, arcsh\, (x/a) + (T_0/E)x , \hspace{1cm} y^2= \sqrt{a^2+x^2}+ (T_0/2Ea)x^2$$ At the limit $T\rightarrow \infty$, we get $y^2=a\, ch(y^1/a)$ as expected. However, we may also find the same result by looking at the extremum of the potential energy under the differential constraint $(ds/dx)-1 =0$, that is by studying the variational problem:\
$$\delta {\int}_C(m_0gy^2 + \lambda (\frac{ds}{dx}-1))dx=0$$ with [*Lagrange multiplier*]{} $\lambda$ and getting the usual identification $\lambda=T$, though we notice that the founding principles are quite different.\
[**4) FRENET-SERRET FORMULAS REVISITED** ]{}\
Though the Frenet (1847) and Serret (1851) formulas are well known by anybody studying curves in ${\mathbb{R}}^3$, we shall revisit them in the light of the differential Galois theory as they will provide one of the best examples of the criteria for automorphic systems already presented. For such a purpose, let $X=\mathbb{R}$ with local coordinates $(x)$ and $Y={\mathbb{R}}^3$ with cartesian coordinates $(y^1,y^2,y^3)$ as usual. We consider the Lie group of rigid motions $G=(A,B)$ of dimension $3+3=6$ where $A$ is a [*constant*]{} orthogonal matrix and $B$ is a [*constant*]{} vector. The action and its successive prolongations are defined by the formulas:\
$$\bar{y}=A\, y + B \Rightarrow {\bar{y}}_x=A \, y_x, {\bar{y}}_{xx}= A\, y_{xx}, {\bar{y}}_{xxx}=A \, y_{xxx}, ...$$ Compared to the previous examples, the rather striking fact is that now the only differential invariant of order one is:\
$$\Omega \equiv {\vec{y}}_x.{\vec{y}}_x\equiv (y^1_x)^2 +(y^2_x)^2 + (y^3_x)^2=\omega$$ and there are thus [*only two*]{} differential invarints at order two, namely:\
$$\Gamma\equiv {\vec{y}}_x.{\vec{y}}_{xx}\equiv y^1_xy^1_{xx}+y^2_xy^2_{xx}+y^3_xy^3_{xx}=\gamma= \frac{1}{2}{\partial}_x\omega ,$$ $$\Sigma \equiv {\vec{y}}_{xx}.{\vec{y}}_{xx}\equiv (y^1_{xx})^2+(y^2_{xx})^2+(y^3_{xx})^2=\sigma$$ Indeed, another possibility could be:\
$$R \equiv \mid {\vec{y}}_x\wedge {\vec{y}}_{xx} {\mid }^2 \equiv (y^2_xy^3_{xx}-y^3_xy^2_{xx})^2 + (y^3_xy^1_{xx}-y^1_xy^3_{xx})^2 +
(y^1_xy^2_{xx}-y^2_xy^1_{xx})^2 = \rho$$ but it is easy to check the identity:\
$$R= \Omega \Sigma - (\Gamma)^2 \Rightarrow \rho=\omega \sigma -{\gamma}^2$$ but the resulting equations for the symbol of order two should be:\
$$((y^3_xy^1_{xx}-y^1_xy^3_{xx})y^3_x -(y^1_xy^2_{xx} - y^2_xy^1_{xx})y^2_x ) v^1_{xx} + ... =0$$ where the sum is done on the permutations of $(1,2,3)$ and these equations are linear combunations of the two other ones, namely:\
$${\delta}_2\Gamma \equiv y^1_xv^1_{xx} +y^2_xv^2_{xx}+y^3_xv^3_{xx}=0, \, \, {\delta}_2\Sigma \equiv y^1_{xx}v^1_{xx} +y^2_{xx}v^2_{xx}+y^3_{xx}v^3_{xx}=0$$ because we have $ {\delta}_2 R =\Omega \, {\delta}_2\Sigma - 2 \Gamma \,{\delta}_2\Gamma $ in $V(J_2(X\times Y))$.\
In the present situation, ${\cal{A}}_1$ is clearly [*not*]{} a PHS for $G$ as it has fiber dimension equal to $3+2=5 < 6$ while ${\cal{A}}_2$ is a PHS for $G$ with fiber dimension $3+2+1=6$ but is [*not*]{} an automorphic system. Indeed, using one prolongation, [*we only get two third order OD equations*]{} and the correct automorphic system ${\cal{A}}_3$ is defined by adding the following [*three*]{} third order equations:\
$$\Phi \equiv {\vec{y}}_x.{\vec{y}}_{xxx}=y^1_xy^1_{xxx}+y^2_xy^2_{xxx}+y^3_xy^3_{xxx}=\varphi={\partial}_x\gamma- \sigma,$$ $$\Psi\equiv {\vec{y}}_{xx}.{\vec{y}}_{xxx}=y^1_{xx}y^1_{xxx}+y^2_{xx}y^2_{xxx}+y^3_{xx}y^3_{xxx}=\psi= \frac{1}{2}{\partial}_x\sigma,$$ $$\Upsilon \equiv ({\vec{y}}_x \wedge {\vec{y}}_{xx}).{\vec{y}}_{xxx} \equiv ({\vec{y}}_x,{\vec{y}}_{xx}, {\vec{y}}_{xxx})= \upsilon$$ with no CC involving $\upsilon$ , that is only $(\omega,\sigma, \upsilon)$ can be given arbitrarily.
In order to recover the classical formulas, we need to introduce the curvilinear abcissa $s$ defined by $ds^2= \omega \,dx^2$ and we get successively:\
$$\frac{d\vec{y}}{ds}= \vec{t}, \,\, \frac{d\vec{t}}{ds}= \kappa \vec{n}, \,\, \frac{d\vec{n}}{ds}= - \kappa \vec{t} + \tau \vec{b},
\,\, \frac{d\vec{b}}{ds}= - \tau \vec{n}$$ where $\kappa$ is called the [*curvature*]{}, $\tau$ is called the [*torsion*]{} and $\vec{b}=\vec{t} \wedge \vec{n}$. As $\mid \vec{t}{\mid}^2=1$, it follows that $\vec{n} $ is orthogonal to $\vec{t}$ and we shall exhibit the relations existing between $(\kappa,\tau)$ and the $(\omega, \sigma, \nu)$ already obtained.\
Now, with $y=f(x),y_x={\partial}_xf(x), y_{xx}={\partial}_{xx}f(x)$, we obtain successively:\
$$y_x=y_s\frac{ds}{dx}, y_{xx}=y_{ss}(\frac{ds}{dx})^2 + y_s \frac{d^2s}{dx^2}, y_{xxx}=y_{sss}(\frac{ds}{dx})^3+ 3y_{ss}\frac{ds}{dx}\frac{d^2s}{dx^2}+y_s\frac{d^3s}{dx^3}, ...$$ $${\vec{y}}_x=\frac{ds}{dx}\vec{t} , {\vec{y}}_{xx}=(\frac{ds}{dx})^2\frac{d\vec{t}}{ds} + \frac{d^2s}{dx^2}\vec{t},
{\vec{y}}_{xxx}= (\frac{ds}{dx})^3\frac{d^2 \vec{t}}{ds^2}+ 3\frac{ds}{dx}\frac{d^2s}{dx^2}\frac{d\vec{t}}{ds}+ \frac{d^3s}{dx^3}\vec{t}, ...$$
$$\omega ={\vec{y}}_x.{\vec{y}}_x=(\frac{ds}{dx})^2 \mid \vec{t}\,{\mid}^2=(\frac{ds}{dx})^2$$ $$\gamma= {\vec{y}}_x.{\vec{y}}_{xx}= \frac{ds}{dx}\frac{d^2s}{dx^2}=\frac{1}{2}\frac{d}{dx}(\frac{ds}{dx})^2=\frac{1}{2}{\partial}_x\omega$$ $$\rho = \mid {\vec{y}}_x\wedge {\vec{y}}_{xx}{\mid}^2=(\frac{ds}{dx})^6 \mid \vec{t}\wedge \frac{d\vec{t}}{ds}{\mid}^2=
{\omega}^3{\kappa}^2\mid \vec{b}\,{\mid}^2={\omega}^3{\kappa}^2$$
$$\sigma ={\vec{y}}_{xx}.{\vec{y}}_{xx}=(\frac{ds}{dx})^4 \frac{d\vec{t}}{ds}.\frac{d\vec{t}}{ds}+ (\frac{d^2s}{dx^2})^2=
{\omega}^2{\kappa}^2 + ( {\gamma}^2/\omega )$$
$$\varphi = {\vec{y}}_x.{\vec{y}}_{xxx}= {\omega}^2\vec{t}.\frac{d^2\vec{t}}{ds^2} + \frac{ds}{dx}\frac{d^3s}{ds^3}= - {\omega}^2{\kappa}^2 + \frac{ds}{dx}\frac{d^3s}{ds^3}= - {\omega}^2{\kappa}^2+{\partial}_x\gamma -({\gamma}^2/{\omega})$$
$$\begin{array}{rcl}
\upsilon & = & ({\vec{y}}_x\wedge {\vec{y}}_{xx}). {\vec{y}}_{xxx} \\
& = & (\frac{ds}{dx})^3(\frac{d\vec{y}}{ds}\wedge \frac{d^2\vec{y}}{ds^2}).((\frac{ds}{dx})^3 \frac{d^3\vec{y}}{ds^3} \\
& = & {\omega}^3(\vec{t}\wedge \frac{d\vec{t}}{ds}). \frac{d^2\vec{t}}{d^2s} \\
& = &{\omega}^3 (\vec{t},\frac{d\vec{t}}{ds},\frac{d^2\vec{t}}{ds^2}) \\
& = & {\omega}^3(\vec{t}\wedge \kappa \vec{n}).(\frac{d\kappa}{ds} \vec{n} + \kappa \frac{d\vec{n}}{ds}) \\
& = & {\omega}^3\kappa \vec{b}.(-{\kappa}^2\vec{t} + \kappa \tau \vec{b}) \\
& = & {\omega}^3{\kappa}^2\tau \\
& = & \rho \tau
\end{array}$$
We now provide the computation for the [*helix*]{} (draw a picture):\
$$y^1=r \,cos(\theta), \hspace{5mm} y^2=r \,sin(\theta), \hspace{5mm} y^3=h \, \theta, \hspace{5mm} r=cst$$ $$dy^1=-r\, sin(\theta) d\theta, \hspace{3mm} dy^2=r \, cos(\theta)d\theta, \hspace{3mm} dy^3=h \, d\theta \Rightarrow (ds)^2=(r^2+h^2)(d\theta)^2$$ $$t^1= - \frac{r\, sin(\theta)}{\sqrt{r^2+h^2}},\hspace{3mm}t^2=\frac{r \, cos(\theta)}{\sqrt{r^2+h^2}},\hspace{3mm}t^3= \frac{h}{\sqrt{r^2+h^2}}$$ $$dt^1= - \frac{r \, cos(\theta)}{\sqrt{r^2+h^2}}d\theta, \hspace{3mm} dt^2= - \frac{r \, sin(\theta)}{\sqrt{r^2+h^2}}d\theta, \hspace{3mm}dt^3=0$$ We obtain therefore $\kappa=\mid\frac{d\vec{t}}{ds}\mid =\frac{r}{r^2+h^2}$ and:\
$$n^1= - cos(\theta), n^2= - sin(\theta), n^3=0 \Rightarrow \frac{dn^1}{ds}=\frac{sin(\theta)}{\sqrt{r^2+h^2}}, \frac{dn^2}{ds}=\frac{cos(\theta)}{\sqrt{r^2+h^2}}, \frac{dn^3}{ds}=0$$ A result leading to:\
$$\frac{dn^1}{ds}+\kappa t^1=\frac{1}{\sqrt{r^2+h^2}} \frac{h^2}{r^2+h^2}sin(\theta), \frac{dn^2}{ds}+\kappa t^2= - \frac{1}{\sqrt{r^2+h^2}} \frac{h^2}{r^2+h^2} cos(\theta),$$ $$\frac{dn^3}{ds}+\kappa t^3=\frac{1}{\sqrt{r^2+h^2}}\frac{hr}{r^2+h^2}$$ and finally to $\tau=\mid \frac{d\vec{n}}{ds}+\kappa \vec{t} \mid=\frac{h}{r^2+h^2}$.\
Studying finally the gauging procedure existing in the differential Galois theory, we may introduce the $3 \times 3$ matrix:\
$$M=\left( \begin{array}{lll}
y^1_x & y^1_{xx} & y^1_{xxx} \\
y^2_x & y^2_{xx} & y^2_{xxx} \\
y^3_x & y^3_{xx} & y^3_{xxx}
\end{array} \right)$$ For two sections $f_3$ and ${\bar{f}}_3$ of ${\cal{A}}_3$, we have $\bar{M}=A\, M$ and thus $A(x)=\bar{M}M^{-1}$ whenever $M$ is invertible, that is $det(M)=\upsilon\neq 0$. Then $B(x)={\bar{f}}(x) - A(x) f(x)$ and, as before, $A$ and $B$ are constant whenever $y=f(x)$ and $\bar{y}={\bar{f}}(x)$ are two solutions of ${\cal{A}}_3 $, a result not evident at first sight.\
[ **5) LIE THEORY OF OD EQUATIONS REVISITED**]{}\
As a very general algebraic problem, if $K$ and $L$ are two abstract extensions of a field $k$, the idea is to look for the rings and field that can be constructed from this only knowledge, in particular we should look for a bigger fields containing $K$ and $L$ in a nice way in order to define the smallest subfield $(K,L)$ containing $K$ and $L$. Coming back to ordinary Galois theory, the basic purpose is to know about all roots of a given polynomial or, in a more abstract way, to split a given polynomial into as many irreducible factors as possible through the knowledge of the roots of another polynomial. For example, if $K/k$ is a Galois extension and $L$ is an arbitrary extension of $k$ such that $K$ and $L$ are contained in a bigger field, then $(K,L)$ is a Galois extension of $L$. Moreover, if $K \cap L=k$, then $K$ and $L$ are linearly disjoint over $k$ in $(K,L)$ (\[20,22\], p 131 or any basic textbook on Galois theory like \[1,28\]).\
As an elementary but tricky counter-example with $k=\mathbb{Q}$, let us consider the polynomials $P\equiv y^3-2\in k[y]$ with residue $y\rightarrow \eta$ and $Q\equiv z^2+z+1\in k[z]$ with residue $z\rightarrow \zeta$. As $P$ and $Q$ are irreducible over $k$, we may consider the fields $K=k(\eta)$ and $L=k(\eta\zeta)$. As $(P,Q)$ is a prime ideal in $k[y,z]$, then we may choose $(K,L)=k(\eta,\zeta)=Q(k[y,z]/(P,Q))=k[y,z]/(P,Q)$. In actual practice, we may choose for $\eta$ the real root $\sqrt[3]{2}$ of $P$ and for $\zeta$ the imaginery cubic root of unity $(-1+i\sqrt{3})/2$ in such a way that $\eta\zeta$ is an imaginary root of $P$. We have of course $K\cap L =k$ but $K$ is [*not*]{} a Galois extension of $k$. Accordingly, $K$ and $L$ are not linearly disjoint over $k$ in $(K,L)$ because we have $(\eta\zeta)^2 \times 1+ (\eta\zeta)\eta + 1 \times {\eta}^2=0$ while $\{1, \eta, {\eta}^2\}$ is a basis of $K$ over $k$. Our aim will be to extend these ideas to the differential framework.\
While studying the integration of the nonlinear OD equation $\frac{dy}{dx}-F(x,y)=0$, namely looking for functions $y=u(x)$ such that ${\partial}_xu(x)-F(x,u(x))\equiv 0$, Lie discovered that the knowledge of a vector field $\theta= \xi (x,y){\partial}_x+\eta (x,y){\partial}_y$ preserving the OD equation brings the integration to a simple [*quadrature*]{}. Indeed, introducing the Lie derivative ${\cal{L}}(\theta)=i(\theta)+di(\theta) $ where now $i( )$ is the interior multiplication by a vector and $d$ is the exterior derivative, the invariant property amounts to the equation ${\cal{L}}(\theta)(dy-Fdx)=A(dy-Fdx)$ where $A$ is a multiplicative factor. Eliminating $A$ among the various factors of $dx$ and $dy$, we get the only condition:\
$$\frac{\partial \eta}{\partial x} + F \frac{\partial \eta}{\partial y} - F \frac{\partial \xi}{\partial x} - F^2 \frac{\partial \xi}{\partial y} - \frac{\partial F}{\partial x}\xi - \frac{\partial F}{\partial y}\eta =0$$ Bringing the terms together in a different way, Lie discovered that this condition can also be written:\
$$\frac{\partial}{\partial x} (\eta - F \xi) + F\frac{\partial}{\partial y}(\eta - F \xi) - \frac{\partial F}{\partial y} (\eta - F \xi) =0$$ Finally, setting $\chi = 1/(\eta - F \xi)$ and $\omega = -F$, we obtain equivalently:\
$$\frac{\partial \chi}{\partial x} - \frac{\partial (\omega \chi)}{\partial y}=0$$ a result proving that $\chi$ is an integrating factor for the $1$-form $dy + \omega dx$, that is to say the $1$-form $\chi(dy+\omega dx)$ is closed.\
It is not evident at all to establish a link between such a problem and the differential Galois theory by extending the previous purely algebraic comment to a differential framework and using the Spencer operator. First of all, [*we must start with an automorphic system*]{}. For this, changing slightly the notations while introducing a manifold $X$, say ${\mathbb{R}}^2$ with local coordinates $(x^1,x^2)$ instead of $(x,y)$, and a manifold $Y$, say $\mathbb{R}$ with local coordinate $(y)$, we may consider the automoprphic system ${\cal{A}}_1$ defined over a differential field $K$:\
$$\Phi \equiv \frac{y_1}{y_2}=\omega \in K, \hspace{5mm} y_2\neq 0$$ for the Lie pseudogroup $\Gamma = aut(Y)$ of invertible transformations $\bar{y}=g(y)$ with $ {\partial}_yg(y)\neq 0$. Then, let us notice that any solution $y=f(x^1,x^2)$ of the preceding system is such that:\
$${\partial}_1f + F(x^1,x^2){\partial}_2 f=0, \hspace{5mm} {\partial}_2f\neq 0$$ while $F\in K$ where the independent variables $(x^1,x^2)$ are not explicitly appearing. If we have any [*first integral*]{} $f(x^1,x^2)=c=cst$, we may use the implicit funxction theorem in order to obtain $x^2=u(x^1,c)$ with an [*identity*]{} $f(x^1,u(x^1,c))\equiv c, \forall x^1$. We obtain therefore:\
$${\partial}_1f(x^1,u(x^1,c)) + {\partial}_2f(x^1, u(x^1,c)){\partial}_1u(x^1,c)\equiv 0 \Rightarrow {\partial}_2f ( {\partial}_1u-F(x^1,u))=0$$ and thus ${\partial}_xu(x,c)-F(x,u(x,c))=0, \forall x,\forall c$. As for the Spencer operator, setting $x^1=x$ and introducing any function $u(x)$, we may define a section $u_1=(u(x),u_x(x))$ of the first jet bundle by choosing $u_x(x)=F(x,u(x))$ and the initial system amounts to $Du_1={\partial}_xu(x) - u_x(x)=0$.\
From the differential algebraic point of view, with $\omega= - F\in K$ when $K$ is a differential field with derivations $({\partial}_1,{\partial}_2)$, we get the differential automorphic extension $L/K=Q(K\{y\})/{\mathfrak{p}})$ where $\mathfrak{p}\subset K\{y\}$ is the prime linear differential ideal generated by the differential polynomial $P=y_1-\omega y_2\in K\{y\}$. The idea is to exhibit another differential extension $M/K$ with $M=Q(K\{z^1,z^2\}/\mathfrak{q})$ and $\mathfrak{q}\subset K\{z^1,z^2\}$ is the differential ideal generated by the differential polynomial:\
$$Q \equiv z^2_1 - \omega z^2_2 + \omega z^1_1 - {\omega}^2z^1_2 + {\partial}_1 \omega z^1 + {\partial}_2 \omega z^2 \in K\{z^1, z^2\}$$ in order to take into account the condition for $\theta = {\xi}^1{\partial}_1 + {\xi}^2 {\partial}_2$ already obtained with different notations. However, contrary to the algebraic case, [*the intersection*]{} $K'=L\cap M$ [*has no meaning at all*]{}, unless we could define such an intersection in a bigger differential field $N$ containing both $L$ and $M$, according to the following commutative diagram of field inclusions:\
$$\begin{array}{ccl}
L & \rightarrow & N \\
\uparrow & & \uparrow \\
\, K' & \rightarrow & M \\
\uparrow & \nearrow & \\
K & &
\end{array}$$ For this, let us consider the differential extension $N/K$ with $N=Q(K\{y,z^1,z^2\}/\mathfrak{r})$ and $\mathfrak{r}\subset K\{y,z^1,z^2\}$ is generated by the two differential polynomials:\
$$y_1 - \omega y_2, \hspace{5mm} y_1z^1 +y_2z^2 - 1$$ We have of course $\mathfrak{r}\cap K\{y\}=\mathfrak{p} \Rightarrow L \subset N$ and it just remains to prove that $\mathfrak{r} \cap K\{z^1,z^2 \}= \mathfrak{q} \Rightarrow M \subset N$. For this, we obtain by substitution the two differential polynomials :\
$$(z^2 + \omega z^1)y_1 - \omega,\hspace{1cm} (z^2 + \omega z^1)y_2 - 1$$ The elimination of $y$ can be done by crossed derivatives and we just find for $(z^1,z^2)$ the integrating factor condition for $\chi=1/(z^2 + \omega z^1)$ already obtained. The last relations prove that $K'$ is indeed generated by $(y_1,y_2)$ modulo $\mathfrak{p}$ and we have the new automorphic extension $L/K'$ for the subpseudogroup ${\Gamma}' \subset \Gamma$ made by the translations $\bar{y}=y+ a$ with $a=cst$. The corresponding automorphic system ${\cal{A}}'_1 \subset {\cal{A}}_1$:\
$${\Psi}^1 \equiv y_1 = \chi \omega, {\Psi}^2 \equiv y_2 = \chi , \hspace{5mm} \chi \in K'$$ is nothing else than a new description of the quadrature concept where we have now $\chi = 1/({\xi}^2 + \omega {\xi}^1)$ and $({\xi}^1,{\xi}^2)$ is the image of $(z^1,z^2)$ under the residue with respect to $\mathfrak{q}$. We finally notice that $L=K'(y)$ ([*care*]{}) while $M=K'<z^1>=K'<z^2>$. It follows that $L$ is [*regular*]{} over $K'$, that is $K'$ is algebraically closed in $L$, while $M$ is differentially transcendental over $K'$. It also follows that $L$ and $M$ are linearly disjoint over $K'$ in $N$. Up to our knowledge, such a Galois type approach has never been provided elsewhere.\
[**6) DRACH-VESSIOT THEORY REVISITED**]{}\
Roughly, in the preceding example we have used $n=2,m=1$ in order to study the [*first order*]{} OD equation $y_x -F(x,y)=0$ and our purpose is now to use $n=3,m=2$ in order to study the [*second order*]{} OD equation $y_{xx} - F(x,y,y_x)=0$. Accordingly, we shall introduce functions $u^k(x,y,y_x)$ for $k=1,2$ and consider the linear homogeneous system:\
$$\frac{\partial u^k}{\partial x}+ y_x \frac{\partial u^k}{\partial y}+ F(x,y,y_x)\frac{\partial u^k}{\partial y_x}=0, \hspace{7mm}
\frac{\partial (u^1,u^2)}{\partial (y,y_x)}\neq 0$$ For any couple of first integrals $u^k(x,y,y_x)=c^k$ with $k=1,2$, setting $c=(c^1,c^2)$, we can use the Jacobian condition and the implicit function theorem in order to solve these two equations with respect to $(y,y_x)$ in order to get:\
$$y=f( x;c), \hspace{1cm} y_x=f_x(x;c)$$ However, differentiating with respect to $x$ the identities $u^k(x,f(x;c),f_x(x;c))\equiv c^k$, we obtain the relations:\
$$\frac{\partial u^k}{\partial x} + \frac{\partial u^k}{\partial y}\frac{\partial f}{\partial x} +
\frac{\partial u^k}{\partial y_x}\frac{\partial f_x}{\partial x} =0$$ Substracting the previous equations in order to eliminate the $\partial u^k/\partial x$ while taking into account the Jacobian condition:\
$$\frac{\partial (u^1,u^2)}{\partial (y,y_x)}\neq 0 \Leftrightarrow \frac{\partial(f,f_x)}{\partial (c^1,c^2)}\neq 0$$ we obtain the [*Spencer operator*]{} through this [*vertical procedure*]{}, namely:\
$$\frac{\partial f}{\partial x} - f_x=0, \hspace{1cm} \frac{\partial f_x}{\partial x} - F=0$$
Changing slightly the notations as before, we may use a manifold $X$ with local coordinates $(x)=(x^1, x^2, x^3)$ instead of $(x,y,y_x)$ with $x^1=x$ and a manifold $Y$ with local coordinates $(y)=(y^1,y^2)$ in order to look for solutions $y^k=f^k(x^1,x^2,x^3)=f^k(x)$ for $k=1,2$ of the linear system:\
$$y^k_1 + x^3 y^k_2 + F(x) y^k_3=0, \hspace{7mm} \frac{\partial (y^1,y^2)}{\partial (x^2,x^3)}\equiv y^1_2y^2_3 - y^1_3y^2_2 \neq 0$$ It is not evident at all and it has been the discovery of Drach (\[9\]) and Vessiot (\[29\]), that this is indeed an automorphic system ${\cal{A}}_1\subset J_1(X\times Y)$ for the Lie pseudogroup $\Gamma =aut(Y)$ made by invertible transformations of the form $\bar{y}=g(y)$ with nonzero jacobian, that can be written:\
$${\Phi}^1 \equiv \frac{ \frac{\partial (y^1,y^2)}{\partial (x^1,x^2)}}{\frac{\partial (y^1,y^2)}{\partial (x^2,x^3)}}={\omega}^1(x)=F(x),\hspace{5mm}{\Phi}^1 \equiv \frac{ \frac{\partial (y^1,y^2)}{\partial (x^3,x^1)}}{\frac{\partial (y^1,y^2)}{\partial (x^2,x^3)}}=
{\omega}^2(x)=x^3$$ We have thus obtained:\
[**THEOREM 6.1**]{}: The search for a family of solutions of the given second order OD equations depending on $2$ parameters is equivalent to the knowlege of one solution of this automorphic system.\
As the two generating first order differential invariants are rational functions of the first jets, we may therefore use the differential Galois theory by introducing a differential field $K$ with derivations $({\partial}_1,{\partial}_2, {\partial}_3)$ in such a way that ${\omega}^1, {\omega}^2 \in K$. Also, the fiber dimension of the system is $2+(3\times 2)-2=6$ while the fiber dimension of the system of Lie equations (made by no equation !) is $2+(2\times 2)=6$ and the automorphic property follows from the fact that ${\cal{A}}_1$ is involutive with no CC.\
Now, following Jacobi, we shall call a function $M(x)$ (Jacobi) [*multiplier*]{} for $\theta={\theta}^i{\partial}_i$ if we have the relation ${\partial}_i(M{\theta}^i)=0$. In particular, if $\bar{x}=\varphi (x)$ is a change of independent variables with jacobian $\Delta (x)=det ({\partial}_i{\varphi}^j(x)\neq 0$, then it is well known that we have the [*identity*]{} (\[21,23,26\]):\
$$\frac{\partial}{\partial {\bar{x}}^j}( \frac{1}{\Delta}\frac{\partial {\varphi}^j}{\partial x^i})\equiv 0, \hspace {1cm} \forall x\in X$$ Setting ${\bar{\theta}}^j={\partial}_i{\varphi}^j{\theta}^i$, we obtain easily:\
$$\frac{\partial}{\partial {\bar{x}}^j}(\frac{M}{\Delta}{\bar{\theta}}^j)=
\frac{\partial}{\partial {\bar{x}}^j} (\frac{M}{\Delta}\frac{\partial {\varphi}^j}{\partial x^i}{\theta}^i)=
\frac{1}{\Delta}\frac{\partial}{\partial x^i}(M{\theta}^i)=0$$ and it follows that $M/\Delta$ is a multiplier for $\bar{\theta}$. In particular, $M=1$ is a multiplier if and only if $\theta$ is divergence free, that is ${\partial}_i{\theta}^i=0$. In the present situation, we should have ${\partial}_1(1) + {\partial}_2 (x^3) + {\partial}_3(F)=0$, that is ${\partial}_3F=0$ and thus $F=F(x^1,x^2)$.\
Coming back to the initial notations, let us consider the second order OD equation $y_{xx} - F(x,y)=0$ and the corresponding system:\
$$\theta.\phi\equiv \frac{\partial \phi}{\partial x} +y_x\frac{\partial \phi}{\partial y}+ F(x,y) \frac{\partial \phi}{\partial y_x}=0$$ If $\phi(x,y,y_x)=c=cst$ is a first integral containing explicily $y_x$, we can use locally the implicit function theorem and find $y_x=\psi(x,y;c)$. Let us prove that the $1$-form $dy - \psi(x,y;c)dx$ has the integrating factor $1/\frac{\partial \phi}{\partial y_x}(x,y,\psi(x,y;c))$ which is also a Jacobi multiplier for the differential system $dx/1=dy/\psi(x,y,c)$, that is let us prove:\
$$\frac{\partial}{\partial x}((1/\frac{\partial \phi}{\partial y_x}) + \frac{\partial}{\partial y}(\psi/\frac{\partial \phi}{\partial y_x})=0$$ whenever $y_x=\psi(x,y;c)$ that is to say:\
$$\frac{{\partial}^2 \phi}{\partial x\partial y_x} +(\frac{\partial \psi}{\partial x} + \psi \frac{\partial \psi}{\partial y})\frac{{\partial}^2\phi}{\partial y_x\partial y_x} + \psi \frac{{\partial}^2 \phi}{\partial y\partial y_x} - \frac{\partial \psi}{\partial y} \frac{\partial \phi}{\partial y_x} =0$$ Indeed, differentiating the identity $\phi(x,y,\psi(x,y;c))\equiv c$ with respect to $c$, $x$ and $y$ successively, we get:\
$$\frac{\partial \phi}{\partial y_x}\frac{\partial \psi}{\partial c}=1 \Rightarrow \frac{\partial \phi}{\partial y_x}\neq 0$$ $$\frac{\partial \phi}{\partial x} + \frac{\partial \phi}{\partial y_x}\frac{\partial \psi}{\partial x} =0, \hspace{5mm}
\frac{\partial \phi}{\partial y} + \frac{\partial \phi}{\partial y_x}\frac{\partial \psi}{\partial y} =0
\, \, \Rightarrow \,\, \frac{\partial \psi}{\partial x}+ \psi \frac{\partial \psi}{\partial y}=F(x,y)$$ and it just remains to differentiate the PD equation satisfied by $\phi$ with respect to $y_x$.\
Among the possible reductions of the Galois pseudogroup $\Gamma$, we may consider the Lie sub-pseudogroup ${\Gamma}'=\{ \bar{y}=g(y) \mid \partial({\bar{y}}^1,{\bar{y}}^2)/\partial (y^1,y^2)=1\}$ leading to the automorphic sub-system ${\cal{A}}'_1\subset {\cal{A}}_1$:\
$${\Psi}^1\equiv \frac{\partial (y^1,y^2)}{\partial (x^2,x^3)}={\psi}^1, \hspace{5mm} {\Psi}^2\equiv \frac{\partial (y^1,y^2)}{\partial (x^3,x^1)}={\psi}^2, \hspace{5mm}{\Psi}^3\equiv \frac{\partial (y^1,y^2)}{\partial (x^1,x^2)}={\psi}^3$$ where ${\psi}^1,{\psi}^2,{\psi}^3 \in K'$ with $K \subset K' \subset L$ and:\
$${\partial}_1{\psi}^1 + {\partial}_2 {\psi}^2 + {\partial}_3 {\psi}^3=0$$ As we must have ${\psi}^2/{\psi}^1= x^3, \, \, {\psi}^3/{\psi}^1=F(x^1,x^2,x^3) $, [*this reduction amounts to the explicit knowledge of a Jacobi multiplier*]{}. In the particular case $F=F(x^1,x^2)$, we may choose ${\psi}^1=1$ and [*this situation is exactly the one obtained by passing from the Lagrangian formalism to the Hamiltonian formalism in analytical mechanics*]{}. We provide the details of this striking result which does not seem to be known.\
With $n=3,m=2$ in this case, let us consider a Lagrangian $L(t,x,\dot{x})$ and the corresponding Euler-Lagrange equations:\
$$\frac{d}{dt}(\frac{\partial L}{\partial \dot{x}}) - \frac{\partial L}{\partial x}=0 \Leftrightarrow \frac{{\partial}^2L}{\partial t \partial \dot{x}}+\dot{x} \frac{{\partial}^2 L}{\partial x\partial \dot{x}}+\ddot{x} \frac{{\partial}^2 L}{\partial \dot{x}\partial \dot{x}}-
\frac{\partial L}{\partial x}=0$$ When the [*Hessian condition*]{} ${\partial}^2L/\partial \dot{x}\partial \dot{x}\neq 0$ is satisfied, we get a second order OD equation of the form $\ddot{x} - F(t,x,\dot{x})=0$ and we may thus introduce the automorphic system:\
$$\frac{\partial y^k}{\partial t} + \dot{x} \frac{\partial y^k}{\partial x} + F(t,x, \dot{x}) \frac{\partial y^k}{\partial \dot{x}}=0, \hspace{5mm}
\forall k=1,2$$ with:\
$$\frac{\partial}{\partial t} (1) +\frac{ \partial}{\partial x} (\dot{x}) +\frac{\partial}{\partial \dot{x}}(F(t,x,\dot{x}))=
\frac{\partial F}{\partial \dot{x}}\neq 0$$ in general. However, if we now consider the corresponding Hamiltonian formalism obtained by setting $p=\frac{\partial L}{\partial \dot{x}}$ and $H=\dot{x} \frac{\partial L}{\partial \dot{x}} - L= H(t,x,p)$, we obtain at once:\
$$dH= \dot{x} dp - \frac{\partial L}{\partial t} dt -\frac{\partial L}{\partial x} dx$$ $$\frac{dp}{dt}= \frac{{\partial}^2 L}{\partial t \partial \dot{x}} + \dot{x} \frac{{\partial}^2 L}{\partial x \partial \dot{x}}+
\ddot{x}\frac{{\partial}^2 L}{\partial \dot{x} \partial \dot{x}}=\frac{\partial L}{\partial x}$$ and thus the well known OD Hamiltonian equations:\
$$\frac{dx}{dt} = \frac{\partial H}{\partial p}, \hspace{1cm} \frac{dp}{dt}= - \frac{\partial H}{\partial x}$$ a result leading to the automorphic system:\
$$\frac{\partial y^k}{\partial t} + \frac{\partial H}{\partial p} \frac{\partial y^k}{\partial x} -
\frac{\partial H}{\partial x}\frac{\partial y^k}{\partial p}=0, \hspace{1cm} \forall k=1,2$$ on which we check:\
$$\frac{\partial}{\partial t} (1) +\frac{\partial }{\partial x} (\frac{\partial H}{\partial p}) + \frac{\partial }{\partial p}(- \frac{\partial H}{\partial x})=0$$ The previous Lagrangian automorphic system had independent variables $(t,x,\dot{x})$ while the new Hamiltonian automorphic system has independent variables $(t,x,p)$. As this latter system admits the Jacobi multiplier $1$, it follows from the general theory explained at the beginning of this example that the Lagrangian system [*must*]{} admit the Jacobi multiplier:\
$$\frac{\partial (t,x,p)}{\partial (t,x,\dot{x})}=\frac{\partial p}{\partial \dot{x}}=\frac{{\partial}^2L}{\partial \dot{x} \partial \dot{x}}$$ Indeed, multiplying respectively $(1,\dot{x}, F(t,x,\dot{x}))$ by the hessian and noticing that:\
$$F(t,x,\dot{x}) \frac{{\partial}^2L}{\partial \dot{x} \partial \dot{x}}=\ddot{x}\frac{{\partial}^2L}{\partial \dot{x} \partial \dot{x}}=
\frac{\partial L}{\partial x} - \frac{{\partial}^2 L}{\partial t \partial \dot{x}}- \dot{x} \frac{{\partial}^2 L}{\partial x \partial \dot{x}}$$ we finally check, after an easy computation, the identity:\
$$\frac{\partial}{\partial t}(\frac{{\partial}^2L}{\partial \dot{x} \partial \dot{x}}) + \frac{\partial}{\partial x} (\dot{x}\frac{{\partial}^2L}{\partial \dot{x} \partial \dot{x}}) + \frac{\partial}{\partial \dot{x}}(\frac{\partial L}{\partial x} - \frac{{\partial}^2 L}{\partial t \partial \dot{x}}- \dot{x} \frac{{\partial}^2 L}{\partial x \partial \dot{x}})\equiv 0$$ The reduction to ${\Gamma}'\subset \Gamma$ becomes respectively:\
$$\frac{\partial (y^1,y^2)}{\partial (x, \dot{x})}=\frac{{\partial}^2L}{\partial \dot{x} \partial \dot{x}} ,\hspace{5mm}
\frac{\partial (y^1,y^2)}{\partial ( \dot{x}, t)}= \dot{x} \frac{{\partial}^2L}{\partial \dot{x} \partial \dot{x}},\hspace{5mm}
\frac{\partial (y^1,y^2)}{\partial ( \dot{x}, t)}=\frac{\partial L}{\partial x} - \frac{{\partial}^2 L}{\partial t \partial \dot{x}}- \dot{x} \frac{{\partial}^2 L}{\partial x \partial \dot{x}}$$ $$\frac{\partial (y^1,y^2)}{\partial (x,p)}=1, \hspace{5mm} \frac{\partial (y^1,y^2)}{\partial (p,t)}=\frac{\partial H}{\partial p}, \hspace{5mm}
\frac{\partial (y^1,y^2)}{\partial (t,x)}= - \frac{\partial H}{\partial x}$$ The knowledge of one first integral, say $y^2=\phi$, reduces the Galois pseudogroup to ${\Gamma}"=\{ {\bar{y}}^1=y^1 + h(y^2), {\bar{y}}^2=y^2 \}$ .\
[**7) HAMILTON-JACOBI EQUATIONS REVISITED**]{}\
We end this list of examples by revisiting the Hamilton-Jacobi equation. This is [*by far*]{} the most difficult example in the sense that no classical approach using exterior calculus can be used in order to exhibit the corresponding automorphic systems involved. At the same time, it uses the first criterion for automorphic systems and thus, in particular, formal integrability or involution become crucial tools that cannot be avoided. This is the reason for which the results we present have not been found during the last century.\
Let $z=f(t,x)$ be a solution of the non-linear PD equation $z_t+H(t,x,z,z_x)=0$ written with jet notations for the single unknown $z$. When dealing with applications, $t$ will be [*time*]{}, $x$ will be [*space*]{}, $z$ will be the [*action*]{} and, as usual, we shall set $p=z_x$ for the [*momentum*]{}. It is important to notice that, in this general setting, $H(t,x,z,p)$ [*cannot be called Hamiltonian as it involves*]{} $z$. By analogy with the preceding example, we shall set (\[9,20-22,29\]):\
[**DEFINITION 7.1**]{}: A [*complete integral*]{} $z=f(t,x;a,b)$ is a family of solutions depending on two constant parameters $(a,b)$ in such a way that the Jacobian condition $\partial (z,p)/\partial (a,b)\neq 0$ whenever $p={\partial}_xf(t,x;a,b)$. Using the implicit function theorem, we may set\
[**THEOREM 7.2**]{}: The search for a complete integral of the PD equation:\
$$z_t+H(t,x,z,z_x)=0$$ is equivalent to the search for a [*single*]{} solution of the automorphic system ${\cal{A}}_1$ with $n=4,m=3$, obtained by eliminating $\rho(t,x,z,p)$ in the Pfaffian system:\
$$dz-pdx+H(t,x,z,p) dt = \rho (dZ-PdX )$$ The corresponding Lie pseudogroup is the pseudogroup $\Gamma$ of [*contact transformations*]{} of $(X,Z,P)$ that reproduces the contact $1$-form $dZ-PdX$ up to a function factor.\
[*Proof*]{}: If $z=f(t,x; a,b)$ is a complete integral, we have:\
$$dz-pdx+H(t,x,z,p)dt= \frac{\partial f}{\partial a}da +\frac{\partial f}{\partial b} db$$ Using the implicit function theorem and the Jacobian condition, we may set:\
$$a =X(t,x,z,p), \, b=Z(t,x,z,p) \Rightarrow \rho(t,x,z,p)=\frac{\partial f}{\partial b}, \,
P(t,x,z,p)=\frac{\partial f}{\partial a}/\frac{\partial f}{\partial b}$$ The converse is left to the reader.\
For another solution denoted wit a “bar”, we have:\
$$dz-pdx+H(t,x,z,p) dt = \bar{\rho} (d\bar{Z}-\bar{P} d\bar{X} )\,\,\Rightarrow \,\,d\bar{Z}-\bar{P} d\bar{X} = \frac{\rho}{\bar{\rho}}(dZ-PdX)$$ Closing this system, we obtain at once:\
$$d\bar{X}\wedge d\bar{Z}\wedge d\bar{P}= (\frac{\rho}{\bar{\rho}})^2 dX\wedge dZ \wedge dP$$ Closing again, we discover that $\rho/\bar{\rho}$ is in fact a function of $(X,Z,P)$, a result bringing the Lie pseudogroup of contact transformations and showing that no restriction must be imposed to $H(t,x,z,p)$.\
Q.E.D.\
It is quite more dificult to exhibit the equations of the above automorphic sytem and the corresponding equations of the Lie pseudogroup $\Gamma$ in Lie form or even as involutive systems of PD equations. From what has been said, we obtain [*at least*]{}:\
$$\frac{\frac{\partial \bar{Z}}{\partial X} - \bar{P}\frac{\partial \bar{X}}{\partial X}}{ \frac{\partial \bar{Z}}{\partial Z} - \bar{P}\frac{\partial \bar{X}}{\partial Z}}= - P ,
\frac{\frac{\partial \bar{Z}}{\partial P} - \bar{P}\frac{\partial \bar{X}}{\partial P}}{ \frac{\partial \bar{Z}}{\partial Z} - \bar{P}\frac{\partial \bar{X}}{\partial Z}}= 0 \Rightarrow \frac{\partial \bar{Z}}{\partial P} - \bar{P}\frac{\partial \bar{X}}{\partial P}=0$$ for defining ${\cal{R}}_1$, that is to say:\
$$\frac{ \partial \bar{Z}}{\partial X} - \bar{P} \frac{\partial \bar{X}}{\partial X} + P (\frac{\partial \bar{Z}}{\partial Z} - \bar{P}\frac{\partial \bar{X}}{\partial Z})=0 , \hspace{1cm} \frac{\partial \bar{Z}}{\partial P} - \bar{P}\frac{\partial \bar{X}}{\partial P}=0$$ Using now letters $(x,z,p)$ instead of the capital letters $(X,Z,P)$ and $(\xi, \eta, \zeta)$ for the corresponding vertical bundles, we obtain by linearization the system of first order infinitesimal Lie equations:\
$$\frac{\partial \xi}{\partial x}- p \frac{ \partial \eta}{\partial x} - \zeta + p ( \frac{\partial \xi}{\partial z}- p\frac{ \partial \eta}{\partial z})=0, \,\,
\frac{\partial \xi}{ \partial p }- p \frac{\partial \eta}{\partial p}=0$$ This system is not involutive as it is not even formally integrable. Using crossed derivatives in $x/p$, we obtain the [*only new first order*]{} equation:\
$$\frac{\partial \eta}{\partial x} - \frac{\partial \xi}{\partial z}+ \frac{\partial \zeta }{\partial p} + 2p \frac{\partial \eta}{\partial z}=0$$ and the resulting system ${\cal{R}}^{(1)}_1$ is involutive with two equations of class $x$ solved with respect to $(\frac{\partial \xi}{\partial x}, \frac{\partial \eta}{\partial x})$ and one equation of class $p$ solved with respecto $\frac{\partial \xi}{\partial p}$, that is $dim_Y({\cal{R}}^{(1)}_1)= (3+ 3 \times 3) - 3=9$. Accordingly, the non-linear system of Lie equations [*must*]{} become involutive by adding [*only one equation in Lie form*]{}, namely:\
$$\frac{ \frac{\partial (\bar{Z},\bar{X}, \bar{P})}{\partial (Z,X,P)}}{ ( \frac{\partial \bar{Z}}{\partial Z} - \bar{P} \frac{\partial \bar{X}}{\partial X})^2} = 1$$ and its linearization jus provides:\
$$\frac{\partial \eta}{\partial x} + \frac{\partial \xi}{\partial z}+ \frac{\partial \zeta }{\partial p}= 2 (\frac{\partial \xi}{\partial z} - p \frac{\partial \eta}{\partial z})$$ that is exactly the previous equation. The following [*Janet board*]{} provides the structure of an involutive [*solved form*]{}:\
$$\left\{ \begin{array}{lcl}
X & \longrightarrow & \left\{ \begin{array}{c} \bar{Z} \\ \bar{X} \end{array}\right. \\
P & \longrightarrow & \, \left\{ \begin{array}{c} \bar{Z} \end{array} \right.
\end{array} \right.
\fbox{ $ \begin{array}{lll}
Z & P & X \\
Z & P & X \\
Z & P & \bullet \\
\end{array}$ }$$
Coming back to the original system and notations, we may suppose $\frac{\partial Z}{\partial z} -P\frac{\partial X}{\partial z}\neq 0$ and introduce the $7=3+4$ equations:\
$$\frac{\partial Z}{\partial x} - P\frac{\partial X}{\partial x} + p ( \frac{\partial Z}{\partial z} - P\frac{\partial X}{\partial z})=0,
\frac{\partial Z}{\partial t} - P\frac{\partial X}{\partial t} - H ( \frac{\partial Z}{\partial z} - P\frac{\partial X}{\partial z})=0,
\frac{\partial Z}{\partial p} - P \frac{\partial X}{\partial p}=0$$ $$\frac{\partial (Z,X,P)}{\partial (z,x,p)} - (\frac{\partial Z}{\partial z} - P\frac{\partial X}{\partial z})^2=0,
\frac{\partial (Z,X,P)}{\partial (z,p,t)} - \frac{\partial H}{\partial p} (\frac{\partial Z}{\partial z} - P\frac{\partial X}{\partial z})^2=0, ...$$
Starting now, the next results [*canot*]{} be obtained by exterior calculus and are therefore not known. Indeed, developping the $ 3 \times 3$ Jacobian determinant, the fourth equation provided can be written as:\
$$\frac{\partial Z}{\partial x}. \frac{\partial (X,P)}{\partial (x,p)} - \frac{\partial Z}{\partial x} . \frac{\partial ( (X,P)}{\partial (z,p)} + \frac{\partial Z}{\partial p}. \frac{\partial (X,P)}{\partial (z,x)} - (\frac{\partial Z}{\partial z} - P\frac{\partial X}{\partial z})^2=0$$ Using the previous equations in order to eliminate $\frac{\partial Z}{\partial x}$ and $\frac{\partial Z}{\partial p}$, we obtain:\
$$\frac{\partial Z}{\partial x}. \frac{\partial (X,P)}{\partial (x,p)} + p(\frac{\partial Z}{\partial z}-P \frac{\partial X}{\partial z}) . \frac{\partial ( (X,P)}{\partial (z,p)} - P\frac{\partial X}{\partial x}. \frac{\partial (X,P)}{\partial (z,p)} + P \frac{\partial X}{\partial p}. \frac{\partial (X,P)}{\partial (z,x)} =$$ $$(\frac{\partial Z}{\partial z} - P \frac{\partial X}{\partial z})( \frac{\partial (X,P)}{\partial (x,p)} + p \frac{\partial (X,P)}{\partial (z,p)}) = (\frac{\partial Z}{\partial z} - P\frac{\partial X}{\partial z})^2$$ and thus:\
$$\frac{\partial (X,P)}{\partial (x,p)} + p \frac{\partial (X,P)}{\partial (z,p)}) - (\frac{\partial Z}{\partial z} - P\frac{\partial X}{\partial z}) =0$$ which is nothing else than the first order equation that can be obtained from the first and third among the previous $7$ equations by using crossed derivatives in $x/p$. It follows that ${\cal{A}}^{(1)}_1$ may be defined by $6$ equations [*only*]{} and we have thus $dim_X ({\cal{A}}^{(1)}_1)= (3 + 4 \times 3)- 6=9$. The following [*Janet board*]{} provides the structure of an involutive [*solved form*]{}:\
$$\left\{ \begin{array}{lcl}
x & \longrightarrow & \left\{ \begin{array}{c} Z \\ X \\ P \end{array}\right. \\
t & \longrightarrow & \left\{ \begin{array}{c} Z \\ X \end{array} \right. \\
p & \longrightarrow & \, \left\{ \begin{array}{c} Z \end{array} \right.
\end{array} \right.
\fbox{ $ \begin{array}{llll}
z & p & t & x \\
z & p & t & x \\
z & p & t & x \\
z & p & t & \bullet \\
z & p & t & \bullet \\
z & p &\bullet & \bullet
\end{array}$ }$$
This result proves that the involutive system ${\cal{A}}^{(1)}_1$ is an automorphic system for the involutive Lie groupoid ${\cal{R}}^{(1)}_1$.\
If $H=H(t,x,p)$ is an Hamiltonian function, then we may look for a complete integral of the form $z=f(t,x;a)+b$ and we have:\
[**COROLLARY 7.3**]{}: The search for such a complete integral of the PD equation:\
$$z_t+H(t,x,z_x)=0$$ is equivalent to the search for a [*single*]{} solution of the automorphic system ${\cal{A}}'_1$ with $n=4,m=3$ described by the Pfaffian system:\
$$dz-pdx+H(t,x,p) dt = dZ-PdX$$ The corresponding Lie pseudogroup is the Lie pseudogroup ${\Gamma}'\subset \Gamma$ of [*unimodular contact transformations*]{} of $(X,Z,P)$ that preserve the contact $1$-form $dZ-PdX$ and we have thus $\partial(\bar{X},\bar{Z},\bar{P})/\partial (X,Z,P)=1$.\
[*Proof*]{}: Now, we have:\
$$p=\frac{\partial f}{\partial x}(t,x;a) \Rightarrow a=X(t,x,p)\Rightarrow b=Z(t,x,z,p)=z-\varphi (t,x,p), \rho(t,x,z,p)=1$$ and thus $$\frac{\partial Z}{\partial z}=1, \frac{\partial X}{\partial z}=0, \frac{\partial f}{\partial b}=1 \Rightarrow P=\frac{\partial f}{\partial a}(t,x;X(t,x,p))\Rightarrow \frac{\partial P}{\partial z}=0.$$ We shall just prove that the Pfaffian system:\
$$dz-pdx+H(t,x,z,p) dt = dZ-PdX$$ is compatible if and only if $\partial H/\partial z=0$ as a new group theoretical justification for revisiting the mathematical foundations of analytical mechanics.\
Indeed, closing the system, we get:\
$$dx\wedge dp +dH\wedge dt=dX\wedge dP$$ Using the exterior multiplication among the coresponding left and right members, we get:\
$$dz\wedge dx\wedge dp + dz\wedge dH \wedge dt- pdx\wedge dH\wedge dt +H dt\wedge dx\wedge dp = dZ\wedge dX \wedge dP$$ Closing again, we finally obtain:\
$$- dp\wedge dx \wedge dH \wedge dt + dH \wedge dt \wedge dx \wedge dp= 2 \frac{\partial H}{\partial z} dt \wedge dx \wedge dz \wedge dp=0$$ and the desired condition on $H$.\
We invite the reader to discover this condition just using CC for the second member of the desired automorphic system and notice that ${\cal{A}}'_1$ [*is neither involutive nor even formally integrable*]{}.\
Q.E.D.\
It is again dificult to exhibit the equations of the above automorphic sytem and the corresponding equations of the Lie pseudogroup ${\Gamma}'$ in Lie form or even as involutives systems of PD equations. From what has been said, we obtain [*at least*]{}:\
$$\frac{\partial \bar{Z}}{\partial Z} - \bar{P}\frac{\partial \bar{X}}{\partial Z}= 1, \frac{\partial \bar{Z}}{\partial X} - \bar{P}\frac{\partial \bar{X}}{\partial X}= - P ,
\frac{\partial \bar{Z}}{\partial P} - \bar{P}\frac{\partial \bar{X}}{\partial P}=0$$ for defining ${\cal{R}}'_1$, that is to say:\
$$\frac{ \partial \bar{Z}}{\partial X} - \bar{P} \frac{\partial \bar{X}}{\partial X} + P (\frac{\partial \bar{Z}}{\partial Z} - \bar{P}\frac{\partial \bar{X}}{\partial Z})=0$$ Now, contrary to the preceding situation, we have the Pfaffian system:\
$$dX\wedge dP =d\bar{X}\wedge d\bar{P}$$ and we may add the $3$ new first order equations:\
$$\frac{\partial (\bar{X},\bar{P})}{\partial (X,P)}=1,\frac{\partial (\bar{X},\bar{P})}{\partial (P,Z)}=0,
\frac{\partial (\bar{X},\bar{P})}{\partial (Z,X)}=0$$ an obtain therefore the $6$ equations:\
$$\frac{\partial \bar{Z}}{\partial Z}=1, \frac{\partial \bar{X}}{\partial Z}=0, \frac{\partial \bar{P}}{\partial Z}=0,
\frac{\partial \bar{Z}}{\partial X} - \bar{P}\frac{\partial \bar{X}}{\partial X}= - P ,\frac{\partial (\bar{X},\bar{P})}{\partial (X,P)}=1,
\frac{\partial \bar{Z}}{\partial P} - \bar{P}\frac{\partial \bar{X}}{\partial P}=0$$ The following [*Janet board*]{} provides the structure of an involutive [*solved form*]{}:\
$$\left\{ \begin{array}{lcl}
Z & \longrightarrow & \left\{ \begin{array}{c} \bar{Z} \\ \bar{X}\\ \bar{P} \end{array}\right. \\
X & \longrightarrow & \left\{ \begin{array}{c} \bar{Z} \\ \bar{X} \\ \end{array} \right. \\
P & \longrightarrow & \, \left\{ \begin{array}{c} \bar{Z} \end{array} \right.
\end{array} \right.
\fbox{ $ \begin{array}{lll}
P & X & Z \\
P & X & Z \\
P & X & Z \\
P & X & \bullet \\
P & X &\bullet \\
P & \bullet & \bullet
\end{array}$ }$$
The resulting system ${\cal{R}}'^{(1)}_1$ is thus involutive with $3$ equations of class $Z$ solved with respect to $(\frac{\partial \bar{Z}}{\partial Z}, \frac{\partial \bar{X}}{\partial Z},\frac{\partial \bar{P})}{\partial Z}$, $2$ equations of class $X$ solved with respect to $(\frac{\partial \bar{Z}}{\partial X}, \frac{\partial \bar{X}}{\partial X})$ and $1$ equation of class $P$ solved with respect to $(\frac{\partial \bar{Z}}{\partial P})$. The characters are $(2,1,0)$ and we have $dim_Y({\cal{R}}'^{(1)}_1)= (3+ 9) - 6=6$.\
Taking into account the initial Pfaffian system $dz-pdx+H(t,x,p)dt=dZ-PdX$ and its exterior closure $dx\wedge dp +dH\wedge dt=dX\wedge dP$, we may proceed as before and find the first order system combining the $6$ equations:\
$$\frac{\partial Z}{\partial z}=1, \frac{\partial X}{\partial z}=0, \frac{\partial P}{\partial z}=0,
\frac{\partial Z}{\partial x} - P \frac{\partial X}{\partial x}= - p ,
\frac{\partial Z}{\partial p} - P \frac{\partial X}{\partial p}=0 , \frac{\partial Z}{\partial t} - P \frac{\partial X}{\partial t}=H$$ with the $3$ equations:\
$$\frac{\partial (X,P)}{\partial (x,p)}=1,\, \, \, \frac{\partial (X,P)}{\partial (x,t)}=\frac{\partial H}{\partial x}, \, \, \, \frac{\partial (X,P)}{\partial (p,t)}=\frac{\partial H}{\partial p}$$ The following [*Janet board*]{} provides the structure of an involutive [*solved form*]{}:\
$$\left\{ \begin{array}{lcl}
z & \longrightarrow & \left\{ \begin{array}{c} Z \\ X \\ P \end{array}\right. \\
x & \longrightarrow & \left\{ \begin{array}{c} Z \\ X \\ P \end{array}\right. \\
p & \longrightarrow & \left\{ \begin{array}{c} Z \\ X \end{array} \right. \\
t & \longrightarrow & \, \left\{ \begin{array}{c} Z \end{array} \right.
\end{array} \right.
\fbox{ $ \begin{array}{llll}
t & p & x & z \\
t & p & x & z \\
t & p & x & z \\
t & p & x & \bullet \\
t & p & x & \bullet \\
t & p & x & \bullet \\
t & p & \bullet & \bullet \\
t & p & \bullet & \bullet \\
t & \bullet & \bullet & \bullet
\end{array}$ }$$ with $3$ equations of class $z$, $3$ equations of class $x$, $2$ equations of class $p$ and $1$ equation of class $t$ giving characters $(2,1,0,0)$ where the non-zero ones coincide with the non-zero ones of ${\cal{R}}^{'(1)}_1$. One must therefore use the involutive automorphic system ${\cal{A}}'^{(1)}_1={\pi}^2_1 ({\cal{A}}'_2)$ where ${\cal{A}}'_2$ is the first prolongation of ${\cal{A}}'_1$. A similar difficulty has been found for the nonlinear Lie equations ${\cal{R}}'_1$ defining the Lie pseudogroup ${\Gamma}'\subset \Gamma$.\
[**COROLLARY 7.4**]{}: The search for a complete integral $z=u(t;a) + v(x;a) +b$ by [*separation of variables*]{} is equivalent to the search for a single solution of the automorphic system obtained by adding $\partial X/\partial t=0$ to the system of the preceding Corollary, provided that:\
$$\frac{\partial H}{\partial z}=0 ,\hspace{1cm} \frac{\partial}{\partial t} ( \frac{\partial H}{\partial x} / \frac{\partial H}{\partial p})=0$$ The corresponding Lie pseudogroup is:\
$${\Gamma}"= \{\bar{X}=g(X), \bar{Z}=Z+ h(X), \bar{P}=(P + \partial h/\partial X)/ (\partial g/ \partial X)\} \subset {\Gamma}' \subset \Gamma$$
[*Proof*]{}: We have $p=\partial v/\partial x \Rightarrow a=X(x,p) \Rightarrow \partial X / \partial t=0 $ and:\
$$b=z-u(t;X(x,p))- v(x;X(x,p))=Z(t,x,p) \Rightarrow \frac{\partial Z}{\partial t}= \frac{\partial u}{\partial t}(t;X(x,p))=H(t,x,p)$$ $$\frac{\partial f}{\partial b}=1\Rightarrow P= \frac{\partial f}{\partial a}= \frac{\partial v}{\partial a}(x; X(x,p)) \Rightarrow \frac{\partial P}{\partial t}=0$$ Using the last three equations of the preceding Corollary, we get:\
$$\frac{\partial X}{\partial t}=\frac{\partial H}{\partial p} / \frac{\partial X}{\partial x} - \frac{\partial H}{\partial x} / \frac{\partial X}{\partial p}=0$$ We obtain therefore the additional differential invariant in Lie form:\
$$\frac{\partial X}{\partial x} / \frac{\partial X}{\partial p} = \frac{\partial H}{\partial x} / \frac{\partial H}{\partial p}$$ and the desired CC by differentiating with respect to $t$. This [*necessary condition*]{} for separating the variables in the integration of the Hamilton-Jacobi equation has been found by T. Levi-Civita in $1904$ (\[17\]) and integrated byA. Huaux in $1976$ (\[12\]).\
It also follows that the corresponding Lie pseudogroup ${\Gamma}"$ must contain transformations $\bar{X}=g(X)$. Then $\partial \bar{Z}/\partial Z=1, \partial \bar{Z}/\partial P=0 \Rightarrow \bar{Z}=Z + h(X)$ and finally:\
$$\frac{\partial \bar{P}}{\partial P}\frac{\partial \bar{X}}{\partial X}=1, \frac{\partial \bar{Z}}{\partial X} - \bar{P}\frac{\partial \bar{X}}{\partial X}= - P \Rightarrow \bar{P}= (P + \partial h/\partial X)/ (\partial g/ \partial X)$$. Q.E.D.\
[**COROLLARY 7.5**]{}: The search for a complete integral $z=u(t) + v(x;a) + at + b$ is equivalent to the search for $1$ solution of the automorphic system obtained by adding $\partial P/\partial t=1$ to the system of the preceding Corollary, provided that:\
$$\frac{\partial H}{\partial z}=0, \hspace{1cm} \frac{{\partial}^2 H}{\partial t\partial x}=0,\hspace{1cm} \frac{{\partial}^2 H}{\partial t \partial p}=0$$ The corresponding Lie pseudogroup is:\
$${\Gamma}'''= \{ \bar{X}=X+c, \bar{Z}=Z + h(X), \bar{P}= P + \partial h/\partial X \}$$
[*Proof*]{}: We have:\
$$p=\partial v / \partial x (x;a) \Rightarrow a=X(x,p), \hspace{5mm} Z=b=z- u(t) - v(x;X(x,p)) -X(x,p)t$$ and:\
$$\frac{\partial f}{\partial b}=1 \Rightarrow P=\frac{\partial v}{\partial a }(x; X(x,p)) + t \Rightarrow \frac{\partial P}{\partial t}=1$$ $$\frac{\partial Z}{\partial t}= - \frac{\partial u}{\partial t}(t) - X(x,p)=H(t,x,p) \Rightarrow \frac{\partial H}{\partial t}= -\frac{{\partial}^2 u}{\partial t^2}(t)$$ Collecting all these results, the system is defied by the $11$ equations;\
$$\frac{\partial Z}{\partial z}=1, \frac{\partial X}{\partial z}=0, \frac{\partial P}{\partial z}=0, \frac{\partial X}{\partial t}=0,
\frac{\partial Z}{\partial t}=H , \frac{\partial P}{\partial t}=1 , \frac{\partial X}{\partial x}=\frac{\partial H}{\partial x}, \frac{\partial X}{\partial p}=\frac{\partial H}{\partial p},$$ $$\frac{\partial Z}{\partial x} - \frac{\partial H}{\partial x}P= - p , \frac{\partial Z}{\partial p} - \frac{\partial H}{\partial p} P=0 ,
\frac{\partial H}{\partial x}\frac{\partial P}{\partial p} - \frac{\partial H}{\partial p}\frac{\partial P}{\partial x}= 1$$ The following [*Janet board*]{} provides the structure of an involutive solved form:\
$$\left\{ \begin{array}{lcl}
z & \longrightarrow & \left\{ \begin{array}{c} Z \\ X \\ P \end{array}\right. \\
x & \longrightarrow & \left\{ \begin{array}{c} Z \\ X \\ P \end{array}\right. \\
t & \longrightarrow & \left\{ \begin{array}{c} Z \\ X \\ P \end{array} \right. \\
p & \longrightarrow & \, \left\{ \begin{array}{c} Z \\ X \end{array} \right.
\end{array} \right.
\fbox{ $ \begin{array}{llll}
p & t & x & z \\
p & t & x & z \\
p & t & x & z \\
p & t & x & \bullet \\
p & t & x & \bullet \\
p & t & x & \bullet \\
p & t & \bullet & \bullet \\
p & t & \bullet & \bullet \\
p & t & \bullet & \bullet \\
p & \bullet & \bullet & \bullet \\
p & \bullet & \bullet & \bullet
\end{array}$ }$$ and the fiber dimension of this non-linear involutive first order system is $(3+4\times 3)-11= 4$ with characters equal to $(1,0,0,0)$.\
As for the non-linear system of finite Lie equations, we have at once from the explicit transformations:\
$$\frac{\partial \bar{Z}}{\partial Z}=1, \frac{\partial \bar{X}}{\partial Z}=0, \frac{\partial \bar{P}}{\partial Z}=0,
\frac{\partial \bar{Z}}{\partial P}=0, \frac{\partial \bar{X}}{\partial P}= 0 ,\frac{\partial \bar{P}}{\partial P}=1,
\frac{\partial \bar{X}}{\partial X}=1, \frac{\partial \bar{Z}}{\partial X}-\bar{P}= - P$$ This system is involutive and the following [*Janet board*]{} provides the structure of an involutive solved form:\
$$\left\{ \begin{array}{lcl}
Z & \longrightarrow & \left\{ \begin{array}{c} \bar{Z} \\ \bar{X}\\ \bar{P} \end{array}\right. \\
P & \longrightarrow & \left\{ \begin{array}{c} \bar{Z} \\ \bar{X} \\ \bar{P} \\ \end{array} \right. \\
X & \longrightarrow & \, \left\{ \begin{array}{c} \bar{Z}\\ \bar{X} \end{array} \right.
\end{array} \right.
\fbox{ $ \begin{array}{lll}
X & P & Z \\
X & P & Z \\
X & P & Z \\
X & P & \bullet \\
X & P & \bullet \\
X & P & \bullet \\
X & \bullet & \bullet \\
X & \bullet & \bullet
\end{array}$ }$$ The fiber dimension at order $1$ is $(3+3\times 3)-8=4$ and the characters are $(1,0,0)$. It follows that we have again an automorphic system. It is worthwhile to notice that the Janet boards of a system and a subsystem may be quite different.\
We end this list of examples with a situation providing an intransitive groupoid in the case of a cyclic variable and let the reader treat it as an exercise.\
[**COROLLARY 7.6**]{}: The search for a complete integral $z=v(x;a)+at +b$ is equivalent to the search for $1$ solution of the automorphic system obtained by adding $X=H $ to the system of the preceding Corollary, provided that:\
$$\frac{\partial H}{\partial z}=0, \hspace{1cm} \frac{\partial H}{\partial t}=0$$ The corresponding intransitive Lie pseudogroup is:\
$${\Gamma}''''= \{ \bar{X}=X, \bar{Z}=Z + h(X), \bar{P}= P + \partial h/\partial X \} \subset {\Gamma }''' \subset \Gamma$$
For more details on these topics of analytical mechanics, the interested reader may look at the recapitulating board in (\[21\], p 506). As a striking conclusion, there are as many specific situations reflected by the hamiltonian as the number of Lie subpseudogroups of the Lie pseudogroup of contact transformations.\
[**REFERENCES**]{}\
\[1\] Artin, E.: Galois Theory, Notre Dame Mathematical Lectures, 2, 1942, 1997.\
\[2\] Bonnet,O.: Mémoire sur la Théorie des Surfaces Applicables sur une Surface Donnée, Journal de l’Ecole Polytechnique, 25 (1867) 31-151.\
\[3\] Byalinicki-Birula,A.: On the field of rational functions of Algebraic Groups, Pacific Journal of Mathematics, 11 (1961) 1205-1210.\
\[4\] Bialynicki-Birula, A.: On Galois Theory of Fields with Operators, Amer. J. Math., 84 (1962) 89-109.\
\[5\] Cartan,E.: Sur la Possibilité de Plonger une Surface Riemannienne dans un Espace Euclidien, Ann. Soc. Pol. Math., 6 (1927) 1-7.\
\[6\] Chevalley, C., Samuel, P.: Twoo Proofs of a Theorem on Algebraic groups, Proc. Amer. Math. Soc., 2 (1951) 126-134;\
\[7\] Ciarlet, P.G., Larsonneur, F.P.: On the Ricovery of a Surface with Prescribed First and Second Forms, J. Math. Pures Appl., 81 (2002) 167-185.\
\[8\] Codazzi, D.: Sulle Coordinate Curvilinee d’una Superficiedello Spazio, Ann. Math. Pura Applicata, 2 (1868-1869) 101-19\
\[9\] Drach, J.: Thèse de Doctorat, Ann. Ec. Normale Sup. (3) 15 (1898) 243-384.\
\[10\] Gauss, K.F.: Disquitiones Generales circa Superficies Curvas, Comm. Soc. Gott., 6 (1828).\
\[11\] Han, Q.: Global Isometric Embedding of Surfaces in ${\mathbb{R}}^3$, Chapter 2 in “ Differential geometry and Coninuum mechanics”, Springer, 2015.\
\[12\] Huaux, A.: Sur la Séparation des Variables dans l’Equation aux dérivées partielles de Hamilton-Jacobi, Annali di Matematica Pura ed Applicata, 108 (1976) 251-282.\
\[13\] Janet, M: Sur la Possibilité de Plonger une Surface Riemannienne dans un Espace Euclidien, Ann. Soc. Pol. Math., 5 (1926) 38-43.\
\[14\] Kaplanski, I.: An Introduction to Differential Algebra, Hermann, 1957, 1976.\
\[15\] Kolchin,, E.R. KOLCHIN: Differential Algebra and Algebraic groups, Academic Press, 1973.\
\[16\] Kolchin, E.R.: Differential Algebraic Groups, Pure and Applied Mathematics, 114, Academic Press, 1985.\
\[17\] Levi-Civitta; : Sulla Integrazione delle Equazione di Hamilton-Jacobi per Separazione di Variabili, Math. Annalen, 59,(1904) 383 (Also “Opere Matematische”, 2 (1901-1907) 395-410).\
\[18\] Mainardi, G.: Su la Theoria Generale delle Superficie, Giornale dell’ Instituto Lombardo, 9 (1856) 385-404.\
\[19\] Pommaret, J.F.: Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach, New York, 1978 (Russian translation by MIR, Moscow, 1983).\
\[20\] : Pommaret, J.F.Differential Galois Theory, Gordon and Breach, New York, 1983 (750 pp).\
\[21\] Pommaret, : Lie Pseudogroups and Mechanics, Gordon and Breach, New York, 1988 (590 pp).\
\[22\] Pommaret, J.F.: Partial Differential Equations and Group Theory: New Perspectives for Applications, Kluwer, 1994.\
http://dx.doi.org/10.1007/978-94-017-2539-2\
\[23\] Pommaret, J.F.: Partial Differential Control Theory, Kluwer, 2001 (957 pp).\
\[24\] Pommaret, J.F.: Algebraic Analysis of Control Systems Defined by Partial Differential Equations, in Advanced Topics in Control Systems Theory, Lecture Notes in Control and Information Sciences LNCIS 311, Chapter 5, Springer, 2005, 155-223.\
\[25\] Pommaret, J.F.: Relative Parametrization of Linear Multidimensional Systems, Multidim. Syst. Sign. Process., 26 (2015) 405-437.\
DOI 10.1007/s11045-013-0265-0\
\[26\] Pommaret, J.F.: Deformation Theory of Algebraic and Geometric Structures, Lambert Academic Publisher (LAP), Saarbrucken, Germany, 2016.\
\[27\] Ritt, J.F.: Differential Algebra, Dover, 1950, 1966.\
\[28\] Stewart, I.: Galois Theory, Chapman and Hall, 1973.\
\[29\] Vessiot, E.: Sur la Théorie de Galois et ses Diverses Généralisations, Ann.Ec. Normale Sup., 21 (1904) 9-85 (Can be obtained from http://numdam.org).\
\[30\] Zariski, O., Samuel, P: Commutative Algebra, Van Nostrand, 1958.\
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We give a formula for the parity of the Maslov index of a triple of Lagrangian subspaces of a skew symmetric bilinear form over ${{\mathbb{R}}}$. We define an index two subcategory (the even subcategory) of a 3-dimensional cobordism category. The objects of the category are surfaces are equipped with Lagrangian subspaces of their real first homology. This generalizes a result of the first author where surfaces are equipped with Lagrangian subspaces of their rational first homology.'
address: |
Department of Mathematics\
Louisiana State University\
Baton Rouge, LA 70803-4918
author:
- 'Patrick M. Gilmer'
- Khaled Qazaqzeh
date: 'April 12, 2005'
title: The parity of the maslov index and the even cobordism category
---
[^1]
introduction
============
In [@G], the first author considered a cobordism category ${{\mathcal{C}}}.$ This category can be described roughly as follows. The objects of ${{\mathcal{C}}}$ are closed surfaces equipped with Lagrangian subspaces of their rational first homology. A morphism of ${{\mathcal{C}}}$ between $N:{{\Sigma}}{{\rightarrow}}{{\Sigma}}'$ is a cobordism between $ {{\Sigma}}$ and ${{\Sigma}}'$. Also, the first author defined a subcategory ${{\mathcal{C}}}^+$ of ${{\mathcal{C}}}$ of index two. It would be more consistent with other work [@T] [@W] to consider a similarly defined cobordism category ${{\mathfrak{C}}}$ where the extra data of a Lagrangian subspace is a subspace of the real first homology. The main goal of this article is define an analogous index two subcategory ${{\mathfrak{C}}}^+$ of ${{\mathfrak{C}}}$. We call ${{\mathfrak{C}}}^+$ the even cobordism category. If one restricts to this ‘index two’ cobordism subcategory, one may obtain functors, related to the TQFT functors defined by Turaev with initial data a modular category, but without taking a quadratic extension of the ground ring of the modular category as is sometimes needed in [@T p.76].
It is not possible to simply modify the proof given in [@G] for the existence of ${{\mathcal{C}}}^+$ to obtain a proof for the existence of ${{\mathfrak{C}}}^+$. This is because not every real Lagrangian subspace can be realized as the kernel of the map induced on first homology by the inclusion of a surface to a 3-manifold which has the surface as its boundary. Only the subspaces which are completions of subspaces of the rational homology can be so realized. So another approach has to be used. We actually reduce the problem to the one already solved for ${{\mathcal{C}}}$ but this requires some new algebraic results. These algebraic results may be of independent interest.
We prove the algebraic results in $\S2$. This section is written without any appeal to topology. It can be read independently of the rest of the paper. We derive the following congruence for the Maslov index, denoted $\mu$:
\[maincong\] Let $V$ be a symplectic vector space and $\lambda_{1},\lambda_{2}$, and $\lambda_{3}$ be any three Lagrangian subspaces, then we have $$\begin{aligned}
\label{E:fivth} \notag
\mu(\lambda_{1},\lambda_{2},\lambda_{3}) &\equiv \dim (\lambda_{1}) +
\sum_{1 \leq i < j \leq 3} \dim (\lambda_{i}\cap \lambda_{j})\quad \bmod(2)\\
\notag &\equiv \dim (\lambda_{1}) + \sum_{1 \leq i < j \leq 3}
\dim (\lambda_{i} + \lambda_{j}) \quad \bmod(2)\end{aligned}$$
If $\lambda_{1} \cap \lambda_{2}=\lambda_{2} \cap \lambda_{3}=\lambda_{1} \cap \lambda_{3}=0$, this result follows from [@LV 1.5.7] which gives a formula for the Maslov index in terms of a special form these Lagrangians must take in this case. We give a very diffferent proof. Theorem \[maincong\] will be the key to proving that the morphisms of ${{\mathfrak{C}}}^+$ are closed under composition. In §3, we describe the weighted cobordism categories ${{\mathcal{C}}}$ and ${{\mathfrak{C}}}$ in greater detail. In §4, ${{\mathfrak{C}}}^+$ is defined.
lagrangian subspaces and the maslov index
==========================================
Let $V$ be a symplectic vector space, i.e. $V$ is finite dimensional over $\mathbb{R}$ and endowed with a skew symmetric bilinear form $\psi$. We do not require that the form is nondegenerate. If $A$ is a subspace of $V$, its annihilator, $\operatorname{Ann}(A)$, is the set of elements which pair under the form with all of $A$ to give zero. If $A$ and $A'$ are two subspaces, then [@T IV.3.1.a, IV.3.1.1 ] $$\label{ann} \operatorname{Ann}(A + A') = \operatorname{Ann}(A) \cap \operatorname{Ann}(A')$$ $$\label{ann2} \operatorname{Ann}(A \cap A') = \operatorname{Ann}(A) + \operatorname{Ann}(A')$$ A subspace $A\subset V$ is said to be a Lagrangian subspace if $A$ = $\operatorname{Ann}(A$). We have not been able to find the following result in the literature.
\[T:firstt\] Let $(V, \psi )$ be a symplectic vector space and $\lambda_{1},\lambda_{2}$, and $\lambda_{3}$ be three Lagrangian subspaces. Then we have $$\begin{aligned}
\notag
\dim (\lambda_{1} + \lambda_{2} + \lambda_{3}) \equiv
\dim (\lambda_{1}\cap\lambda_{2}\cap\lambda_{3})\quad \bmod(2)\end{aligned}$$
We have an skew symmetric bilinear form $\psi$ on $V$. Now define a form $\{\, , \}$ on $(\lambda_{1}+\lambda_{2}+\lambda_{3})/(\lambda_{1}\cap\lambda_{2}\cap\lambda_{3})$ by $\{a,b\}=\psi(a,b)$ where $a,b\in
(\lambda_{1}+\lambda_{2}+\lambda_{3})/(\lambda_{1}\cap\lambda_{2}\cap\lambda_{3})$. To show that this new form is well-defined, let $a_{1}, a_{2}\in
(\lambda_{1}+\lambda_{2}+\lambda_{3})$ such that $\bar{a_{1}}=\bar{a_{2}}$, i.e. $a_{1}-a_{2}\in (\lambda_{1}\cap\lambda_{2}\cap\lambda_{3})$. It follows $\psi(a_{1}-a_{2},b)=0$ for all $b\in
(\lambda_{1}+\lambda_{2}+\lambda_{3})$, so $\psi(a_{1},b)=\psi(a_{2},b) $. Hence $\{a_{1},b\}=\{a_{2},b\}$ for all $b\in
(\lambda_{1}+\lambda_{2}+\lambda_{3})$ that $\{\, , \}$ is well-defined. Since $\psi$ is skew symmetric bilinear form, so is $\{\, , \}$. We now wish to show that $\{\, , \}$ is non-degenerate. So let $a\in(\lambda_{1}+\lambda_{2}+\lambda_{3})/(\lambda_{1}\cap\lambda_{2}\cap\lambda_{3})$ such that $\{a,b\}=0$ for all $b\in(\lambda_{1}+\lambda_{2}+\lambda_{3})/(\lambda_{1}\cap\lambda_{2}\cap\lambda_{3})$, i.e. $\psi(a,b)=0$ for all $b\in \lambda_{1}+\lambda_{2}+\lambda_{3}$, it implies that $a\in \operatorname{Ann}(\lambda_{1}+\lambda_{2}+\lambda_{3}$). By equation $$\begin{aligned}
\notag
\operatorname{Ann}(\lambda_{1}+\lambda_{2}+\lambda_{3}) &= \operatorname{Ann}(\lambda_{1} +
\lambda_{2})\cap \operatorname{Ann}(\lambda_{3})\\ \notag & = (\operatorname{Ann}(\lambda_{1})\cap
\operatorname{Ann}(\lambda_{2})) \cap \lambda_{3}\\ \notag & =
\lambda_{1}\cap\lambda_{2}\cap\lambda_{3}.\end{aligned}$$
So $a\in\lambda_{1}\cap\lambda_{2}\cap\lambda_{3}$, i.e. $a=0$ in $(\lambda_{1}+\lambda_{2}+\lambda_{3})/(\lambda_{1}\cap\lambda_{2}\cap\lambda_{3})$. As is well-known, A non-degenerate symplectic vector space must be even dimensional. Hence $(\lambda_{1}+\lambda_{2}+\lambda_{3})/(\lambda_{1}\cap\lambda_{2}\cap\lambda_{3})$ is of even dimension, so we get $$\begin{aligned}
\notag \dim (\lambda_{1} + \lambda_{2} + \lambda_{3}) \equiv
\dim (\lambda_{1}\cap\lambda_{2}\cap\lambda_{3}) \quad \bmod(2)\end{aligned}$$
We have the following well-known proposition [@T IV.3.5]
Let $\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3}$ be three Lagrangian subspaces of $V$. Define a bilinear form $\langle\, , \rangle$ on $(\lambda_{1}+\lambda_{2})\cap \lambda_{3}$ by $$\begin{aligned}
\label{E:second}
\langle a,b\rangle=\psi(a_{2},b)\end{aligned}$$ where $a, b\in (\lambda_{1}+\lambda_{2})\cap \lambda_{3}$ and $a=a_{1}+a_{2}$. $\langle\, , \rangle$ is a well-defined symmetric bilinear form.
To show is $\langle\, , \rangle$ is well-defined, note that the decomposition $a=a_{1}+a_{2}$, where $a_1 \in A_1$ and $a_2 \in A_2$, is unique up to an element in $\lambda_{1}\cap\lambda_{2}$, and this element annihilates $b$ for all $b\in\lambda_{1}+\lambda_{2}$. So the form is well-defined. As $\psi$ is bilinear, $\langle\, , \rangle$ is bilinear.
Let $a$ be as before and $b=b_{1}+b_{2}$ where $b_{1}\in\lambda_{1}, b_{2}\in\lambda_{2}$ and $b\in\lambda_{3}$. Since $\lambda_{i} = \operatorname{Ann}(\lambda_{i})$ for $i$ = 1, 2, 3 and $\psi$ is skew symmetric, we have $$\begin{aligned}
\notag \psi(a_{2},b)&=\psi(a-a_{1},b)& \\
\notag &=\psi(a,b)-\psi(a_{1},b) \\
\notag &=\psi(b,a_{1}) \\
\notag &=\psi(b_{1}+b_{2},a_{1}) \\
\notag &=\psi(b_{1},a_{1})+\psi(b_{2},a_{1})+\psi(b_{2},a_{2})\\
\notag &=\psi(b_{2},a).\end{aligned}$$ Hence the form is symmetric.
The Maslov index $\mu(\lambda_{1},\lambda_{2},\lambda_{3})$ of the triple $(\lambda_{1},\lambda_{2},\lambda_{3})$ is the signature of the form $\langle\, , \rangle$ defined above.
In general, $\langle\, , \rangle$ is degenerate. In fact, it is known that its annihilator contains $(\lambda_{1}\cap\lambda_{3})$ + $(\lambda_{2}\cap\lambda_{3})$ [@T p.182-183]. If $\lambda_1 \cap \lambda_2=0$, it is known that the annhilator is $(\lambda_{1}\cap\lambda_{3})$ + $(\lambda_{2}\cap\lambda_{3})$ [@LV 1.5.6]. We show this is true in general.
\[T:secondt\] Let ($V$,$\psi$) be a symplectic vector space and $\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$ be three Lagrangian subspaces, then the induced form $\langle\, , \rangle$ on ($\lambda_{1} + \lambda_{2}$) $\cap
\lambda_{3}$ given in (\[E:second\]) has annihilator equal to $(\lambda_{1}\cap\lambda_{3}) +
(\lambda_{2}\cap\lambda_{3})$.
Let $W$ denote the annihilator of this form. It is clear that $\lambda_{1}\cap\lambda_{3} \subset W $, also $\lambda_{2}\cap\lambda_{3} \subset W$. Hence $(\lambda_{1}\cap\lambda_{3})$ + $(\lambda_{2}\cap\lambda_{3})\subset W$. Now to prove the other containment, let $a\in W$, so $\langle a,b\rangle$ = 0 for all $b\in (\lambda_{1} + \lambda_{2}) \cap \lambda_{3}$. In other words; if $a=a_{1}+a_{2}\in \lambda_{3}$ where $a_{1}\in\lambda_{1}$ and $a_{2}\in\lambda_{2}$, then we have $\psi(a_{2},b)$ = 0. It follows that $a_{2}\in \operatorname{Ann}((\lambda_{1} + \lambda_{2}$) $\cap \lambda_{3})$ in $V$. Using equations and , we have that $$\begin{aligned}
\notag \operatorname{Ann}((\lambda_{1} + \lambda_{2})\cap\lambda_{3})
\notag & = \operatorname{Ann}(\lambda_{1} + \lambda_{2})+ \operatorname{Ann}(\lambda_{3})\\
\notag & = ( \operatorname{Ann}(\lambda_{1} ) \cap \operatorname{Ann}(\lambda_{2}) )+ \lambda_{3} \\
\notag & =(\lambda_{1}\cap\lambda_{2})+ \lambda_{3} \end{aligned}$$ Thus $a_{2}\in
(\lambda_{1}\cap\lambda_{2})+\lambda_{3}$. So we could write $a_{2}=c+d$ where $c\in\lambda_{1}\cap\lambda_{2}$ and $d\in\lambda_{3}$. It follows that $a=(a_{1}+c)+d$ where $a_{1}+c\in\lambda_{1} $ and $d\in \lambda_{3}$. Now since we have $a_{2}, c\in\lambda_{2}$ we get $d\in\lambda_{2}$. Since $a, d\in\lambda_{3}$ we get $ a_{1}+c\in\lambda_{3}$. Hence $d\in\lambda_{2}\cap\lambda_{3}$ and $a_{1}+c\in\lambda_{1}\cap\lambda_{3}$. So $a=(a_{1}+c)+d\in
(\lambda_{1}\cap\lambda_{3})+(\lambda_{2}\cap\lambda_{3})$. Thus $W\subset (\lambda_{1}\cap\lambda_{3})+(\lambda_{2}\cap\lambda_{3})$. So $W=(\lambda_{1}\cap\lambda_{3})+(\lambda_{2}\cap\lambda_{3})$, i.e. the annihilator of the form $\langle\, , \rangle$ is equal to $(\lambda_{1}\cap\lambda_{3})+(\lambda_{2}\cap\lambda_{3})$.
\[R:firstr\] For any pair of Lagrangian subspaces $\lambda_{1}$, and $\lambda_{2}$ we have $$\dim (\lambda_{1}) = \dim (\lambda_{2})$$ and $$\begin{aligned}
\label{E:newequation}
\dim (\lambda_{1} + \lambda_{2}) \equiv \dim (\lambda_{1}\cap\lambda_{2})
\quad \bmod(2).\end{aligned}$$
The first formula follows by reducing it to the nonsingular case and $$\begin{aligned}
\notag
\dim (A) = \dim (V) - \dim (\operatorname{Ann}(A)).\end{aligned}$$ We obtain the second congruence from $$\begin{aligned}
\label{E:third}
\dim (A + B) = \dim (A) + \dim (B) - \dim (A\cap B)\end{aligned}$$ and the first formula.
\[cor\] $$\begin{aligned}
\notag
\mu(\lambda_{1},\lambda_{2},\lambda_{3})\equiv \dim ((\lambda_{1} +
\lambda_{2}) \cap
\lambda_{3})+\dim ((\lambda_{1}\cap\lambda_{3})+(\lambda_{2}\cap\lambda_{3}))
\quad \bmod(2).\end{aligned}$$
Since the annihilator of the form is $(\lambda_{1}\cap\lambda_{3})$ + $(\lambda_{2}\cap\lambda_{3})$, it follows that the rank of the form is $$\dim ((\lambda_{1} + \lambda_{2}) \cap
\lambda_{3})-\dim ((\lambda_{1}\cap\lambda_{3})+(\lambda_{2}\cap\lambda_{3}))
.$$ The result follows from the fact that the signature and the rank of a nondegenerate form agree modulo two.
By equation (\[E:third\]), we have $$\begin{aligned}
\notag
\dim (\lambda_{1} + \lambda_{2} + \lambda_{3}) \equiv \dim (\lambda_{1}) +
\dim (\lambda_{2} + \lambda_{3}) + \dim (\lambda_{1} \cap (\lambda_{2} +
\lambda_{3})) \quad \bmod(2)\end{aligned}$$ and also have $$\begin{aligned}
\notag
\dim (\lambda_{1}\cap\lambda_{2}\cap\lambda_{3}) \equiv
\dim (\lambda_{1}\cap\lambda_{2}) + \dim (\lambda_{1}\cap\lambda_{3}) +
\dim ((\lambda_{1}\cap\lambda_{2}) + (\lambda_{1}\cap \lambda_{3}))
\quad \bmod(2).\end{aligned}$$
Hence by Theorem (\[T:firstt\]), the left hand sides of these two congruences are congruent. So their right hand sides must be congruent as well: $$\begin{aligned}
\notag \dim (\lambda_{1}) + \dim (\lambda_{2} + \lambda_{3}) + \dim (\lambda_{1}
\cap( \lambda_{2} + \lambda_{3})) \equiv
\dim (\lambda_{1}\cap\lambda_{2}) + \dim (\lambda_{1}\cap\lambda_{3})\\
\notag + \dim ((\lambda_{1}\cap\lambda_{2}) + (\lambda_{1}\cap \lambda_{3}))
\quad \bmod(2).\end{aligned}$$ The last equation is equivalent to $$\begin{aligned}
\notag
\dim (\lambda_{1} \cap( \lambda_{2} + \lambda_{3})) +
\dim ((\lambda_{1}\cap\lambda_{2}) + (\lambda_{1}\cap \lambda_{3})) \equiv
\dim (\lambda_{1}) + \dim (\lambda_{2} + \lambda_{3})\\ \notag +
\dim (\lambda_{1}\cap\lambda_{2}) + \dim (\lambda_{1}\cap\lambda_{3}) \quad
\bmod(2).\end{aligned}$$ The left hand side of this last equation is congruent to the Maslov index by Corollary \[cor\], and hence the first formula follows. The second formula follows by equation .
the weighted cobordism categories
=================================
All 3-manifolds and surfaces in this paper are assumed to be oriented and compact. We define a weighted cobordism category ${{\mathfrak{C}}}$ whose objects are surfaces ${{\Sigma}}$ without boundary equipped with a Lagrangian subspace $\lambda \subset H_{1}(\Sigma;{{\mathbb{R}}})$. We will denote objects by pairs $({{\Sigma}}, \lambda)$. A cobordism from $(\Sigma ,\lambda) $ to $(\Sigma',\lambda')$ is a 3-manifold together with an orientation preserving homeomorphism (called its boundary identification) from its boundary to $-\Sigma\sqcup\Sigma'$. Here, and elsewhere, $-\Sigma$ denotes $\Sigma$ with the opposite orientation. Two cobordisms are equivalent if there is an orientation preserving homeomorphism between the underlying 3-manifolds that commutes with the boundary identifications. A morphism $M: (\Sigma ,\lambda) {{\rightarrow}}(\Sigma',\lambda')$ is an equivalence class of cobordisms from $(\Sigma ,\lambda) $ to $(\Sigma',\lambda')$ together with an integer weight. We denote morphisms by a single letter. We let $w(M)$ denote the weight of $M$. We let $\flat M$ denote the underlying 3-manifold of a representative cobordism. This is well defined up to homeomorphism respecting the boundary identifications. We call $(\Sigma ,\lambda)$ the source of $M$ and $(\Sigma' ,\lambda' )$ the target of $M$. We let $j_{M}$ denote the inclusion $\Sigma$ into $\flat M,$ and $j^{M}$ denote the inclusion $\Sigma'$ into $\flat M.$ Here and sometimes below we ignore the boundary identifications for simplicity and we write as if $\Sigma \coprod \Sigma'$ were the boundary of $\flat M$.
We let ${M}_{*}(\lambda)$ denote the Lagrangian subspace [@T p188-189] of $H_1(\Sigma',{{\mathbb{R}}})$ given by $ \left({ j^M_*}\right) ^{-1} \left( {{j_{M}}_ *}(\lambda) \right)$. Similarly we have the Lagrangian subspace of $H_1(\Sigma,{{\mathbb{R}}})$: $ M^{*}(\lambda')= \left( {j_{M}}_*\right) ^{-1} \left( {j^M_ *}(\lambda') \right).$
If $M_1: (\Sigma ,\lambda) {{\rightarrow}}(\Sigma',\lambda')$ and $M_2: (\Sigma' ,\lambda') {{\rightarrow}}(\Sigma',\lambda'')$ are two morphisms we define $\flat (M_2 \circ M_1)$ by gluing $\flat M_2$ to $\flat M_1$ by identifying the target of $M_1$ to the source of $M_2.$ The boundary of this new 3-manifold is equipped with a boundary identification in the obvious way. The weight of the composition is given by the formula [^2]. $$\begin{aligned}
\label{wofcomp}
w(M_2 \circ M_1)= w(M_1) + w(M_2)
- \mu({M_{1}}_{*}(\lambda),\lambda',{{M}_{2}}^{*}(\lambda'')) \end{aligned}$$
The identity $\operatorname{id}_{(\Sigma,\lambda)}: (\Sigma,\lambda) {{\rightarrow}}(\Sigma,\lambda)$ is given by $ {{\Sigma}}\times I$ with the weight zero and the standard boundary identification. Any morphism $C:(\Sigma,\lambda) {{\rightarrow}}(\Sigma,\lambda') $ with $ {{\Sigma}}\times I$ as the underlying 3-manifold, and with the standard boundary identification will be called a pseudo-cylinder over $\Sigma$.
\[L:thirdl\] Pseudo-cylinders are invertible in ${{\mathfrak{C}}}.$ The inverse of $C:(\Sigma,\lambda) {{\rightarrow}}(\Sigma,\lambda') $ is the pseudo-cylinder from $(\Sigma,\lambda')$ to $(\Sigma,\lambda)$ with weight $-w(C)$.
This follows immediately from the definitions. One needs that the Maslov index vanishes when two of the three lagrangians coincides [@T p183]
If we make the same definitions but using Lagrangians in $H_1(\Sigma, {{\mathbb{Q}}})$, we obtain the cobordism category ${{\mathcal{C}}}$ studied in [@G]. As $H_1(\Sigma, {{\mathbb{Q}}}) \otimes {{\mathbb{R}}}$ is naturally isomorphic to $H_1(\Sigma, {{\mathbb{R}}})$, a Lagrangian in $H_1(\Sigma, {{\mathbb{Q}}})$ determines one in $H_1(\Sigma, {{\mathbb{R}}}).$ A Lagrangian of $H_1(\Sigma, {{\mathbb{R}}})$ which arises in this way is called rational. In this way, we obtain a functor ${{\mathcal{C}}}{{\rightarrow}}{{\mathfrak{C}}}.$
the even cobordism category
===========================
We repeat a definition from [@G] except now we apply it to morphisms of ${{\mathfrak{C}}}$ instead of ${{\mathcal{C}}}$. We denote $\beta_i(\flat M)$ by $\beta_i(M).$
\[evendef\] A cobordism $M:(\Sigma,\lambda) \rightarrow (\Sigma',\lambda')$ of ${{\mathfrak{C}}}$ is even if and only if $$\begin{aligned}
\notag
w(M) \equiv \dim \left({j_{M}}_{*}(\lambda) + j^M_{*}(\lambda' ) \right)+\beta_1(M)+\beta_0(M) +
\beta_0(\Sigma)+ \frac {\beta_1(\Sigma')} 2+ \epsilon(M) \quad \bmod(2)\end{aligned}$$ where $\epsilon (M)$ is one if exactly one of $\Sigma$ and $\Sigma
'$ is nonempty and otherwise $\epsilon (M)$ is zero. If a cobordism is not even, it is called odd.
We note that the inverse of an even pseudo-cylinder is even.
The first author showed that the composite of two even morphisms of ${{\mathcal{C}}}$ is again even [@G Theorem 7.2]. The subcategory ${{\mathcal{C}}}^+$ was defined to be the category with the same objects as ${{\mathcal{C}}}$ but with only even morphisms. In the rest of this section, we generalize this result to morphisms in ${{\mathfrak{C}}}.$ Given this result, we define the subcategory ${{\mathfrak{C}}}^+$ to be the category with the same objects as ${{\mathfrak{C}}}$ but with only the even morphisms. We would also get a subcategory if we left the $\epsilon(N)$ term out of Definition \[evendef\]. However the definition that we give is more natural from some points of view [@G].
A pseudo-cylinder $C:(\Sigma,\lambda) {{\rightarrow}}(\Sigma,\lambda') $ is even if and only if $$\begin{aligned}
\notag
w(C) \equiv \frac{\beta_{1}(\Sigma)}{2}+\dim (\lambda+\lambda') \quad
\bmod(2)\end{aligned}$$
Apply the definition above.
\[L:secondl\] Let $M:(\Sigma,\lambda) \rightarrow (\Sigma',\lambda')$ be an even morphism. If $C:(\Sigma, \hat \lambda) \rightarrow (\Sigma, \lambda)$ and $C':(\Sigma',\lambda') \rightarrow (\Sigma', \tilde \lambda)$ be an even pseudo-cylinders, then $M \circ C$ and $C' \circ M$ are even.
We first show that $M \circ C$ is even. We need to show $$\begin{aligned}
\label{E:Mc}
w(M \circ C) \equiv
\dim (j_{M *}(\hat \lambda) + j_{*}^{M}(\lambda' ) )+\beta_1(M)+ \beta_0(M) + \beta_0(\Sigma)+ \frac {\beta_1(\Sigma_{1})} 2+ \epsilon(M)
\bmod(2)\end{aligned}$$
By Equation \[wofcomp\], $$\begin{aligned}
\label{we}
w(M \circ C) \equiv w(M) + w(C) +\mu(\hat \lambda,\lambda,M^{*}(\lambda')) \bmod(2)\end{aligned}$$
By assumption, we have that: $$\begin{aligned}
\label{E:M}
w(M) \equiv \dim (j_{M *}(\lambda) + {j_{*}}^{M}(\lambda'))+\beta_1(M)+\beta_0(M)+ \beta_0(\Sigma) + \frac{\beta_1(\Sigma_{1})} 2+
\epsilon(M) \bmod(2)\end{aligned}$$ and, $$\begin{aligned}
\label{E:C}
w (C) \equiv \frac{\beta_{1}(\Sigma)}{2}+\dim ( \hat \lambda+\lambda) \quad
\bmod(2)\end{aligned}$$ So after we substitute , and into , we conclude that we need only prove: $$\begin{aligned}
\notag
\dim (j_{M *}(\hat \lambda) + {j_{*}}^{M}(\lambda' ) ) +
\mu(\hat \lambda,\lambda, M^{*}(\lambda')) + \dim (\hat \lambda + \lambda) +
\frac{\beta_1(\Sigma)} 2 \equiv \\ \notag \dim (j_{M *}(\lambda) +
{j_{*}}^{M}(\lambda' ) ) \quad \bmod(2)\end{aligned}$$ Given Theorem \[maincong\], this last congruence becomes:
$$\begin{aligned}
\label{con2}
\dim (j_{M *}(\hat \lambda) + {j_{*}}^{M}(\lambda' ) ) + \dim (\hat \lambda +
M^{*}(\lambda')) \equiv \dim (j_{M *}(\lambda) + { j_{*}}^{M}(\lambda' ) )\ +
\\ \notag \dim (\lambda + M^{*}(\lambda')) \quad \bmod(2)\end{aligned}$$
For any subspace $\delta$ of $H_1(M,{{\mathbb{R}}})$, we have that $$\delta + M^*(\lambda') = \left( j_{M *} \right) ^{-1} \left( j_{M *} ( \delta) + j_*^M (\lambda') \right)$$ as $$j_{M *} \left( \delta + M^*(\lambda') \right) = j_{M *} ( \delta) + j_*^M (\lambda')$$ and kernel of $j_{M*} $ is a subset of $M^*(\lambda').$ Thus we have that $\dim (\delta + M^{*}(\lambda')) = \dim (j_{M*}(\delta) +
j^M_{*}(\lambda')) + n$ where $n$ is the dimension of kernel of $j_{M*} $. Thus both sides of are congruent to $n$. Hence, we obtain (\[E:Mc\]).
The proof that $C' \circ M$ is even follows formally from the first part, if we consider how the parity of a cobordism changes when we reverse the orientation of the underlying 3-manifold and reverse the roles of source and target.
\[P:firstp\] If there are even pseudo-cylinders $C$ and $C'$ over $\Sigma$, and $\Sigma'$ such that $ C \circ M \circ C'$ is even, then $M$ is an even cobordism in ${{\mathfrak{C}}}$ from $(\Sigma,\lambda)$ to $(\Sigma',\lambda')$.
It follows by lemma (\[L:thirdl\]) that we can factor $M$ as $C^{-1}\circ C \circ M \circ C' \circ C'^{-1}$. Hence $M$ is even by two applications of Lemma (\[L:secondl\]).
The composition of two even morphisms of ${{\mathfrak{C}}}$ is again even.
Let $M_{1}, M_{2}$ be two even morphisms and adopt the notations associated to $M_1$ and $M_2$ in §3. We need to show that $M_{2}\circ M_{1}$ is an even cobordism. It suffices to show $C'' \circ M_{2}\circ M_{1}\circ
C$ is even for some even pseudo-cylinders over $C$ and $C''$ over $\Sigma$ and $\Sigma''$ with rational Lagrangians for $\Sigma$ and $\Sigma''$ . On the other hand we can write $M_{2}\circ M_{1}$ as $ M_{2}\circ C' \circ {C'} ^{-1} \circ M_{1}$ where $C'$ is an even pseudo-cylinder over $\Sigma'$ whose the target has a rational Lagrangian. We have that $$C''\circ M_{2}\circ M_{1}\circ C= C'' \circ M_{2}\circ C' \circ {C'} ^{-1} \circ M_{1}\circ C = N_2 \circ N_1$$ where $N_2=C'' \circ M_{2}\circ C'$ and $N_1={C'} ^{-1} \circ M_{1}\circ C.$ By Lemma \[L:secondl\], $N_1$, $N_2$ are even morphisms. By Theorem 7.2 in [@G], it follows that $N_2 \circ N_1$ is even. Hence $M_{2}\circ
M_{1}$ is even.
[W]{}
P. Gilmer, [*Integrality for TQFTs*]{}, Duke Math. J., 125. ,389-413, 2004.
G. Lion, M. Vergne, [*The Weil representation, Maslov index and Theta series,* ]{} Progress in Mathematics 6, Birkhauser, 1980
V. Turaev, [*Quantum invariants of knots and 3-manifolds*]{}, de Gruyer studies in mathematics 18, 1994
Preprint 1991\
http://canyon23.net/math/
[^1]: partially supported by NSF-DMS-0203486
[^2]: As in [@G], we adopt the sign convention of [@W] rather than [@T] for the sign of the Maslov index term in this formula. It makes no real difference for this paper.
| {
"pile_set_name": "ArXiv"
} |
---
bibliography:
- 'library.bib'
---
| {
"pile_set_name": "ArXiv"
} |
---
abstract: '[ There are many ways of calculating photon statistics in quantum optics in general and single molecule spectroscopy in particular such as the generating function method, the quantum jump approach or time ordering methods. In this paper starting with the optical Bloch equation, within the paths interpretation of Zoller, Marte and Walls we obtain the photon statistics from a sequence of laser pulses expressed by means of quantum trajectories. We find general expressions for $P_n(t)$ - the probability of emitting n photons up to time t, discuss several consequences and show that the interpretation of the quantum trajectories (i) emphasizes contribution to the photon statistics of the coherence paths accumulated in the delay interval between the pulses and (ii) allows simple classification of the terms negligible under certain physical constraints . Applying this method to the concrete example of two square laser pulses we find the probabilities of emitting 0,1 and 2 photons, examine several limiting cases and investigate the upper and lower bounds of $P_0(t)$, $P_1(t)$ and $P_2(t)$ for a sequence of two strong pulses in the limit of long measurement times. Implication to single molecule non-linear spectroscopy and theory of pairs of photons on demand are discussed briefly.]{}'
author:
- |
F. Shikerman$^{1}$, Y. He$^{1,2}$, E. Barkai$^{1}$\
$^{1}$Department of Physics, Bar Ilan University, Ramat-Gan 52900, Israel\
$^2$School of Physics Science and Technology, Central South University, Changsha 410083, China\
title: Single Molecule Photon Statistics from a Sequence of Laser Pulses
---
Introduction
============
The interaction of matter with a sequence of laser pulses is a powerful tool frequently used for the investigation of a wide variety of chemical, physical and biological systems [@MukamelB]. This field of research called non-linear spectroscopy uses clever design of laser pulses for the investigation of fast dynamics (e.g. pico - seconds) of ensembles of molecules in the condensed phase. Recently van Dijk et al [@vanDijk] reported the first experimental study of an ultra-fast pump-probe single molecule system. Unlike the previous approaches to such non-linear spectroscopy where only the ensemble average response to the external fields is resolved, the new approach yields direct information on single molecule dynamics, gained through the analysis of photon counting statistics [@SB]. In [@SB] we considered the non-linear spectroscopy for a single molecule undergoing stochastic spectral diffusion process. Here we neglect all dephasing and spectral diffusion effects, and concentrate on the effect of the laser field parameters on the photon statistics.\
Another related application is the generation of two indistinguishable photons using two short laser pulses interacting with a single molecule or atom [@Santori]. Numerous applications for such photon sources have been proposed [@Hong; @Knill; @Shih; @Katz; @Bouwmeester] for the investigation of entangled states between identical photons and quantum properties of light, in the field of quantum information and quantum computation requiring consecutive photons to have identical wave packets. Usually in the mentioned experiments the single emitter is requested to supply one or two photons within as short as possible time interval. Although it is well established how to generate two photons from two ideal $\pi$-pulses if the delay interval between the pulses is very long, we can never produce two photons with probability equal 1, when the interaction time is finite. Thus, the information on the upper and lower bounds of the probabilities of emitting 0, 1 and 2 photons as a function of the laser field parameters, obtained in the manuscript, can be very useful.\
Although the theory of single particle photon statistics is well established [@Zoller; @Mollow], it remained unapplied due to the absence of experimental ability to check the results. Recent experimental achievements [@vanDijk; @Santori; @Katz; @Orrit] allowed the investigation of the interaction of a single quantum system with an external laser field inspiring further development of theoretical methods [@SB; @BarkaiPRL; @BarkaiRev; @Mukamel; @Cao; @Xie; @Goppich]. Today there are several approaches to photon counting statistics such as generating function method [@Brown] or quantum jump approach [@Plenio] suitable for analytical predictions and numerical calculations. In this paper we follow the path interpretation approach of Mollow and Zoller, Marte and Walls [@Zoller; @Mollow] of the optical Bloch equations [@CT], and show that this method is very useful for the analysis of single molecule non-linear spectroscopy.\
In what follows we consider a two level molecule interacting with two laser pulses and obtain general expressions for $P_n(t)$ - the probability of emitting n photons in interval $(0,t)$ by means of quantum trajectories. We discuss the influence of the coherence on the photon statistics. Also the explicit calculation of $P_0$, $P_1$ and $P_2$ - the probabilities of emitting 0,1 and 2 photons in the limit of long measurement times $( t\to\infty )$ is investigated in detail using the example of two identical square pulses. Some technical details skipped in the text are given in Appendixes A, B and C.\
Optical Bloch Equations
=======================
Interaction of an atom or a molecule with a radiation field is described by the optical Bloch equations under well established conditions [@CT], and we remind the reader some of the basic assumptions. First (i) the laser field is intense, so that it can be modeled classically. Here the external electric field is ${\bf E}(t) = {\bf E_0} f(t)$, where the amplitude ${\bf E_0}$ is independent of time. (ii) The electronic states of the single emitter are modeled based on the two level system approximation. This assumption is excellent when the laser is resonating with a particular absorption frequency of the molecule, the latter being well separated from other natural frequencies of the emitter. Most single molecules have a triplet state, however the life time of the triplet is much longer than the time scales under consideration in this manuscript, and it can be neglected. (iii) The spontaneous emission process is described by the Markovian approximation, where the inverse life time of the excited state is $\Gamma$. (iv) We neglect thermal dephasing, spectral diffusion and interaction of the emitter with a thermal bath, which was partially treated in [@SB]. (v) Finally, we will assume that the dipole moment of the excited and ground state of the single emitter is zero, so that only the transition dipole moment of the particle is important. Assumptions (i,ii,iii,iv) are physical assumptions which are justified in many single molecule experiments at least at low temperatures [@Orrit; @Rozkov], and condition (v) is not limiting since our technique could be modified in principle to the case where excited and ground states of the molecule have a dipole.\
The two level system is described by a vector composed of the density matrix elements: $\sigma=(\sigma_{\rm ee},
\sigma_{\rm gg}, \sigma_{\rm ge}, \sigma_{\rm eg})^{T}$. Here $\sigma_{\rm ee}$ and $\sigma_{\rm gg}$ represent the populations of the excited and ground states respectively and $\sigma_{\rm
ge},\sigma_{\rm eg}$ describe the coherences, namely the off diagonal matrix elements of the density matrix, and obey $\sigma_{\rm eg} ^{*}=\sigma_{\rm ge}$. The optical Bloch equation is [@CT] $$\dot{\sigma} = L\left( t \right) \sigma + \hat{\Gamma}\sigma,
\label{eqBloch}$$ with $$L(t) = \left(
\begin{array}{c c c c}
- \Gamma & 0 & - i \Omega f(t) & i \Omega f(t) \\
0 & 0 & i \Omega f(t) & - i \Omega f(t) \\
- i \Omega f(t) & i \Omega f(t) & i \omega_0 - \Gamma/2 & 0 \\
i \Omega f(t) & - i \Omega f(t) & 0 & - i \omega_0 - \Gamma/2
\end{array}
\right) \label{eqOBE}$$ and $$\hat{\Gamma} = \left(
\begin{array}{c c c c}
0 & 0 & 0 & 0 \\
\Gamma & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}
\right), \label{eqLoOBE}$$\
where the Rabi frequency is $ \Omega=-\frac{1 }{\hbar} {\bf d} _{ge}
\cdot {\bf E} _0 $ and ${\bf d} _{ge}$ is the transition dipole moment of the two level system. The operator $\hat{\Gamma}$ Eq. (\[eqLoOBE\]) describes direct transition from the excited to the ground state, and hence is associated with the spontaneous emission of a single photon.\
The optical Bloch equation Eq. (\[eqBloch\]) does not yield a direct method for calculating the probabilities of the number of emitted photons. However starting with [@Zoller; @Mollow] an interpretation of the optical Bloch formalism yields a tool for the calculation of photon statistics, based on the n-photon-propagators (see details below). The formal solution to Eq. (\[eqBloch\]) may be given by the infinite iterative expansion in $\hat{\Gamma}$ [@Brown; @Mukamel]: $$\sigma_{(t)} = {\cal G}(t,0) \sigma_{(0)} + \int_0 ^t {\rm d t_1} {\cal G} (t,t_1) \hat{\Gamma} {\cal G}(t_1, 0) \sigma_{(0)} +$$ $$+\int_0 ^{t} {\rm d} t_2 \int_0 ^{t_2} {\rm d} t_1 {\cal G}( t, t_2)
\hat{\Gamma} {\cal G}(t_2 , t_1) \hat{\Gamma} {\cal G} (t_1 , 0 )
\sigma_{(0)} + \cdots, \label{eqformal}$$\
where $\sigma_{(0)}$ is the initial condition, and the Green function describing the evolution of the system in the absence of spontaneous transitions into the ground state (i.e. without $\hat{\Gamma}$ ) is $${\cal G} (t,t') = \hat{T} \exp\left[ \int_{t'} ^ t L(t_1) {\rm d}
t_1 \right],
\label{eqGreenf}$$ where $\hat{T}$ is the time ordering operator. The first term in the expansion Eq. (\[eqformal\]) does not include $\hat{\Gamma}$ at all, and hence describes the process where no photons are emitted, the second term includes $\hat{\Gamma}$ just once and describes the process where one photon is emitted etc. It is therefore useful to define the conditional state $\sigma^{(n)}_{(t)}$, where $n$ is an index for the number of photons emitted in the time interval $(0,t)$. Then by definition $$\sigma^{(n)}_{(t)} = U^{(n)}_{(t,0)} \sigma_{(0)},
\label{eqsign}$$ where the n-photon-propagator [@Mukamel] is $$U^{(n)}_{(t,t')} =
\int_{t'} ^t {\rm d} t_n \cdots \int_{t'} ^{t_2} {\rm d} t_1~ {\cal
G}(t,t_n) \hat{\Gamma} \cdots\hat{\Gamma} {\cal G}(t_1,t') .
\label{eqUn}$$ The physical origin of the n-photon-propagator defined by Eq. (\[eqUn\]) is simple and intuitive: the system evolves interacting with the laser field without photon emissions until time $t_1$, it then emits a single photon and continues the evolution without emissions until time $t_2$ when it emits the second photons and so on. At this point it is convenient to choose a four-dimensional orthonormal basis to work with: $| e \rangle=(1,0,0,0)^{T}$, $|g \rangle= (0,1,0,0)^{T}$, $| c \rangle=(0,0,1,0)^{T}$ and $| c* \rangle= (0,0,0,1)^{T}$. According to the matrix form of the Bloch equation Eq. (\[eqBloch\]) the first two vectors correspond to pure excited and ground states respectively. The last two vectors, however, do not represent any real physical state and should be simply considered as convenient mathematical way to include all possible quantum paths going through superposition of the pure physical states $|e\rangle$ and $|g\rangle$, thus representing the contribution of the coherence effect. Using this notation the main equation for calculating the probability of $n$ emission events up to time $t$ is [@remark] $$P_n(t) = (\langle e|+\langle g|)\sigma^{(n)}_{(t)} \rangle=(\langle
e|+\langle g|)U^{(n)}_{(t,0)}|\sigma_{(0)}\rangle. \label{eqPn}$$ For example the probability of emitting zero photons is $$P_0(t) = \sum_{i = e,g} \langle i | {\cal G} (t,0) |
\sigma_{(0)}\rangle,$$ and the probability of emitting a single photon is $$P_1(t) = \sum_{i = e,g} \langle i | \int_0 ^t {\rm d} t_1 {\cal G}
\left( t , t_1 \right) \hat{\Gamma} {\cal G} \left( t_1 , 0 \right)
| \sigma_{(0)} \rangle.$$\
Consider a laser field interacting with the molecule in the time interval $(t',t)$ and choose a fixed point $t_a$ inside this interval. Such a partitioning of the time axis is useful for the analysis of sequence of pulses investigated in the following section, when we distinguish between time intervals where the laser is turned on and off. First, let’s split the integration over $t_n$ in Eq. (\[eqUn\]) into two parts: $$U^{(n)}_{(t,t')}=\int_{t'} ^{t}{\rm d} t_n\cdots=\int_{t'} ^{t_a}{\rm d}
t_n\cdots+\int_{t_a} ^{t}{\rm d} t_n\cdots.$$ Using the fact that in the first interval $(t',t_a)$ $t_n \leq t_a$ and replacing the Green function ${\cal G}(t,t_n)$ by the product ${\cal G}(t,t_a){\cal G}(t_a,t_n)$ one easily finds $$\int_{t'} ^{t_a}{\rm d} t_n\cdots \int_{t'} ^{t_2}{\rm d} t_1{\cal
G}(t,t_n) \hat{\Gamma}\cdots \hat{\Gamma} {\cal
G}(t_1,t')=U^{(0)}_{(t,t_a)}U^{(n)}_{(t_a,t')}.$$ Now, left with the integral over the second range $(t_a,t)$, we repeat exactly the same procedure as we did with the initial expression, but this time we split the integration over $t_{n-1}$ into $(t',t_a)$ and $(t_a,t_n)$. Similarly, using $t_{n-1} \leq t_a$ and replacing ${\cal G}(t_n,t_{n-1})$ by ${\cal G}(t_n,t_a){\cal
G}(t_a,t_{n-1})$ inside the first interval we find $$\int_{t_a} ^{t}{\rm d} t_n\int_{t'} ^{t_a}{\rm d} t_{n-1}\cdots
\int_{t'} ^{t_2}{\rm d} t_1{\cal G}(t,t_n) \hat{\Gamma}\cdots
\hat{\Gamma} {\cal G}(t_1,t')=$$ $$=U^{(1)}_{(t,t_a)}U^{(n-1)}_{(t_a,t')}.$$ Repeating this algorithm n times it is easy to prove that $$U^{(n)}_{(t,t')}=\sum_{\alpha=0}^{n}U^{(\alpha)}_{(t,t_a)}U^{(n-\alpha)}_{(t_a,t')}.
\label{eqSumUU}$$ Eq. (\[eqSumUU\]) means that the propagator corresponding to n emission events in $(t',t)$ can be decomposed into $\alpha$ emission events in $(t',t_a)$ and $n-\alpha$ emission events in $(t_a,t)$. The extension to more than one time point such as $t_a$ is trivial and leads to summation over all possible permutations of the n photons propagators resulting in n emission events.\
Turning back to the Eq. (\[eqPn\]) for $P_n(t)$ and inserting the closure relation $$\sum_{j=e,g,c,c*}|j \rangle\langle j|=1, \label{closure}$$ we find $$P_n(t) = \sum_{i=e,g} \sum_{j=e,g,c,c*} \sum_{\alpha=0} ^n \langle
i | U^{n-\alpha}_{(t,t_a)}| j \rangle \langle j|
U^{\alpha}_{(t_a,0)}| \sigma_{(0)} \rangle \label{eqPnt1}$$ Eq. (\[eqPnt1\]) describes the summation over all possible paths resulting in n emission events and suggests the following classification : the paths going through the pure states $|j\rangle=| e \rangle,| g \rangle$ may be identified as semiclassical, whereas the paths going through the states $|j\rangle=| c \rangle,| c^* \rangle$ describe the contribution of the coherence.
Two Pulses
==========
Now we focus on the case of two laser pulses separated by a window $\Delta$ in which the laser is turned off. The initial time is $t=0$, the time $t_1$ is the moment when the first pulse is switched off. The amplitude of the external field remains equal zero $f(t)=0$ for the delay period $t_1 < t<t_1+\Delta$. At time $t_2=t_1
+ \Delta$ the laser is turned on again, and then again turned off for $t>t_3$ ($f(t)=0$ for $t>t_3)$. Schematically the sequence is represented in Fig. 1 for square pulses, however we emphasize that the results obtained in this section are valid for pulses of any shape. Our goal is the derivation of general expressions for $P_n(t)$ from two pulses in the limit of the long measurement time $t\rightarrow\infty$ when we know that eventually the system is in the ground state. We assume that the molecule is always in the ground state at the beginning of the experiment. If we divide the time axis into four distinct intervals : two intervals when the laser is turned on and two others when the laser is turned off, the most general expression for $U^{(n)}_{(t,0)}$ following from the extension of Eq. (\[eqSumUU\]) is $$U^{(n)}_{(t,0)}=
U^{(n-\alpha-\beta-\gamma)}_{(t,t_3)}U^{(\gamma)}_{(t_3,t_2)}U^{(\beta)}_{(t_2,t_1)}U^{(\alpha)}_{(t_1,0)},
\label{eqPngen}$$ where the superscripts $\alpha$, $\beta$ and $\gamma$ are all non negative integer values leading to n photons (i.e. $n-\alpha-\beta-\gamma\geq0$). The Einstein’s summation rule from 0 to n must be applied to every superscript appearing twice. Inside time intervals $(t,t_3)$ and $(t_2,t_1)$, when the laser is turned off, the Rabi frequency is equal zero $\Omega=0$, and the calculation of the Green function ${\cal G} (t,t')$ Eq. (\[eqGreenf\]) becomes nearly trivial. For the delay interval $\Delta$ we find only two non-zero n-photon-propagators: $$U^{(0)}_{(t_1+\Delta,t_1)} =\left(
\begin{array}{c c c c}
e^{-\Gamma\Delta} & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & e^{ (i \omega_0 - \Gamma/2)\Delta} & 0 \\
0 & 0 & 0 & e^{ -(i \omega_0 + \Gamma/2)\Delta}
\end{array}
\right),\label{eqU0}$$ $$U^{(1)}_{(t_1+\Delta,t_1)} =\left(
\begin{array}{c c c c}
0 & 0 & 0 & 0 \\
1-e^{-\Gamma\Delta} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}
\right) \label{eqU1}$$ and $U^{(n)}_{(t_1+\Delta,t_1)}=0$ for $n>1$. This result is definitely expected , since if nothing excites the molecule, there is no chance to get more than a single photon. The matrix representation of the propagators in the interval $(t>t_3)$ when $t\rightarrow\infty$, are found by taking the limit $\Delta\rightarrow\infty$ of Eqs. (\[eqU0\]),(\[eqU1\]). Now inserting the closure relation Eq. (\[closure\]) between each two propagators of Eq. (\[eqPngen\]) and using the just obtained matrix elements Eqs. (\[eqU0\]),(\[eqU1\]) we find :
$$P_n=\lim_{t\rightarrow\infty}P_n(t)=P_n^{\rm Cla}+
e^{-\Delta\frac{\Gamma}{2}}(e^{i \Delta\omega_0}A_n^{\rm
Coh}+C.C.),\label{eqPn1}$$
where $$P_n^{\rm Cla}=\sum_{\alpha=0}^{n}\left\{\langle g|U^{(n-\alpha)}_{(t_3,t_2)}|g\rangle
\langle g|U^{(\alpha)}_{(t_1,0)}|g\rangle +
e^{-\Delta\Gamma}\langle g|U^{(n-\alpha)}_{(t_3,t_2)}|e\rangle
\langle
e|U^{(\alpha)}_{(t_1,0)}|g\rangle\right\}+\sum_{\alpha=0}^{n-1}(1-e^{-\Delta\Gamma})\langle
g|U^{(n-\alpha-1)}_{(t_3,t_2)}|g\rangle \langle
e|U^{(\alpha)}_{(t_1,0)}|g\rangle$$ $$+ \sum_{\alpha=0}^{n-1}\left\{\langle
e|U^{(n-\alpha-1)}_{(t_3,t_2)}|g\rangle \langle
g|U^{(\alpha)}_{(t_1,0)}|g\rangle +e^{-\Delta\Gamma}\langle
e|U^{(n-\alpha-1)}_{(t_3,t_2)}|e\rangle \langle
e|U^{(\alpha)}_{(t_1,0)}|g\rangle\right\}+\sum_{\alpha=0}^{n-2}(1-e^{-\Delta\Gamma})\langle
e|U^{(n-\alpha-2)}_{(t_3,t_2)}|g\rangle \langle
e|U^{(\alpha)}_{(t_1,0)}|g\rangle\label{Pncla}$$ and $$A_n^{\rm Coh}=\sum_{\alpha=0}^{n}\langle
g|U^{(n-\alpha)}_{(t_3,t_2)}|c\rangle \langle
c|U^{(\alpha)}_{(t_1,0)}|g\rangle + \sum_{\alpha=0}^{n-1}\langle
e|U^{(n-\alpha-1)}_{(t_3,t_2)}|c\rangle \langle
c|U^{(\alpha)}_{(t_1,0)}|g\rangle.\label{Ancoh}$$
Eqs. (\[eqPn1\]),(\[Pncla\]) and (\[Ancoh\]) summarize all possible paths resulting in n photon emission events and allow simple identification of negligible terms, when particular physical constraints are taken into account. It is easy to see, that the first two terms of $P_n^{\rm Cla}$ Eq. (\[Pncla\]) (those with $\sum_{\alpha=0}^{n}$) describe processes where all n photons are emitted during the pulses and none in the delay interval or after the second pulse. The third term of this expression represents the processes, where a single photon is emitted in the delay period (with probability $\left[1-e^{-\Delta\Gamma}\right]$) and n-1 photons during the pulses events. Similarly, the next two terms originate from the processes, where a single photon is emitted after the second pulse and zero photons in the delay interval, and finally, the last term describes situations, where one photon is emitted in the delay interval and another after the second pulse. This interpretation may be used to simplify the calculations, as for instance in the case of the short pulses considered below, where we neglect the trajectories with photons emitted during the pulse events.\
Although Eqs. (\[eqPn1\]),(\[Pncla\]) and (\[Ancoh\]) are very general, they already contain interesting physical information. First of all, we pay attention to the fact, that the coherence terms $A_n^{\rm Coh}$, describing the processes where the molecule is left in the superposition of the pure states at the end of the first pulse (i.e. the paths going trough the states $|c\rangle$ and $|c^*\rangle$), never include trajectories where a photon is emitted within the delay interval $\Delta$. Mathematically this follows from Eqs. (\[eqU0\]) and (\[eqU1\]), and physically it makes sense, because spontaneous collapse into the ground state destroys the coherence. Secondly, we see, that the coherence terms are multiplied by the exponentially decaying factor $e^{-\Gamma\Delta/2}$, responsible for the dephasing effect, and oscillate in $\Delta$ with orbital frequency $\omega_0$ (see the $e^{i\Delta\omega_0}$ term in Eq. (\[eqPn1\])). In optics $\omega_0$ is much larger than the inverse of $\tau$ - the minimum time resolution of the measurement device : $\tau\omega_0\gg1$. Therefore, in order to match our results for the probability of emitting n photons to those observed by an experimentalist, it is essential to treat the coherence terms as stochastic variables - i.e. it is reasonable to replace them with their time average, which is equal zero. However, it should not be forgotten that : (i) in the limit $\Delta\to0$, when the pulses are attached together, the coherence contribution $A_n^{\rm Coh}$ becomes non-oscillating and non-negligible part of $P_n$, and (ii) in non-optical microwave experiments, where the absorption frequency is comparable with the time resolution of the measurement device [@Katz], the influence of the coherence trajectories is important.\
It is possible to derive another useful expression for the probability of emitting n photons from two pulses. First we note, that according to Eqs. (\[eqPn\]), (\[eqU0\]) and (\[eqU1\]) the probability of emitting n photons from any single pulse or sequence of pulses of total length T in the limit of infinitely long measurement time $t\to\infty$ is $$P_n=\langle g|U^{(n)}_{(T,0)}|g\rangle + \langle
e|U^{(n-1)}_{(T,0)}|g\rangle, \label{SinglePulse}$$ (where for n=0 the second term $\langle
e|U^{(-1)}_{(T,0)}|g\rangle=0$). From the physical point of view the second term of Eq. (\[SinglePulse\]) expresses the fact, that the molecule left in the pure excited state eventually decays to the ground state by spontaneous photon emission. Simple rearrangement of Eqs. (\[Pncla\]) and (\[Ancoh\]), with details given in Appendix A, results in :
$$P_n=\sum_{\alpha=0}^{n}P_{n-\alpha}^{I_2}P_{\alpha}^{I_1}+
e^{-\Delta\Gamma}\left\{P_{n}^{I_2I_1}-\sum_{\alpha=0}^{n}P_{n-\alpha}^{I_2}P_{\alpha}^{I_1}
+\left[(e^{\Delta(\Gamma/2+i\omega_0)}-1)A_n^{\rm
Coh}+C.C.\right]\right\}. \label{Pncorrel}$$
Here $P_n^{I_1}=\langle g|U^{(n)}_{(t_1,0)}|g\rangle + \langle
e|U^{(n-1)}_{(t_1,0)}|g\rangle$ is the probability of emitting n photons only from the first pulse [@Yong; @Yong1], similarly $P_n^{I_2}=\langle g|U^{(n)}_{(t_3,t_2)}|g\rangle + \langle
e|U^{(n-1)}_{(t_3,t_2)}|g\rangle$ designates the probability of emitting n photons only from the second pulse, and $$P_n^{I_2I_1}=\sum_{\alpha=0}^n\langle
g|U^{(n-\alpha)}_{(t_3,t_2)}U^{(\alpha)}_{(t_1,0)}|g\rangle +
\sum_{\alpha=0}^{n-1}\langle
e|U^{(n-1-\alpha)}_{(t_3,t_2)}U^{(\alpha)}_{(t_1,0)}|g\rangle
\label{Pi2i1}$$ is the probability of emitting n photons from the two pulses produced one immediately after another (i.e. with zero delay). This formulation of $P_n$ Eq. (\[Pncorrel\]) shows, that the first term $\sum_{\alpha=0}^{n}P_{n-\alpha}^{I_2}P_{\alpha}^{I_1}$ represents the sum of all possible ways of emitting n photons from the both pulses, as if the consequences of the interaction of the molecule with the first pulse had no influence on the state of the system at the beginning of the second pulse, i.e. like if the treatment of each pulse could be done independently. Nevertheless, since such an influence exists, it is reasonable to define the rest of the terms on the righthand side of Eq. (\[Pncorrel\]) as a correlation: $$C(\Delta)=P_n-\sum_{\alpha=0}^{n}P_{n-\alpha}^{I_2}P_{\alpha}^{I_1}=
e^{-\Delta\Gamma}\left\{P_{n}^{I_2I_1}-\sum_{\alpha=0}^{n}P_{n-\alpha}^{I_2}P_{\alpha}^{I_1}+\right.$$ $$\left.+\left[(e^{\Delta(\Gamma/2+i\omega_0)}-1)A_n^{\rm
Coh}+C.C.\right]\right\}. \label{correl}$$ Note that Eq. (\[Pncorrel\]) makes perfect physical sense in the limits $\Delta\rightarrow\infty$ and $\Delta\rightarrow0$ where we find trivially expected results. In the first case only the first term on the righthand side of Eq. (\[Pncorrel\]) survives - i.e. this limit describes the situation where the interaction of the molecule with the first pulse indeed has no influence on the interaction of the molecule with the second pulse, since all coherence effects have enough time to decay completely. And the second limit gives $P_n=P_{n}^{I_2I_1}$. We emphasize, that in the first case, once the probabilities of emitting n photons from each single pulse are known, the efforts needed for the calculations are considerably reduced. However, care must be taken while using Eq. (\[Pncorrel\]) for the calculation of the second limit $\Delta\rightarrow0$, since the continuity of the laser’s phase plays important role, as demonstrated on the example of two square pulses in the subsequent section.
Example : Two Square Pulses
===========================
In this section we apply our general results to the concrete example of two identical square laser pulses. Consider the sequence :
![Two square laser pulses as modeled in Eq. (\[Figure1\]). T is the length of a single pulse and $\Delta$ is the delay interval between the pulses. The dashed arrows show four cases out of eight possibilities of emission of two photons. For example: (a) two photons are emitted within the duration of the first pulse.[]{data-label="model1"}](fig1.eps "fig:"){width="\columnwidth"}\
$$f(t) = \left\{
\begin{array}{l l}
\cos(\omega_L t + \phi_1 ) & \ \ 0<t<t_1\\
0 & \ \ t_1< t<t_2\\
\cos\left[ \omega_L (t-t_2) + \phi_2 \right] & \ \ t_2< t<t_3 \\
0 & \ \ t_3 < t
\end{array}
\right.
\label{Figure1}$$
where $t_1=t_3-t_2=T$ - is the pulse’s duration, $t_2-t_1=\Delta$ - is the delay period between the pulses and $\phi_1$ and $\phi_2$ are the initial phases of each pulse. For simplicity we assume, that the laser frequency $\omega_L = \omega_0$, namely we consider the case of zero detuning, and also the initial phase of the first pulse is zero $\phi_1=0$. The time dependence of the Rabi frequency $\Omega$, describing the interaction, is schematically illustrated in Fig. 1. The propagators acting on the molecule during the square pulses are found using the Rotating Wave Approximation (RWA). In Appendixes B and C we show, that within RWA Eq. (\[eqPn1\]) can be rewritten in the form $$P_n=p_n^{\rm Cla}+
e^{-\Delta\frac{\Gamma}{2}}(e^{i (\Delta+T)\omega_{0}-\phi_{2}}a_n^{\rm Coh}+C.C.),\label{eqPnRWA}$$\
where the definition of $p_n^{\rm Cla}$ and $a_n^{\rm Coh}$ follows from $P_n^{\rm Cla}$ and $A_n^{\rm Coh}$ Eqs. (\[Pncla\]),(\[Ancoh\]) by replacing all n-photon-propagators Eq. (\[eqUn\]) by $\tilde U^{(n)}_{(t,t')}$ - the n-photon-propagators calculated within RWA (see the derivation of Eq. (\[eqtildUn\]) in Appendix B) . All mathematical manipulations and calculations were obtained with the help of Mathematica 5.0.\
First we consider in detail the probability of emitting zero photons. For the calculation of $P_0(t)$ we need the zero photon propagator. This is the simplest case, since there is only one possible permutation. According to Eq. (\[eqPngen\]) we have : $$U^{(0)}_{(t,0)}=U^{(0)}_{(t,t_3)}U^{(0)}_{(t_3,t_2)}U^{(0)}_{(t_2,t_1)}U^{(0)}_{(t_1,0)}.
\label{eqP0}$$ Inserting the closure relation Eq. (\[closure\]) and applying RWA lead to : $$p^{\rm Cla}_0 = e^{ -
\Gamma \Delta}\langle {\rm g}|\tilde U^{(0)}_{(t_3,t_2)}| {\rm e}
\rangle \langle {\rm e} |\tilde U^{(0)}_{(t_1,0)} | {\rm
g}\rangle+$$ $$+ \langle {\rm g}|\tilde U^{(0)}_{(t_3,t_2)}| {\rm g} \rangle
\langle {\rm g}|\tilde U^{(0)}_{(t_1,0)}| {\rm g} \rangle
,\label{pzeroCla}$$
$$a^{\rm Coh}_0=\langle {\rm g} | \tilde U^{(0)}_{(t_3,t_2)} | {\rm
c} \rangle \langle c |\tilde U^{(0)}_{(t_1,0)}| {\rm g} \rangle,
\label{pzeroCoh}$$
\
Calculating the matrix elements of Eqs. (\[pzeroCla\]) and (\[pzeroCoh\]) we find the following explicit expression for $P_0=\lim_{t\to\infty}P_0(t)$:
$$P_0 = {16 \Omega^4 e^{ - T - \Delta} \over \left( 1 - 4 \Omega^2\right)^2} \sinh^4 \left( { \sqrt{1 - 4 \Omega^2} \over 4} T \right) +$$ $${e^{-T} \over \left( 1 - 4 \Omega^2\right)^2} \left[ \left( 1 - 2 \Omega^2\right) \cosh\left( { \sqrt{ 1 - 4 \Omega^2} \over 2} T \right) +
\sqrt{1 - 4 \Omega^2} \sinh\left( {\sqrt{1 - 4 \Omega^2} \over 2} T
\right) - 2 \Omega^2 \right]^2$$ $$- {8 \Omega^2 \over \left(1 - 4 \Omega^2\right)^2 } e^{ - T -
\Delta/2 } \sinh^2\left({\sqrt{1 - 4 \Omega^2} \over 4} T \right)
\left[ \sqrt{ 1 - 4 \Omega^2} \cosh\left( { \sqrt{ 1 - 4 \Omega^2}
\over 4 } T \right) + \sinh\left( { \sqrt{1 - 4 \Omega^2} \over 4 }
T \right) \right]^2 \cos[\omega_0
\left(\Delta+T\right)-\phi_2],\label{P0exact}$$ where we set $\Gamma=1$ for simplicity. The last term, exhibiting oscillations due to the $\cos[\omega_0
\left(\Delta+T\right)-\phi_2]$, results from the quantum paths going through the $|c\rangle$ and $|c*\rangle$, thus representing the coherence effect. Of course, when $T=0$ or $\Omega=0$, $P_0=1$ since no photons are emitted, and if $T \to \infty$, $P_0 =0$ since many photons are emitted. Similar calculations were made also for $P_1$ and $P_2$ - see Eqs. (\[P1\]), (\[P2\]) for the final results in Appendix C.\
For very intense laser fields, when the Rabi frequency is much larger than the inverse life time of the excited state, taking the limit $\Omega \gg1$ of Eq. (\[P0exact\]) we obtain : $$\lim_{\Omega\gg1}P_0 \sim e^{ - T} \left\{ e^{-\Delta}
\sin^4\left( { \Omega T \over 2} \right) + \cos^4 \left( { \Omega T
\over 2} \right) - {1 \over 2} e^{ - \Delta/2} \sin^2 \left( \Omega
T \right) \cos [\omega_0 \left(\Delta+T\right)-\phi_2]\right\}.
\label{eqppzz}$$ And using Eqs. (\[P1\]), (\[P2\]): $$\lim_{\Omega \gg1}P_1 = \frac{e^{-T}}{8} \left \{4 \sin^2(\Omega T)+2 T \left[1-\sin^2\left(\frac{\Omega T}{2}\right)\left(1-e^{-\Delta}\right) \right]+T \sin^2(\Omega T)\left(1+e^{-\Delta}\right) \right\}$$ $$+ \frac{e^{-(T+{\Delta/2})}}{4} (T+2) \sin ^2(\Omega T)
\cos[\omega_0 \left(\Delta+T\right)-\phi_2], \label{P1exact}$$ $$\lim_{\Omega \gg1} P_2 = e^{-T} \left \{\left[ \sin ^4\left(\frac{\Omega T}{2}\right)+\frac{ T^2}{64}\cos ( \Omega T) \right] \left(1-e^{-\Delta}\right) + \frac{T}{4} \left[\cos ^2(\Omega T)+2\right]\right \}$$ $$+\frac{ e^{-T} }{32} T^2 \left( \cos ^2(\Omega T)+4
\right)\left(1+e^{-\Delta}\right)-\frac{e^{-(T+{\Delta/2})}}{16} T
(T+4) \sin ^2(\Omega T) \cos [\omega_0
\left(\Delta+T\right)-\phi_2]. \label{P2exact}$$\
Finally, we would like to investigate the limiting behavior of $P_0$, $P_1$ and $P_2$ within the strong fields approximation in the case of long $\Delta\rightarrow\infty$ and short $\Delta\rightarrow0$ delay intervals. As shown in [@Yong; @Yong1], the probabilities of emission 0, 1 and 2 photons from a single square pulse of length T (see Eq. (\[SinglePulse\])) are given by $$\lim_{\Omega\gg1}P_0^{I}=\lim_{\Omega\gg1}P_0^{I_1}=\lim_{\Omega\gg1}P_0^{I_2}\sim
e^{ - T/2} \cos^2 \left( { \Omega T \over 2} \right) ,
\label{p0single}$$ $$\lim_{\Omega\gg1}P_1^{I}\sim \frac{e^{ - T/2}}{8}\left[
4+2T-(4+T)\cos(\Omega T ) \right] \label{p1single}$$ and $$\lim_{\Omega\gg1}P_2^{I}\sim \frac{e^{ - T/2}}{64}T\left[
4T+16+(8+T)\cos(\Omega T ) \right] . \label{p2single}$$ Taking the limit $\Delta\gg1$ of Eq. (\[P0exact\]) we find $$\lim_{\Omega\gg1,\Delta\gg1}P_0=(P_0^{I})^2\sim e^{ - T} \cos^4
\left( { \Omega T \over 2} \right) , \label{p0deltainfty}$$ which is equal precisely to the product of the probabilities of emitting zero photons from two single square pulses Eq. (\[p0single\]) and completely agrees with Eq. (\[Pncorrel\]). Now using Eqs. (\[Pncorrel\]), (\[p1single\]), (\[p2single\]) and the fact, that the two pulses are identical, we can easily obtain the limit $\Delta\rightarrow\infty$ of $P_1$ and $P_2$: $$\lim_{\Omega\gg1,\Delta\gg1}P_1=2P_1^IP_0^I=$$ $$\frac{e^{ - T}}{4} \left[ (8+3T)\cos^2 \left( { \Omega T \over 2}
\right)-2(4+T)\cos^4 \left( { \Omega T \over 2} \right)\right]
\label{p1deltaifnty}$$ and $$\lim_{\Omega\gg1,\Delta\gg1}P_2=2P_0^{I}P_2^{I}+(P_1^{I})^2=\frac{e^{-T}}{64}\left[24 + T(40 + 9T) +\right.$$ $$\left.(-32 +
T^2)\cos(\Omega T) +(8 + T(8 + T))\cos(2\Omega T)\right].
\label{p2deltaifnty}$$\
Considering the opposite limit $\Delta\to0$ we remind, that the contribution of the coherence paths going through the states $|c\rangle$ and $|c^*\rangle$ must not be neglected. Moreover, it is essential to take into account, that two attached square pulses of the same Rabi frequency are equal to a single long square pulse, only if $\phi_2=\phi_1+\omega_{0}(\Delta+T)$ - i.e. the pulse is continuous. Assuming for simplicity, that this is the case, when $\Delta\rightarrow0$ from Eq. (\[P0exact\]) we find for $P_0$ : $$\lim_{\Omega\gg1,\Delta\ll1}P_0\sim e^{ - T} \cos^2(\Omega T) ,
\label{p0deltazero}$$ which once again agrees with Eq. (\[Pncorrel\]), since it is exactly the result of replacing T with 2T in Eq. (\[p0single\]). Similarly the limits $\Delta\to0$ of $P_1$ and $P_2$ may be obtained by replacing T with 2T in Eqs. (\[p1single\]) and (\[p2single\]).\
In Fig. 2, neglecting the fast oscillating coherence paths, we plotted the semiclassical terms $p_0^{\rm Cla}$, $p_1^{\rm Cla}$ and $p_2^{\rm Cla}$ for the relatively long $\Delta=3$ and short $\Delta=0.5$ delay intervals for the case of strong laser field $\Omega=10$. Comparing the graphs one may see, that the dependence of $p_1^{\rm
Cla}$ on $\Delta$ is visibly weaker than those of $p_0^{\rm Cla}$ and $p_2^{\rm Cla}$ (we explain this effect later - see the discussion below Table 1). When $\Gamma T\gg1$, we expect that : (i) $p_0^{\rm Cla}, p_1^{\rm Cla}$ and $p_2^{\rm Cla}$ are all small, since many photons are expected to be emitted during the pulses, and (ii) independent of $\Delta$, since the contribution of the photons emitted during the pulse event is much larger than the contribution of the photons emitted in the delay interval. Such a behavior is clearly seen for $p_2^{\rm Cla}$ (compare Figs. $2.{\bf c}$ and $2.{\bf f}$) where the difference between the case $\Delta=3$ and $\Delta=0.5$ is stronger for short T. Below we prove this in a Poissonian limit.
$$\begin{aligned}
\includegraphics[height=1.5in, width=2in]{fig2a.eps}\textbf{a}&
\includegraphics[height=1.5in, width=2in]{fig2b.eps}\textbf{b}&
\includegraphics[height=1.5in, width=2in]{fig2c.eps}\textbf{c}\\
\includegraphics[height=1.5in, width=2in]{fig2d.eps}\textbf{d}&
\includegraphics[height=1.5in, width=2in]{fig2e.eps}\textbf{e}
&
\includegraphics[height=1.5in, width=2in]{fig2f.eps}\textbf{f}\\\end{aligned}$$
\[Correl\]
Strong and Short Pulses
-----------------------
The sequence of two very short and strong pulses is important due to its numerous practical applications. Mathematically we define this limit as $T\rightarrow0$, $\Omega\rightarrow\infty$ in such a way, that the product $\Omega T$ stays of the order of unity $\Omega T\sim1$ (automatically leading also to $\Omega\gg\Gamma$), otherwise the molecule will never reach the excited state, and the probability to obtain non-zero results for $P_1$ and $P_2$ will be negligible. As a consequence, the spontaneous emission process during the pulse event may be neglected, and therefore, it is reasonable to approximate the behavior of the system by simple Shrödinger evolution with well-known Rabi oscillations. In this limit the photons can be emitted only in the delay interval or after the second pulse, while the only non-zero propagator acting on the molecule during the pulses within RWA has the following matrix representation : $$\lim_{\begin{array}{c}
_{T\rightarrow0,}\\
_{\Omega\rightarrow\infty}\end{array}} \tilde{U}^{(0)}_{(T,0)} =
\left(
\begin{array}{c c c c}
\cos^2 {\Omega T \over 2} & \sin^2 { \Omega T \over 2} & - i {\sin \Omega T \over 2} & i {\sin \Omega T \over 2} \\
\sin^2 { \Omega T \over 2} & \cos^2 { \Omega T \over 2} & i {\sin \Omega T \over 2} & - i {\sin \Omega T \over 2} \\
-i {\sin \Omega T \over 2} & i {\sin \Omega T \over 2} & \cos^2 {\Omega T \over 2} & \sin^2 { \Omega T \over 2} \\
i {\sin \Omega T \over 2} & - i {\sin \Omega T \over 2} & \sin^2 {\Omega T \over 2} & \cos^2 { \Omega T \over 2}
\end{array}
\right), \label{RabiOscl}$$ which is independent of $\Gamma$. Note, that since this zero-photon-propagator Eq. (\[RabiOscl\]) describes the conservative evolution of the system, the transformation is unitary, all the elements exhibit Rabi oscillations, and symmetry and reversibility of the matrix elements are found: $\langle {\rm
e}|\tilde{U}^{(0)}_{(T,0)}|{\rm e}\rangle=\langle {\rm g} |
\tilde{U}^{(0)}_{(T,0)}| g\rangle$, $\langle {\rm e}
|\tilde{U}^{(0)}_{(T,0)}|{\rm g}\rangle=\langle{\rm g}
|\tilde{U}^{(0)}_{(T,0)}|{\rm e}\rangle$ etc.\
Now we consider $P_0$, $P_1$ and $P_2$ in the limit of the short and strong pulses and demonstrate how this physical constrain can help in reducing the number of paths appearing in Eqs. (\[eqPn1\]),(\[Pncla\]),(\[Ancoh\]). The exact expressions for $U^{(1)}_{(t,0)}$ and $U^{(2)}_{(t,0)}$, according to Eq. (\[eqPngen\]) consist of the sum of 4 and 10 terms respectively - see Eqs. (\[eqU1exact\]), (\[eq2exact\]) in Appendix C. After neglecting the trajectories where photons are emitted within the pulse events (since $T\to0)$) we have : $$\lim_{T\to0,\Omega\to\infty}U^{(1)}_{(t,0)}=U^{(1)}_{(t,t_3)}U^{(0)}_{(t_3,t_2)}U^{(0)}_{(t_2,t_1)}U^{(0)}_{(t_1,0)}$$ $$+U^{(0)}_{(t,t_3)}U^{(0)}_{(t_3,t_2)}U^{(1)}_{(t_2,t_1)}U^{(0)}_{(t_1,0)}
\label{U1shortpulses}$$ and $$\lim_{T\to0,\Omega\to\infty}U^{(2)}_{(t,0)}=U^{(1)}_{(t,t_3)}U^{(0)}_{(t_3,t_2)}U^{(1)}_{(t_2,t_1)}U^{(0)}_{(t_1,0)}.
\label{U2shortpulses}$$ This is one example where the formulation of photon statistics based on quantum trajectories is very convenient, since we can identify the underlying physical processes and make approximations. Inserting the closure relation Eq. (\[closure\]) between every two propagators of Eqs. (\[U1shortpulses\]), (\[U2shortpulses\]) and using the matrix elements of the zero-photon-propagator Eq. (\[RabiOscl\]), we obtain the leading semiclassical and coherence terms of $P_0$, $P_1$ and $P_2$ in the limit of the short and strong pulses. The results are summarized in Table 1 below and they are valid for any sequence of short pulses.
$$\mbox{
\begin{tabular}{|c| c| c|}
\hline
$n$ & $p_n^{{\rm Cla}}$ & $a_n ^{{\rm Coh}}$ \\
\hline $ 0$ & $ \langle {\rm g} | \tilde U^{(0)}_{(t_3,t_2)} | {\rm
g} \rangle\langle {\rm g} | \tilde U^{(0)}_{(t_1,0)}| {\rm g}
\rangle + e^{-\Gamma\Delta}\langle {\rm g} | \tilde
U^{(0)}_{(t_3,t_2)}| {\rm e} \rangle \langle {\rm e} |\tilde
U^{(0)}_{(t_1,0)}| {\rm g} \rangle$&
$ \langle {\rm g} | \tilde U^{(0)}_{(t_3,t_2)}|c\rangle \langle c | \tilde U^{(0)}_{(t_1,0)}| {\rm g} \rangle $ \\
\ & \ & \\
\hline $ 1$ & $ \langle {\rm g} | \tilde U^{(0)}_{(t_3,t_2)} | {\rm
g} \rangle\langle {\rm e} | \tilde U^{(0)}_{(t_1,0)}| {\rm g}
\rangle + \langle {\rm e} | \tilde U^{(0)}_{(t_3,t_2)}| {\rm g}
\rangle \langle {\rm g} |\tilde U^{(0)}_{(t_1,0)}| {\rm g} \rangle$&
$ \langle {\rm e} | \tilde U^{(0)}_{(t_3,t_2)}|c\rangle \langle c | \tilde U^{(0)}_{(t_1,0)}| {\rm g} \rangle $ \\
\ & \ & \\
\hline $ 2$ & $ \langle {\rm e} |\tilde U^{(0)}_{(t_3,t_2)} | {\rm
g} \rangle \langle {\rm e} |\tilde U^{(0)}_{(t_1,0)}| {\rm g}
\rangle\left(1 - e^{ - \Gamma \Delta} \right) $ &
$ 0 $ \\
& \ & \ \\
\hline
\end{tabular}}$$ $$\mbox{\textbf{Table 1:} Photon statistics for short pulses.}$$ Note, that the coherence terms of $P_2$ vanish, since emission of a photon in the delay interval, necessary for emitting two photons in the limit of very short pulses, destroys the coherence . Using the symmetry and reversibility of the zero-photon-propagator matrix elements Eq. (\[RabiOscl\]), it is easy to show that $a_{0}^{\rm
Coh}=-a_{1}^{\rm Coh}$ and $ p_0^{\rm Cla}+ p_1 ^{\rm Cla} + p_2
^{\rm Cla}=1$, i.e. the semiclassical paths conserve probability. Finally, we bring attention to the fact, that the semiclassical paths of $P_1$ do not depend neither on the spontaneous emission rate $\Gamma$ not on the delay interval duration $\Delta$ (see Fig. 2.[**b**]{} and Fig. 2.[**e**]{}). Comparing the two non-negligible trajectories of $p_1^{\rm Cla}$ with Eq. (\[Pncorrel\]), we see that they correspond to the first term of the righthand side of this equation - i.e to the product of probabilities of emitting 0 and 1 photons, related to each one of the pulses independently.\
Using the matrix elements Eq. (\[RabiOscl\]) and Table 1, after some algebra we obtain explicitly : $$\lim_{T\to0,\Omega\to\infty}P_0^{\rm} =
e^{-\Delta}\sin^4\left({\Omega T \over 2}\right) + \cos^4
\left({\Omega T \over 2}\right)- {1\over 2} e^{-\Delta/2} \sin^2
\left(\Omega T\right)
\cos\left[\omega_0\left(T+\Delta\right)-\phi_2\right]
\label{standshP0}$$ which is the $\Omega\rightarrow\infty$, $T \to 0$ limit of Eq. (\[P0exact\]). $$\lim_{T\to0,\Omega\to\infty}P_1^{\rm } = {1 \over 2} \sin^2 (\Omega
T )\left\{ 1 + e^{ - {\Delta/2} } \cos\left[ \omega_0 \left( T +
\Delta\right)-\phi_2 \right]\right\} \label{Eqpooo}$$
and $$\lim_{T\to0,\Omega\to\infty}P_2^{\rm}= \left(1 - e^{ -
\Delta}\right) \sin^4 \left({\Omega T \over 2}\right) \label{Eqpttt}$$ It is easy to see, that when $T\rightarrow0$ the Eqs. (\[P1exact\]), (\[P2exact\]) reduce to Eqs. (\[Eqpooo\]), (\[Eqpttt\]) as expected.\
For a $\pi$-pulse defined by $\Omega T = \pi + 2 \pi n$ [@Yong; @Yong1], where $n$ is a non negative integer, in the strong field limit we obtain : $$\lim_{T\to0,\Omega\to\infty}P_0 \sim e^{ - \Delta}.$$ This behavior may be easily understood for : substituting $\Omega T
= \pi$ into Eq. (\[RabiOscl\]) we have $$\lim_{\begin{array}{c}
_{T\rightarrow0,\Omega\rightarrow\infty},\\_{\Omega
T=\pi}\end{array}} \tilde{U}^{(0)}_{(T,0)} = \left(
\begin{array}{c c c c}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0\\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0
\end{array}
\right) \label{Pipulse}$$ The physical meaning of this propagator follows directly from its matrix representation : as well-known, the ideal $\pi$-pulse simply switches the state of the molecule from the excited to the ground state and vice versa. Thus, the first $\pi$-pulse of the sequence pumps the molecule from the ground state to the excited state. If the delay between the pulses $\Delta$ is long, the molecule will emit a photon before the arrival of the second pulse, and then $\lim_{T\to0,\Omega\to\infty}P_0 = 0$. Contrary, if $\Delta\ll 1$, the second $\pi$-pulse pushes the molecule back to the ground state before the emission of a photon, and then $\lim_{T\to0,\Omega\to\infty}P_0 = 1$. We also notice, that the interference $\cos \left[ \omega_0 \left( T + \Delta \right)-\phi_2
\right]$ term vanishes for the $\pi$-pulses, since according to Eq. (\[Pipulse\]) the off-diagonal matrix elements giving raise to this term are equal zero. On the opposite, for a $\pi/2$-pulse, defined by $\Omega T = \pi/2 + 2 \pi n $, the influence of the coherence on the photon statistics generally does not vanish: $$\lim_{\Omega\gg1}P_0 \sim {1 \over 4} \left\{ e^{ - \Delta} + 1 - 2
e^{- \Delta/2} \cos\left[ \omega_0(\Delta+T)\right] \right\}$$ Once again we remind, that in optics in many cases the ideal $\pi$ and $\pi/2$-pulses are considered where the interaction time $\Gamma
T \to 0$, and then $e^{ - T} = 1$, but $\omega_0 T$ is not a small number, especially because we work under the assumption $\Omega\ll\omega_0$, which is essential for the two level model approximation of the molecule and for the assumption, that the spontaneous emission rate $\Gamma$ is not effected by the presence of the laser field [@CT].\
Weak and Long Pulses
--------------------
Here we consider the case of very long and weak pulses. In this limit the delay interval has a negligible effect on $P_n$. In this limit we expect, that according to Eq. (\[Pncorrel\]) the probability of emitting n photons from two separated pulses may be approximated by $P_n^{I_2I_1}$ - the probability of emitting n photons from two attached pulses with zero delay : $$\lim_{T\rightarrow\infty}P_n=P_n^{I_2I_1}=\sum_{\alpha=0}^n\langle
g|U^{(\alpha)}_{(2T)}|g\rangle + \sum_{\alpha=0}^{n-1}\langle
e|U^{(\alpha)}_{(2T)}|g\rangle \label{LongT}$$ Taking the limit $\Omega\rightarrow0$, $T\rightarrow\infty$ in such a way that $\Omega^{2}T$ remains finite of the exact solution for $P_0$, $P_1$ and $P_2$ Eqs. (\[P0exact\]),(\[P1\]),(\[P2\]), we find $$\lim_{\Omega\to0,T\to\infty}P_0 = e^{-2 \Omega^2 T }\label{P0weak}$$ $$\lim_{\Omega\to0,T\to\infty}P_1 = 2 \Omega^2 T e^{-2 \Omega^2 T }
\label{P1weak}$$ $$\lim_{\Omega\to0,T\to\infty}P_2 = {(2 \Omega^2 T )^2 \over 2! }
e^{-2 \Omega^2 T }\label{P2weak}$$ $$\lim_{\Omega\to0,T\to\infty}P_n = {( 2 \Omega^2 T )^n \over n! }
e^{-2 \Omega^2 T } \label{Poissonian}$$\
Indeed one can show, that Eq. (\[LongT\]) and the limiting behavior Eqs.(\[P0weak\]-\[Poissonian\]) of the exact results are identical. Note, that Eq. (\[Poissonian\]) corresponds to Poissonian statistics. This behavior originates from the fact, that because of the long pulses duration and weak laser field, the leading terms of $P_n$ are those, where photon emissions are well separated one from another on the time axis, and therefore photon statistics is described by nearly uncorrelated emission events.
The upper and lower bounds for strong pulses
---------------------------------------------
Finally we investigate the upper and lower bounds of $P_0$, $P_1$ and $P_2$ within the strong field approximation. Calculating the first and the second order partial derivatives with respect to T, we find the extremum of Eqs. (\[eqppzz\]), (\[P1exact\]) and (\[P2exact\]), and neglecting the ultra-fast oscillating coherence terms, obtain the bounds of semiclassical parts of $P_0$, $P_1$ and $P_2$ : $$e^{-(T+\Delta)} \leq p_0^{\rm Cla}\leq e^{-T},
\label{boundP0}$$ $${1 \over 4} T e^{-(T+\Delta)} \leq p_1^{\rm Cla} \leq {e^{-T} \over 8} \left[ 4 + 3 T \left( 1 + e^{-\Delta} \right )
\right],
\label{boundP1}$$ $${ 3 \over 4} T e^{-T} \leq p_2^{\rm Cla} \leq e^{-T} \left( 1 - e^{-\Delta} + { 3 \over 4} T
\right).
\label{boundP2}$$ where once again $\Gamma=1$.\
The origin of Eqs. (\[boundP0\]),(\[boundP1\]) and (\[boundP2\]), although non-trivial for a finite T, can be easily understood in the limit of the short pulses $T\to0$. The upper bound of $p_0^{\rm Cla}$ is obvious, since if the interaction time is zero, no photons will be emitted for sure. Further, since we neglect the probability of emitting photons during the pulses, $p_0^{\rm
Cla}$ may only be decreased by the probability of not emitting a photon during the delay interval - $e^{-\Delta}$. Considering $p_1^{\rm Cla}$ we see, that Eq. (\[Eqpooo\]) reaches the maximum value - $\frac{1}{2}$, when $\Omega T=\pi/2$. Hence, the maximization of $p_1^{\rm Cla}$ corresponding to the interaction with a sequence of two pulses is achieved by applying two ideal $\pi/2$-pulses. Finally, from Eq. (\[Eqpttt\]) for $\lim_{T\to0,\Omega\to\infty}P_2$ follows, that maximum of this expression is found for $\Omega T=\pi$, and hence the optimization of $P_2$ is achieved by two ideal $\pi$-pulses.\
Learning from this simple example, although not rigorously, we extend it to the conclusion, that the maximum of $P_n$ ($n>1$) for any fixed interaction time is optimized by producing n equally separated $\pi$-pulses. This statement follows from the following argument : as far as we work under assumption, that the laser field does not effect the spontaneous emission process, the maximization of $P_n$ in any limited time interval may be achieved by minimization of induced emission, which is guaranteed by the strong and short ideal $\pi$-pulses better than any by any others.\
In Fig. 3 we show the maximum of $p_2^{\rm Cla}$ from a sequence of two equal square pulses as a function of the interaction’s strength $\Omega$ and pulse’s duration $T$ for three fixed values of the total interaction time $2T+\Delta$ . The curves were obtained using the extremum conditions of the exact expression for $P_2$ Eq. (\[P2\]). From Fig. 3 we see, that as the interaction time $\Gamma T$ becomes shorter, the delay period $\Delta$ longer and the Rabi frequency $\Omega$ larger, the probability of emitting 2 photons is getting close to 1. Although it becomes equal exactly 1 only for two ideal $\pi$-pulses separated by infinite delay, the graph shows, that starting from some range of parameters the increasing of $p_2^{\rm Cla}$ slows down, so that further increasing $\Omega$ does not contribute much. Finally, we note that for short delay intervals $\Delta<1$ the maximum of $p_2^{\rm Cla}$ is much less than 1 as expected.\
![The maximum of the probability of emission of two photons in a two-pulse laser field. The $t_3$ of the end time of second pulse is fixed at $0.5$ (solid curve), $2.0$ (dashed curve), $4.5$ (dot-dashed curve), respectively. The delay time between two pulses, $\Delta= t_3-2T$. The star gives the asymptotic behavior of Eq. (\[P2\]) in the limit of strong fields ($\Omega$ =50) and short pulse with $\Omega T = \pi$, $\Gamma=1$.[]{data-label="p2max"}](fig3.eps "fig:"){width="\columnwidth"}\
Summary
=======
We obtained general expressions for the probability of n photon emission events for a two level system interacting with two laser pulses separated by a delay interval $\Delta$. The photon statistic was represented as summation over quantum trajectories, which allowed simple intuitive physical interpretation of the final results Eqs. (\[eqPn1\]), (\[Pncla\]), (\[Ancoh\]). In particular, the contribution of the coherence effect, resulting from the quantum trajectories going through the superposition of the pure states at the end of the first pulse, was discussed. Although in optics it might be difficult to detect this effect experimentally, since the coherence paths oscillate in $\Delta$ with extremely large molecule absorption frequency $\omega_0$, nevertheless in microwave spectroscopy, dealing with lower range of absorption frequencies, the coherence effect is important [@Katz]. In addition, the correlation function $C(\Delta)$ Eq. (\[correl\]) was suggested as a measure of the photon statistics deviation from a treatment where the sequence of pulses is considered as if the pulses were independent. This correlation might be a useful tool to quantify the coherence and “memory" of single molecules, atoms or quantum dots through the measured photon statistics.\
The application of our general results was demonstrated on detailed calculation of $P_0$, $P_1$ and $P_2$ - the probabilities of emitting 0, 1 and 2 photons from the sequence of two square laser pulses Eqs. (\[P0exact\]), (\[P1\]) and (\[P2\]) in the limit of long measurement times $t\to\infty$. The physical interpretation of the quantum paths was shown to be useful in reducing the complexity of calculations in the limit of short and strong pulses (e.g. neglecting the paths where the photons are emitted within the pulse events). Finally, the non-trivial upper and lower bounds for the strong square pulses with finite duration were obtained. This kind of information is useful in experiments, where pulses are neither infinitely short not infinitely strong. Our approach can be applied to the theoretical study of other types of non-linear spectroscopy such as three level systems, systems undergoing stochastic dynamics [@SB], or Josephson junction qubits [@Katz] controlled by microwave radiation, where the strong dependence on the contribution of coherence was already experimentally proved.\
[**Acknowledgment:**]{} This work was supported by the Israel Science Foundation.\
Appendix A
==========
In this appendix we give the detailed derivation of Eq. (\[Pncorrel\]). Consider the semiclassical trajectories Eq. (\[Pncla\]) of $P_n$ from the two pulses . First let us sort the terms according to $$P_n^{\rm Cla}=\sum_{\alpha=0}^{n}\langle g|U^{(n-\alpha)}_{(t_3,t_2)}|g\rangle
\langle
g|U^{(\alpha)}_{(t_1,0)}|g\rangle+\sum_{\alpha=0}^{n-1}\langle
e|U^{(n-\alpha-1)}_{(t_3,t_2)}|g\rangle \langle
g|U^{(\alpha)}_{(t_1,0)}|g\rangle +\sum_{\alpha=0}^{n-1}\langle
g|U^{(n-\alpha-1)}_{(t_3,t_2)}|g\rangle \langle
e|U^{(\alpha)}_{(t_1,0)}|g\rangle$$ $$+\sum_{\alpha=0}^{n-2}\langle
e|U^{(n-\alpha-2)}_{(t_3,t_2)}|g\rangle \langle
e|U^{(\alpha)}_{(t_1,0)}|g\rangle +e^{-\Delta\Gamma}\left\{
\sum_{\alpha=0}^{n}\langle g|U^{(n-\alpha)}_{(t_3,t_2)}|e\rangle
\langle e|U^{(\alpha)}_{(t_1,0)}|g\rangle+
\sum_{\alpha=0}^{n-1}\langle e|U^{(n-\alpha-1)}_{(t_3,t_2)}|e\rangle
\langle e|U^{(\alpha)}_{(t_1,0)}|g\rangle\right.$$ $$\left. -\sum_{\alpha=0}^{n-1}\langle
g|U^{(n-\alpha-1)}_{(t_3,t_2)}|g\rangle \langle
e|U^{(\alpha)}_{(t_1,0)}|g\rangle-\sum_{\alpha=0}^{n-2}\langle
e|U^{(n-\alpha-2)}_{(t_3,t_2)}|g\rangle \langle
e|U^{(\alpha)}_{(t_1,0)}|g\rangle \right\}
\label{Pncla0.1}$$ Now we consider the sum of the first two terms $$\sum_{\alpha=0}^{n}\langle g|U^{(n-\alpha)}_{(t_3,t_2)}|g\rangle
\langle
g|U^{(\alpha)}_{(t_1,0)}|g\rangle+\sum_{\alpha=0}^{n-1}\langle
e|U^{(n-\alpha-1)}_{(t_3,t_2)}|g\rangle \langle
g|U^{(\alpha)}_{(t_1,0)}|g\rangle$$ $$=\left\{\langle
g|U^{(n)}_{(t_3,t_2)}|g\rangle +\langle
e|U^{(n-1)}_{(t_3,t_2)}|g\rangle\right\} \langle
g|U^{(0)}_{(t_1,0)}|g\rangle+\left\{\langle
g|U^{(n-1)}_{(t_3,t_2)}|g\rangle +\langle
e|U^{(n-2)}_{(t_3,t_2)}|g\rangle\right\} \langle g|U^{(1}_{(t_1,0)}|g\rangle+\cdots$$\
$$=P_n^{I_2}\langle g|U^{(0)}_{(t_1,0)}|g\rangle+ P_{n-1}^{I_2}\langle
g|U^{(1)}_{(t_1,0)}|g\rangle+\cdots=\sum_{\alpha=0}^n
P_{n-\alpha}^{I_2}\langle g|U^{(\alpha)}_{(t_1,0)}|g\rangle,
\label{Pncla0.2}$$ Similar manipulations with the second two terms lead to : $$\sum_{\alpha=0}^{n-1}\langle
g|U^{(n-\alpha-1)}_{(t_3,t_2)}|g\rangle \langle
e|U^{(\alpha)}_{(t_1,0)}|g\rangle+\sum_{\alpha=0}^{n-2}\langle
e|U^{(n-\alpha-2)}_{(t_3,t_2)}|g\rangle \langle
e|U^{(\alpha)}_{(t_1,0)}|g\rangle = \sum_{\alpha=0}^{n-1}
P_{n-1-\alpha}^{I_2}\langle e|U^{(\alpha)}_{(t_1,0)}|g\rangle
\label{Pncla0.3}$$ Combining Eqs. (\[Pncla0.2\]) and (\[Pncla0.3\]) we get : $$\sum_{\alpha=0}^n P_{n-\alpha}^{I_2}\langle
g|U^{(\alpha)}_{(t_1,0)}|g\rangle+\sum_{\alpha=0}^{n-1}
P_{n-1-\alpha}^{I_2}\langle e|U^{(\alpha)}_{(t_1,0)}|g\rangle=$$ $$P_{n}^{I_2}\langle
g|U^{(0)}_{(t_1,0)}|g\rangle+\left\{P_{n-1}^{I_2}\langle
e|U^{(0)}_{(t_1,0)}|g\rangle+P_{n-1}^{I_2}\langle
g|U^{(1)}_{(t_1,0)}|g\rangle\right\}+\left\{P_{n-2}^{I_2}\langle
e|U^{(1)}_{(t_1,0)}|g\rangle+P_{n-2}^{I_2}\langle
g|U^{(2)}_{(t_1,0)}|g\rangle\right\}+\cdots =\sum_{\alpha=0}^n
P_{n-\alpha}^{I_2}P_{\alpha}^{I_1} \label{Pncla0.4}$$ Now we concentrate on the terms multiplied by the factor $e^{-\Gamma\Delta}$ in Eq. (\[Pncla0.1\]). Let’s add and subtract the two following terms :$$\sum_{\alpha=0}^{n}\langle
g|U^{(n-\alpha)}_{(t_3,t_2)}|g\rangle \langle
g|U^{(\alpha)}_{(t_1,0)}|g\rangle+\sum_{\alpha=0}^{n-1}\langle
e|U^{(n-\alpha-1)}_{(t_3,t_2)}|g\rangle \langle
g|U^{(\alpha)}_{(t_1,0)}|g\rangle.$$ We get $$e^{-\Delta\Gamma}\left\{\sum_{\alpha=0}^{n}\langle
g|U^{(n-\alpha)}_{(t_3,t_2)}|e\rangle \langle
e|U^{(\alpha)}_{(t_1,0)}|g\rangle+\sum_{\alpha=0}^{n-1}\langle
e|U^{(n-\alpha-1)}_{(t_3,t_2)}|e\rangle \langle
e|U^{(\alpha)}_{(t_1,0)}|g\rangle+ \sum_{\alpha=0}^{n}\langle
g|U^{(n-\alpha)}_{(t_3,t_2)}|g\rangle \langle
g|U^{(\alpha)}_{(t_1,0)}|g\rangle\right.$$ $$+\sum_{\alpha=0}^{n-1}\langle
e|U^{(n-\alpha-1)}_{(t_3,t_2)}|g\rangle \langle
g|U^{(\alpha)}_{(t_1,0)}|g\rangle -\sum_{\alpha=0}^{n}\langle
g|U^{(n-\alpha)}_{(t_3,t_2)}|g\rangle \langle
g|U^{(\alpha)}_{(t_1,0)}|g\rangle
-\sum_{\alpha=0}^{n-1}\langle
e|U^{(n-\alpha-1)}_{(t_3,t_2)}|g\rangle \langle
g|U^{(\alpha)}_{(t_1,0)}|g\rangle$$ $$\left. -\sum_{\alpha=0}^{n-1}\langle
g|U^{(n-\alpha-1)}_{(t_3,t_2)}|g\rangle \langle
e|U^{(\alpha)}_{(t_1,0)}|g\rangle-\sum_{\alpha=0}^{n-2}\langle
e|U^{(n-\alpha-2)}_{(t_3,t_2)}|g\rangle \langle
e|U^{(\alpha)}_{(t_1,0)}|g\rangle\right\}
\label{Pncla0.5}$$ But the first 4 paths of Eq. (\[Pncla0.5\]) are just the semiclassical part of probability of emitting n photons from the two pulses attached together $$P_n^{Cla,I_2I_1}=\sum_{\alpha=0}^n\langle
g|U^{(n-\alpha)}_{(t_3,t_2)} \left(|g \rangle \langle g|+ |e \rangle
\langle e| \right) U^{(\alpha)}_{(t_1,0)}|g\rangle +
\sum_{\alpha=0}^{n-1}\langle e|U^{(n-1-\alpha)}_{(t_3,t_2)}\left(|g
\rangle \langle g|+ |e \rangle \langle e| \right)
U^{(\alpha)}_{(t_1,0)}|g\rangle \label{pi2i1cla}$$ (compare with Eq. (\[eqPnt1\])). And the last 4 paths of Eq. (\[Pncla0.5\]) are equal to Eq. (\[Pncla0.2\])+Eq. (\[Pncla0.3\]). Putting all this information together we obtain $$P_n=\sum_{\alpha=0}^{n}P_{n-\alpha}^{I_2}P_{\alpha}^{I_1}+
e^{-\Delta\Gamma}\left\{P_{n}^{Cla,
I_2I_1}-\sum_{k=0}^{n}P_{n-\alpha}^{I_2}P_{\alpha}^{I_1}+
(e^{\Delta(\Gamma/2+i\omega_0)}A_n^{\rm Coh}+C.C.)\right\}
\label{Pncla0.6}$$ Finally, by addition and substraction of the coherence trajectories: $$e^{-\Delta\Gamma}\left\{A_n^{Coh,I_2I_1}+C.C.\right\}=e^{-\Delta\Gamma}\left\{\sum_{\alpha=0}^n\langle
g|U^{(n-\alpha)}_{(t_3,t_2)} \left(|c \rangle \langle c|+ |c^*
\rangle \langle c^*| \right) U^{(\alpha)}_{(t_1,0)}|g\rangle +
\sum_{\alpha=0}^{n-1}\langle e|U^{(n-1-\alpha)}_{(t_3,t_2)}\left(|c
\rangle \langle c|+ |c^*\rangle \langle c^*| \right)
U^{(\alpha)}_{(t_1,0)}|g\rangle\right\}$$ $$=e^{-\Delta\Gamma}\left\{A_n^{\rm Coh}+C.C.\right\}$$ to $P_{n}^{Cla,I_2I_1}$ we obtain Eq. (\[Pncorrel\]). Using Eq. (\[Pncorrel\]) it should be taken into account however that not all the coherence paths now oscillate in $\Delta$ with $\omega_0$.
Appendix B
==========
The Rotating Wave Approximation (RWA) [@CT] consists of neglecting the non-resonant processes of rising from $|g\rangle$ to $|e\rangle$ by emitting a photon and falling from $|e\rangle$ to $|g\rangle$ by absorbing a photon. Switching to the rotating frame by applying the transformation $A_{(t-t',\phi,\omega_L)}$ defined below, it is possible to suppress any time dependence in the Bloch equation Eq. (\[eqBloch\]). As a result the following time independent equation is obtained $$\dot{\tilde{\sigma}}_{(t)} = \left[\tilde L+
\hat{\Gamma}\right]\tilde{\sigma}_{(t)}\label{eqBloch1}$$ where $$\tilde{\sigma}_{(t)}=A_{(t-t',\phi,\omega_L)}\sigma_{(t)},
\label{tildsig}$$ $t'$ - is the initial moment, $$\tilde L = \left(
\begin{array}{c c c c}
-\Gamma & 0 & {-i \Omega \over 2} & {i \Omega \over 2} \\
0 & 0 & {i \Omega \over 2} & {-i \Omega \over 2} \\
{-i \Omega \over 2} & {i \Omega \over 2} & - {\Gamma\over 2} - i \delta_L & 0 \\
{i \Omega \over 2} & {-i \Omega \over 2} & 0 & - {\Gamma\over 2} + i \delta_L
\end{array}
\right) \label{eqLRWA}$$ and the transformation $A_{(t-t',\phi,\omega_L)}$ is given by $$A_{(t-t',\phi,\omega_L)} = \left(
\begin{array}{c c c c}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & e^{-i (\omega_L(t-t') + \phi)} & 0 \\
0 & 0 & 0 & e^{i (\omega_L(t-t') + \phi)}
\end{array}
\right), \label{eqA}$$ where $\phi$ is the phase of the laser at the initial moment $t'$. In the new representation the calculation of the Green function is straightforward (see Eq. (\[eqtildeGreenf\])). Representing the solution to the time independent Bloch equation Eq. (\[eqBloch1\]) in the rotating frame as the infinite iterative expansion in $\hat
\Gamma $ we find the following expression for n-photon-propagator within RWA $$\tilde{U}^{(n)}_{(t-t')}=\int_{t'} ^t \cdots \int_{t'} ^{t_2}
\tilde{{\cal G}}(t-t_n) \hat{\Gamma} \cdots \hat{\Gamma}
\tilde{{\cal G}}(t_{1}-t') {\rm d} t_1 \cdots {\rm d} t_n.
\label{eqtildUn}$$ Hence $$\tilde{\sigma}^{(n)}_{(t)}=\tilde{U}^{(n)}_{(t-t')}\tilde{\sigma}_{(t')}.
\label{sigmaRWA}$$ where $\tilde{\sigma}_{(t')}=A_{(0,\phi,\omega_L)}\sigma_{(t')}$ is the initial condition. Finally for obtaining $\sigma^{(n)}_{(t)}$ we have to apply the inverse transformation $$\sigma^{(n)}_{(t)}= A^{-1}_{(t-t',\phi,\omega_L)}\tilde{U}^{
(n)}_{(t-t')}A_{(0,\phi,\omega_L)}\sigma_{(t')} \label{sigRWA}$$ From Eq. (\[sigRWA\]) one easily makes the following conclusions: (i) the initial state of the molecule is multiplied by $A_{(0,\phi_1,\omega_L)}$, where $\phi_1$ is the initial phase of the laser at the beginning of the first pulse, which shifts the initial coherence phase by $-\phi_1$ (ii) the delay period propagators are now multiplied by $A_{(0,\phi_2,\omega_L)}$ from the left due to the second pulse and by $A^{-1}_{(T,\phi_1,\omega_L)}$ from the right due to the first pulse ($\phi_2$ is the initial laser phase at the beginning of the second pulse). Clearly, this leads only to an additional phase shift $(T\omega_L +\phi_1-\phi_2)$ of the coherence terms. Therefore calculating the photon statistics for the square pulses we rewrite the Eqs. (\[eqPn1\]),(\[Pncla\]),(\[Ancoh\]) with the following modifications:\
1) In the definition of the n-photon-propagator Eq. (\[eqUn\]) the Green function defined as the time ordered exponential are replaced by $$\tilde{{\cal G}} (t-t')= \exp\left[(t-t') \tilde L
\right]\label{eqtildeGreenf}$$ which are Green functions for the time intervals inside the pulses within RWA.\
2) The initial state of the system must be replaced by $A_{(0,\phi,\omega_L)}\sigma_{(0)} $\
3) The coherent terms $A_n^{\rm Coh}$ are multiplied by additional phase factor $e^{i(\omega_L T+\phi_1-\phi_2)}$.\
*Remark : If in experiments the initial phases of the pulses are random variables, it’s necessary to replace all the phase factors with their ensemble averages.*\
Thus summarizing we have $$P_n=p_{n}^{\rm Cla}+
e^{-\Delta\Gamma/2}\left[e^{i
(\Delta+T)\omega_{0}+\phi_{1}-\phi_{2}}
a_n^{\rm Coh}+C.C.\right],\label{eqPnRWA}$$\
where
$$p_n^{\rm Cla}=\sum_{\alpha=0}^{n}\left\{\langle g|\tilde U^{(n-\alpha)}_{(t_3,t_2)}|g\rangle
\langle g|\tilde U^{(\alpha)}_{(t_1,0)}|\tilde g\rangle +
e^{-\Delta\Gamma}\langle g|\tilde
U^{(n-\alpha)}_{(t_3,t_2)}|e\rangle \langle e|\tilde
U^{(\alpha)}_{(t_1,0)}|\tilde
g\rangle\right\}+\sum_{\alpha=0}^{n-1}(1-e^{-\Delta\Gamma})\langle
g|\tilde U^{(n-\alpha-1)}_{(t_3,t_2)}|g\rangle \langle e|\tilde
U^{(\alpha)}_{(t_1,0)}|\tilde g\rangle$$ $$+ \sum_{\alpha=0}^{n-1}\left\{\langle
e|\tilde U^{(n-\alpha-1)}_{(t_3,t_2)}|g\rangle \langle g|\tilde
U^{(\alpha)}_{(t_1,0)}|\tilde g\rangle +e^{-\Delta\Gamma}\langle
e|\tilde U^{(n-\alpha-1)}_{(t_3,t_2)}|e\rangle \langle e|\tilde
U^{(\alpha)}_{(t_1,0)}|\tilde
g\rangle\right\}+\sum_{\alpha=0}^{n-2}(1-e^{-\Delta\Gamma})\langle
e|\tilde U^{(n-\alpha-2)}_{(t_3,t_2)}|g\rangle \langle e|\tilde
U^{(\alpha)}_{(t_1,0)}|\tilde g\rangle\label{PnclaRWA}$$ and $$a_n^{\rm Coh}=\sum_{\alpha=0}^{n}\langle
g|\tilde U^{(n-\alpha)}_{(t_3,t_2)}|c\rangle \langle c|\tilde
U^{(\alpha)}_{(t_1,0)}|\tilde g\rangle +
\sum_{\alpha=0}^{n-1}\langle e|\tilde
U^{(n-\alpha-1)}_{(t_3,t_2)}|c\rangle \langle c|\tilde
U^{(\alpha)}_{(t_1,0)}|\tilde
g\rangle,\label{AncohRWA}$$ with $|\tilde g\rangle = A_{(0,\phi_1,\omega_L)}| g\rangle$.
Appendix C
==========
Here we show the derivation of exact expressions for $P_1$ and $P_2$ for two equal square pulses obtained within RWA .\
**For $P_1$ :** The one-photon-propagator in (0,t) may be decomposed to the sum of 4 different terms $$U^{(1)}_{(t,0)}=\sum_{k=0}^1 \left[
U^{(\alpha)}_{(t,t_3)}U^{(1-\alpha)}_{(t_3,t_2)}U^{(0)}_{(t_2,t_1)}U^{(0)}_{(t_1,0)}
+
U^{(0)}_{(t,t_3)}U^{(0)}_{(t_3,t_2)}U^{(\alpha)}_{(t_2,t_1)}U^{(1-\alpha)}_{(t_1,0)}
\right] \label{eqU1exact}$$ Inserting the closure relation Eq. (\[closure\]) between every two propagators of Eq. (\[eqU1exact\]) and applying RWA we obtain the probability of emitting a single photon using the trajectories notation $$p^{\rm Cla}_1 = \sum_{\alpha=0} ^1 \left\{ \langle g |\tilde U^{(\alpha)}_{t_3,t_2} | g \rangle \langle g |\tilde U^{(1-\alpha)}_{t_1,0} | g \rangle
+ \langle g |\tilde U^{(\alpha)}_{t_3,t_2}| e \rangle \langle e
|\tilde U^{(1-\alpha)}_{t_1,0} | g \rangle e^{- \Gamma \Delta }
\right\}$$ $$+\langle g |\tilde U^{(0)}_{t_3,t_2} | g \rangle\langle e |\tilde U^{(0)}_{t_1,0} | g \rangle \left( 1 - e^{ - \Gamma \Delta } \right)
+\langle e |\tilde U^{(0)}_{t_3,t_2} | e \rangle \langle e |\tilde
U^{(1-\alpha)}_{t_1,0}| g \rangle e^{ - \Gamma \Delta} + \langle e
|\tilde U^{(0)}_{t_3,t_2}| g\rangle \langle g |\tilde
U^{(0)}_{t_1,0} | g \rangle. \label{eqponeCla}$$ $$a^{\rm Coh}_1= \sum_{\alpha=0} ^1 \langle g|\tilde
U^{(\alpha)}_{t_3,t_2} | c \rangle \langle c |\tilde
U^{(1-\alpha)}_{t_1,0} | g \rangle + \langle e |\tilde
U^{(0)}_{t_3,t_2} |c\rangle \langle c | \tilde U^{(0)}_{t_1,0} | g
\rangle. \label{eqponeCoh}$$ After some tedious algebra using Eqs. (\[eqLoOBE\]), (\[eqLRWA\]), (\[eqtildUn\]), (\[eqtildeGreenf\]) we finally obtain : $$P_1 = a_1+b_1 e^{-\Delta}+ c_1
e^{-{\Delta \over 2}} \cos \left [\omega_0 \left (T + \Delta \right ) \right ].
\label{P1}$$ Where $$a_1 = \frac{e^{-T}}{16 y^7} \left(1-y^2\right) \left[a_{11}+a_{12} \cosh \left(\frac{T
y}{2}\right)+a_{13}\sinh \left(\frac{T y}{2}\right) +a_{14}\cosh (T y) + a_{15}\sinh (T y)\right].$$ $$y = \sqrt{1-4 \Omega^2},$$ $$\begin{aligned}
&&a_{11}= y\left(y^2-1\right) \left[4 \left(y^2-3\right)+3 T\left(y^2-1\right)\right], a_{12}= -2 y \left(T y^4+8y^2-T-16\right),a_{13}= 4 \left[-(T+3) y^4+(T+4) y^2+3 \right], \\
&&a_{14}= (T+4) y^5+6 T y^3+(T-20) y, a_{15}= 2 \left[(2 T+5)y^4+2 (T-5) y^2-3\right]. \\\end{aligned}$$
$$b_1 = \frac{e^{-T}}{16 y^7}\left(1-y^2\right)^3 \left[-3 T y+2 T y
\cosh \left(\frac{T
y}{2}\right) +12 \sinh \left(\frac{Ty}{2}\right) +T y \cosh (T y) -6 \sinh (T y)\right].$$
And $$c_1 = \frac{e^{-T}}{16 y^7} \left(1-y^2\right) \left[c_{11}+
c_{12}\cosh \left(\frac{T y}{2}\right)+ c_{13}\sinh \left(\frac{T
y}{2}\right) + c_{14}\cosh (T y)+ c_{15}\sinh(T y)\right].$$ $$\begin{aligned}
&&c_{11}= -2 y \left[2\left(y^4-3\right)+T \left(y^4+2y^2-3\right)\right],c_{12}= 4 y \left[(T+4) y^2-T-8\right],c_{13}=4 \left[Ty^4+(2-T) y^2-6\right], \\
&&c_{14}=2 y \left((T+2)y^4-8 y^2-T+10\right),c_{15}= 4 \left[T y^4 -(T+1) y^2 + 3\right]. \\\end{aligned}$$
**For $P_2$ :**\
The two-photon-propagator may be decomposed to the sum of 10 terms. $$U^{(2)}_{(t,0)}=\sum_{\alpha=0}^2 \left[
U^{(0)}_{(t,t_3)}U^{(\alpha)}_{(t_3,t_2)}U^{(0)}_{(t_2,t_1)}U^{(2-\alpha)}_{(t_1,0)}+
U^{(\alpha)}_{(t,t_3)}U^{(0)}_{(t_3,t_2)}U^{(2-\alpha)}_{(t_2,t_1)}U^{(0)}{(t_1,0)}\right]
+ \sum_{\beta,\alpha=0}^1
U^{(\beta)}_{(t,t_3)}U^{(\alpha)}_{(t_3,t_2)}U^{(1-\beta)}_{(t_2,t_1)}U^{(1-\alpha)}_{(t_1,0)}
\label{eq2exact}$$ ( Since $U^{(2)}_{t_2,t_1}=0$ there are only 8 non-zero terms.) This leads to the following expressions for $P^{\rm Cla}_2$ and $A^{\rm Coh}_2$ : $$p_2^{\rm Cla}= \sum_{k=0} ^2 \left\{ \langle g |\tilde U^{(k)}_{(t_3,t_2)} | g \rangle \langle g |\tilde U^{(2 - k)} _{(t_1,0)} | g \rangle + \langle g |\tilde U^{(k)}_{(t_3,t_2)} | e \rangle \langle e |\tilde U^{(2 - k)} _{(t_1,0)} | g \rangle e^{- \Gamma \Delta }
\right\}+ \langle e | \tilde U^{(0)}_{(t_3,t_2)} | g \rangle\langle e |\tilde U^{(0)} _{(t_1,0)} | g \rangle \left( 1 - e^{ - \Gamma \Delta } \right)$$\
$$+ \sum_{k=0} ^1 \left\{\langle g |\tilde U^{(k)}_{(t_3,t_2)}| g \rangle \langle e |\tilde U^{(1 - k)} _{(t_1,0)} | g \rangle \left( 1 - e^{ - \Gamma \Delta } \right)+ \langle e | \tilde U^{(k)}_{(t_3,t_2)}| g \rangle \langle g |\tilde U^{(1 - k)} _{(t_1,0)} | g \rangle + \langle e |\tilde U^{(k)}_{(t_3,t_2)}| e \rangle \langle e | \tilde U^{(1 - k) }_{(t_1,0)} | g \rangle e^{- \Gamma \Delta } \right\}$$ $$a_2^{\rm Coh}=\sum_{k=0} ^2 \langle g|\tilde U^{(k)}_{(t_3,t_2)}| c
\rangle \langle c |\tilde U^{(2 - k)} _{(t_1,0)} | g \rangle +
\sum_{k=0} ^1 \langle e|\tilde U^{(k)}_{(t_3,t_2)} | c \rangle \langle c |\tilde U^{(1 - k)} _{(t_1,0)} | g \rangle + \mbox{C.C.}
\label{eqpone}$$ Calculating the matrix elements with the help of Mathematica yields: $$P_2 = a_2+ b_2 e^{-\Delta}+ c_2e^{-{\Delta \over 2}} \cos
\left[\omega_0 \left (T + \Delta \right ) \right]. \label{P2}$$ where $$a_2 = \frac{e^{-T}}{64 y^{10}}
\left(y^2-1\right)^2\left[a_{21}+a_{22} \cosh
\left(\frac{Ty}{2}\right)+ a_{23} \sinh\left(\frac{T
y}{2}\right)+a_{24} \cosh (T y)+ a_{25} \sinh (Ty)\right].$$ $$\begin{aligned}
&&a_{21} = \left(9 T^2+40 T+24\right) y^6-2 \left(9 T^2+56T+17\right) y^4+9 \left(T^2+8 T-12\right) y^2+126, \\
&&a_{22} = \left(T^2-32\right) y^6+4 (T+32) y^4-\left(T^2+20T+64\right) y^2-192, \\
&&a_{23} = 2 y \left[\left(y^2-1\right) T^2 y^2+T \left(3 y^4-8y^2-3\right)-4 \left(8 y^4-39 y^2+51\right)\right], \\
&&a_{24} = \left(T^2+8 T+8\right) y^6+2 \left(3 T^2-6 T-47\right)y^4+\left(T^2-52 T+172\right) y^2+66,\\
&&a_{25} = y \left[4 \left(y^2+1\right) T^2 y^2+\left(17 y^4-58y^2-15\right) T-4 \left(4 y^4+9 y^2-51\right)\right]. \\\end{aligned}$$
$$b_2 = \frac{e^{-T}}{64 y^{10}} \left(y^2-1\right)^2 \left[b_{21}+
b_{22} \cosh \left(\frac{Ty}{2}\right)+ b_{23} \sinh\left(\frac{T
y}{2}\right)+ b_{24} \cosh (T y)+b_{25} \sinh (Ty)\right].$$
$$\begin{aligned}
&&b_{21} = 3\left(3 T^2-8\right) y^6-18 \left(T^2-7\right) y^4+9\left(T^2-28\right) y^2+126,b_{22} = -\left(T^2-32\right) y^6+2 \left(T^2-96\right)y^4-\left(T^2-384\right) y^2-192, \\
&&b_{23} = -6 T y \left(y^2-1\right)^2,b_{24} = \left(T^2-8\right) y^6-2 \left(T^2-33\right)y^4+\left(T^2-132\right) y^2+66,b_{25} = -15 T y \left(y^2-1\right)^2. \\\end{aligned}$$
And $$c_2 = \frac{e^{-T}}{32 y^{10}} \left(y^2-1\right)^2\left[c_{21}+
c_{22} \cosh \left(\frac{Ty}{2}\right)+ c_{23} \sinh\left(\frac{T
y}{2}\right)+ c_{24} \cosh (T y)+ c_{25} \sinh (Ty)\right],$$ $$\begin{aligned}
&&c_{21} = -T (T+4) y^6+2 \left(5 T^2+12 T-15\right) y^4-3 \left(3T^2+12 T-44\right) y^2-126, \\
&&c_{22} = -\left(T^2+2 T-32\right) y^4+\left(T^2+10 T-160\right)y^2+192, \\
&&c_{23} = -\left(T^2-32\right) y^5+\left(T^2+2 T-156\right) y^3+6(T+34) y,\\
&&c_{24} = T (T+4) y^6-2 (11 T+1) y^4+\left(-T^2+26 T+28\right)y^2-66, \\
&&c_{25} = y \left[2 \left(y^2-1\right)T^2 y^2+T \left(-3 y^4-4y^2+15\right)-2 \left(8 y^4-39 y^2+51\right)\right]. \\\end{aligned}$$
[99]{}
S. Mukamel [*Principles of Nonlinear Optical Spectroscopy*]{} Oxford Univ. Press. Oxford (1995).
Erik M.H.P van Dijk et al [*Phys. Rev. Lett.*]{} [**94**]{} 078302 (2005).
F. Shikerman, E. Barkai [ *http://arxiv.org/abs/0705.4028*]{} (2007).
C. Santori, D. Fattal, J. Vu$\hat{c}$kovi$\acute{c}$, G. S. Solomon and Y. Yamamoto [*Nature*]{} [**419**]{}, 594-597 (2002).
C.K. Hong, Z. Y. Ou, L. Mandel [*Phys. Rev. Lett.*]{} [**59**]{} 2044-2046 (1987).
E. Knill, R. Laflamme and G. J. Milburn [*Nature*]{} [**409**]{} 46-52 (2001).
Y.H. Shih, C. O. Alley [*Phys. Rev. Lett.*]{} [**61**]{} 2921-2024 (1988).
N. Katz, M. Ansmann, R. C. Bialczak, E. Lucero, R. McDermott, M. Neeley, M. Steffen, E. M. Weig, A. N. Korotkov [*Sience*]{} [**312**]{}, 1498-1500 (2006).
D. Bouwmeester, A. Ekert, A. Zelinger [*the Physics of Quantum Information*]{}49-92 Springer, Berlin (2000).
P. Zoller, M. Marte, and D. F. Walls [*Phys. Rev. A*]{} [**35**]{} 198 (1987).
B. R. Mollow, [*Phys. Rev. A*]{} [**12**]{}, 1919 (1975).
C. Brunel, B. Lounis, P. Tamarat and M. Orrit [*Phys. Rev. Lett.*]{} [**83**]{} 2722 (1999).
E. Barkai, Y.J. Jung, and R. Silbey, [*Phys. Rev. Lett.*]{} [**87**]{}, 207403 (2001).
E. Barkai, Y. Jung and R. Silbey [*Annual Review of Physical Chemistry*]{} [**55**]{}, 457 (2004).
S. Mukamel [*Phys. Rev. A*]{} [**68**]{} 063821 (2003).
J.S. Cao [*J. of Phys. Chem. B*]{} [**110**]{} 19040 (2006)
H. Yang, X.C. Xie [*J. of Chem. Phys.*]{} [**117**]{} 10965 ( 2002).
I. Gopich, A. Szabo [*J. of Chemical Physics*]{} [**122**]{} 014707 (2005).
Y. Zheng, F. L. H. Brown [*Phys. Rev. Lett.*]{} [**90**]{} 238305 (2003).
M. B. Plenio and P. L. Knight, [*Rev. of Mod. Phys.*]{} [**70**]{} 101-144 (1998).
C. C. Tannoudji, J. Dupont-Roc, G. Grynberg [*Atom-Photon Interactions*]{}, John Wiley (1992).
I. Rozhkov, E. Barkai [*J. of Chem. Phys.*]{} [**123**]{} 074703 (2005).
Reperesenting $\sigma^{(n)}$ as a $2\times 2$ matrix, Eq. (\[eqPn\]) is just the trace of the matrix. While some authors prefer the presentation of $\sigma^{(n)}$ as a matrix, we follow Zheng and Brown [@Brown] presentation of $\sigma^{(n)}$ as a vector.
Y. He, E. Barkai [*Physical Chemistry Chemical Physics*]{} [**8**]{}, 5056 (2006).
Y. He, E. Barkai [*Phys. Rev. A*]{} [**74**]{}, 011803(R) (2006).
| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
Halina Abramowicz$^a$, Angel Abusleme$^b$, Konstantin Afanaciev$^c$, Jonathan Aguilar$^d$, Prasoon Ambalathankandy$^d$, Philip Bambade$^e$, Matthias Bergholz$^{f,1}$, Ivanka Bozovic-Jelisavcic$^g$, Elena Castro$^f$, Georgy Chelkov$^h$, Cornelia Coca$^i$, Witold Daniluk$^j$, Angelo Dragone$^k$, Laurentiu Dumitru$^i$, Konrad Elsener$^l$, Igor Emeliantchik$^c$, Tomasz Fiutowski$^d$, Mikhail Gostkin$^h$, Christian Grah$^{f,2}$, Grzegorz Grzelak$^{j,3}$, Gunther Haller$^k$, Hans Henschel$^f$, Alexandr Ignatenko$^{c,4}$, Marek Idzik$^d$, Kazutoshi Ito$^m$, Tatjana Jovin$^g$, Eryk Kielar$^j$, Jerzy Kotula$^j$, Zinovi Krumstein$^h$, Szymon Kulis$^d$, Wolfgang Lange$^f$, Wolfgang Lohmann$^{f,1}$[^1], Aharon Levy$^a$, Arkadiusz Moszczynski$^j$, Uriel Nauenberg$^n$, Olga Novgorodova$^{f,1}$, Marin Ohlerich$^{f,1}$, Marius Orlandea$^i$, Gleb Oleinik$^n$, Krzysztof Oliwa$^j$, Alexander Olshevski$^h$, Mila Pandurovic$^g$, Bogdan Pawlik$^j$, Dominik Przyborowski$^d$, Yutaro Sato$^m$, Iftach Sadeh$^a$, Andre Sailer$^l$, Ringo Schmidt$^{f,1}$, Bruce Schumm$^o$, Sergey Schuwalow$^f$, Ivan Smiljanic$^g$, Krzysztof Swientek$^d$, Yosuke Takubo$^m$, Eliza Teodorescu$^i$, Wojciech Wierba$^j$, Hitoshi Yamamoto$^m$, Leszek Zawiejski$^j$ and Jinlong Zhang$^p$\
Tel Aviv University, Tel Aviv, Israel\
Stanford University, Stanford, USA\
NCPHEP, Minsk, Belarus\
AGH University of Science & Technology, Cracow, Poland\
Laboratoire de l Accelerateur Lineaire, Orsay, France\
DESY, Zeuthen, Germany\
Vinca Institute of Nuclear Sciences, University of Belgrade, Serbia\
JINR, Dubna, Russia\
IFIN-HH, Bucharest, Romania\
INP PAN, Cracow, Poland\
SLAC, Menlo Park, USA\
CERN, Geneva, Switzerland\
Tohoku University, Sendai, Japan\
University of Colorado, Boulder, USA\
UC California, Santa Cruz, USA\
ANL, Argonne, USA\
also at Brandenburg University of Technology, Cottbus, Germany\
now at BTO Consulting AG, Berlin, Germany\
also at University of Warsaw, Poland\
now at DESY, Hamburg, Germany\
E-mail:\
title: Forward Instrumentation for ILC Detectors
---
Introduction and challenges
===========================
A high energy e$^+$e$^-$ linear collider is considered to be the future research facility complementary to the LHC collider. Whereas LHC has a higher potential for discoveries, an e$^+$e$^-$ collider will allow precision measurements to explore in detail the mechanism of electroweak symmetry breaking and the properties of the physics beyond the Standard Model, should it be found at the LHC. Two concepts of an e$^+$e$^-$ linear collider are presently considered, the ILC [@ILC_pub] and CLIC [@clic_info]. For the ILC, with superconducting cavities, an engineering design report will be issued in 2012. The centre-of-mass energy will be 500 GeV, with the possibility of an upgrade to 1 TeV. CLIC is based on conventional cavities. A conceptional design report is foreseen in 2011. CLIC will allow to collide electrons and positrons up to energies of 3 TeV.
An R&D program is ongoing to develop the technologies for detectors for precision measurements in this new energy domain. Letters of Intent have been submitted for detectors at the ILC in 2009. Two detectors, the ILD [@ILD_pub] and the SiD [@SiD_pub], are reviewed and validated. In both detectors two specialised calorimeters are foreseen in the very forward region, LumiCal for the precise measurement of the luminosity and BeamCal for a fast estimate of the luminosity and for the control of beam parameters [@ieee1]. Both will also improve the hermeticity of the detector. To support beam-tuning an additional pair-monitor will be positioned just in front of BeamCal.
With LumiCal the luminosity will be measured using Bhabha scattering, ${\rm{e}}^+{\rm{e}}^- \rightarrow {\rm{e}}^+{\rm{e}}^-(\gamma)$, as a gauge process. To match the physics benchmarks, an accuracy of better than 10$^{-3}$ is needed at a centre-of-mass energy of 500 GeV [@ILD_pub]. For the GigaZ option, where the ILC will be operated for precision measurements at centre-of-mass energies around the Z boson, an accuracy of 10$^{-4}$ would be required [@klaus]. To reach these accuracies, a precision device is needed, with particularly challenging requirements on the mechanics and position control.
BeamCal is positioned just outside the beam-pipe. At ILC energies we have to tackle here a new phenomenon – the beamstrahlung. When electron and positron bunches collide, the particles are accelerated in the magnetic field of the bunches towards the bunch centre. This so called pinch effect enhances the luminosity. However, electrons and positrons may radiate photons. A fraction of these photons converts in the Coulomb field of the bunch particles creating low energy ${\rm{e}}^+{\rm{e}}^-$ pairs. A large amount of these pairs will deposit their energy after each bunch crossing in BeamCal. These depositions, useful for a bunch-by-bunch luminosity estimate and the determination of beam parameters [@grah1], will lead, however, to a radiation dose of about one MGy per year in the sensors at lower polar angles. Hence radiation hard sensors are needed to instrument BeamCal. BeamCal is supplemented by a pair monitor, consisting of a layer of pixel sensors positioned just in front of it to measure the density of beamstrahlung pairs and give additional information for the beam parameter determination.
All detectors in the very forward region have to tackle relatively high occupancy, requiring special front-end electronics.
A small Molière radius is of importance for both calorimeters. It ensures high energy electron veto capability for BeamCal even at small polar angles. This is essential to suppress background in searches for new particles for which the signature consists of large missing energy and momentum. In LumiCal the precise reconstruction of electron, positron and photon showers in Bhabha events is facilitated. Both calorimeters also shield the inner tracking detectors from back-scattered particles induced by beamstrahlung pairs hitting the downstream beam-pipe and magnets.
![The very forward region of the ILD detector. LumiCal, BeamCal and LHCal are carried by the support tube for the final focusing quadrupole and the beam-pipe. LHCal extends the coverage of the hadron calorimeter to the polar angle range of LumiCal. TPC denotes the central track chamber, ECAL the electromagnetic and HCAL the hadron calorimeter.\[fig:Forward\_structure\]](figures/forward_design.eps){width="0.6\columnwidth"}
Summary
=======
A design for the instrumentation of the very forward region of a detector at the International Linear collider is presented. Two calorimeter are planned, LumiCal to measure precisely the luminosity and BeamCal, supplemented by a pair monitor, for a fast luminosity estimate and beam tuning. Both calorimeters extend the coverage of the detector to small polar angles. Parameters relevant for the physics program have been estimated by Monte Carlo simulations and found to match the requirements for the chosen geometry. Prototypes of the major components such as sensors, front-end ASICs and ADC ASICs are developed, produced and tested. Their measured performance fulfils the specifications derived from the Monte Carlo simulations. The results presented here demonstrate that the sensors and the ASICs are ready to be integrated into a fully functional prototype detector and to perform, as the next step, tests of fully assembled sensor plane prototypes.
Acknowledgments
===============
This work is supported by the Commission of the European Communities under the 6$^{th}$ Framework Program “Structuring the European Research Area”, contract number RII3-026126. Tsukuba University is supported in part by the Creative Scientific Research Grant No. 18GS0202 of the Japan Society for Promotion of Science. The AGH-UST is supported by the Polish Ministry of Science and Higher Education under contract Nr. 372/6.PRUE/2007/7. The INP PAN is supported by the Polish Ministry of Science and Higher Education under contract Nr. 141/6.PR UE/2007/7. IFIN-HH is supported by the Romanian Ministry of Education, Research and Innovation through the Authority CNCSIS under contract IDEI-253/2007. The VINCA group is benefiting from the project “Physics and Detector R$\&$D in HEP Experiments” supported by the Ministry of Science of the Republic of Serbia. J. Aguilar, P. Ambalathankandy and O. Novgorodova are supported by the 7th Framework Programme “Marie Curie ITN”, grant agreement number 214560.
[9]{} *International Linearcollider Reference Report* (2007) http://www.linearcollider.org/about/Publications/Reference-Design-Report.
*The Compact Linear Collider Study*, http://clic-study.web.cern.ch/clic-study/.
T. Abe , *The International Large Detector: Letter of Intent*, *FERMILAB-LOI-2010-01, FERMILAB-PUB-09-682-E, DESY-2009-87, KEK-REPORT-2009-6, arXiv:1006.3396* (2010).
E. L. Berger , *SiD Letter of Intent* (2009) https://confluence.slac.stanford.edu/display/SiD/home.
H. Abramowicz , *Instrumentation of the very forward region of a linear collider detector*, [*IEEE Trans.Nucl.Sci.* [**51**]{} (2004) 2983.]{}
K. Mönig, *Physics needs for the forward region*, *V. Workshop: Instrumentation of the Forward Region of a Linear Collider Detector*, August, 26–28 (2004) DESY, Zeuthen, Germany, http://www-zeuthen.desy.de/lcdet/Aug\_04\_WS/aug\_04\_ws.html.
Ch. Grah and A. Sapronov, *Beam parameter determination using beamstrahlung photons and incoherent pairs*, .
R. Bonciani and A. Ferroglia, *Bhabha Scattering at NNLO*, [*Nucl.Phys.Proc.Suppl.181-182* (2008) 259]{}, T. Becher and K. Melnikov, [*JHEP 0706:084* (2007)]{}, S. Actis, M. Czakon, J. Gluza and T. Riemann, *Fermionic NNLO contributions to Bhabha scattering*, [*Acta Phys.Polon.* B[**38**]{} (2007) 3517]{}, A.A. Penin, *Two-loop photonic corrections to massive Bhabha scattering*, [*Nucl. Phys.* B [**734**]{} (2006) 185]{}, M. Czakon, J. Gluza and T. Riemann, *The Planar four-point master integrals for massive two-loop Bhabha scattering*, [*Nucl. Phys. B751* (2006) 1.]{}
S. Jadach, W. Placzek and B.F.L. Ward, *BHWIDE 1.00: O(alpha) YFS exponentiated Monte Carlo for Bhabha scattering at wide angles for LEP1/SLC and LEP2*, [*Phys. Lett. B* [**[390]{}**]{} (1997) 298-308.]{}
S. Agostinelli , *Geant4- a simulation toolkit*, [*Nucl. Inst. and Meth. A* [**506**]{} (2003) 250-303.]{}
*MOKKA, A simulation program for linear collider detectors*, http://polzope.in2p3.fr:8081/MOKKA/.
T.C. Awes, F.E. Obenshain, F. Plasil, S. Saini, S.P. Sorensen and G.R. Young, *A simple method of shower localization and identification in laterally segmented calorimeters*, [*Nucl. Inst. Meth. A* [**311**]{} (1992) 130.]{}
I. [S]{}adeh, *Luminosity measurement at the International Linear Collider*, [*arXiv:1010.5992*]{} (2010).
H. [A]{}bramowicz , *Redefinition of the geometry of the luminosity calorimeter*, [*EUDET-Memo-2008-09* (2008)]{}, http://www.eudet.org.
H. [A]{}bramowicz , *Revised requirements on the readout of the luminosity calorimeter*. [*EUDET-Memo-2008-08* (2008)]{} http://www.eudet.org.
J. Brau , *ILC Reference Design Report* (2007) http://arxiv.org/abs/0712.1950.
D. Schulte, *Beam-beam simulations with guinea-pig*, *CERN-PS-99-014LPCLIC-Note 387* (1998).
A. Seryi, T. Maruyama and B. Parker, *IR optimization and anti-DID*, *SLAC-PUB-11662* (2006).
P. Bambade, V. Drugakov and W. Lohmann, *The impact of Beamcal performance at different ILC beam parameters and crossing angles on stau searches*, [*Pramana J. Phys.* [**69**]{} (2007) 1123.]{}
A. Heikkinen and N. Stepanov, *Bertini Intra-nuclear Cascade implementation in Geant4*, [*Proceedings of CHEP03*, nucl-th0306008 (2003) La Jolla, California.]{}
C. Coca , *Expected electromagnetic and neutron doses for the BeamCal at ILD* [*Rom. J.Phys* [**55**]{} (2010) 687.]{}
T. Tauchi and K. Yokoya, *Nanometer beam-size measurement during collisions at linear colliders*, [*Phys.Rev.E* [**51**]{} (1995) 6119-6126.]{}
T. Tauchi, K. Yokoya and P. Chen, *Pair creation from beam-beam interaction in linear colliders*, [*Part. Accel.* [**41**]{} (1993) 29-39.]{}
K.Ito, *Study of Beam Profile Measurement at Interaction Point in International Linear Collider*, *arXive: 0901.4151* (2009).
A. Elagin, *The optimized sensor segmentation for the very forward calorimeter*, in proceedings of *the 2005 International Linear Collider Physics and Detector Workshop*, ECONF C0508141:ALCPG0719 (2005) Snowmass, Colorado.
J. Blocki , *LumiCal new mechanical structure*, *EUDET-Memo-2009-10* (2009) http://www.eudet.org/.
J. Blocki , *Laser alignment system for LumiCal*, *EUDET-Report-2008-05* (2008) http://www.eudet.org.
C. Rimbault, P. Bambade, K. Monig and D. Schulte, *Impact of beam-beam effects on precision luminosity measurements at the ILC*, .
W. Kilian, [*WHIZARD: A generic Monte-Carlo integration and event generation package for multi-particle processes, LC-TOOL-2001-039* (2001).]{}
V. N. Pozdnyakov, *Two-photon interactions at LEP*, [*Phys. Part. Nucl. Lett.* [**[4]{}**]{} (2007) 289-303.]{}
S. Jadach, E. Richter-Was, B.F.L. Ward and Z. Was, *Monte Carlo program BHLUMI for Bhabha scattering at low polar angle with Yennie-Frautschi-Suura exponentiation*, [*Comp. Phys. Commun.* [**[70]{}**]{} (1992) 305-344.]{}
B. Pawlik , *BARBIE V4.3, Simulation-package of the LumiCal Detector*, http://www.ifj.edu.pl/dept/no1/nz13/barbi.php.
I. Smiljanic , *TOWARDS A FINAL SELECTION FOR LUMINOSITY MEASUREMENT*, [*Proceedings of the Workshop of the Collaboration on Forward Calorimetry at ILC* (2008), Belgrade Serbia]{}.
Ch. Grah , *Report to the Detector R&D Panel– Instrumentation of the Very Forward Region*, http://www.desy.de/prc/docs\_rd/prc\_rd\_02\_01\_update\_05\_07.pdf (2007) Hamburg, Germany.
S. Boogert , *Polarimeters and Energy Spectrometers for the ILC Beam Delivery System*, .
*S-DALINAC: Superconducting DArmstadt LInear ACcelerator*, http://www.ikp.tu-darmstadt.de/beschleuniger\_1/S-DALINAC.de.jsp.
Ch. Grah , *Radiation hard sensor for the BeamCal of the ILD detector*, *Proceedings of the IEEE conference*, October 27 – November 3 (2007) Honolulu, USA.
Ch. Grah , *Polycrystalline CVD Diamonds for the Beam Calorimeter of the ILC*, [*IEEE Trans.Nucl.Sci.* [**56**]{} (2009) 462.]{}
J. Blocki , *Silicon Sensors Prototype for LumiCal Calorimeter*, [*EUDET-Memo-2009-07* (2009)]{}. http://www.eudet.org.
M. Idzik , *Status of VFCAL*, [*EUDET-memo-2008-01* (2008)]{}. http://www.eudet.org.
E. Beuville . *AMPLEX, a low-noise, low-power analog CMOS signal processor for multi-element silicon particle detectors*, [*Nucl. Instr. and Meth.*, A[**[288a]{}**]{} (1990) 157-167]{}.
M. Idzik, Sz. Kulis and D. Przyborowski, *Development of front-end electronics for the luminosity detector at ILC*, [*Nucl. Instr. and Meth.*, A [**[608]{}**]{} (2009) 169-174.]{}
http://www.home.agilent.com.
M. Idzik, K. Swientek and Sz. Kulis, *Development of a Pipeline ADC for the Luminosity Detector at ILC*, .
*IEEE standard for terminology and test methods for analog-to-digital converters*, *IEEE-STD-1241* (2000).
J.L. McCreary and P.R. Gray, *All-MOS charge redistribution analog-to-digital conversion techniques. I*. [*Solid-State Circuits, IEEE Journal of*, [**[10]{}**]{}(6) (1975) 371–379.]{}
H. Spieler, *Semiconductor Detector Systems*. Oxford University Press (2005) Oxford, UK.
R. Gregorian, K.W. Martin, and G.C. Temes, *Switched-capacitor circuit design*, [*Proceedings of the IEEE*, 71(8), (1983),941–966.]{}
F. Silveira, D. Flandre, and P.G.A. Jespers, *A $g_{m}/I_{D}$ based methodology for the design of CMOS analog circuits and its application to the synthesis of a silicon-on-insulator micropower OTA*. [*Solid-State Circuits, IEEE Journal of*, [**[31]{}**]{}(9) (1996) 1314–1319.]{}
S. Rabii, *Design of Low-Voltage Low-Power Sigma-Delta Modulators*, *PhD thesis, Stanford University*, (1998).
Y. Arai, *Electronics and sensor study with the OKI SOI process*, in proceedings of the *Topical workshop on electronics for particle physics (TWEPP-07)*, September 3–7 (2007) Prague, Czech Republic.
[^1]: Corresponding author.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Storage allocation affects important performance measures of distributed storage systems. Most previous studies on the storage allocation consider its effect separately either on the success of the data recovery or on the service rate (time) where it is assumed that no access failure happens in the system. In this paper, we go one step further and incorporate the access model and the success of data recovery into the service rate analysis. In particular, we focus on quasi-uniform storage allocation and provide a service rate analysis for both fixed-size and probabilistic access models at the nodes. Using this analysis, we then show that for the case of exponential waiting time distribution at individuals storage nodes, minimal spreading allocation results in the highest system service rate for both access models. This means that for a given storage budget, replication provides a better service rate than a coded storage solution.'
author:
- |
Moslem Noori, Emina Soljanin, Masoud Ardakani\
Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada\
Department of Electrical and Computer Engineering, Rutgers University, Piscataway, NJ, USA
bibliography:
- 'IEEEabrv.bib'
- 'moslembib.bib'
title: On Storage Allocation for Maximum Service Rate in Distributed Storage Systems
---
Conclusion {#Sec Conc}
==========
Content allocation throughout a distributed storage system affects the probability that the content can be recovered when there is uncertainty in the number, identity, and/or availability of the storage nodes queried for service. So far the concern has been only that the stored data can eventually be downloaded, and not how long that process might take. To the best of our knowledge, this paper is the first attempt to understand how content allocation affects the download service rate. We showed that under certain assumptions, the minimal spreading allocation maximizes the service rate for the commonly assumed content access models specified by the number, identity, and/or availability of the storage nodes queried for service. Therefore, storing data through replication results in faster service for the incoming download requests than a coded storage with the same storage budget. Our assumption was that the service time at the storage nodes follows an exponential distribution, and is identically distributed and independent for all users. A more advanced model should involve other distributions (in particular, the shifted exponential as in [@chen2014queueing] and [@joshi2012coding]) as well as fork-join queuing considerations.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors were in part supported by Alberta Innovates Technology Futures (AITF) and Natural Sciences and Engineering Research Council of Canada (NSERC), and would also like to thank A. Badr, G. Joshi, and K. Mahdaviani for valuable discussions at the Banff International Research Station (BIRS).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Physical processes involving charge transfer, spin exchange, or excitation exchange often occur in conditions of resonant scattering. We show that the $s$-wave contribution can be used to obtain a good approximation for the full cross section. This approximation is found to be valid for a wide range of scattering energies, including high above the Wigner regime, where many partial waves contribute. We derive an analytical expression for the exchange cross section and demonstrate its relationship to the Langevin cross section. We give examples for resonant charge transfer as well as spin-flip and excitation exchange. Our approximation can be used to gain information about the $s$-wave regime from data obtained at much higher temperatures, which would be advantageous for systems where the ultracold quantum regime is not easily reachable.'
author:
- Robin Côté and Ionel Simbotin
bibliography:
- 'langevin-ref.bib'
title: 'Signature of the $s$-wave regime high above ultralow temperatures'
---
In recent years, rapid progress has been made to increase the number of systems which can be studied at ultralow temperatures, including atomic species [@Cold-atoms-review], and also molecular [@Carr-NJP-2009; @Dulieu-Review-2011; @Cote1997-mol; @Cote1999-mol] and ionic species [@Cote-review-ion; @RMP-ion]. In many cases, the quantum regime where $s$-wave scattering dominates is still outside the reach of today’s experimental techniques, such as in atom-ion hybrid system [@vladan09; @kohl2010a; @denschlag2010; @kohl2010b; @Ratschbacher-2013; @Haze-2015; @Idziaszek-2011; @Tomza-2015; @Gacesa-Be-2017]. However, a large class of physical systems is characterized by two states that are asymptotically degenerate, and for which an initial scattering state can be described by a superposition of those states; interference between the two possible interaction paths may lead to resonant exchange between the two states. Such processes have been studied in the scattering of neutral atoms, e.g., spin-flip in alkali atom collisions [@Cote-Li-1994; @Cote-Na-1994] with singlet and triplet potential curves, as well as in $S$-$P$ excitation exchange for identical atoms [@Bouledroua-2001], and charge transfer between an atomic ion and its neutral parent atom [@Cote-Dalgarno-2000; @Cote-2000-mobility]. Moreover, in cases involving quasi-resonant scattering, e.g., when considering different isotopes, the resonant approximation adequately describes the behavior of the system if the scattering energy is higher than the energy splitting between the asymptotic states [@Peng-Li; @Peng-Be].
In this Letter, we study the resonant exchange process $$X^\alpha + X^{\alpha'} \longrightarrow X^{\alpha'}+ X^\alpha,
\label{eq:exchange-process}$$ where $\alpha$ and $\alpha'$ denote internal states. For example, in charge transfer ($X + X^+ \longrightarrow X^+ + X$) $\alpha$ denotes the charge, with $\alpha=0$ and $\alpha'=+1$, while for excitation exchange $\alpha=S$ and $\alpha'=P$ are the electronic states. For such resonant exchange processes, the cross section reads [@Cote-Dalgarno-2000; @Cote-review-ion; @mott-massey] $$\sigma_{\rm exc} (E)= \frac{\pi}{k^2} \sum_{\ell =0}^\infty (2\ell +1)
\sin^2 (\eta_\ell^a - \eta_\ell^b ),
\label{eq:csec-def}$$ where $k=\sqrt{2\mu E/\hbar^2}$ is the center of mass wave number for the scattering of a pair of particle of reduced mass $\mu$ and collision energy $E$. Here, $\eta_\ell^{a(b)}$ is the scattering phase shift of the $\ell^{\rm th}$ partial wave along the interaction potential $V_{a(b)}$, which correspond to the two asymptotically degenerate channels
For energies high above the Wigner regime, where many partial waves are contributing, we can regard $\ell$ as a continuous variable and use the semi-classical expression [@mott-massey; @Harald-book] $$\frac{\partial \eta_\ell}{\partial \ell} \approx \frac{\pi}{2}
+ \int_{r_0(J)}^\infty dr \frac{\partial}{\partial J}
\left[ 2\mu (E-V(r)) -\frac{J^2}{r^2} \right]^{1/2} \;,
\label{eq:eta-one}$$ where $J=(\ell+\frac{1}{2})\hbar$, and $r_0(J)$ is the inner classical turning point. Although $\ell$ can be large, we assume that the centrifugal term $J^2/r^2$ is a small perturbation on the potential $V(r)$, [*i.e.*]{}, $J^2/r^2 \ll 2\mu [E-V(r)]$. Under such conditions, the scattering wave function still probes the inner region, and $r_0$ depends weakly on $J$; we thus take it to be independent of $J$ and equal to the $s$-wave turning point, i.e., $
r_0 (J) \equiv r_0$. We now take the partial derivative out of the integral (\[eq:eta-one\]) and expand the integrand in small powers of $J^2/r^2$ to obtain $$\frac{\partial \eta_\ell}{\partial \ell} \approx \frac{\pi}{2}
-J \int_{r_0}^\infty \frac{dr}{r^2} \frac{1}{\sqrt{2\mu (E-V)}}
\equiv \frac{\pi}{2} -J \frac{A}{\hbar}\;,
\label{eq:alpha-approx}$$ where $A$ is an integral independent of $J$. Using $J=(\ell+\frac{1}{2})\hbar$, we have $\partial \eta_\ell/\partial \ell
\approx \pi/2 + A (\ell + 1/2) + {\cal O} (\ell^2)$, which after integration over $\ell$ gives $$\eta_\ell \approx \eta_0 + \frac{\pi}{2}\ell + \frac{A}{2} \ell (\ell +1)
+ {\cal O} (\ell^3) \;.
\label{eq:eta-four}$$ Using the Levinson theorem [@mott-massey; @Harald-book], we have $\eta_0 = N\pi +
\delta_0$, where $\delta_0$ is the $s$-wave phase shift modulo $\pi$ and $N$ is the number of bound states. Therefore, the phase shift difference $\Delta\eta_\ell \equiv \eta_\ell^a - \eta_\ell^b$ reads $$\Delta \eta_\ell \approx \pi \Delta N\ + \Delta \delta_0
+ \ell (\ell +1) \frac{\Delta A}{2} + {\cal O} (\ell^3) \;,
\label{eq:eta-diff}$$ where $\Delta N = N_a-N_b$, $\Delta \delta_0 = \delta^a_0-\delta^b_0$, and $\Delta A = A_a - A_b$, and we obtain $$\sin^2(\Delta \eta_\ell) \approx \sin^2 \left[ \Delta \delta_0
+ \ell (\ell +1) \frac{\Delta A}{2} \right]\;.
\label{eq:eta-diff-two}$$
We now approximate the sum in Eq. (\[eq:csec-def\]) with an integral, $\sigma_{\rm exc} \approx \frac{\pi}{k^2} \int_0^\infty d\ell (2\ell
+1) \sin^2 (\eta^a_\ell - \eta^b_\ell)$, which yields $$\sigma_{\rm exc} \!\! \approx \!\!\frac{\pi}{k^2} \int_0^{L} \!\! d\ell (2\ell +1)
\sin^2 \!\!\left[\Delta \delta_0 \!+\!\ell (\ell +1) \frac{\Delta A}{2}\right]\!.$$ In the integral above, the upper limit $L$ is set to a suffieciently large value of $\ell$ such that the two phase shifts become equal; thus, there will be no further contribution to the integral for $\ell>L$. This occurs when the centrifugal barrier becomes dominant for both potential curves [@Cote-Dalgarno-2000]. Changing variable to $x\equiv \Delta \delta_0 +\ell (\ell +1) \frac{\Delta
A}{2}$, our integral simply becomes $\sigma_{\rm exc} \simeq
\frac{\pi}{k^2} \frac{2}{\Delta A}\int_{x_0}^{x_L} dx\;\sin^2 x =
\frac{\pi}{k^2} \frac{2}{\Delta A} \left[ \frac{x}{2} -
\frac{1}{4}\sin (2x) \right]_{x_0}^{x_L}$, with $x_0=\Delta
\delta_0$ and $x_L = \Delta \delta_0 + L(L+1)\Delta A/2$, giving $$\begin{aligned}
\sigma_{\rm exc} & \simeq & \frac{\pi}{k^2} \frac{1}{\Delta A} \left[ L(L+1)\frac{\Delta A}{2}
+ \frac{1}{2}\sin (2 \Delta \delta_0)\right. \nonumber \\
& & \left. \hspace{.45in} - \frac{1}{2}\sin (2 \Delta \delta_0 + L(L+1)\Delta A)
\right].
\label{eq:csec-app-1}\end{aligned}$$
Finally, we can simplify our result if we employ the approximation $L(L+1)\Delta A \ll 1$, which can be justified if we examine the parameter $A_i$ given by Eq.(\[eq:alpha-approx\]), [*i.e.*]{} $$A_i = \frac{\hbar}{\sqrt{2\mu}} \int_{r^i_0(E)}^\infty
\frac{dr}{r^2} \frac{1}{\sqrt{E-V_i(r)}} \; ,
\label{eq:A-def}$$ where the inner turning point $r^i_0$ for the potential $V_i$ depends on the scattering energy $E$. In resonant processes where the long-range tail of each $V_i$ is the same, only their shorter range difference contribute to $\Delta A$. Typically, $\Delta A$ varies little with $E$, and is of the order 0.01–0.001 for the physical systems considered in this Letter, with $\Delta A$ smaller for heavier systems due the $2\mu$ factor. We now return to Eq. (\[eq:csec-app-1\]) and we use the approximation $\sin [2 \Delta
\delta_0 + L(L+1)\Delta A ]\approx L (L+1)\Delta A \cos (2 \Delta
\delta_0)+\sin (2 \Delta \delta_0)$ to obtain $$\begin{aligned}
\sigma_{\rm exc}(E) & \simeq &
\frac{\pi}{k^2} \frac{1}{\Delta A} L (L+1) \frac{\Delta A}{2}
[ 1- \cos (2 \Delta \delta_0)] \;, \nonumber \\
& \approx & \frac{\pi}{k^2} L^2
\sin^2 \Delta \delta_0 (E)\;,
\label{eq:csec-app-2}\end{aligned}$$ where we assume $L(L+1)\approx L^2$.
The expression above can be related to the Langevin cross section $\sigma_L$. Indeed, we defined $L$ as the maximum $\ell$ for which the phase shift difference is non-negligible, which corresponds to the height of the centrifugal barrier slightly larger than $E$. This critical value of $\ell$ also defines $\sigma_L$, which is determined by the impact parameter $b_{\rm max}$ still allowing penetration in the inner region where the exchange process occurs with unit probability [@mott-massey; @Cote-Dalgarno-2000; @Cote-review-ion]. With the impact parameter $b\equiv (\ell + \frac{1}{2})/k$, we obtain $$\sigma_L (E)= \pi b_{\rm max}^2 \simeq \frac{\pi}{k^2} L^2 \;,
\label{eq:csec-Langevin-def}$$ where we assume $L+\frac{1}{2} \approx L$ for large $L$. We remark that $L$ has the same value for both potentials $V_a$ and $V_b$, which is a valid assumption for the energy range dominated by the long range tail (which is the same for both potentials). For potentials with an asymptotic behavior $V(r)\sim -C_n/r^n$, the location of the top of the barrier is $r_{\rm top} = \left(\frac{\mu n C_n}{\ell
(\ell+1)\hbar^2}\right)^{\frac{1}{n-2}} $, and $L(E)$ is obtained from $E=V (r_{\rm top})$, which yields $$L(L+1) = \frac{1}{\hbar^2} \left( \frac{n}{n-2}\right)^{\frac{n-2}{n}}
(\mu n C_n)^{\frac{2}{n}} (2\mu E)^{\frac{n-2}{n}}\;.
\label{eq:L(L+1)}$$ Again, assuming $L(L+1)\approx L^2$, we can write the Langevin cross section $\sigma_L = \frac{\pi}{k^2} L^2$ as $$\sigma_L (E)= \pi
\left( \frac{n}{n-2}\right)^{\frac{n-2}{n}}
(n C_n)^{\frac{2}{n}} (2 E)^{-\frac{2}{n}} \;.
\label{eq:Langevin-general}$$ The expressions for the most common long range inverse power-law potentials are listed in Table \[tab1\], with $n=3$ appearing in dipole allowed excitation exchange, $n=4$ in polarization potentials between atoms and ions, and $n=6$ in van der Waals interactions between ground state atoms.
$n$ 3 4 6
-------------- -- ----------------------------------------------------------- -- ------------------------------------------------------- -- ------------------------------------------------------------------
$ \sigma_L $ $\displaystyle 3 \pi \left(\frac{ C_3}{2E}\right)^{2/3}$ $\displaystyle 2\pi \left(\frac{C_4}{E}\right)^{1/2}$ $\displaystyle \frac{3\pi}{2}\left(\frac{2 C_6}{E}\right)^{1/3}$
: Langevin cross section $\sigma_L$ for various $n$.[]{data-label="tab1"}
The simple expression $\sigma_{\rm exc} = \sigma_L \sin^2 (\Delta
\delta_0)$ is obtained by combining Eqs.(\[eq:csec-Langevin-def\]) and (\[eq:csec-app-2\]). However, for energies low enough that $L^2<1$, $\sigma_{\rm exc}$ will decrease rapidly and not capture the asymptotically constant $s$-wave cross section of the Wigner regime as $E\rightarrow 0$. This is remedied by adding the low energy contribution, so that we finally can write the resonant exchange cross section as $$\sigma_{\rm exc} (E)= \left[\frac{\pi}{k^2}+\sigma_L(E) \right] \sin^2 \Delta \delta_0 (E).
\label{eq:csec-final}$$ This equation explicitly shows how the $s$-wave regime modulates the Langevin cross section, leading to a signature of the $s$-wave regime at higher temperatures.
To illustrate the effect of the $s$-wave regime at higher energies, we first consider charge transfer between Yb and Yb$^+$ for a variety of isotopes. As noted in a previous article on resonant charge transfer [@Peng-Yb-2009], the cross section exhibits an intermediate “modified" Langevin regime where the $\sigma_{\rm exc}$ seems to be affected by the ultracold behavior, even if many partial waves contribute to its overall value. In fact, one could notice a “correlation" between the cross section at ultralow energy and at much higher energies. The expression above provides the explanation for the correlation noted in [@Peng-Yb-2009]: if the $s$-wave phase shifts corresponding to states $a$ and $b$ happen to have nearly equal values, their difference remains small even at higher scattering energy. This can be seen from the WKB approximation, or equivalently Eq.(\[eq:eta-four\]), as $\eta_\ell$ varies slowly with $\ell$. If $\Delta\eta_0$ is small, the phase difference for higher $\ell$ will also remain small for a wide range of partial waves, due to the phase shift “locking" described above, and will result in a reduced cross section. Naturally, the applicability of Eq.(\[eq:csec-final\]) depends on the details of the potentials and validity of the approximations involved in its derivation.
![(Color on line) Resonant charge transfer $\sigma_{\rm exc}$ between various isotopes of Yb and Yb$^+$ vs. scattering energy $E$. The numerical results (black line) are compared to Eq.(\[eq:csec-final\]) (magenta line), together with its components; the $s$-wave contribution $\frac{\pi}{k^2}\sin^2\Delta\delta_0$ alone (black dashed line) and the $\sigma_L \sin^2\Delta\delta_0$ alone (solid blue line). The standard $\sigma_L$ (blue dot-dashed line) is shown for comparison purposes. Isotopes 168 in (a), 174 in (e), and 176 in (f) show significant suppression when compared to $\sigma_L$, while 170 in (b), 172 in (c), and 173 in (d), $\sigma_{\rm exc}\approx \frac{1}{2}\sigma_L$ over a wide range of energies. []{data-label="fig:Yb"}](Yb-2x3panels.pdf){width="1\linewidth"}
In Fig. \[fig:Yb\], we compare the simple expression (\[eq:csec-final\]) to the full numerical results computed using the approach described in [@Peng-Yb-2009]. The potentials $V_g$ and $V_u$ corresponding to the $^2\Sigma_g^+$ and $^2\Sigma_u^+$ of Yb$_2^+$ behave as $-C_4/r^{-4}$ with $C_4= 72.5$ a.u. at large separation. The cross section $\sigma_{\rm
exc}(E)$ depends strongly on the atomic mass of the Yb isotopes. In each plot of Fig. \[fig:Yb\], the “standard" Langevin cross section $\sigma_L$ is included to emphasize the effect of the $s$-wave phase shifts. In some cases, like plots (b), (c), and (d), corresponding to isotopes 170, 172, and 173, both $\sigma_L$ and $\sigma_{\rm exc}$ give similar values, [*i.e.*]{} the $s$-wave has no sizable effect on the cross section. Actually, $\sigma_{\rm exc}$ is roughly $\frac{1}{2}\sigma_L$, which is to be expected in general since $\langle \sin^2\Delta\delta_0\rangle = 1/2$ if the phase shift difference $\Delta \delta_0$ is random. However, in other cases, like for isotopes 168, 174, or 176 in (a), (e) and (f) respectively, the signature of the s-wave regime is noticeable, with a reduction of two orders of magnitude for (a) and (e), and one for (f). As mentioned above, this is due to the accidental proximity in values of the residual phase shifts $\delta_0^{g(u)}$ corresponding to $V_{g(u)}$, and the phase shift locking as $E$ increases. We note that according to Eq.(\[eq:L(L+1)\]), $L^2<1$ when $E$ becomes smaller than roughly $10^{-13}$ a.u. for the Yb systems above, at which point the $s$-wave contribution (negligible at higher $E$) satisfying the Wigner regime kicks in. It is worth noting that when the $s$-wave suppression of $\sigma_{\rm exc}$ is significant, as in Figs. \[fig:Yb\] (a) and (e), the underlying shape resonances become more apparent as the background cross section diminishes. Naturally, these resonances are absent from our WBK-treatment in Eq.(\[eq:csec-final\]), which reproduces the general trend of the numerical results over a large range of $E$.
![(Color on line). Same as Fig. \[fig:Yb\] for spin-flip in collision of Na and various isotopes of Ca$^+$, 40 in (a), 42 in (b), 43 in (c), and 44 in (d). Significant suppression occurs in (b), while $\sigma_{\rm exc}$ is close to $\sigma_L$ for the other isotopes. []{data-label="fig:NaCa"}](NaCa-2x2panels.pdf){width="1\linewidth"}
Atom-ion scattering can also lead to a resonant spin-flip process, such as in [@Makarov-2003; @Smith2014], where a ground state Na atom approaching a Ca$^+$ ion can interact via a singlet A$^1\Sigma^+$ or a triplet a$^3\Sigma^+$ electronic state described by the singlet (triplet) potential $V_{S(T)}$ with corresponding residual phase shift $\delta_\ell^{S(T)}$. In that work, $\sigma_{\rm exc}$ was found to be roughly $\frac{1}{2}\sigma_L$. Recent experiments on Yb$^+$+$^{87}$Rb [@Ratschbacher-2013], Yb$^+$+$^6$Li [@Tomza-2018], and $^{88}$Sr$^+$+Rb [@Sikorsky-2018; @Timur-arXiv-2018] have explored spin-flip dynamics. In Fig. \[fig:NaCa\], we explore the effect of the $s$-wave scattering on the spin-flip in Ca$^+$+Na, using $V_{S(T)}$ described in [@Makarov-2003; @Gacesa-NaCa-2016] (behaving as $r^{-4}$ at large $r$) for four isotopes of Ca, namely 40, 42, 43, and 44. Again, the simple expression (\[eq:csec-final\]) agrees with the numerical cross sections over a wide range of energy. Fig. \[fig:NaCa\] shows a variety of behavior of $\sigma_{\rm exc}$. For example, in (a) and (d), $\sigma_{\rm exc}\approx \sigma_L$ at higher energies, corresponding to $\Delta\delta_0 =\delta_0^S-\delta_0^T \approx
\pi/2$, while (c) depicts a small suppression by a factor of about 1.5. The case of $^{40}$Ca leads to a substantial reduction of about 200, again revealing the underlying shape resonances.
Spin-flip collisions have also been studied between neutral atoms, especially alkali atoms, such as in Li [@Cote-Li-1994] or Na [@Cote-Na-1994]. Again, the ground state atoms approach each other in a superposition of singlet X$^1\Sigma_g^+$ and triplet a$^3\Sigma_u^+$ states, behaving asymptotically as $-C_6/r^6$. To illustrate the effect of the $s$-wave regime on $\sigma_{\rm exc}$, we consider a system for which the scattering lengths are known to be close to each other, namely $^{87}$Rb. Using the potential curves described in [@Timur-Rb2-2008], we computed $\sigma_{\rm exc}$ for pure $^{87}$Rb, $^{85}$Rb, and their mixture. The results are shown in Fig. \[fig:Rb\]; $\sigma_{\rm exc}$ for the mixture in (a) follows roughly $\sigma_L$ away from ultracold temperatures. As expected, for $^{87}$Rb in (b) with both singlet and triplet scattering lengths almost equal ($a_S\approx a_T\approx 100$ a.u.), the $s$-wave suppression is drastic, with shape resonances emerging from the suppressed background. Although not perfect, the simple expression (\[eq:csec-final\]) tracks the overall reduction of a factor of $10^4$ in $\sigma_{\rm exc}$. Much more surprising is the result for $^{85}$Rb (c) with scattering lengths ($a_S\approx 2500$ a.u. and $a_T\approx -390$ a.u.) which are very different. In this case, one could have expected the system to follow the Langevin case. However, a closer look at the $s$-wave phase shifts explains the seemingly unusual cross section. The large positive and negative values of $a_S$ and $a_T$ imply rapid changes of $\delta_0^{S(T)}$ with energy in the Wigner regime, as shown in Fig. \[fig:Rb\](d). The large initial value of the phase shift difference $\Delta\delta_0$ (mod $\pi$) quickly evolves into a much smaller value, comparable to the case of $^{87}$Rb.
![(Color on line). Same as Fig. \[fig:Yb\] for spin-flip in collision Rb atoms, with $^{85}$Rb+$^{87}$Rb mixture in (a), pure $^{87}$Rb in (b), and pure $^{85}$Rb in (c). Both pure cases show extreme suppression compared to $\sigma_L$, with resonances revealed by the small background. Corresponding $\delta_0^{S(T)}$ are depicted in (d). []{data-label="fig:Rb"}](Rb-2x2.pdf){width="1\linewidth"}
As a final example, we consider a system for which the interaction potentials behave as $r^{-3}$ at long range. Many examples occur in nature, such as excitation exchange [@Bouledroua-2001], or in the scattering of metastable atoms, like H(2s)+H(2s) [@Forrey-PRL-2000; @Jonsell-PRA-2002], or the excitation exchange in metastable helium He(1$^1$S)+He$^*$(2$^3$P) [@Vrinceanu-2010; @Peach-2017]. Here, we focus our attention on Cs$^+$+Cs$(6p)$ which can lead to the exchange of the $6p$ excitation onto Cs$^+$. Four excited electronic states are involved if we neglect spin-orbit coupling, two $\Sigma^+_{g(u)}$ and two $\Pi_{g(u)}$, each correlated to the Cs$_2^+(6p)$ asymptote, and described by potential curves $V^\Sigma_{g(u)}$ and $V^\Pi_{g(u)}$ and corresponding residual phase shifts $\delta_{\Sigma,\ell}^{g(u)}$ and $\delta_{\Pi,\ell}^{g(u)}$. The cross section is [@mott-massey; @Bouledroua-2001] $$\sigma_{\rm exc} = \frac{\pi}{3k^2} \sum_{\ell =0}^\infty (2\ell +1)
\left[\sin^2 \Delta \delta_\ell^\Sigma
+ 2\sin^2\Delta\delta_\ell^\Pi \right],$$ where $\Delta \delta_\ell^\Sigma \equiv \delta_{\Sigma,\ell}^{g} -
\delta_{\Sigma,\ell}^{u}$ and $\Delta \delta_\ell^\Pi \equiv
\delta_{\Pi,\ell}^{g} - \delta_{\Pi,\ell}^{u}$. Since the $\Sigma$ and $\Pi$ curves have different $C_3$ values, $L$ for both sets is different. Using our approximations, $\sigma_{\rm exc}$ becomes $$\sigma_{\rm exc} \!=\! \frac{1}{3}\! \left[\frac{\pi}{k^2} \!+\! \sigma_L^\Sigma \right]\! \sin^2\! \Delta \delta^\Sigma_0
\!+\! \frac{2}{3}\! \left[\frac{\pi}{k^2} \!+\! \sigma_L^\Pi \right]\! \sin^2 \! \Delta \delta^\Pi_0,$$ where $\sigma_L^{\Sigma(\Pi)}$ is obtained with the appropriate value of $C_3$.
The results shown in Fig. \[fig:Cs\] were obtained with the $^2\Sigma^+_{g(u)}$ and $^2\Pi^+_{g(u)}$ from Jraij [*et al.*]{} [@Jraij-Cs2]. The $\Pi$ curves are repulsive at large separation behaving as $+C_3^\Pi/r^3$ with $C_3^\Pi = -13.95$ a.u.; the [*gerade*]{} and [*ungerade*]{} phase shifts are basically equal for all $\ell$, their cancellation leading to a negligible $\Pi$ contribution. The two $\Sigma$ curves are attractive, and were matched at large separation to $-C_4/r^4 -C_3/r^3$ with $C_4=1082$ a.u., and $C_3^\Sigma = 27.9$ a.u. Since there is only one stable isotope of Cs, we rescaled its mass to model a different isotope. For the real mass of Cs, the cross section is roughly half the Langevin cross section, while choosing $m_{\rm Cs} = 132.75$ u, the cross section is reduced by a factor of 20, again exposing the resonances as in previous examples.
![(Color on line). Same as Fig. \[fig:Yb\] for excitation exchange in Cs$^+$+Cs($6p$), for the real mass in (a), and a fictitious mass of 132.75 u in (b), to illustrate the $s$-wave suppression. []{data-label="fig:Cs"}](Cs.pdf){width="1\linewidth"}
In conclusion, we derived a simple expression for resonant scattering processes, relating the cross section to the Langevin cross section and the $s$-wave regime. By relying on the WKB approximation, we derived the expression for the exchange cross section, and showed that it has wide applicability, as long as the pairs of potential curves have the same long range tail. We illustrated the range of applicability using various resonant systems such as charge transfer, spin-flip, and excitation exchange, and for a variety of long range inverse power-law tail behaving as $r^{-n}$ covering the most common powers. The expression points to the signature of the $s$-wave regime at higher temperatures, and how the $s$-wave phase shift “locking" actually modulates the cross section. The results presented here also provide a diagnostic tool particularly relevant to system for which ultracold temperatures are not easily achievable, such as atom-ion hybrid systems for which the nK regime remains a challenge. In fact, by measuring the cross section or rate for a resonant process, e.g., charge transfer or spin-flip, at higher temperatures more easily accessible, one can gain information about the $s$-wave regime. If a sizable suppression is observed as compared to $\sigma_L$, this implies that the $s$-wave phase shifts are close to each other. In addition, the suppression helps revealing shape resonances otherwise submerged which, together with the $s$-wave suppression, can help determining the potential curves more accurately down to the $s$-waves. Finally, the expression should be applicable to quasi-resonant processes as well [@mott-massey], such as charge transfer in systems with mixed isotopes [@Peng-Li; @Peng-Be], or in reactions involving isotope substitutions, as long as the scattering energy is larger than the energy gap between the asymptotes of the relevant potentials.
This work was partially supported by the National Science Foundation Grant PHY-1415560 (IS) and by the MURI US Army Research Office Grant No. W911NF-14-1-0378 (RC).
| {
"pile_set_name": "ArXiv"
} |
=10000 =10000 =10000 6.4in
hep-th/0406127\
BHU-SNB/Preprint
1.7cm
[****]{}
2.5cm
[**R.P.Malik**]{}\
[*S. N. Bose National Centre for Basic Sciences,*]{}\
[*Block-JD, Sector-III, Salt Lake, Calcutta-700 098, India*]{}\
.2cm
and\
.2cm
[*Centre of Advanced Studies, Physics Department,*]{}\
[*Banaras Hindu University, Varanasi-221 005, India*]{}\
[**E-mail address: malik@bhu.ac.in**]{}
2.5cm
[**Abstract**]{}: An appropriate definition of the Hodge duality $\star$ operation on any arbitrary dimensional supermanifold has been a long-standing problem. We define a working rule for the Hodge duality $\star$ operation on the $(2 + 2)$-dimensional supermanifold parametrized by a couple of even (bosonic) spacetime variables $x^\mu (\mu = 0 , 1)$ and a couple of odd (fermionic) variables $\theta$ and $\bar\theta$ of the Grassmann algebra. The Minkowski spacetime manifold, hidden in the supermanifold and parametrized by $x^\mu (\mu = 0, 1)$, is chosen to be a flat manifold on which a two $(1 + 1)$-dimensional (2D) free Abelian gauge theory, taken as a prototype field theoretical model, is defined. We demonstrate the applications of the above definition (and its further generalization) for the discussion of the (anti-)co-BRST symmetries that exist for the field theoretical models of 2D- and 4D free Abelian gauge theories considered on the four $(2 + 2)$- and six $(4 + 2)$-dimensional supermanifolds, respectively.\
PACS numbers: 11.15.-q, 12.20.-m, 03.70.+k\
[*Keywords*]{}: 2D- and 4D Abelian gauge theories, geometrical superfield formalism, supermanifolds, Hodge duality operation, dual-horizontality condition, (anti-)co-BRST symmetries
[**1 Introduction**]{}\
The geometrical superfield formalism is one of the most intuitive approaches to gain an insight into some of the physical and mathematical ideas behind the Becchi-Rouet-Stora-Tyutin (BRST) formalism which plays a very important role in (i) the covariant canonical quantization of the gauge theories that are endowed with the first-class constraints in the language of Dirac’s prescription for the classification of constraints (see, e.g., \[1,2\]), (ii) the proof of unitarity of the “quantum” gauge theories at any arbitrary order of perturbative computations (see, e.g., \[3,4,5\]), and (iii) providing a deep connection between the physics of gauge theories with the mathematical ideas behind the cohomology (see, e.g., \[6-9\]) of the differential geometry. In the usual superfield approach \[10-17\] to the $p$-form ($ p = 1, 2,
3....)$ Abelian gauge theories, defined on the $D$-dimensional spacetime manifold, a $(p + 1)$-form super curvature $\tilde F^{(p
+ 1)} = \tilde d \tilde A^{(p)}$ is constructed from the super exterior derivative $\tilde d = dx^\mu
\partial_\mu + d \theta \partial_\theta + d \bar\theta
\partial_{\bar\theta}$ (with $\tilde d^2 = 0$) and the super $p$-form connection $\tilde A^{(p)}$ on the $(D + 2)$-dimensional supermanifold parametrized by $D$-number of even (bosonic) spacetime coordinates $x^\mu (\mu = 0, 1,2......D-1)$ and a couple of odd (fermionic) elements $\theta, \bar\theta$ (with $\theta^2 =
\bar\theta^2 = 0, \theta\bar\theta + \bar\theta \theta = 0$) of the Grassmann algebra which constitute the superspace variable $Z^M = (x^\mu, \theta, \bar \theta)$. This $(p + 1)$-form super curvature is subsequently equated, due to the so-called horizontality condition [^1], to the ordinary $(p + 1)$-form curvature $F^{(p +
1)} = d A^{(p)}$ constructed from the ordinary exterior derivative $d = dx^\mu \partial_\mu$ (with $d^2 = 0$) and the ordinary $p$-form connection $A^{(p)}$ on the ordinary $D$-dimensional Minkowskian spacetime manifold parametrized by the bosonic spacetime variables $x^\mu$ only. This restriction [^2] provides the geometrical origin and interpretation for (i) the nilpotent (anti-)BRST symmetry transformations (and the corresponding nilpotent and conserved charges) as the translation generators $(\partial/\partial\theta)
\partial/\partial\bar\theta$ along the Grassmannian directions of the supermanifold, (ii) the nilpotency of the above transformations (and the corresponding nilpotent generators) as a couple of successive translations (i.e. $(\partial/\partial\theta)^2 = (\partial/\partial\bar\theta)^2 = 0$) along the Grassmannian directions of the supermanifold, and (iii) the anticommutativity of the (anti-)BRST transformations (and the corresponding conserved and nilpotent charges) as the anticommutativity $(\partial/\partial\theta) (\partial/\partial\bar\theta)
+ (\partial/\partial\bar\theta) (\partial/\partial\theta) = 0$ of the translation generators along the Grassmannian directions of the $(D + 2)$-dimensional supermanifold.
It is obvious from the above discussions that, in the horizontality condition, only one (i.e. $(\tilde d) d$) of the existing three (super) de Rham cohomological operators ($(\tilde
d) d, (\tilde \delta)\delta, (\tilde \Delta)\Delta$) is exploited for the geometrical interpretations of some of the key properties associated with the nilpotent (anti-)BRST transformations and the corresponding conserved charges. To clarify the above notations, it is worthwhile to be more specific about the de Rham cohomological operators of the differential geometry defined on an ordinary spacetime manifold without a boundary. On such a manifold, the operators $d = dx^\mu \partial_\mu$, $\delta = \pm *
d *$ and $\Delta = (d + \delta)^2$ form a set of de Rham cohomological operators where $(\delta)d$ are the nilpotent (co-)exterior derivatives, $\Delta$ is the Laplacian operator and $*$ is the Hodge duality operation on the manifold. These operators obey an algebra: $d^2 = \delta^2 = 0, \Delta = \{d,
\delta \}, [\Delta, d] = [\Delta, \delta] = 0$ showing that $\Delta$ is the Casimir operator for the whole algebra (see, e.g., \[6-9\] for details). It has been a long-standing problem to exploit the other nilpotent (i.e. $\tilde \delta^2 = 0, \delta^2 = 0$) mathematical entities $(\tilde \delta)\delta$ of the (super) de Rham cohomological operators in the context of the dual-horizontality condition ($\tilde \delta \tilde A^{(p)} =
\delta A^{(p)}$) and study its consequences on a $p$-form gauge theory in the framework of the geometrical superfield approach to BRST formalism. Here $\tilde \delta = - \star \tilde d \star$ and $ \delta = - * d *$ are the super co-exterior derivative and ordinary co-exterior derivative, respectively. The mathematical symbols $\star$ and $*$ stand for the Hodge duality operations on the $(D + 2)$-dimensional supermanifold and $D$-dimensional ordinary manifold, respectively, and the super Laplacian operator is defined as $\tilde \Delta = (\tilde d + \tilde \delta)^2$. To tap the mathematical power of $\tilde \delta = - \star \tilde d
\star$, it is clear that the definition of the Hodge duality $\star$ operation on the $(D + 2)$-dimensional supermanifold is quite important.
A consistent and systematic definition of the Hodge duality $*$ operation on an ordinary spacetime manifold of any arbitrary dimensionality is already quite well-known in the literature (see, e.g., \[6-9\] for details). In fact, the existence of the totally symmetric metric tensor and the totally antisymmetric Levi-Civita tensor on the spacetime manifold plays a crucial role in such a consistent and systematic definition of the duality operation ($*$). However, such a consistent, precise and elaborate definition of the Hodge duality $\star$ operation on a supermanifold, to the best of our knowledge, is not well-known in the literature (see, e.g., \[18-26\] for details). The purpose of our present paper is to provide a working rule for the definition of the Hodge duality $\star$ operation on the four $(2 + 2)$- and six $(4 + 2)$-dimensional supermanifolds on which the 2D- and 4D free 1-form ($A^{(1)} = dx^\mu A_\mu$) Abelian gauge theories are defined for the derivation of the nilpotent (anti-)co-BRST symmetry transformations in the framework of superfield approach to BRST formalism. We exploit this working rule for the definition of the $\star$ operation in the context of the dual-horizontality condition ($\tilde \delta \tilde A^{(1)} = \delta A^{(1)}$) where the action of the super co-exterior derivative $\tilde \delta = -
\star \tilde d \star$ on the super connection 1-form $\tilde
A^{(1)}$ does require, the action of the Hodge duality $\star$ operations (in $\tilde \delta \tilde A^{(1)} = - \star \tilde d
\star \tilde A^{(1)}$) for the derivations of the (anti-)co-BRST symmetry transformations. To be more precise, for the case of the 4D Abelian gauge theory, defined on the six $(4 + 2)$-dimensional supermanifold, the $\star$ operation is defined (i) on the super 1-form $\tilde A^{(1)}$ to produce $(\star\; \tilde A^{(1)})$ as a super 5-form, and subsequently (ii) on the super 6-form $(\tilde d
\star \tilde A^{(1)})$ to produce a super 0-form $(\star \tilde d
\star \tilde A^{(1)})$ to obtain explicitly $\tilde \delta \tilde
A^{(1)} = - \star \tilde d \star \tilde A^{(1)}$. In exactly similar fashion, the $\star$ operations could be defined for the 2D free Abelian gauge theory, considered on the four $(2 +
2)$-dimensional supermanifold, for the derivation of the nilpotent (anti-)co-BRST symmetries. Towards the above goals in mind, we propose, in a systematic manner, the Hodge duality $\star$ operations on all the possible super forms that could be defined on the $(2 + 2)$-dimensional supermanifold (cf. Section 2.2)) as well as on the $(4 + 2)$-dimensional supermanifold (cf. Section 3.2). These definitions are subsequently exploited for the derivation of the nilpotent (anti-)co-BRST symmetries in the framework of superfield formalism (cf. Sections 2.3 and 3.3 below). Our present study is essential on three counts. First and foremost, it has been a long-standing problem to exploit the potential and power of the (super) co-exterior derivatives $\tilde \delta = - \star \tilde d
\star$ and $\delta = - * d *$ in the context of the derivations of some specific nilpotent symmetries for the BRST formulation of the gauge theories. We find that the above (super) cohomological operators do play a set of decisive roles in the context of the derivations of the nilpotent (anti-)co-BRST symmetry transformations for the 2D- and 4D free Abelian gauge theories. Second, in our recent works \[27-32\], we have been able to exploit $(\tilde \delta)\delta$ in the dual-horizontality condition ($\tilde \delta \tilde A^{(1)} = \delta A^{(1)}$) but the precise expressions for the $\star$ operations on all the super forms, defined for some suitable supermanifolds, have not yet been obtained. Finally, our present study [*might*]{} turn out to be useful for the discussion of an interacting gauge theory \[33,34\] which has been shown to provide (i) the field theoretical model for the Hodge theory, and (ii) a model for the interacting topological field theory where topological $U(1)$ field couples with the matter (Dirac) fields \[33,34\].
The contents of our present paper are organized as follows.
In Section 2, we very briefly recapitulate the bare essentials of the (anti-)BRST- and (anti-)co-BRST symmetry transformations for the 2D free Abelian gauge theory in the Lagrangian formulation. We also derive the nilpotent (anti-)BRST symmetry transformations in the framework of superfield formalism by exploiting the horizontality condition on the $(2 + 2)$-dimensional supermanifold and provide the geometrical interpretation for the nilpotent (anti-)BRST charges $Q_{(a)b}$ (cf. Section 2.1). For the derivation of the (anti-)co-BRST symmetry transformations in the superfield formalism, we discuss the dual-horizontality condition and define the Hodge duality $\star$ operation, in a systematic way, for all the (super)forms defined on the four $(2 + 2)$-dimensional supermanifold on which a 2D free Abelian gauge theory is considered. The double Hodge duality $\star$ operations are also defined for all the (super)forms that are supported by the $(2 + 2)$-dimensional supermanifold.
Section 3 is devoted to (i) a concise synopsis of the local, covariant, continuous and nilpotent (anti-)BRST- and non-local, non-covariant, continuous and nilpotent (anti-)co-BRST symmetry transformations for the free 4D Abelian theory in the Lagrangian formulation, (ii) a brief discussion for the derivation of the (anti-)BRST symmetry transformations in the usual superfield formalism and its key points of differences with such a derivation for the 2D free Abelian theory, (iii) a systematic definition of the single Hodge duality $\star$ operation (and the double Hodge duality $\star$ operations) for all the (super)forms defined on the six $(4 + 2)$-dimensional supermanifold, and (iv) the derivation of the nilpotent (anti-)co-BRST symmetries by exploiting the $\star$ operation in the context of the dual-horizontality condition.
Finally, in Section 4, we make some concluding remarks.\
[**2 (Anti-)BRST- and (anti-)co-BRST symmetries for 2D theory: a brief sketch**]{}\
Let us begin with the BRST- and anti-BRST invariant Lagrangian density ${\cal L}_b$ for the two $(1 + 1)$-dimensional [^3] (2D) free Abelian gauge theory in the Feynman gauge \[3,4,18,35\] $$\begin{array}{lcl}
{\cal L}^{(2)}_b &=& -
{\displaystyle \frac{1}{4}}\; F^{\mu\nu} F_{\mu\nu} + B (\partial \cdot A)
+ {\displaystyle \frac{1}{2}}\;
B^2 - i \partial_\mu \bar C \partial^\mu C, \nonumber\\
&\equiv&
{\displaystyle \frac{1}{2}}\; E^2 + B (\partial \cdot A)
+ {\displaystyle \frac{1}{2}}\; B^2 - i \partial_\mu \bar C \partial^\mu C,
\end{array} \eqno(2.1)$$ where $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ is the antisymmetric field strength (curvature) tensor derived from the 2-form $F^{(2)} = d A^{(1)} = \frac{1}{2} (dx^\mu \wedge
dx^\nu) F_{\mu\nu}$. The latter is constructed by the application of the exterior derivative $d = dx^\mu \partial_\mu$ (with $ d^2 =
0$) on the 1-form connection $A^{(1)} = dx^\mu A_\mu$ which defines the vector potential $A_\mu$ for the Abelian gauge theory. Thus, the operation of $d$ on 1-form increases the degree by $+1$. It will be noted that $F_{\mu\nu}$ has only electric component and the magnetic component of $F_{\mu\nu}$ is zero in 2D. The Nakanishi-Lautrup auxiliary field $B$ has been introduced to linearize the gauge-fixing term $- \frac{1}{2} (\partial \cdot
A)^2$ and the fermionic $(\bar C^2 = C^2 = 0, C \bar C + \bar C C
= 0$) (anti-)ghost fields $(\bar C)C$ are required to maintain the unitarity and quantum gauge (i.e. BRST) invariance together for a given physical process allowed by the theory. At this stage, it is worth emphasizing that the gauge-fixing term $(\partial\cdot A)$ owes its origin to the other nilpotent $(\delta^2 = 0$) cohomological operator $\delta$ because the operation of the latter ($ \delta A^{(1)} = - * d * A^{(1)} = (\partial \cdot A))$ on the 1-form $A^{(1)}$ produces it. The operator $\delta = - * d
*$, which decreases the degree of a form by $1$, is known as the co-exterior derivative and $*$ is the Hodge duality operation on the 2D spacetime manifold. The action of the Laplacian operator $\Delta$ on the 1-form $A^{(1)}$ (i.e. $\Delta A^{(1)} = dx^\mu
\Box A_\mu$) leads to the derivation of the equation of motion $\Box A_\mu = 0$ for the gauge field $A_\mu$ if we demand the validity of the Laplace equation $\Delta A^{(1)} = 0$. The degree of a form remains intact under the operation of $\Delta$. Thus, we note that all the three de Rham cohomological operators ($d,
\delta, \Delta$) of differential geometry play very important roles in the description of the gauge theories. One can linearize the kinetic energy term $\frac{1}{2} E^2$ of (2.1) by introducing another auxiliary field ${\cal B}$ as $$\begin{array}{lcl}
{\cal L}^{(2)}_B &=& {\cal B} E - \frac{1}{2} {\cal B}^2 + B (\partial \cdot A)
+ \frac{1}{2} B^2 - i \partial_\mu \bar C \partial^\mu C.
\end{array} \eqno(2.2)$$ For the special case of 2D free Abelian gauge theory, the auxiliary field ${\cal B}$ is analogous to the Nakanishi-Lautrup field $B$. In fact, the former linearizes of the kinetic energy term $\frac{1}{2} E^2$ in exactly the same manner as the latter linearizes the gauge-fixing term $- \frac{1}{2} (\partial \cdot A)^2$. The above Lagrangian density (2.2) is endowed with the following local, off-shell nilpotent $(s_{(a)b}^2 = 0)$ and anticommuting ($s_b s_{ab} + s_{ab} s_b = 0$) (anti-)BRST $s_{(a)b}$ transformations [^4] $$\begin{array}{lcl}
s_b A_\mu &=& \partial_\mu C, \qquad s_b C = 0, \qquad
s_b \bar C = i B, \quad s_b B = 0, \qquad s_b {\cal B} = 0, \quad
s_b E = 0, \nonumber\\
s_{ab} A_\mu &=& \partial_\mu \bar C, \;\quad s_{ab} \bar C = 0, \quad
s_{ab} C = - i B, \quad s_{ab} B = 0, \quad s_{ab} {\cal B} = 0,
\quad s_{ab} E = 0.
\end{array} \eqno(2.3)$$ The key point to be noted, at this stage, is the fact that the kinetic energy term (more precisely the electric field itself), owing its origin to the exterior derivative $d$ and $A^{(1)}$, remains invariant under the (anti-)BRST transformations. In contrast, under the following local, off-shell nilpotent ($s_{(a)d}^2 = 0$) and anticommuting ($s_d s_{ad} + s_{ad} s_d =
0$) (anti-)co-BRST (or (anti-)dual-BRST) transformations $s_{(a)d}$ $$\begin{array}{lcl}
s_d A_\mu &=& -\varepsilon_{\mu\nu} \partial^\nu \bar C,
\quad s_d \bar C = 0, \quad
s_d C = - i {\cal B}, \nonumber\\
s_d B &=& 0, \quad s_d {\cal B} = 0, \quad
s_d (\partial \cdot A) = 0,
\nonumber\\
s_{ad} A_\mu &=& -\varepsilon_{\mu\nu} \partial^\nu C,
\quad s_{ad} C = 0, \quad
s_{ad} \bar C = + i {\cal B}, \nonumber\\
s_{ad} B &=& 0, \qquad s_{ad} {\cal B} = 0,
\qquad s_{ad} (\partial \cdot A) = 0,
\end{array} \eqno(2.4)$$ it is the gauge-fixing term (more precisely $(\partial\cdot A)$ itself), owing its origin to the co-exterior derivative $\delta$ and $A^{(1)}$, remains invariant. The anticommutator $(s_w = \{
s_b, s_d \} = \{s_{ab}, s_{ad} \})$ of the (anti-)BRST- and (anti-)co-BRST transformations leads to the existence of a non-nilpotent $s_w^2 \neq 0$ bosonic symmetry transformation in the theory \[36-38\] under which the (anti-)ghost fields do not transform at all. This bosonic symmetry is the analogue of the Laplacian operator of the differential geometry. There exists a global ghost scale symmetry transformation: $ s_g A_\mu = 0, s_g B
= 0, s_g {\cal B} = 0, s_g C = -\Lambda C, s_g \bar C = +\Lambda
\bar C$, under which, the Lagrangian density (2.2) remains invariant. Here $\Lambda$ is an infinitesimal spacetime independent (global) parameter. All the above six symmetry transformations can be concisely expressed, in terms of the generic local field $\Sigma (x) = A_\mu (x), C (x), \bar C (x), B
(x), {\cal B} (x)$, as $$\begin{array}{lcl}
s_r\; \Sigma (x) = - i \; [\; \Sigma (x) , Q_r\; ]_{\pm}, \qquad\;\;
r = b, ab, d, ad, w, g,
\end{array} \eqno(2.5)$$ where $(+)-$ signs on the square brackets stand for the (anti-)commutator for the generic local field $\Sigma$ being (fermionic)bosonic in nature. Here $Q_r$ are the generator of transformations which can be derived from the Noether’s theorem. Their exact form is not required for our present discussion but their explicit and exact form can be found in \[36-38\].\
[**2.1 Superfield formulation of (anti-)BRST symmetries: a concise review**]{}\
We begin here with a four ($2 + 2$)-dimensional supermanifold parametrized by the superspace coordinates $Z^M =
(x^\mu, \theta, \bar \theta)$ where $x^\mu (\mu = 0, 1)$ are the two even (bosonic) spacetime coordinates and $\theta$, $\bar
\theta$ are the two odd (Grassmannian) coordinates (with $\theta^2
= \bar \theta^2 = 0, \theta \bar \theta + \bar \theta \theta =
0)$. On this supermanifold, one can define a supervector 1-form superfield $\tilde A^{(1)} = d Z^M \tilde A_M (x, \theta,
\bar\theta)$ with the following component multiplet superfields (see, e.g., \[13,12\]) $$\begin{array}{lcl}
\tilde A_M (x,\theta,\bar\theta) = \Bigl ( B_{\mu} (x, \theta, \bar \theta),
\;\Phi (x, \theta, \bar \theta),
\;\bar \Phi (x, \theta, \bar \theta) \Bigr ).
\end{array} \eqno(2.6)$$ It will be noted that component superfields $B_\mu (x,\theta,\bar\theta), \Phi (x,\theta,\bar\theta),
\bar \Phi (x, \theta, \bar\theta)$ are the generalization of the basic local fields $A_\mu (x), C (x), \bar C (x)$, defined on the 2D ordinary spacetime manifold, to the four $(2 + 2)$-dimensional supermanifold. The most general expansion of these superfields along the Grassmannian directions of the supermanifold, is \[13,27-34\] $$\begin{array}{lcl}
B_{\mu}\; (x, \theta, \bar \theta) &=& A_{\mu} (x)
+ \;\theta\; \bar R_\mu (x)
+ \;\bar \theta\; R_\mu (x)
+ i \;\theta \;\bar \theta \;S_{\mu} (x), \nonumber\\
\Phi\; (x, \theta, \bar \theta) &=& C (x) \;+ i \; \theta\; \bar B (x)
\;- i \;\bar \theta\; {\cal B} (x)
\;+ i \;\theta \;\bar \theta \; s (x),
\nonumber\\
\bar \Phi\; (x, \theta, \bar \theta) &=& \bar C (x)
\;- i \;\theta\;\bar {\cal B} (x)) \;+ i\;\bar \theta\; B (x)
\;+ i\;\theta\;\bar \theta \; \bar s (x),
\end{array} \eqno(2.7)$$ where $(+)-$ signs in the above expansion have been chosen for the algebraic convenience. It should be noted that (i) in the limit $\theta \to 0,
\bar\theta \to 0$, we get back the local basic fields $A_\mu (x), C (x),
\bar C (x)$ of the theory from the superfields $B_\mu (x,\theta,\bar\theta), \Phi (x,\theta,\bar\theta),
\bar \Phi (x,\theta, \bar\theta)$. (ii) The fermionic degrees of freedom ($ C, \bar C, R_\mu, \bar R_\mu, s, \bar s)$ match with that of the bosonic $(A_\mu, S_\mu, B, \bar B, {\cal B}, \bar {\cal B})$ degrees of freedom so that the expansion can be consistent with the basic tenets of supersymmetry. (iii) All the fields on the r.h.s. of the expansion are the local functions of spacetime $x^\mu$ alone.
The secondary fields (i.e. $R_\mu, \bar R_\mu, S_\mu, s, \bar s$) can be expressed in terms of the basic fields (i.e. $A_\mu, C,
\bar C, B, {\cal B}$) of the Lagrangian density (2.2) by exploiting the horizontality condition ($\tilde F^{(2)} =
F^{(2)}$) where the super curvature 2-form $\tilde F^{(2)} =
\tilde d \tilde A^{(1)}$, defined on the $(2 + 2)$-dimensional supermanifold, is equated with the ordinary 2-form curvature $F^{(2)} = d A^{(1)}$, defined on the 2D ordinary flat Minkowskian spacetime manifold. The explicit expressions for these forms are $$\begin{array}{lcl}
\tilde F^{(2)} = \frac{1}{2}\; (d Z^M \wedge d Z^N)\; \tilde
F_{MN} = \tilde d \tilde A^{(1)}, \qquad
F^{(2)} = \frac{1}{2}\; (dx^\mu \wedge d x^\nu)\;
F_{\mu\nu} = d A^{(1)},
\end{array} \eqno(2.8)$$ where the exact expressions for $\tilde d$, $\tilde A^{(1)}$ and $\tilde d \tilde A^{(1)} = \tilde F^{(2)}$ (constructed by $\tilde d, \tilde A^{(1)}$), are $$\begin{array}{lcl}
\tilde d &=& \;d Z^M \;\partial_{M} = d x^\mu\; \partial_\mu\; +
\;d \theta \;\partial_{\theta}\; + \;d \bar \theta
\;\partial_{\bar \theta}, \nonumber\\ \tilde A^{(1)} &=& d Z^M\;
\tilde A_{M} = d x^\mu \;B_{\mu} (x , \theta, \bar \theta) + d
\theta\; \bar \Phi (x, \theta, \bar \theta) + d \bar \theta\; \Phi
( x, \theta, \bar \theta),
\end{array}\eqno(2.9)$$ $$\begin{array}{lcl}
\tilde F^{(2)} &=& \tilde d \tilde A^{(1)} = (d x^\mu \wedge d
x^\nu)\; (\partial_{\mu} B_\nu) - (d \theta \wedge d \theta)\;
(\partial_{\theta} \bar \Phi) + (d x^\mu \wedge d \bar \theta)
(\partial_{\mu} \Phi - \partial_{\bar \theta} B_{\mu})\nonumber\\
&+& (d x^\mu \wedge d \theta) (\partial_{\mu} \bar \Phi -
\partial_{\theta} B_{\mu}) - (d \bar\theta \wedge d \bar\theta)
(\partial_{\bar\theta} \Phi) - (d \theta \wedge d \bar\theta)
(\partial_{\theta} \Phi + \partial_{\bar\theta} \bar \Phi).
\end{array}\eqno(2.10)$$ Ultimately, the application of soul-flatness (horizontality) condition ($ \tilde d \tilde A^{(1)} = d A^{(1)}$), leads to the following restrictions (cf. (2.11) ) and thereby the ensuing relationships (cf. (2.12)) $$\begin{array}{lcl}
\partial_\mu \Phi &=& \partial_{\bar\theta} B_\mu, \qquad
\partial_\mu \bar \Phi = \partial_{\theta} B_\mu, \qquad
\partial_{\theta} \bar \Phi = \partial_{\bar\theta} \Phi = 0, \nonumber\\
\partial_\mu R_\nu &=& \partial_{\nu } R_\mu, \qquad
\partial_\mu \bar R_\nu = \partial_{\nu } \bar R_\mu, \qquad
\partial_{\theta} \Phi + \partial_{\bar\theta} \bar \Phi = 0,
\end{array} \eqno(2.11)$$ $$\begin{array}{lcl}
R_{\mu} \;(x) &=& \partial_{\mu}\; C(x), \qquad
\bar R_{\mu}\; (x) = \partial_{\mu}\;
\bar C (x), \qquad
S_{\mu}\; (x) = \partial_{\mu} B\; (x),
\nonumber\\
{\cal B}\; (x) &=&
\bar {\cal B}\; (x) = 0,
\qquad \bar s\;(x) = s\; (x) = 0, \qquad B (x) + \bar B (x) = 0.
\end{array} \eqno(2.12)$$ The insertion of all the above values into the most general expansion (2.7) on the $(2 + 2)$-dimensional supermanifold leads to the derivation of the off-shell nilpotent (anti-)BRST transformations for the most basic fields $A_\mu, C, \bar C$ as expressed below $$\begin{array}{lcl}
B_{\mu}\; (x, \theta, \bar \theta) &=& A_{\mu} (x)
+ \;\theta\; (s_{ab} A_{\mu} (x))
+ \;\bar \theta\; (s_{b} A_{\mu} (x))
+ \;\theta \;\bar \theta \;(s_{b} s_{ab} A_{\mu} (x)), \nonumber\\
\Phi\; (x, \theta, \bar \theta) &=& C (x) \;+ \; \theta\; (s_{ab} C (x))
\;+ \;\bar \theta\; (s_{b} C (x))
\;+ \;\theta \;\bar \theta \;(s_{b}\; s_{ab} C (x)),
\nonumber\\
\bar \Phi\; (x, \theta, \bar \theta) &=& \bar C (x)
\;+ \;\theta\;(s_{ab} \bar C (x)) \;+\bar \theta\; (s_{b} \bar C (x))
\;+\;\theta\;\bar \theta \;(s_{b} \;s_{ab} \bar C (x)).
\end{array} \eqno(2.13)$$ It should be noted that (i) the third- and the fourth terms in the above expansion of $\Phi (x,\theta,\bar\theta)$ and the second- and the fourth terms of the above expansion of $\bar\Phi (x,\theta,\bar\theta)$ are exactly equal to zero because $s_b C = 0, s_{ab} \bar C = 0, s_{(a)b} B = 0$. (ii) A comparison with (2.5) establishes the geometrical interpretation for the nilpotent ($Q_{(a)b}^2 = 0$) (anti-)BRST charges $Q_{(a)b}$ as the translation generators along $(\theta)\bar\theta$-directions of the supermanifold. In fact, there exists a mapping $$\begin{array}{lcl}
s_{b} \leftrightarrow \mbox{Lim}_{\theta \to 0}
{\displaystyle \frac{\partial}{\partial \bar\theta}} \leftrightarrow Q_b,
\qquad
s_{ab} \leftrightarrow \mbox{Lim}_{\bar \theta \to 0}
{\displaystyle \frac{\partial}{\partial \theta}} \leftrightarrow Q_{ab},
\end{array} \eqno(2.14)$$ among the (anti-)BRST transformations $s_{(a)b}$, the translation generators along $(\theta)\bar\theta$-direction of the supermanifold and the nilpotent (anti-)BRST charges $Q_{(a)b}$. (iii) The geometrical interpretation of the nilpotency property is contained in the translations generators which satisfy $(\partial/\partial\theta)^2 =
(\partial/\partial\bar\theta)^2 = 0$. (iv) The anticommutativity properties of the transformations $s_b s_{ab} + s_{ab} s_b = 0$ and their corresponding generators $Q_b Q_{ab} + Q_{ab} Q_b = 0$ are reflected in the specific property of the translation generators $(\partial/\partial\theta) (\partial/\partial\bar\theta)
+ (\partial/\partial\bar\theta) (\partial/\partial\theta) = 0$. (v) Under the (anti-)BRST transformations, the superfields $(\bar\Phi)\Phi$ convert themselves from the general superfields (cf. (2.7)) to the (anti-)chiral superfields (i.e. $\Phi (x,\theta,\bar\theta) = C (x) - i\; \theta\; B (x),
\bar \Phi (x,\theta,\bar\theta) = \bar C (x) + i\; \bar \theta \;B (x)$) because these satisfy $\partial_{\bar\theta} \Phi = 0,
\partial_\theta \bar \Phi = 0$.\
[**2.2 Hodge duality operation on $(2 + 2)$-dimensional supermanifold**]{}\
It is evident from the previous Section that we have been able to derive the local, covariant, nilpotent ($s_{(a)b}^2 =
0$) and anticommuting ($s_b s_{ab} + s_{ab} s_b = 0$) (anti-)BRST symmetry transformations $s_{(a)b}$ without any recourse to the definition of the Hodge duality operation. This is primarily due to the fact that we have exploited only the (super) exterior derivatives $(\tilde d) d$ and the (super) 1-form connections $(\tilde A^{(1)}) A^{(1)}$ in the horizontality condition $\tilde
d \tilde A^{(1)} = d A^{(1)}$ where the Hodge duality operation plays [*no*]{} role at all. For the derivation of the local, covariant, nilpotent ($s_{(a)d}^2 = 0$) and anticommuting ($s_d
s_{ad} + s_{ad} s_d = 0$) (anti-)co-BRST symmetry transformations $s_{(a)d}$, we have to tap the potential and power of the super co-exterior derivative $\tilde \delta = - \star \tilde d \star$ and the ordinary co-exterior derivative $\delta = - * d *$ in the dual-horizontality condition $\tilde \delta \tilde A^{(1)} =
\delta A^{(1)}$, where (i) $\star$ and $*$ are the Hodge duality operations, and (ii) $\tilde A^{(1)}$ and $A^{(1)}$ are the (super) 1-form connections on the supermanifold and ordinary manifold, respectively. On the four $(2 + 2)$-dimensional supermanifold, there exist three independent 4-forms (and their linear combinations are also allowed). These independent 4-forms are $$\begin{array}{lcl}
&&\phi_{1} = \frac{1}{2!}
(dx^\mu \wedge dx^\nu \wedge d \theta \wedge d \bar\theta)\;
{\cal F}_{\mu\nu\theta\bar\theta}, \quad
\phi_{2} = \frac{1}{2!}
(dx^\mu \wedge dx^\nu \wedge d \theta \wedge d \theta)\;
{\cal F}_{\mu\nu\theta\theta}, \nonumber\\
&&\phi_{3} = \frac{1}{2!}
(dx^\mu \wedge dx^\nu \wedge d \bar\theta \wedge d \bar\theta)\;
{\cal F}_{\mu\nu\bar\theta\bar\theta}.
\end{array} \eqno(2.15)$$ It will be noted that (i) the wedge product between the pure Grassmannian differentials is symmetric (i.e. $d\theta \wedge d\theta = d \theta \wedge d\theta,
d\theta \wedge d\bar \theta = d \bar \theta \wedge d\theta,
d\bar \theta \wedge d\bar \theta = d \bar \theta \wedge d\bar \theta$), the wedge product between the pure spacetime differentials is antisymmetric (i.e. $dx^\mu \wedge dx^\nu = - dx^\nu \wedge dx^\mu$), and the wedge product between the mixed differentials is also antisymmetric (i.e $dx^\mu \wedge d\theta = - d \theta \wedge dx^\mu,
dx^\mu \wedge d\bar\theta = - d \bar\theta \wedge d x^\mu$). Accordingly, the covariant indices of ${\cal F}$’s will also be symmetric as well as antisymmetric corresponding to our specific choice of these indices. (ii) On the $(2 + 2)$-dimensional supermanifold, more than two spacetime- as well as two Grassmannian differentials (e.g. $dx^\mu \wedge dx^\nu
\wedge dx^\lambda \wedge d \theta, dx^\mu \wedge d \theta
\wedge d \theta \wedge d \bar\theta$ etc.) are not allowed. (iii) For the present supermanifold, the overall numerical factors (e.g. $\frac{1}{2!}$), present in the definition of the superforms (e.g. (2.15)), correspond to such numerical factors present in the definition of ordinary forms on the ordinary spacetime manifold. (iv) The Hodge duality $\star$ operation for some selected super-forms on a six $(4 + 2)$-dimensional supermanifold have been defined in our earlier work \[42\]. However, some ad-hoc assumptions have been made in \[42\]. No such assumptions have been made in our present Hodge duality $\star$ definitions. (v) The operation of the Hodge duality on a given form does not affect ${\cal F}$’s [*per se*]{}. However, the wedge products, present in the above forms, are affected by the Hodge duality operation. For instance, a single Hodge duality $\star$ operation on the wedge product of the above cited differentials of the 4-forms, on the $(2 + 2)$-dimensional supermanifold, is $$\begin{array}{lcl}
&&\star\; (dx^\mu \wedge dx^\nu \wedge d \theta \wedge d \bar\theta)
= \varepsilon^{\mu\nu}, \qquad
\star\; (dx^\mu \wedge dx^\nu \wedge d \theta \wedge d \theta)
= \varepsilon^{\mu\nu} \; s^{\theta\theta}, \nonumber\\
&&\star\; (dx^\mu \wedge dx^\nu \wedge d \bar \theta \wedge d \bar\theta)
= \varepsilon^{\mu\nu}\; s^{\bar\theta\bar\theta},
\end{array} \eqno(2.16)$$ which, ultimately, imply the following zero-forms: $$\begin{array}{lcl}
\star\; \phi_{1} = \frac{1}{2!}\; \varepsilon^{\mu\nu}\;
{\cal F}_{\mu\nu\theta\bar\theta}, \qquad
\star \; \phi_{2} = \frac{1}{2!}\; \varepsilon^{\mu\nu} s^{\theta\theta}\;
{\cal F}_{\mu\nu\theta\theta}, \qquad
\star \; \phi_{3} = \frac{1}{2!}\;
\varepsilon^{\mu\nu} s^{\bar\theta\bar\theta}\;
{\cal F}_{\mu\nu\bar\theta\bar\theta}.
\end{array} \eqno(2.17)$$ At this juncture, a few comments are in order. First, in contrast to the ordinary spacetime differentials where $(dx^\mu \wedge dx^\mu) = 0$, the Grassmann differentials of the form $(d\theta \wedge d \theta)$ and $(d \bar\theta \wedge d \bar\theta)$ are non-zero on the supermanifold. Second, the coordinates $x^0, x^1, \theta, \bar\theta$ correspond to the [*four*]{} linearly independent directions on the ($2 + 2$)-dimensional supermanifold. This is why, a single $\star$ operation on ($dx^\mu \wedge dx^\nu \wedge d \theta \wedge d \bar\theta$) yields only $\varepsilon^{\mu\nu}$ on the r.h.s. The same does not happen when we take a single $\star$ operation on $(dx^\mu \wedge dx^\nu \wedge d \theta \wedge d \theta)$ and $(dx^\mu \wedge dx^\nu \wedge d \bar\theta \wedge d \bar \theta)$ because $(d\theta \wedge d \theta)$ and $(d \bar\theta \wedge d \bar\theta)$ do not incorporate the linearly independent differentials $d\theta$ and $d \bar\theta$ together. Third, the symmetric quantities $s^{\theta\theta}$ and $s^{\bar\theta\bar\theta}$ have been introduced so that one can keep track of the Grassmannian wedge products when a second Hodge duality operation is applied on a given form. For instance, two successive $\star$ operations on the wedge products corresponding to the independent 4-forms, yield the following $$\begin{array}{lcl}
&&\star\;
[\;\star\; (dx^\mu \wedge dx^\nu \wedge d \theta \wedge d \bar\theta)\;]
= - \; (dx^\mu \wedge dx^\nu \wedge d \theta \wedge d \bar\theta), \nonumber\\
&&\star\;
[\;\star\; (dx^\mu \wedge dx^\nu \wedge d \theta \wedge d \theta)\;]
= -\; (dx^\mu \wedge dx^\nu \wedge d \theta \wedge d \theta),
\nonumber\\
&&\star\;
[\;\star\; (dx^\mu \wedge dx^\nu \wedge d \bar \theta \wedge d \bar\theta)\;]
= -\; (dx^\mu \wedge dx^\nu \wedge d \bar \theta \wedge d \bar\theta),
\end{array} \eqno(2.18)$$ where we have used the following inputs while taking the second $\star$ operation $$\begin{array}{lcl}
&&\star\; \bigl [\;\varepsilon^{\mu\nu}\;\bigr ] =
\frac{1}{2!}\; \varepsilon_{\sigma\rho}\; (dx^\sigma \wedge d x^\rho
\wedge d \theta \wedge d \bar\theta)\;\varepsilon^{\mu\nu}, \nonumber\\
&&\star\;\bigl [\;\varepsilon^{\mu\nu}\; s^{\theta\theta}\;\bigr ] =
\frac{1}{2!}\; \varepsilon_{\sigma\rho}\; (dx^\sigma \wedge d x^\rho
\wedge d \theta \wedge d \theta)
\;\varepsilon^{\mu\nu}, \nonumber\\
&&\star\;\bigl [\;\varepsilon^{\mu\nu}\; s^{\bar\theta\bar\theta}\;\bigr ]
= \frac{1}{2!}\; \varepsilon_{\sigma\rho}\; (dx^\sigma \wedge d x^\rho
\wedge d \bar \theta \wedge d \bar\theta)\;
\varepsilon^{\mu\nu}.
\end{array} \eqno(2.19)$$ Thus, it is clear that the presence of the constant symmetric factors $s^{\theta\theta}, s^{\bar\theta\bar\theta}$ in (2.16) do provide a kind of guidance for the operation of a couple of Hodge duality $\star$ operations on a given wedge product (see, e.g., (2.18) and (2.19)). The double $\star$ operations are essential because our $\star$ definition should comply with the general requirements of a duality invariant theory where $\star (\star G) = \pm G$ is true for any arbitrary form $G$ (see, e.g., \[39\]).
Let us concentrate now on the 3-forms. These independent forms are five in number on the $(2 + 2)$-dimensional supermanifold. These are as given below $$\begin{array}{lcl}
&& \tau_{1} = \frac{1}{2!}\; (dx^\mu \wedge dx^\nu \wedge d \theta) \;
T_{\mu\nu\theta}, \;\;\;\qquad\;\;\;
\tau_{2} = \frac{1}{2!}\; (dx^\mu \wedge dx^\nu \wedge d\bar\theta)\;
T_{\mu\nu\bar\theta}, \nonumber\\
&&
\tau_{3} =
(d x^\mu \wedge d \theta \wedge d\theta)
\; T_{\mu\theta\theta}, \;\;\;\;\;\;\qquad\;\;\;\;\;\;
\tau_{4} =
(d x^\mu \wedge d \bar \theta \wedge d\bar\theta)\;
T_{\mu\bar\theta\bar\theta}, \nonumber\\
&&
\tau_{5} =
(d x^\mu \wedge d \theta \wedge d\bar\theta)\;
\;T_{\mu\theta\bar\theta}.
\end{array} \eqno(2.20)$$ As discussed earlier, the operation of the Hodge duality would affect the wedge products. This is why, we shall obtain a set of 1-forms as the dual to the above 3-forms. The explicit expressions for a single $\star$ operation on the wedge products corresponding to 3-forms, are $$\begin{array}{lcl}
&& \star\;(dx^\mu \wedge dx^\nu \wedge d \theta) =
\varepsilon^{\mu\nu}\;(d \bar\theta), \;\;\;\;\;\;\qquad\;\;
\star\; (dx^\mu \wedge dx^\nu \wedge d\bar\theta)\; =
\varepsilon^{\mu\nu}\;(d\theta), \nonumber\\ && \star\;(d x^\mu
\wedge d \theta \wedge d\bar\theta) = \varepsilon^{\mu\nu}\;(d
x_\nu), \;\;\;\qquad\;\;\;\; \star\; (d x^\mu \wedge d \bar \theta
\wedge d\bar\theta)\; = \varepsilon^{\mu\nu}\;(d x_\nu)
s^{\bar\theta\bar\theta}, \nonumber\\ && \star\; (d x^\mu \wedge d
\theta \wedge d\theta)\; = \varepsilon^{\mu\nu}\;(d x_\nu)
s^{\theta\theta}.
\end{array} \eqno(2.21)$$ Application of (2.21) to (2.20) (with inputs as the analogue of (2.19)) imply $$\begin{array}{lcl}
&& \star \;\tau_{1} = \frac{1}{2!}\; \varepsilon^{\mu\nu}\; (d \bar \theta) \;
T_{\mu\nu\theta}, \;\;\;\qquad\;\;\;\;\;\;
\star\;\tau_{2} = \frac{1}{2!}\;
\varepsilon^{\mu\nu}\;(d \theta)\;
T_{\mu\nu\bar\theta}, \nonumber\\
&&
\star\; \tau_{3} = \varepsilon^{\mu\nu} \;s^{\theta\theta}\;
(d x_\nu)\; T_{\mu\theta\theta}, \;\;\;\qquad\;\;\;
\star\; \tau_{4} =
\varepsilon^{\mu\nu}\;s^{\bar\theta\bar\theta}\;(d x_\nu)\;
T_{\mu\bar\theta\bar\theta}, \nonumber\\
&&
\star\; \tau_{5} =
\varepsilon^{\mu\nu}\; (d x_\nu)\;
\;T_{\mu\theta\bar\theta},
\end{array} \eqno(2.22)$$ which are dual to the 3-forms given in (2.20). The double $\star$ operation on the wedge products corresponding to 3-forms, are $$\begin{array}{lcl}
&&\star\;
[\; \star\; (dx^\mu \wedge dx^\nu \wedge d \theta)\;] = -
(dx^\mu \wedge dx^\nu \wedge d \theta), \nonumber\\
&& \star\;[ \;\star\; (dx^\mu \wedge dx^\nu \wedge d \bar\theta)\;] = -
(dx^\mu \wedge dx^\nu \wedge d \bar\theta),
\nonumber\\
&& \star\; [\; \star\; (dx^\mu \wedge d \theta \wedge d \bar\theta)\;] =
+\; (dx^\mu \wedge d \theta \wedge d \bar\theta), \nonumber\\
&&\star\; [\; \star\; (dx^\mu \wedge d \theta \wedge d \theta)\; ] =
+ (dx^\mu \wedge d \theta \wedge d \theta), \nonumber\\
&& \star\; [\; \star\; (dx^\mu \wedge d \bar \theta \wedge d \bar \theta)\; ]
= +\;
(dx^\mu \wedge d \bar \theta \wedge d \bar \theta).
\end{array} \eqno(2.23)$$ There exist six independent 2-forms on the four $(2 + 2)$-dimensional supermanifold as $$\begin{array}{lcl}
&& \chi_{1} = \frac{1}{2!}\; (dx^\mu \wedge dx^\nu) \;
S_{\mu\nu}, \;\;\qquad\;
\chi_{2} = (d \theta \wedge d \bar\theta)\;
S_{\theta\bar\theta}, \nonumber\\
&&
\chi_{3} = (d\theta \wedge d \theta) \;
S_{\theta\theta}, \;\;\;\;\;\qquad\;\;\;\;
\chi_{4} =
(d\bar\theta \wedge d \bar\theta)\;
S_{\bar\theta\bar\theta}, \nonumber\\
&&\chi_{5} = (dx^\mu \wedge d\theta)
\;S_{\mu\theta}, \;\;\;\qquad \;\;\;\;
\chi_{6} = (dx^\mu \wedge d\bar\theta)
\;S_{\mu\bar\theta}.
\end{array} \eqno(2.24)$$ A single $\star$ operation on the wedge products corresponding to the above 2-forms are as $$\begin{array}{lcl}
&& \star\; (dx^\mu \wedge dx^\nu) = \varepsilon^{\mu\nu}\;
(d \theta \wedge d \bar\theta), \;\;\qquad
\star\; (d \theta \wedge d\bar\theta)
= \frac{1}{2!} \varepsilon^{\mu\nu} (dx_\mu \wedge dx_\nu), \nonumber\\
&& \star\; (d \theta \wedge d\theta) = \frac{1}{2!} s^{\theta\theta}
\varepsilon^{\mu\nu} (dx_\mu \wedge dx_\nu), \quad
\star\; (d \bar\theta \wedge d\bar\theta)
= \frac{1}{2!} s^{\bar\theta\bar\theta}
\varepsilon^{\mu\nu} (dx_\mu \wedge dx_\nu), \nonumber\\
&& \star\;(dx^\mu \wedge d \theta) = \varepsilon^{\mu\nu}\; (dx_\nu \wedge
d \bar\theta), \;\;\qquad\;
\star\;(dx^\mu \wedge d \bar \theta) = \varepsilon^{\mu\nu}\; (dx_\nu \wedge
d \theta),
\end{array} \eqno(2.25)$$ which clearly establish the fact that the dual of 2-forms (cf. 2.24) are 2-forms on a four $(2 + 2)$-dimensional supermanifold as listed below $$\begin{array}{lcl}
&& \star\; \chi_{1} = \frac{1}{2!}\; \varepsilon^{\mu\nu} \;
(d \theta \wedge d \bar\theta)\;
S_{\mu\nu},\;\; \;\;\;\qquad\;\;\;\;\;\;
\star\; \chi_{2} = \frac{1}{2!}\; \varepsilon_{\sigma\rho}\;
(dx^\sigma \wedge dx^\rho)\;
S_{\theta\bar\theta}, \nonumber\\
&&
\star\;\chi_{3} = s^{\theta\theta} \;
\frac{1}{2!}\;
\varepsilon_{\mu\nu}\; (dx^\mu \wedge dx^\nu)
\;S_{\theta\theta}, \qquad
\star\;\chi_{4} =
s^{\bar\theta\bar\theta}\;
\frac{1}{2!}\;
\varepsilon_{\mu\nu}\; (dx^\mu \wedge dx^\nu)
\;S_{\bar\theta\bar\theta}, \nonumber\\
&&\star\; \chi_{5} = \varepsilon^{\mu\nu}\; (dx_\nu \wedge d \bar\theta)\;
\;S_{\mu\theta}, \;\;\;\qquad \;\;\;\;\;\;\;
\star\;\chi_{6} = \varepsilon^{\mu\nu} \;(dx_\nu \wedge d\theta)
\;S_{\mu\bar\theta}.
\end{array} \eqno(2.26)$$ The double $\star$ operation on the wedge products corresponding to the six independent 2-forms on the $(2 + 2)$-dimensional supermanifold is $$\begin{array}{lcl}
&& \star\; [\;\star\; (dx^\mu \wedge dx^\nu)\;] = -\;
(dx^\mu \wedge dx^\nu),\qquad
\star\; [\; \star\; (d \theta \wedge d\bar\theta)\;]
= -\;
(d \theta \wedge d\bar\theta),
\nonumber\\
&& \star\; [\; \star\; (d \theta \wedge d\theta)] = -\;
(d \theta \wedge d\theta), \;\;\;\qquad\;\;\;\;\;
\star\; [\; \star\; (d \bar\theta \wedge d\bar\theta)\;] = -\;
(d \bar\theta \wedge d\bar\theta),
\nonumber\\
&& \star\; [\;\star\;(dx^\mu \wedge d \theta)\;] =
+\; (dx^\mu \wedge d \theta), \;\;\qquad\;\;
\star\; [\;\star\;(dx^\mu \wedge d \bar \theta)\;] =
+\; (dx^\mu \wedge d \bar \theta).
\end{array} \eqno(2.27)$$ It is straightforward to guess that there exist only three independent 1-forms on the four $(2 + 2)$-dimensional supermanifold as [^5] $$\begin{array}{lcl}
\alpha_{1} =
(dx^\mu)\;
{\cal A}_{\mu}, \qquad
\alpha_{2} =
(d \theta)\;
{\cal A}_{\theta}, \qquad
\alpha_{3} =
(d \bar\theta)\;
{\cal A}_{\bar\theta}.
\end{array} \eqno(2.28)$$ A single $\star$ operation on the above independent 1-forms would lead to the 3-forms on the four $(2 + 2)$-dimensional supermanifold. The operation of the single Hodge duality on the independent 1-form differentials are $$\begin{array}{lcl}
&& \star\; (dx^\mu) = \varepsilon^{\mu\nu}\; (d x_\nu \wedge d \theta
\wedge d \bar\theta), \qquad
\star\; (d \theta) = \frac{1}{2!}\; \varepsilon^{\mu\nu}\;
(dx_\mu \wedge d x_\nu \wedge d \bar\theta), \nonumber\\
&& \star\; (d \bar\theta) = \frac{1}{2!}\; \varepsilon^{\mu\nu} \;
(dx_\mu \wedge d x_\nu \wedge d \theta),
\end{array} \eqno(2.29)$$ which finally imply the following independent 3-forms corresponding to the independent 1-forms of equation (2.28), defined on the $(2 + 2)$-dimensional supermanifold, namely; $$\begin{array}{lcl}
&&\star\; \alpha_{1} =
\varepsilon^{\mu\nu} \;(dx_\nu \wedge d \theta \wedge d \bar\theta)\;
{\cal A}_{\mu}, \;\;\;\qquad\;\;\;
\star\;\alpha_{2} =
\frac{1}{2!}\; \varepsilon_{\sigma\rho}\;
(dx^\sigma \wedge dx^\rho \wedge d \bar\theta)\;
{\cal A}_{\theta}, \nonumber\\
&&\star\;\alpha_{3} =
\frac{1}{2!}\; \varepsilon_{\sigma\rho}\;
(dx^\sigma \wedge dx^\rho \wedge d \theta)\;
{\cal A}_{\bar\theta}.
\end{array} \eqno(2.30)$$ The result of a couple of successive $\star$ operations on the differentials, corresponding to the 1-forms on the $(2 + 2)$-dimensional supermanifold, is given by $$\begin{array}{lcl}
\star \; [\;\star\; (dx^\mu)\;] = +\; (d x^\mu), \qquad
\star \; [\;\star\; (d \theta)\;] = -\; (d \theta), \qquad
\star \; [\;\star\; (d \bar\theta)\;] = -\; (d \bar\theta).
\end{array} \eqno(2.31)$$ We shall be exploiting the above Hodge duality operations on the wedge products of the differentials of a given form in the forthcoming Section 2.3 in the context of the derivation of the (anti-)co-BRST symmetries for the 2D free 1-form Abelian gauge theory considered on a four $(2 + 2)$-dimensional supermanifold.\
[**2.3 Superfield formulation of (anti-)co-BRST symmetries for 2D theory**]{}\
It is clear from the symmetry transformations (2.4) that the local, covariant, continuous, nilpotent ($s_{(a)d}^2 = 0$) and anticommuting ($s_d s_{ad} + s_{ad} s_d = 0$) (anti-)co-BRST symmetries $s_{(a)d}$ exist for the Lagrangian density (2.2) describing the free (non-interacting) Abelian gauge theory on the flat 2D Minkowskian spacetime manifold. Exploiting the dual-horizontality condition $\tilde \delta \tilde A^{(1)} =
\delta A^{(1)}$ with the following inputs $$\begin{array}{lcl}
\tilde \delta \tilde A^{(1)} = - \star\; \tilde d\; \star \tilde
A^{(1)}, \qquad \delta A^{(1)} = -\; *\; d *\; A^{(1)} = (\partial
\cdot A),
\end{array} \eqno(2.32)$$ we expect to obtain all the secondary fields of the super expansion (2.7) in terms of the basic fields of the Lagrangian density (2.2) of the theory. Towards this goal in mind, we first explicitly compute $\tilde \delta \tilde A^{(1)} = - \star \;
\tilde d\; \star \; \tilde A^{(1)}$ taking the help of the definitions (2.9) and the Hodge duality operations discussed earlier. First, the dual ($\star \tilde A^{(1)})$ of the super 1-form connection $\tilde A^{(1)} = d Z^M \tilde A_M$ is a 3-form on the $(2 + 2)$-dimensional supermanifold. The explicit expression for this 3-form (i.e. dual to the 1-form super connection $\tilde A^{(1)}$) is $$\begin{array}{lcl}
\star\; \tilde A^{(1)} &=& \varepsilon^{\mu\nu} \; (dx_\nu \wedge
d \theta \wedge d \bar\theta)\; B_\mu + \frac{1}{2!}\;
\varepsilon_{\sigma\rho}\; (dx^\sigma \wedge dx^\rho \wedge d
\bar\theta)\; \bar \Phi \nonumber\\ &+& \frac{1}{2!}\;
\varepsilon_{\sigma\rho}\; (dx^\sigma \wedge dx^\rho \wedge d
\theta)\; \Phi,
\end{array} \eqno(2.33)$$ where we have used the definition of the 1-form super connection $\tilde A^{(1)}$ from (2.9) and the Hodge duality operations on the 1-forms from (2.29). We apply now the super exterior derivative $\tilde d = d Z^M \partial_M$ from (2.9) on the 3-form dual super connection (2.33), the outcome is $$\begin{array}{lcl}
\tilde d\; (\star \tilde A^{(1)}) &=& \varepsilon^{\mu\nu}\;
(dx_\xi \wedge dx_\nu \wedge d\theta \wedge d \bar\theta)\;
(\partial^\xi B_\mu) - \frac{1}{2!} \varepsilon_{\sigma\rho}\;
(dx^\sigma \wedge dx^\rho \wedge d\theta \wedge d\bar\theta)\;
(\partial_\theta \bar \Phi)\nonumber\\ &-&
\frac{1}{2!} \varepsilon_{\sigma\rho}\;
(dx^\sigma \wedge dx^\rho \wedge d \bar\theta \wedge d\bar\theta)\;
(\partial_{\bar\theta} \bar \Phi)
- \frac{1}{2!} \varepsilon_{\sigma\rho}\;
(dx^\sigma \wedge dx^\rho \wedge d\theta \wedge d \bar \theta)\;
(\partial_{\bar\theta} \Phi)\nonumber\\
&-& \frac{1}{2!} \varepsilon_{\sigma\rho}\;
(dx^\sigma \wedge dx^\rho \wedge d\theta \wedge d \theta)\;
(\partial_\theta \Phi).
\end{array} \eqno(2.34)$$ A few remarks are in order. First, all the wedge products with more than two spacetime differentials- as well as Grassmannian differentials are dropped out because a $(2 + 2$)-dimensional supermanifold cannot support such forms. Second, the negative signs, in the above, have cropped up because $(d \theta \partial_\theta) (dx^\mu \wedge dx^\nu \wedge d \theta) \bar \Phi
= - (dx^\mu \wedge dx^\nu \wedge d\theta \wedge d \theta)\;
\partial_\theta \bar\Phi$, etc. The stage is now set for the application of the $(- \star)$ on the above super 4-forms which will lead to the derivation of a 0-form (superscalar) on the supermanifold. Exploiting the Hodge duality operation, defined in (2.16), we obtain the following expression $$\begin{array}{lcl}
\tilde \delta \tilde A^{(1)} = - \star \; \tilde d \; \star \;
\tilde A^{(1)} = (\partial \cdot B) - (\partial_\theta \bar\Phi +
\partial_{\bar \theta} \Phi) - s^{\theta\theta}\; (\partial_\theta
\Phi) - s^{\bar\theta\bar\theta}\; (\partial_{\bar\theta} \bar
\Phi),
\end{array} \eqno(2.35)$$ where we have used $\varepsilon_{\sigma\rho}
\varepsilon^{\sigma\rho} = - 2!, \varepsilon^{\mu\nu}
\varepsilon_{\xi\nu} = - \delta^\mu_\xi$, etc. When the above superscalar is equated with the ordinary scalar (i.e. $\delta
A^{(1)} = - * d * A^{(1)} = (\partial \cdot A)$) due to the requirement of the dual-horizontality condition (i.e. $\tilde
\delta \tilde A^{(1)} = \delta A^{(1)}$), we obtain the following restrictions $$\begin{array}{lcl}
(\partial \cdot B) - (\partial_\theta \bar \Phi +
\partial_{\bar\theta} \Phi) = (\partial \cdot A), \qquad
\partial_\theta \Phi = 0, \qquad \partial_{\bar\theta} \bar \Phi = 0.
\end{array} \eqno(2.36)$$ The insertion of the most general super expansions (cf. (2.7)) on the $(2 + 2)$-dimensional supermanifold for the superfields $B_\mu (x,\theta,\bar\theta), \Phi (x,\theta,\bar\theta),
\bar \Phi (x,\theta, \bar\theta)$ into the above restriction leads to $$\begin{array}{lcl}
&&(\partial \cdot R) (x) = 0, \;\;\qquad\; (\partial \cdot \bar R) (x) = 0,
\;\;\qquad\;
(\partial \cdot S) (x) = 0, \nonumber\\
&& s (x) = \bar s (x) = B (x) = \bar B (x) = 0, \;\;\;\qquad \;\;
{\cal B} (x) + \bar {\cal B} (x) = 0.
\end{array} \eqno(2.37)$$ It is worthwhile to mention that, unlike the horizontality condition where the secondary fields are expressed explicitly and exactly in terms of the basic fields of the Lagrangian density (2.2), the dual-horizontality condition provides only the restrictions that are quoted in (2.37). For the 2D free Abelian gauge theory, the local, covariant and continuous solutions for the above restrictions exist as given below [^6] $$\begin{array}{lcl}
R_\mu = - \varepsilon_{\mu\nu} \partial^\nu \bar C, \qquad
\bar R_\mu = - \varepsilon_{\mu\nu} \partial^\nu C, \qquad
S_\mu = + \varepsilon_{\mu\nu} \partial^\nu {\cal B}.
\end{array} \eqno(2.38)$$ Substitution of the above values into the most general super expansion in (2.7) leads to the following expression for the expansion [*vis-[à]{}-vis*]{} the off-shell nilpotent (anti-)co-BRST transformations of equation (2.4): $$\begin{array}{lcl}
B_{\mu}\; (x, \theta, \bar \theta) &=& A_{\mu} (x)
+ \;\theta\; (s_{ad} A_{\mu} (x))
+ \;\bar \theta\; (s_{d} A_{\mu} (x))
+ \;\theta \;\bar \theta \;(s_{d} s_{ad} A_{\mu} (x)), \nonumber\\
\Phi\; (x, \theta, \bar \theta) &=& C (x) \;+ \; \theta\; (s_{ad} C (x))
\;+ \;\bar \theta\; (s_{d} C (x))
\;+ \;\theta \;\bar \theta \;(s_{d}\; s_{ad} C (x)),
\nonumber\\
\bar \Phi\; (x, \theta, \bar \theta) &=& \bar C (x)
\;+ \;\theta\;(s_{ad} \bar C (x)) \;+\bar \theta\; (s_{d} \bar C (x))
\;+\;\theta\;\bar \theta \;(s_{d} \;s_{ad} \bar C (x)).
\end{array} \eqno(2.39)$$ This equation is the analogue of the expansion in (2.13) where (anti-)BRST symmetry transformations have been derived. It is clear from (2.39) (which produces the (anti-)co-BRST transformations for the basic fields $A_\mu, C, \bar C$) that (anti-)co-BRST nilpotent charges $Q_{(a)d}$, similar to the (anti-)BRST charges $Q_{(a)b}$, correspond to the translation generators $(\partial/\partial\theta,
\partial/\partial\bar\theta)$ along the Grassmannian directions $(\theta)\bar\theta$ of the supermanifold. However, there is a clear-cut distinction between these two sets of charges when it comes to the discussion of the nilpotent transformations for the (anti-)ghost fields corresponding to the fermionic superfields $\Phi$ and $\bar\Phi$. For instance, under the nilpotent anti-BRST transformations, the superfield $\Phi$ becomes anti-chiral (i.e. $\Phi = C + \theta\; (s_{ab} C (x))$) but the same superfield becomes chiral $(i.e. \
Phi = C (x) + \bar\theta \; (s_d C (x))$) due to the co-BRST transformations. Similar arguments and interpretations can be provided for the nature of the superfield $\bar \Phi$ as far as the off-shell nilpotent BRST and anti-co-BRST transformations are concerned.\
[**3 (Anti-)BRST and (anti-)co-BRST symmetries for 4D theory: a brief synopsis**]{}\
Let us start off with the analogue of the Lagrangian density (2.1) for the 4D free Abelian gauge theory defined on the four dimensional [^7] ordinary flat Minkowski spacetime manifold $$\begin{array}{lcl}
{\cal L}^{(4)}_b &=& -
{\displaystyle \frac{1}{4}}\; F^{\mu\nu} F_{\mu\nu} + B (\partial \cdot A)
+ {\displaystyle \frac{1}{2}}\;
B^2 - i \partial_\mu \bar C \partial^\mu C, \nonumber\\
&\equiv&
{\displaystyle \frac{1}{2}}\; ({\bf E}^2 - {\bf B}^2) + B (\partial \cdot A)
+ {\displaystyle \frac{1}{2}}\; B^2 - i \partial_\mu \bar C \partial^\mu C,
\end{array} \eqno(3.1)$$ where the field strength tensor $F_{\mu\nu} = \partial_\mu
A_\nu - \partial_\nu A_\mu$, constructed from $d = dx^\mu
\partial_\mu$ and the 1-form $A^{(1)} = dx^\mu A_\mu$ through $ F^{(2)} = d
A^{(1)} = \frac{1}{2} (dx^\mu \wedge dx^\nu) F_{\mu\nu}$, has the electric ($F_{0i} = E_i = {\bf E}$) and the magnetic ($F_{ij} =
\epsilon_{ijk} B_k, B_i = {\bf B} = - \frac{1}{2} \epsilon_{ijk}
F_{jk}$) components and the gauge-fixing term $(\partial \cdot A)
= \partial_0 A_0 - \partial_i A_i$ is constructed by the application of the nilpotent ($\delta^2 = 0$) co-exterior derivative $\delta = - * d *$ on the 1-form $A^{(1)} = dx^\mu
A_\mu$ (i.e. $\delta A^{(1)} = (\partial \cdot A)$). Here the Hodge duality $*$ operation is defined on the 4D Minkowskian flat spacetime manifold. All the other symbols carry the same meaning as discussed in Section 2. The above Lagrangian density can be linearized by introducing a couple of vector auxiliary fields ${\bf b^{(1)}}, {\bf b^{(2)}}$ as \[42\] $$\begin{array}{lcl}
{\cal L}^{(4)}_B &=& b^{(1)}_i E_i - \frac{1}{2} ({\bf b^{(1)}})^2
- b^{(2)}_i B_i + \frac{1}{2} ({\bf b^{(2)}})^2 + B (\partial \cdot A)
+ \frac{1}{2} B^2 - i \partial_\mu \bar C \partial^\mu C.
\end{array} \eqno(3.2).$$ The above Lagrangian density respects the following local, covariant, continuous, off-shell nilpotent ($s_{(a)b}^2 = 0$) and anticommuting ($s_b s_{ab} + s_{ab} s_b = 0$) (anti-)BRST $(s_{(a)b})$ symmetry transformations \[42\] $$\begin{array}{lcl}
s_b A_\mu &=& \partial_\mu C, \qquad s_b C = 0, \qquad
s_b \bar C = i B, \qquad s_b B = 0, \nonumber\\
s_b {\bf B} &=& 0, \qquad s_b {\bf b^{(1)}} = 0, \qquad s_b {\bf b^{(2)}} = 0,
\qquad s_b {\bf E} = 0, \nonumber\\
s_{ab} A_\mu &=& \partial_\mu \bar C, \qquad s_{ab} \bar C = 0, \qquad
s_{ab} C = - i B, \qquad s_{ab} B = 0, \nonumber\\
s_{ab} {\bf B} &=& 0, \qquad s_{ab} {\bf b^{(1)}} = 0, \qquad
s_{ab} {\bf b^{(2)}} = 0, \qquad
\quad s_{ab} {\bf E} = 0,
\end{array} \eqno(3.3)$$ because (3.2) transforms to a total derivative under the above transformations. Furthermore, the same Lagrangian density is endowed with the following non-local, non-covariant, continuous, off-shell nilpotent $(s_{(a)d}^2 = 0)$ and anticommuting ($s_d s_{ad} + s_{ad} s_d = 0$) (anti-)co-BRST symmetry transformations $(s_{(a)d})$ (see, e.g., \[40-42\] for details) $$\begin{array}{lcl}
s_d A_0 &=& i \bar C, \qquad s_d A_i = i
{\displaystyle \frac{\partial_0 \partial_i}{\nabla^2}} \bar C,
\qquad s_d \bar C = 0, \qquad s_d {\bf B} = 0,
\qquad s_d {\bf b^{(1)}} = 0,
\nonumber\\
s_d C &=& + {\displaystyle \frac{\partial_i b^{(1)}_i} {\nabla^2}},
\;\;\qquad s_d B = 0, \;\;\qquad s_d {\bf b^{(2)}} = 0, \;\;\qquad
s_d (\partial \cdot A) = 0,
\nonumber\\
s_{ad} A_0 &=& i C, \qquad s_{ad} A_i = i
{\displaystyle \frac{\partial_0 \partial_i}{\nabla^2}} C,
\qquad s_{ad} C = 0, \qquad s_{ad} {\bf B} = 0,
\qquad s_{ad} {\bf b^{(1)}} = 0,\nonumber\\
s_{ad} \bar C &=& - {\displaystyle \frac{\partial_i b^{(1)}_i} {\nabla^2}},
\;\;\qquad s_{ad} B = 0, \;\;\qquad s_{ad} {\bf b^{(2)}} = 0, \;\;\qquad
s_{ad} (\partial \cdot A) = 0,
\end{array} \eqno(3.4)$$ where $\nabla^2 = \partial_i \partial_i = (\partial_1)^2
+ (\partial_2)^2 + (\partial_3)^2$. At this stage, a few comments are in order. (i) It is clear that the (anti-)BRST symmetry transformations are local, covariant, continuous, nilpotent and anticommuting. In contrast, the (ant-)co-BRST symmetry transformations are non-local, non-covariant, continuous, nilpotent and anticommuting. (ii) The nilpotent (anti-)BRST as well as (anti-)co-BRST transformations keep the magnetic field ${\bf B}$ invariant. (iii) Under the (anti-)BRST and (anti-)co-BRST transformations, the 2-form $F^{(2)} = d A^{(1)}$ and the 0-form $(\partial\cdot A) = \delta A^{(1)}$ remain invariant, respectively. (iv) It is evident that the anticommutator $\{ s_d, s_b \} = \{s_{ab}, s_{ad} \} = s_w$ leads to the definition of a non-nilpotent bosonic symmetry $s_w$. However, the exact expressions for these transformations are not essential for our present discussions. (v) The global scale transformations on the (anti-)ghost fields define the ghost symmetry in the theory. The corresponding conserved charge is the ghost charge $Q_g$. (vi) The above conserved Noether charges generate the transformations (2.5).\
[**3.1 Superfield formulation of (anti-)BRST symmetries for 4D theory**]{}\
We consider the free four $(3 + 1)$-dimensional (4D) Abelian gauge theory on a six $(4 + 2)$-dimensional supermanifold parametrized by the four spacetime $x^\mu (\mu = 0, 1, 2, 3)$ bosonic variables and a couple of odd ($\theta^2 = \bar\theta^2 = 0,
\theta \bar\theta + \bar\theta \theta = 0$) Grassmannian variables $\theta$ and $\bar\theta$. The local basic fields $(A_\mu (x), C (x), \bar C (x))$ of the Lagrangian density (3.1) are now generalized to the superfields $(B_\mu (x,\theta,\bar\theta), \Phi (x,\theta,\bar\theta),
\bar\Phi (x,\theta,\bar\theta)$ on the six dimensional supermanifold. These latter superfields can be expanded in terms of the basic fields as given in (2.7). However, there is a subtle difference between the expansion on the four $(2 + 2)$-dimensional (cf. Section 2) and the six $(4 + 2)$-dimensional supermanifold. For instance, in the following $$\begin{array}{lcl}
B_{\mu}\; (x, \theta, \bar \theta) &=& A_{\mu} (x)
+ \;\theta\; \bar R_\mu (x)
+ \;\bar \theta\; R_\mu (x)
+ i \;\theta \;\bar \theta \;S_{\mu} (x), \nonumber\\
\Phi\; (x, \theta, \bar \theta) &=& C (x) \;+ i \; \theta\; \bar B (x)
\;- i \;\bar \theta\; {\cal B} (x)
\;+ i \;\theta \;\bar \theta \; s (x),
\nonumber\\
\bar \Phi\; (x, \theta, \bar \theta) &=& \bar C (x)
\;- i \;\theta\;\bar {\cal B} (x)) \;+ i\;\bar \theta\; B (x)
\;+ i\;\theta\;\bar \theta \; \bar s (x),
\end{array} \eqno(3.5)$$ the auxiliary scalar fields ${\cal B}$ and $\bar {\cal B}$ are [*not*]{} the ones that have been written for the 2D free Abelian gauge theory. In particular, the auxiliary scalar field ${\cal B}$ appears explicitly in the Lagrangian density (2.2) for the 2D theory. However, it does not appear explicitly in the Lagrangian density of the 4D theory. All the rest of the steps are exactly the same (see, e.g., equations (2.8)–(2.12)) as discussed in the sub-section 2.1 for the discussion of the 2D Abelian theory on a $(2 + 2)$-dimensional supermanifold. Finally, the horizontality condition $\tilde d \tilde A^{(1)} = d A^{(1)}$ leads to the derivation of the nilpotent (anti-)BRST symmetry transformations (3.3) for the 4D free Abelian gauge theory as expressed below in the language of the superfield expansion on the six $(4 +
2)$-dimensional supermanifold (see, e.g., \[42\] for details) $$\begin{array}{lcl}
B_{\mu}\; (x, \theta, \bar \theta) &=& A_{\mu} (x)
+ \;\theta\; (s_{ab} A_{\mu} (x))
+ \;\bar \theta\; (s_{b} A_{\mu} (x))
+ \;\theta \;\bar \theta \;(s_{b} s_{ab} A_{\mu} (x)), \nonumber\\
\Phi\; (x, \theta, \bar \theta) &=& C (x) \;+ \; \theta\; (s_{ab} C (x))
\;+ \;\bar \theta\; (s_{b} C (x))
\;+ \;\theta \;\bar \theta \;(s_{b}\; s_{ab} C (x)),
\nonumber\\
\bar \Phi\; (x, \theta, \bar \theta) &=& \bar C (x)
\;+ \;\theta\;(s_{ab} \bar C (x)) \;+\bar \theta\; (s_{b} \bar C (x))
\;+\;\theta\;\bar \theta \;(s_{b} \;s_{ab} \bar C (x)).
\end{array} \eqno(3.6)$$ The above equation establishes the geometrical interpretation for the off-shell nilpotent (anti-)BRST charges $Q_{(a)b}$ as the translation generators $(\partial/\partial\theta)\partial/\partial\bar\theta$ along the $(\theta)\bar\theta$-directions of the six $(4 + 2)$-dimensional supermanifold. In fact, the process of translations of the superfields $(B_\mu, \Phi, \bar \Phi)$ along $(\theta)\bar\theta$-directions of the supermanifold produces the internal (anti-)BRST symmetry transformations $s_{(a)b}$ (cf. (3.3)) for the local fields $(A_\mu, C, \bar C)$.\
[**3.2 Hodge duality on $(4 + 2)$-dimensional supermanifold**]{}\
To obtain the nilpotent ($s_{(a)d}^2 = 0$) and anticommuting $(s_d s_{ad} + s_{ad} s_d = 0$) (anti-)co-BRST transformations $s_{(a)d}$ for the basic fields $(A_\mu, C, \bar
C)$ of the 4D free Abelian gauge theory, we have to exploit the dual-horizontality condition $\tilde \delta \tilde A^{(1)} =
\delta A^{(1)}$ where $\tilde \delta = - \star \tilde d \star$ and $\delta = -
* d *$ are the super co-exterior derivative and the ordinary co-exterior derivative, respectively. These derivatives are defined on the six $(4 + 2)$-dimensional supermanifold and the ordinary 4D Minkowskian spacetime manifold. As discussed earlier, $\tilde d$ is the super exterior derivative (see, e.g., for the definition, equation (2.9)) and $\star$ and $*$ are the Hodge duality operations on the supermanifold and the ordinary manifold, respectively. The $(4 + 2)$-dimensional supermanifold can support only three (super) 1-forms as given below $$\begin{array}{lcl}
{\cal O}_{1} = dx^\mu\; P_\mu, \qquad
{\cal O}_{2} = d \theta\; P_\theta, \qquad
{\cal O}_{3} = d\bar\theta \;P_{\bar\theta}.
\end{array} \eqno(3.7)$$ However, as will become clear later, a triplet of (super) 1-forms can be constructed from the differential $dx^\mu$ that is present in the definition of ${\cal O}_1$. The Hodge duality $\star$ operation on the above 1-forms produces the following 5-forms $$\begin{array}{lcl}
&& \star\; {\cal O}_1 = \frac{1}{3!}\;
\varepsilon^{\mu\nu\lambda\zeta}\; (d x_\nu \wedge dx_\lambda \wedge
dx_\zeta \wedge d \theta \wedge d \bar\theta)\; P_\mu , \nonumber\\
&& \star\; {\cal O}_2 = \frac{1}{4!}\; \varepsilon^{\mu\nu\lambda\zeta}\;
(dx_\mu \wedge d x_\nu \wedge dx_\lambda \wedge dx_\zeta
\wedge d \bar\theta)\; P_\theta, \nonumber\\
&& \star\; {\cal O}_3 = \frac{1}{4!}\; \varepsilon^{\mu\nu\lambda\zeta} \;
(dx_\mu \wedge d x_\nu \wedge d x_\lambda \wedge dx_\zeta \wedge d \theta)
\; P_{\bar\theta}.
\end{array} \eqno(3.8)$$ It will be noted, in the above, that (i) the $\star$ operation acts basically on the differentials and it does not act on $P$’s. (ii) A linear combination of the 1-forms of (3.7) can also be considered as 1-form. (iii) The double $\star$ operation on the above 1-forms yields $$\begin{array}{lcl}
\star\; [\;\star\;{\cal O}_{1}\;] = + \; {\cal O}_1, \qquad
\star\; [\; \star\; {\cal O}_{2}\;] = - \; {\cal O}_2, \qquad
\star \; [\;\star\; {\cal O}_{3}\;] = - \; {\cal O}_3,
\end{array} \eqno(3.9)$$ where we have used $\varepsilon^{\mu\nu\lambda\zeta}
\varepsilon_{\nu\lambda\zeta\rho} = + 3! \delta^\mu_\rho,
\varepsilon^{\mu\nu\lambda\zeta}
\varepsilon_{\mu\nu\lambda\zeta} = - 4!$. Furthermore, we have used the $\star$ operation on the 5-forms which are found to be dual to 1-forms. In fact, the six $(4 + 2)$-dimensional supermanifold can support five independent 5-forms: $$\begin{array}{lcl}
&&\tilde \phi_{1} = \frac{1}{3!}\;
(dx^\mu \wedge dx^\nu \wedge dx^\lambda
\wedge d \theta \wedge d \bar\theta)\;
\tilde {\cal F}_{\mu\nu\lambda\theta\bar\theta}, \nonumber\\
&&\tilde \phi_{2} = \frac{1}{3!}\;
(dx^\mu \wedge dx^\nu \wedge dx^\lambda \wedge
d \theta \wedge d \theta)\;
\tilde {\cal F}_{\mu\nu\lambda\theta\theta}, \nonumber\\
&&\tilde \phi_{3} = \frac{1}{3!}\;
(dx^\mu \wedge dx^\nu \wedge dx^\lambda
\wedge d \bar\theta \wedge d \bar\theta)\;
\tilde {\cal F}_{\mu\nu\lambda\bar\theta\bar\theta},\nonumber\\
&& \tilde \phi_4 = \frac{1}{4!}\;
(dx^\mu \wedge dx^\nu \wedge dx^\lambda \wedge dx^\zeta
\wedge d \theta)\;
\tilde {\cal F}_{\mu\nu\lambda\zeta\theta}, \nonumber\\
&& \tilde \phi_5 = \frac{1}{4!}\;
(dx^\mu \wedge dx^\nu \wedge dx^\lambda \wedge dx^\zeta
\wedge d \bar\theta)\;
\tilde {\cal F}_{\mu\nu\lambda\zeta\bar\theta}.
\end{array} \eqno(3.10)$$ The Hodge duality $\star$ operation on the wedge products of the differentials, present in the above 5-forms, yields the following 1-form differentials [^8] $$\begin{array}{lcl}
&&\star\;
(dx^\mu \wedge dx^\nu \wedge dx^\lambda
\wedge d \theta \wedge d \bar\theta)\;
= \varepsilon^{\mu\nu\lambda\zeta}\; (dx_\zeta), \nonumber\\
&&\star\;
(dx^\mu \wedge dx^\nu \wedge dx^\lambda \wedge
d \theta \wedge d \theta)\;
= s^{\theta\theta}\;
\varepsilon^{\mu\nu\lambda\zeta}\; (dx_\zeta), \nonumber\\
&&\star\;
(dx^\mu \wedge dx^\nu \wedge dx^\lambda
\wedge d \bar\theta \wedge d \bar\theta)\;
= s^{\bar\theta\bar\theta}\;
\varepsilon^{\mu\nu\lambda\zeta}\; (dx_\zeta), \nonumber\\
&& \star\;
(dx^\mu \wedge dx^\nu \wedge dx^\lambda \wedge dx^\zeta
\wedge d \theta)\;
= \varepsilon^{\mu\nu\lambda\zeta}\; (d\bar\theta), \nonumber\\
&& \star\;
(dx^\mu \wedge dx^\nu \wedge dx^\lambda \wedge dx^\zeta
\wedge d \bar\theta)\;
= \varepsilon^{\mu\nu\lambda\zeta}\; (d\bar\theta).
\end{array} \eqno(3.11)$$ The presence of the symmetric constants, $s^{\theta\theta}$ and $s^{\bar\theta\bar\theta}$ on the r.h.s. of (3.11), enforces the following Hodge duality $\star$ operation $$\begin{array}{lcl}
&&\star\; [\;s^{\theta\theta}\; (dx^\mu)\;]
= \frac{1}{3!}\;\varepsilon^{\mu\nu\lambda\zeta}\;
(dx_\nu\wedge dx_\lambda\wedge dx_\zeta \wedge d\theta \wedge d\theta),
\nonumber\\
&&\star\; [\;s^{\bar\theta\bar\theta}\; (dx^\mu)\;]
= \frac{1}{3!}\;\varepsilon^{\mu\nu\lambda\zeta}\;
(dx_\nu\wedge dx_\lambda\wedge dx_\zeta
\wedge d\bar\theta \wedge d\bar\theta).
\end{array} \eqno(3.12)$$ Taking into account (3.8), (3.11) and (3.12), it is clear that the double $\star$ operation on the 5-forms in (3.10) leads to $$\begin{array}{lcl}
&&\star\;[\;\star\;\tilde \phi_{1}\;] = + \; \tilde \phi_1, \qquad
\star\;[\; \star\;\tilde \phi_{2}\;] = +\; \tilde \phi_2,
\qquad \star\; [\; \star\; \tilde \phi_{3}\;] = +\; \tilde \phi_3,
\nonumber\\
&& \star\;[\;\star\;\tilde \phi_4 \;] = \; - \; \tilde \phi_4, \;\;\qquad\;\;
\star\; [\; \star\; \tilde \phi_5 \;] = \; -\; \tilde \phi_5.
\end{array} \eqno(3.13)$$ The six $(4 + 2)$-dimensional supermanifold can support six 2-forms analogous to (2.24). Their explicit expressions are as under $$\begin{array}{lcl}
&& \tilde \chi_{1} = \frac{1}{2!}\; (dx^\mu \wedge dx^\nu) \;
\tilde S_{\mu\nu}, \;\qquad\;
\tilde \chi_{2} = (d \theta \wedge d \bar\theta)\;
\tilde S_{\theta\bar\theta}, \nonumber\\
&&
\tilde \chi_{3} = (d\theta \wedge d \theta) \;
\tilde S_{\theta\theta}, \;\;\;\;\;\qquad\;\;\;\;
\tilde \chi_{4} =
(d\bar\theta \wedge d \bar\theta)\;
\tilde S_{\bar\theta\bar\theta}, \nonumber\\
&&\tilde \chi_{5} = (dx^\mu \wedge d\theta)
\;\tilde S_{\mu\theta}, \;\;\;\qquad \;\;\;\;
\tilde \chi_{6} = (dx^\mu \wedge d\bar\theta)
\;\tilde S_{\mu\bar\theta}.
\end{array} \eqno(3.14)$$ It is clear that the components $\tilde S_{\mu\nu}, \tilde S_{\mu\theta}, \tilde S_{\mu\bar\theta}$ are antisymmetric. However, the components with the Grassmannian indices $\tilde S_{\theta\theta}, \tilde S_{\bar\theta\bar\theta},
\tilde S_{\theta\bar\theta}$ are symmetric. On the above supermanifold, the operation of a single Hodge duality $\star$ operation leads to the definition of 4-forms which are dual to the above 2-forms. In fact, a single $\star$ operation on the wedge products of the differentials of the above 2-forms, are $$\begin{array}{lcl}
&& \star\; (dx^\mu \wedge dx^\nu) = \frac{1}{2!}
\;\varepsilon^{\mu\nu\lambda\zeta}\;
(dx_\lambda \wedge dx_\zeta \wedge d \theta \wedge d \bar\theta), \nonumber\\
&& \star\; (d \theta \wedge d\bar\theta)
= \frac{1}{4!} \varepsilon^{\mu\nu\lambda\zeta}
(dx_\mu \wedge dx_\nu\wedge dx_\lambda \wedge dx_\zeta), \nonumber\\
&& \star\; (d \theta \wedge d\theta) = \frac{1}{4!} s^{\theta\theta}
\varepsilon^{\mu\nu\lambda\zeta} (dx_\mu \wedge dx_\nu \wedge dx_\lambda
\wedge dx_\zeta), \nonumber\\
&& \star\; (d \bar\theta \wedge d\bar\theta)
= \frac{1}{4!} s^{\bar\theta\bar\theta}
\varepsilon^{\mu\nu\lambda\zeta} (dx_\mu \wedge dx_\nu
\wedge dx_\lambda \wedge dx_\zeta), \nonumber\\
&& \star\;(dx^\mu \wedge d \theta) =
\frac{1}{3!} \varepsilon^{\mu\nu\lambda\zeta}\; (dx_\nu \wedge dx_\lambda
\wedge dx_\zeta \wedge d \bar\theta), \nonumber\\
&& \star\;(dx^\mu \wedge d \bar \theta) =
\frac{1}{3!} \varepsilon^{\mu\nu\lambda\zeta}\; (dx_\nu \wedge dx_\lambda
\wedge dx_\zeta \wedge d \theta).
\end{array} \eqno(3.15)$$ This shows that the wedge products of the differentials corresponding to the 4-forms in the above equations are Hodge dual to the wedge products of the differentials corresponding to 2-forms considered (cf. (3.14)) on the six $(4 + 2)$-dimensional supermanifold. The total number of the independent 4-forms on the above supermanifold are $$\begin{array}{lcl}
&& \tilde \tau_{1} = \frac{1}{2!} (dx^\mu \wedge dx^\nu \wedge d
\theta \wedge d \bar\theta) \tilde T_{\mu\nu\theta\bar\theta},
\qquad \tilde \tau_{2} = \frac{1}{2!} (dx^\mu \wedge dx^\nu \wedge
d \bar\theta \wedge d \bar\theta) \tilde
T_{\mu\nu\bar\theta\bar\theta}, \nonumber\\ && \tilde \tau_{3} =
\frac{1}{2!}
(dx^\mu \wedge dx^\nu \wedge d\theta \wedge d \theta) \;
\tilde T_{\mu\nu\theta\theta}, \qquad
\tilde \tau_{4} = \frac{1}{3!}
(dx^\mu \wedge d x^\nu \wedge dx^\lambda \wedge d \bar\theta)\;
\tilde T_{\mu\nu\lambda\bar\theta}, \nonumber\\
&&\tilde \tau_{5} = \frac{1}{3!}
(dx^\mu \wedge dx^\nu \wedge dx^\lambda \wedge d\theta)
\;\tilde T_{\mu\nu\lambda\theta}, \quad\;
\tilde \tau_6 = \frac{1}{4!}
(dx^\mu \wedge dx^\nu \wedge dx^\lambda \wedge dx^\zeta)
\; \tilde T_{\mu\nu\lambda\zeta}.
\end{array} \eqno(3.16)$$ It will be noted that the 4-forms with the wedge products $(dx_\mu \wedge dx_\nu \wedge dx_\lambda\wedge dx_\zeta),
(dx_\mu \wedge dx_\nu \wedge dx_\lambda\wedge dx_\zeta)\; s^{\theta\theta},
(dx_\mu \wedge dx_\nu \wedge dx_\lambda\wedge dx_\zeta)\;
s^{\bar\theta\bar\theta}$ are different because their dual 2-forms are different as can be seen from (3.15). However, for the sake of brevity, we have chosen only one [^9] of these in (3.16). A single $\star$ operation on the wedge products of the differentials corresponding to 4-forms are $$\begin{array}{lcl}
&& \star\; (dx^\mu \wedge dx^\nu \wedge d \theta \wedge d
\bar\theta) = \frac{1}{2!} \varepsilon^{\mu\nu\lambda\zeta}\;
(dx_\lambda \wedge d x_\zeta), \nonumber\\ &&\star\; (dx^\mu
\wedge dx^\nu \wedge d \bar\theta \wedge d \bar\theta)\; =
s^{\bar\theta\bar\theta}\; \frac{1}{2!}\;
\varepsilon^{\mu\nu\lambda\zeta} (dx_\lambda \wedge dx_\zeta),
\nonumber\\ && \star\;(dx^\mu \wedge dx^\nu \wedge d\theta \wedge
d \theta) \; = s^{\theta\theta} \frac{1}{2!}
\varepsilon^{\mu\nu\lambda\zeta} (dx_\lambda \wedge dx_\zeta),
\nonumber\\ &&\star \;(dx^\mu \wedge d x^\nu \wedge dx^\lambda
\wedge d \bar\theta)\; = \varepsilon^{\mu\nu\lambda\zeta}
(dx_\zeta \wedge d \theta), \nonumber\\ &&\star\; (dx^\mu \wedge
dx^\nu \wedge dx^\lambda \wedge d\theta) =
\varepsilon^{\mu\nu\lambda\zeta} (dx_\zeta \wedge d \bar\theta),
\nonumber\\ &&\star\;(dx^\mu \wedge dx^\nu \wedge dx^\lambda
\wedge dx^\zeta) = \varepsilon^{\mu\nu\lambda\zeta} \; (d\theta
\wedge d\bar\theta).
\end{array} \eqno(3.17)$$ It is clear from (3.15)–(3.17) that one can compute now the double $\star$ operations on the 2-forms as well as 4-forms. Finally, we focus on the independent 3-forms that can be supported on the $(4 + 2)$-dimensional supermanifold. There exist six such forms: $$\begin{array}{lcl}
&& \sigma_{1} = \frac{1}{2!} (dx^\mu \wedge dx^\nu \wedge d \theta) \;
R_{\mu\nu\theta}, \;\;\;\qquad\;\;\;
\sigma_{2} = \frac{1}{2!} (dx^\mu \wedge dx^\nu \wedge d \bar\theta)\;
R_{\mu\nu\bar\theta}, \nonumber\\
&&
\sigma_{3} = (dx^\mu \wedge d\theta \wedge d \theta) \;
R_{\mu\theta\theta}, \;\;\;\;\qquad\;\;\;\;\;
\sigma_{4} =
(dx^\mu \wedge d\bar\theta \wedge d \bar\theta)\;
R_{\mu\bar\theta\bar\theta}, \nonumber\\
&&\sigma_{5} = (dx^\mu \wedge d\theta \wedge d \bar\theta)
\;R_{\mu\theta\bar\theta}, \;\;\;\;\qquad\;\;\;\;
\sigma_{6} = \frac{1}{3!} (dx^\mu \wedge dx^\nu \wedge d x^\lambda) \;
R_{\mu\nu\lambda}.
\end{array} \eqno(3.18)$$ A single Hodge duality $\star$ operation on the above 3-forms will lead to the derivation of the dual 3-forms on the six $(4 + 2)$-dimensional supermanifold. Such an operation will affect the wedge products of the differentials as given below $$\begin{array}{lcl}
&& \star\; (dx^\mu \wedge dx^\nu \wedge d \theta) =
\frac{1}{2!}\;\varepsilon^{\mu\nu\lambda\zeta}\;
(dx_\lambda \wedge dx_\zeta \wedge d \bar\theta), \nonumber\\
&&\star\; (dx^\mu \wedge dx^\nu \wedge d \bar\theta) =
\frac{1}{2!}\;\varepsilon^{\mu\nu\lambda\zeta}\;
(dx_\lambda \wedge dx_\zeta \wedge d \theta), \nonumber\\
&& \star\; (dx^\mu \wedge d \theta \wedge d \bar\theta)
= \frac{1}{3!}\;\varepsilon^{\mu\nu\lambda\zeta}\;
(d x_\nu \wedge dx_\lambda \wedge dx_\zeta), \nonumber\\
&&\star\; (dx^\mu \wedge d \theta \wedge d \theta)
= \frac{1}{3!}\;\varepsilon^{\mu\nu\lambda\zeta}\;
(d x_\nu \wedge dx_\lambda \wedge d x_\zeta)\; s^{\theta\theta}, \nonumber\\
&& \star\; (dx^\mu \wedge d \bar \theta \wedge d \bar \theta)
= \frac{1}{3!}\;\varepsilon^{\mu\nu\lambda\zeta}\;
(d x_\nu \wedge dx_\lambda \wedge dx_\zeta)\; s^{\bar\theta \bar\theta},
\nonumber\\
&& \star\; (dx^\mu \wedge dx^\nu \wedge d x^\lambda) =
\;\varepsilon^{\mu\nu\lambda\zeta}\;
(dx_\zeta \wedge d \theta \wedge d \bar\theta).
\end{array} \eqno(3.19)$$ As expected, there are three 3-forms constructed by the wedge products of the spacetime differentials $(dx^\mu \wedge dx^\nu \wedge dx^\lambda),
(dx^\mu \wedge dx^\nu \wedge dx^\lambda) s^{\theta\theta},
(dx^\mu \wedge dx^\nu \wedge dx^\lambda) s^{\bar\theta\bar\theta}$ whose Hodge duals are different 3-forms as given below $$\begin{array}{lcl}
&&\star \; (dx^\mu \wedge dx^\nu \wedge dx^\lambda)
= \varepsilon^{\mu\nu\lambda\zeta}\;
(dx_\zeta \wedge d\theta \wedge d \bar\theta), \nonumber\\
&&\star \; [\; (dx^\mu \wedge dx^\nu \wedge dx^\lambda)\; s^{\theta\theta}\; ]
= \varepsilon^{\mu\nu\lambda\zeta}\;
(dx_\zeta \wedge d\theta \wedge d \theta), \nonumber\\
&&\star \; [\; (dx^\mu \wedge dx^\nu \wedge dx^\lambda)\;
s^{\bar\theta\bar\theta} \;]
= \varepsilon^{\mu\nu\lambda\zeta}\;
(dx_\zeta \wedge d\bar\theta \wedge d \bar\theta).
\end{array} \eqno(3.20)$$ The above considerations allow us to define the following triplet of $\sigma_6$ of (3.18) $$\begin{array}{lcl}
&& \sigma_6^{(1)} = \frac{1}{3!} (dx^\mu \wedge dx^\nu \wedge dx^\lambda)\;
R^{(1)}_{\mu\nu\lambda}, \qquad\;
\sigma_6^{(2)} = \frac{1}{3!}
(dx^\mu \wedge dx^\nu \wedge dx^\lambda)\;s^{\theta\theta}
\;R^{(2)}_{\mu\nu\lambda}, \nonumber\\
&& \sigma_6^{(3)} = \frac{1}{3!}
(dx^\mu \wedge dx^\nu \wedge dx^\lambda)\;
s^{\bar\theta\bar\theta}\;
R^{(3)}_{\mu\nu\lambda}.
\end{array} \eqno(3.21)$$ However, for the sake of brevity, we have taken only one of the above triplets in equation (3.18). It is now straightforward to check that the double Hodge duality $\star$ operations on the 3-forms of (3.18) yield the following $$\begin{array}{lcl}
&& \star\; [\; \star \;\sigma_{1}\;] = \;- \; \sigma_1,
\;\;\;\qquad\;\;\;
\star\; [\; \star\;\sigma_{2}\;]
= \;-\; \sigma_2, \nonumber\\
&&
\star\; [\; \star\; \sigma_{3}\;] = \;+\; \sigma_3,
\;\;\;\qquad\;\;\;
\star \; [\; \star\; \sigma_{4}\;] = \; +\; \sigma_4,
\nonumber\\
&&
\star\; [\; \star\; \sigma_{5}\;]
=\; +\; \sigma_5, \;\;\;\qquad\;\;\;
\star\; [\; \star\; \sigma_{6}\;] =\; + \; \sigma_6.
\end{array} \eqno(3.22)$$ Finally, we do know that the six $(4 + 2)$-dimensional supermanifold can support three 6-forms, modulo some constant factors, as given below $$\begin{array}{lcl}
&&\tilde \Psi_{1} = \frac{1}{4!}\;
(dx^\mu \wedge dx^\nu \wedge dx^\lambda \wedge dx^\zeta
\wedge d \theta \wedge d \bar\theta)\;
\tilde {\cal G}_{\mu\nu\lambda\zeta\theta\bar\theta}, \nonumber\\
&&\tilde \Psi_{2} = \frac{1}{4!}\;
(dx^\mu \wedge dx^\nu \wedge dx^\lambda \wedge d x^\zeta \wedge
d \theta \wedge d \theta)\;
\tilde {\cal G}_{\mu\nu\lambda\zeta\theta\theta}, \nonumber\\
&&\tilde \Psi_{3} = \frac{1}{4!}\;
(dx^\mu \wedge dx^\nu \wedge dx^\lambda \wedge dx^\zeta
\wedge d \bar\theta \wedge d \bar\theta)\;
\tilde {\cal G}_{\mu\nu\lambda\zeta\bar\theta\bar\theta}.
\end{array} \eqno(3.23)$$ A single Hodge duality $\star$ operation on the above 6-forms produces 0-form scalars on the six dimensional supermanifold. Such an operation on the wedge products of the differentials are $$\begin{array}{lcl}
&&\star\;
(dx^\mu \wedge dx^\nu \wedge dx^\lambda \wedge dx^\zeta
\wedge d \theta \wedge d \bar\theta)\;
= \varepsilon^{\mu\nu\lambda\zeta}, \nonumber \\
&&\star\;
(dx^\mu \wedge dx^\nu \wedge dx^\lambda \wedge d x^\zeta \wedge
d \theta \wedge d \theta)\;
= \varepsilon^{\mu\nu\lambda\zeta}\; s^{\theta\theta}, \nonumber\\
&&\star\;
(dx^\mu \wedge dx^\nu \wedge dx^\lambda \wedge dx^\zeta
\wedge d \bar\theta \wedge d \bar\theta)\;
= \varepsilon^{\mu\nu\lambda\zeta} \;s^{\bar\theta\bar\theta}.
\end{array} \eqno(3.24)$$ Two consecutive $\star$ operation on the 6-forms of (3.23) leads to $$\begin{array}{lcl}
\star\; [\; \star\; \Psi_{1}\;] = -\; \Psi_1, \qquad
\star\; [\; \star\; \Psi_{2}\;] = -\; \Psi_2, \qquad
\star\; [\; \star\; \Psi_{3}\;] = -\; \Psi_3.
\end{array} \eqno(3.25)$$ It is evident that we have collected, in the present section, all the possible super-forms, their single Hodge dual- as well as their double Hodge dual superforms, etc., that could be defined on the $(4 + 2)$-dimensional supermanifold.\
[**3.3 Superfield formulation of (anti-)co-BRST symmetries for 4D theory**]{}\
As evident from (3.4) that the non-local, non-covariant, continuous, off-shell nilpotent and anticommuting ($s_d s_{ad} +
s_{ad} s_d = 0$) (anti-)co-BRST symmetries $s_{(a)d}$ do exist for the 4D free Abelian gauge theory. To obtain these symmetries in the framework of superfield formulation, we have to exploit the dual-horizontality condition $\tilde \delta \tilde A^{(1)} =
\delta A^{(1)}$ on the six $(4 + 2)$-dimensional supermanifold. It is clear that the r.h.s. of the above condition (i.e $\delta
A^{(1)} = - * d * A^{(1)} = (\partial \cdot A)$) is the usual gauge-fixing term on the ordinary 4D spacetime manifold. For the computation of the l.h.s. $\tilde \delta \tilde A^{(1)} = - \star
\delta \star \tilde A^{(1)}$, we first concentrate on the dual $(\star \tilde A^{(1)} = \star dZ^M \tilde A_M)$ of the super 1-form connection $\tilde A^{(1)}$. The ensuing expression for $(\star \tilde A^{(1)})$, due to the Hodge duality operation given in (3.8) and definition (2.9), is $$\begin{array}{lcl}
\star\; \tilde A^{(1)} &=&
\frac{1}{3!}\;\varepsilon^{\mu\nu\lambda\zeta} \; (dx_\nu \wedge
dx_\lambda \wedge dx_\zeta \wedge d \theta \wedge d \bar\theta)\;
B_\mu (x,\theta,\bar\theta)\nonumber\\ &+& \frac{1}{4!}\;
\varepsilon_{\mu\nu\lambda\zeta}\; (dx^\mu \wedge dx^\nu \wedge
dx^\lambda \wedge dx^\zeta \wedge d \bar\theta)\; \bar \Phi
(x,\theta, \bar\theta) \nonumber\\ &+& \frac{1}{4!}\;
\varepsilon_{\mu\nu\lambda\zeta}\; (dx^\mu \wedge dx^\nu \wedge
dx^\lambda \wedge dx^\zeta \wedge d \theta)\; \Phi
(x,\theta,\bar\theta),
\end{array} \eqno(3.26)$$ which is nothing but the 5-form defined on the six $(4 + 2)$-dimensional supermanifold. Applying now the super exterior derivative $\tilde d
= dZ^M \partial_M$ on the above 5-form, we obtain the following 6-form $$\begin{array}{lcl}
\tilde d\; (\star \tilde A^{(1)}) &=& \frac{1}{3!}
\varepsilon^{\mu\nu\lambda\zeta}\; (dx_\rho \wedge dx_\nu \wedge
dx_\lambda \wedge d x_\zeta \wedge d\theta \wedge d \bar\theta)\;
(\partial^\rho B_\mu) (x,\theta,\bar\theta) \nonumber\\ &-&
\frac{1}{4!} \varepsilon_{\mu\nu\lambda\zeta}\; (dx^\mu \wedge
dx^\nu \wedge dx^\lambda \wedge dx^\zeta \wedge d\theta \wedge
d\bar\theta)\; (\partial_\theta \bar \Phi) (x,\theta,\bar\theta)
\nonumber\\ &-&
\frac{1}{4!} \varepsilon_{\mu\nu\lambda\zeta}\;
(dx^\mu \wedge dx^\nu \wedge dx^\lambda \wedge dx^\zeta
\wedge d \bar\theta \wedge d\bar\theta)\;
(\partial_{\bar\theta} \bar \Phi) (x,\theta,\bar\theta) \nonumber\\
&-& \frac{1}{4!} \varepsilon_{\mu\nu\lambda\zeta}\;
(dx^\mu \wedge dx^\nu \wedge dx^\lambda \wedge dx^\zeta
\wedge d\theta \wedge d \bar \theta)\;
(\partial_{\bar\theta} \Phi) (x,\theta,\bar\theta) \nonumber\\
&-& \frac{1}{4!} \varepsilon_{\mu\nu\lambda\zeta}\;
(dx^\mu \wedge dx^\nu \wedge dx^\lambda \wedge dx^\zeta
\wedge d\theta \wedge d \theta)\;
(\partial_\theta \Phi) (x,\theta,\bar\theta).
\end{array} \eqno(3.27)$$ It should be noted that all the wedge products with more than four spacetime differentials and two Grassmannian differentials have been dropped out because on a $(4 + 2$)-dimensional supermanifold one cannot define such kind of differential forms. On (3.27), we now apply another $(- \star)$ to obtain a super 0-form (superscalar) by exploiting the Hodge duality operation defined in (3.24). Such a superscalar is $$\begin{array}{lcl}
\tilde \delta \tilde A^{(1)} = - \star \; \tilde d \; \star \;
\tilde A^{(1)} = (\partial \cdot B) - (\partial_\theta \bar\Phi +
\partial_{\bar \theta} \Phi) - s^{\theta\theta}\; (\partial_\theta
\Phi) - s^{\bar\theta\bar\theta}\; (\partial_{\bar\theta} \bar
\Phi).
\end{array} \eqno(3.28)$$ Equating the above superscalar with the ordinary scalar $\delta
A^{(1)}$, due to the dual-horizontality condition ($\tilde \delta
\tilde A^{(1)} = \delta A^{(1)})$, we obtain the following relationships $$\begin{array}{lcl}
(\partial \cdot B) - (\partial_\theta \bar \Phi +
\partial_{\bar\theta} \Phi) = (\partial \cdot A), \qquad
\partial_\theta \Phi = 0, \qquad \partial_{\bar\theta} \bar \Phi = 0.
\end{array} \eqno(3.29)$$ The insertion of the most general super expansions (cf. (3.5)) leads to the following restrictions on the secondary fields of expansion (3.5): $$\begin{array}{lcl}
&&(\partial \cdot R) (x) = 0, \;\;\qquad\; (\partial \cdot \bar R) (x) = 0,
\;\;\qquad\;
(\partial \cdot S) (x) = 0, \nonumber\\
&& s (x) = \bar s (x) = B (x) = \bar B (x) = 0, \;\;\;\qquad \;\;
{\cal B} (x) + \bar {\cal B} (x) = 0.
\end{array} \eqno(3.30)$$ Consistent with the statements made after (3.5), the following choices of the secondary fields in terms of the basic fields (see, e.g., \[42\] for details) $$\begin{array}{lcl}
&& R_0 = i \bar C, \qquad
R_i = i {\displaystyle
\frac{\partial_0\partial_i}{\nabla^2}} \bar C,
\qquad \bar R_0 = i C,
\qquad
\bar R_i = i {\displaystyle
\frac{\partial_0\partial_i}{\nabla^2}} C, \nonumber\\
&&{\cal B} = + i {\displaystyle
\frac{\partial_i b^{(1)}_i}{\nabla^2}}, \qquad
\bar {\cal B} = - i {\displaystyle
\frac{\partial_i b^{(1)}_i}{\nabla^2}}, \qquad
S_0 = {\displaystyle \frac{\partial_i b^{(1)}_i}{\nabla^2}}, \qquad
S_i = {\displaystyle \frac{\partial_0\partial_i}{\nabla^2}}\;\Bigl (
{\displaystyle \frac{\partial_j b^{(1)}_j}{\nabla^2}} \Bigr ),
\end{array} \eqno(3.31)$$ do satisfy all the above conditions (3.30), emerging from the application of the dual-horizontality condition. It is worth emphasizing, at this juncture, that the auxiliary field ${\bf b^{(2)}}$ has [*not*]{} been taken into account in the expansion (3.5) as well as in the choices (3.31) because this field (and its equivalent magnetic field ${\bf B}$) do not appear in any transformations listed in (3.3) and (3.4). Furthermore, this field, on its own, does not transform under (co-)BRST transformations. In terms of the transformations in (3.4) and expressions (3.31), we obtain the following expansions $$\begin{array}{lcl}
B_{0}\; (x, \theta, \bar \theta) &=& A_{0} (x)
+ \;\theta\; (s_{ad} A_{0} (x))
+ \;\bar \theta\; (s_{d} A_{0} (x))
+ \;\theta \;\bar \theta \;(s_{d} s_{ad} A_{0} (x)), \nonumber\\
B_{i}\; (x, \theta, \bar \theta) &=& A_{i} (x)
+ \;\theta\; (s_{ad} A_{i} (x))
+ \;\bar \theta\; (s_{d} A_{i} (x))
+ \;\theta \;\bar \theta \;(s_{d} s_{ad} A_{i} (x)), \nonumber\\
\Phi\; (x, \theta, \bar \theta) &=& C (x) \;+ \; \theta\; (s_{ad} C (x))
\;+ \;\bar \theta\; (s_{d} C (x))
\;+ \;\theta \;\bar \theta \;(s_{d}\; s_{ad} C (x)),
\nonumber\\
\bar \Phi\; (x, \theta, \bar \theta) &=& \bar C (x)
\;+ \;\theta\;(s_{ad} \bar C (x)) \;+ \;\bar \theta\; (s_{d} \bar C (x))
\;+\;\theta\;\bar \theta \;(s_{d} \;s_{ad} \bar C (x)).
\end{array} \eqno(3.32)$$ The above expansion does establish the geometrical interpretation for the conserved and nilpotent (anti-)co-BRST charges $Q_{(a)d}$ as the translation generators along the Grassmannian directions of the six $(4 + 2)$-dimensional supermanifold. In fact, there exists some inter-connections among the nilpotent transformations $s_{(a)d}$, the translations generators along the Grassmannian directions of the supermanifold and the nilpotent charges $Q_{(a)d}$, as $$\begin{array}{lcl}
s_{d} \leftrightarrow \mbox{Lim}_{\theta \to 0}
{\displaystyle \frac{\partial}{\partial \bar\theta}} \leftrightarrow Q_d,
\qquad
s_{ad} \leftrightarrow \mbox{Lim}_{\bar \theta \to 0}
{\displaystyle \frac{\partial}{\partial \theta}} \leftrightarrow Q_{ad}.
\end{array} \eqno(3.33)$$ The above relationship is the analogue of exactly the same kind of relation existing in the context of the nilpotent (anti-)BRST symmetries (cf. (2.14)).\
[**4 Conclusions**]{}\
In our present investigation, we have been able to define a consistent Hodge duality $\star$ operation on (i) the four $(2 + 2)$-dimensional supermanifold, and (ii) the six $(4 +
2)$-dimensional supermanifold. These definitions are essential for the derivation of the nilpotent ($s_{(a)d}^2 = 0$) (anti-)co-BRST symmetries $s_{(a)d}$ for (i) the two $(1 + 1)$-dimensional (2D) free Abelian gauge theory, and (ii) the four $(3 + 1)$-dimensional (4D) free Abelian gauge theory in the framework of superfield formulation. In fact, the above 2D- and 4D free Abelian gauge theories (described by the local fields that take values on the 2D and 4D flat Minkowskian spacetime manifold) are considered on the four $(2 + 2)$-dimensional- and six $(4 + 2)$-dimensional supermanifolds, respectively. Our study on these supermanifolds (described by the superfields that take values on the supermanifold parametrized by the superspace variables $Z^M=
(x^\mu, \theta, \bar\theta)$) does provide the geometrical origin and interpretation for the nilpotent (anti-)BRST- and (anti-)co-BRST symmetries (and the corresponding nilpotent generators). The physical application of a consistent definition of the Hodge duality $\star$ operation turns up in the context of the dual-horizontality condition $\tilde \delta \tilde A^{(1)} =
\delta A^{(1)}$ where the use of the super co-exterior derivative $\tilde \delta = - \star \tilde d \star$ (on the l.h.s.) does require a consistent definition of the $\star$ operation. In fact, the existence of the nilpotent (anti-)co-BRST symmetry transformations owes its origin to the (super) co-exterior derivatives where the definition of $\star$ plays a very decisive role. In the language of physics, it is the gauge-fixing term of the (anti-)BRST invariant Lagrangian density of a gauge theory that remains invariant under the (anti-)co-BRST transformations (cf. Sections 2 and 3). This statement has been shown to be true for both the 2D- and 4D free Abelian gauge theories where there is no interaction between the $U(1)$ gauge field and the matter fields.
One of the novel and the most decisive ingredients in our whole discussion is the introduction of the constant symmetric parameters $s^{\theta\theta}$ and $s^{\bar\theta\bar\theta}$ in the definition of the Hodge duality $\star$ operation on the wedge products of the differentials of some given (super)forms on the $(D + 2)$-dimensional supermanifold. The usefulness of these parameters, in our whole discussion, are primarily four folds. First, these allow us, for instance, to take into account the fact that there are three (super) differentials corresponding to the 1-forms defined on the four $(2 + 2)$-dimensional supermanifold. These are, for the sake of emphasis, once again written as $$\begin{array}{lcl}
(dx^\mu), \qquad \;
(dx^\mu)\; s^{\theta\theta}, \qquad \;
(dx^\mu)\; s^{\bar\theta\bar\theta},
\end{array} \eqno(4.1)$$ whose Hodge duals correspond to wedge products of the differentials corresponding to the 3-forms on the supermanifold as given below $$\begin{array}{lcl}
&& \star\; (dx^\mu) = \varepsilon^{\mu\nu} (dx_\nu \wedge d \theta \wedge
d\bar\theta), \qquad
\star\; [(dx^\mu s^{\theta\theta})] = \varepsilon^{\mu\nu} (dx_\nu
\wedge d \theta \wedge d \theta), \nonumber\\
&& \star\; [(dx^\mu s^{\bar\theta\bar\theta})] = \varepsilon^{\mu\nu} (dx_\nu
\wedge d \bar \theta \wedge d\bar \theta).
\end{array} \eqno(4.2)$$ In fact, the above prescription can be generalized to any $(D + 2)$-dimensional supermanifold. Second, the presence of these parameters facilitate the action of the double $\star$ operations on any arbitrary form $f$ that is supposed to obey $\star (\star f) = \pm f$ \[39\]. For instance, in the above example, the following results turn out automatically $$\begin{array}{lcl}
\star\; [\; \star\; (dx^\mu)\;] = d x^\mu, \quad
\star\; [\; \star\; (dx^\mu)\;s^{\theta\theta}\;]
= d x^\mu\; s^{\theta\theta}, \quad
\star\; [\; \star\; (dx^\mu)\;s^{\bar\theta\bar\theta}\;]
= d x^\mu\; s^{\bar\theta\bar\theta}.
\end{array} \eqno(4.3)$$ Third, it is evident that the Hodge dual of a 2-superform (e.g. $d\theta \wedge
d \theta$) will be a 2-superform on a $(2 + 2)$-dimensional supermanifold. The existence of $s^{\theta\theta}$ does allow such a definition because $\star (d\theta \wedge d\theta) = \frac{1}{2!} \varepsilon^{\mu\nu}
(dx_\mu \wedge dx_\nu) s^{\theta\theta}$. Fourth, the existence of the above parameters is at the heart of the accurate derivation of the nilpotent (anti-)co-BRST symmetry transformations for the gauge field as well as the (anti-)ghost fields of the 2D- and 4D free Abelian gauge theories as is evident from the key equations (2.35), (2.36), (3.28) and (3.29).
It is clear from our present discussion that the geometrical superfield formalism provides an exact and unique way of deriving the local, covariant, continuous, nilpotent and anticommuting (anti-)BRST symmetry transformations. However, this is [*not*]{} the case with the derivation of the (anti-)co-BRST symmetries which are not found to be [*unique*]{}. In fact, for both the 2D- and 4D free 1-form Abelian gauge theories, the dual-horizontality condition ($\tilde \delta \tilde A^{(1)} = \delta A^{(1)}$) leads to the conditions $(\partial \cdot R) = 0, (\partial \cdot \bar R)
= 0, (\partial \cdot S) = 0$ on the secondary fields of the expansions in (2.7) as well as (3.5). For the 2D theory, there exist local, covariant, continuous and nilpotent solutions for $R_\mu, \bar R_\mu, S_\mu$ so that one obtains the (anti-)co-BRST transformations of (2.4). However, for the 4D free Abelian gauge theory, only non-local, non-covariant, continuous and nilpotent solutions exist for $R_\mu, \bar R_\mu, S_\mu$. Furthermore, these solutions for the latter case are not [*unique*]{}. In fact, there has been a whole lot of discussion on the various possibilities of the existence of the dual-BRST symmetry transformations for the Abelian gauge theory in \[41\]. All these possibilities of symmetries are captured by different choices of $R_\mu$ and $\bar
R_\mu$ (see, e.g., \[42\] for details). Thus, in some sense, the superfield formalism with the super co-exterior derivative $\tilde
\delta$ does provide the reasons behind the non-uniqueness of the nilpotent (anti-)co-BRST symmetry transformations where the dual-horizontality condition plays a very decisive role.
It would be an interesting endeavour to generalize our present work to the case of the interacting gauge theories where the gauge fields couple to the matter fields. In fact, one such example, where the $U(1)$ gauge field $A_\mu$ couples with the Dirac fields in 2D, has been shown to present the field theoretical model for the Hodge theory. In this model, the nilpotent (anti-)BRST and (anti-)co-BRST symmetries co-exist together \[33,34\]. In a recent set of papers \[43-47\], the nilpotent symmetries for all the basic fields of (i) the interacting 2D- as well as 4D (non-)Abelian gauge theories, and (ii) a reparametrization invariant theory, have been derived by exploiting the [*augmented*]{} superfield formulation. In this formalism, in addition to the (dual-)horizontality conditions, the invariance of the (super)matter conserved currents on the supermanifold has also been exploited. In fact, the latter restriction yields the nilpotent symmetries for the matter fields of an interacting gauge theory. Furthermore, it would be an interesting venture to generalize our present work to the discussion of the free 4D 2-form Abelian gauge theory where the existence of the local, covariant, continuous and nilpotent (anti-)co-BRST symmetries has been shown \[48,49\]. Yet another direction that could be pursued is to generalize the superfield formalism with only two Grassmann variables (i.e. $\theta$ and $\bar\theta$) to the superfield approach depending upon multiple Grassmann variables (e.g. $\theta_\alpha$ and $\bar\theta_{\dot \alpha}$ with $\alpha, \dot \alpha = 1, 2, 3...$). These are some of the issues that are under investigation and our results would be reported elsewhere \[50\].\
[**Acknowledgements**]{}\
Fruitful discussions with V. P. Nair, K. S. Narain and G. Thompson at AS-ICTP, Trieste, Italy are gratefully acknowledged where this work was initiated. Thanks are also due to the members of the HEP group at AS-ICTP for their warm hospitality and useful conversations.
= 12pt
[99]{}
P. A. M. Dirac, [*Lectures on Quantum Mechanics*]{}, Belfer Graduate School of Science,\
(Yeshiva University Press, New York, 1964).
See, e.g., for a complete review, K. Sundermeyer, [*Constrained Dynamics: Lecture Notes in Physics*]{}, vol. 169 (Springer Verlag, Berlin, 1982).
See, e.g., for an exhaustive review, M. Henneaux and C. Teitelboim, [*Quantization of Gauge Systems*]{} (Princeton University Press, New Jersey, Princeton, 1992).
K. Nishijima, in: [*Progress in Quantum Field Theory*]{}, eds. H. Ezawa and S. Kamefuchi (North-Holland, Amsterdam, 1986) pp. 99-119;\
For an extensive review, see, e.g., K. Nishijima, [*Czech. J. Phys.*]{} [**46**]{}, 1 (1996).
I. J. R. Aitchison and A. J. G. Hey, [*Gauge Theories in Particle Physics: A Practical Introduction*]{} (Adam Hilger, Bristol, 1982).
T. Eguchi, P. B. Gilkey and A. J. Hanson, [*Phys. Rep.*]{} [**66**]{}, 213 (1980).
S. Mukhi and N. Mukunda, [*Introduction to Topology, Differential Geometry and Group Theory for Physicists*]{} (Wiley Eastern Pvt. Ltd., New Delhi, 1990).
J. W. van Holten, [*Nucl. Phys.*]{} [**B 339**]{}, 158 (1990);\
G. F[ü]{}l[ö]{}p and R. Marnelius, [*Nucl. Phys.*]{} [**B 456**]{}, 442 (1995).
K. Nishijima, [*Prog. Theo. Phys.*]{} [**80**]{}, 897 (1988);\
K. Nishijima, [*Prog. Theo. Phys.*]{} [**80**]{}, 905 (1988);\
H. Aratyn, [*J. Math. Phys.*]{} [**31**]{}, 1240 (1990).
J. Thierry-Mieg, [*J. Math. Phys.*]{} [**21**]{}, 2834 (1980).
M. Quiros, F. J. De Urries, J. Hoyos, M. L. Mazon and E. Rodrigues, [*J. Math. Phys.*]{} [**22**]{}, 1767 (1981).
R. Delbourgo and P. D. Jarvis, [*J. Phys. A: Math. Gen.*]{} [**15**]{}, 611 (1981);\
R. Delbourgo, P. D. Jarvis and G. Thompson, [*Phys. Lett.*]{} [**B 109**]{}, 25 (1982).
L. Bonora and M. Tonin, [*Phys. Lett.*]{} [**B 98**]{}, 48 (1981);\
L. Bonora, P. Pasti and M. Tonin, [*Nuovo Cimento*]{} [**63 A**]{}, 353 (1981).
L. Baulieu and J. Thierry-Mieg, [*Nucl. Phys.*]{} [**B 197**]{}, 477 (1982);\
L. Alvarez-Gaum[é]{} and L. Baulieu, [*Nucl. Phys.*]{} [**B 212**]{}, 255 (1983).
D. S. Hwang and C. -Y. Lee, [*J. Math. Phys.*]{} [**38**]{}, 30 (1997).
P. M. Lavrov, P. Yu. Moshin and A. A. Rashetnyak, [*Mod. Phys. Lett.*]{} [**A 10**]{}, 2687 (1995);\
P. M. Lavrov and P. Yu. Moshin, [*Phys. Lett.*]{} [**B 508**]{}, 127 (2001).
N. R. F. Braga and A. Das, [*Nucl. Phys.*]{} [**B 442**]{}, 655 (1995).
N. Nakanishi and I. Ojima, [*Covariant Operator Formalism of Gauge Theories and Quantum Gravity*]{} (World Scientific, Singapore, 1990).
J. Wess and J. Bagger, [*Supersymmetry and Supergravity*]{}, (2nd Edition) (Princeton University Press, New Jersey, Princeton, 1991).
A. S. Galperin, E. A. Ivanov, V. I. Ogievetsky and E. S. Sokatchev, [*Harmonic Superspace*]{} (Cambridge University Press, Cambridge, 2001).
B. DeWitt, [*Supermanifolds*]{} (Cambridge University Press, Cambridge, 1992).
C. Bartocci, U. Bruzzo and D. H. Ruip[é]{}rez, [*The Geometry of Supermanifolds*]{} (Kluwer Academic Publication, Dordrecht, 1991).
G. Sardanashvily, [*Mod. Phys. Lett.*]{} [**A 16**]{}, 1531 (2001).
G. Barnich, F. Brandt and M. Henneaux, [*Phys. Rep.*]{} [**338**]{}, 439 (2000).
D. Franco and C. Polito, [*J. Math. Phys.*]{} [**45**]{}, 1447 (2004).
D. V. Soroka and V. A. Soroka, [*JHEP*]{} [**0303**]{}, 001 (2003);\
V. A. Soroka, [*Phys. Lett.*]{} [**B 451**]{}, 349 (1999).
R. P. Malik, [*Phys. Lett.*]{} [**B 521**]{}, 409 (2001), hep-th/0108105.
R. P. Malik, [*Ann. Phys.*]{} (N.Y.) [**307**]{}, 1 (2003), hep-th/0205135.
R. P. Malik, [*J. Phys. A: Math. Gen.*]{} [**35**]{}, 3711 (2002), hep-th/0106215.
R. P. Malik, [*Mod. Phys. Lett.*]{} [**A 17**]{}, 185 (2002), hep-th/0111253.
R. P. Malik, [*J. Phys. A: Math. Gen.*]{} [**35**]{}, 6919 (2002), hep-th/0112260.
R. P. Malik, [*J. Phys. A: Math. Gen.*]{} [**35**]{}, 8817 (2002), hep-th/0204015.
R. P. Malik, [*Mod. Phys. Lett.*]{} [**A 15**]{}, 2079 (2000), hep-th/0003128.
R. P. Malik, [*Mod. Phys. Lett.*]{} [**A 16**]{}, 477 (2001), hep-th/9711056.
S. Weinberg, [*The Quantum Theory of Fields: Modern Applications*]{} vol. 2 (Cambridge University Press, Cambridge, 1996).
R. P. Malik, [*J. Phys. A: Math. Gen.*]{} [**33**]{}, 2437 (2000), hep-th/9902146.
R. P. Malik, [*J. Phys. A: Math. Gen.*]{} [**34**]{}, 4167 (2001), hep-th/0012085.
R. P. Malik, [*Int. J. Mod. Phys.*]{} [**A 15**]{}, 1685 (2000), hep-th/9808040.
S. Deser, A. Gomberoff, M. Henneaux and C. Titelboim, [*Phys. Lett.*]{} [**B 400**]{}, 80 (1997).
See, e.g., D. McMullan and M. Lavelle, [*Phys. Rev. Lett.*]{} [**71**]{}, 3758 (1993).
See, e.g., V. O. Rivelles, [*Phys. Rev.*]{} [**D 53**]{}, 3247 (1996).
R. P. Malik, [*J. Phys. A: Math. Gen.*]{} [**37**]{}, 1059 (2004), hep-th/0306182.
R. P. Malik, [*Phys. Lett.*]{} [**B 584**]{}, 210 (2004), hep-th/0311001.
R. P. Malik, [*J. Phys. A: Math. Gen*]{} [**37**]{}, 5261 (2004), hep-th/0311193.
R. P. Malik, [*Int. J. Mod. Phys.*]{} [**A 20**]{}, 4899 (2005), hep-th/0402005.
R. P. Malik, [*Mod. Phys. Lett.*]{} [**A 20**]{}, 1767 (2005), hep-th/0402123.
R. P. Malik, [*Int. J. Geom. Meth. Mod. Phys.*]{} [**1**]{}, 467 (2004), hep-th/0403230.
E. Harikumar, R. P. Malik and M. Sivakumar, [*J. Phys. A: Math. Gen.*]{} [**33**]{}, 7149 (2000), hep-th/0004145.
R. P. Malik, [*J. Phys. A: Math. Gen.*]{} [**36**]{}, 5095 (2003), hep-th/0209136.
R. P. Malik, in preparation.
[^1]: This condition is referred to as the soul-flatness condition by Nakanishi and Ojima \[18\] which amounts to setting equal to zero all the Grassmannian components of the (anti-)symmetric curvature tensor that constitutes the $(p
+ 1)$-form super curvature on the $(D + 2)$-dimensional supermanifold.
[^2]: The horizontality condition has also been applied to 1-form 4D non-Abelian gauge theory where the six $(4 + 2)$-dimensional 2-form super curvature $\tilde F^{(2)} = \tilde d \tilde A^{(1)} +
\tilde A^{(1)} \wedge \tilde A^{(1)}$ is equated with the 4D ordinary 2-form curvature $F^{(2)} = d A^{(1)} + A^{(1)} \wedge
A^{(1)}$ leading to the derivation of (anti-)BRST symmetry transformations for the non-Abelian gauge field and the corresponding (anti-)ghost fields (see, e.g., \[18\]).
[^3]: We adopt here the convention and notations such that the flat 2D Minkowskian spacetime manifold is endowed with the flat metric $\eta_{\mu\nu} =$ diag $(+1, -1)$ and $\Box = \eta^{\mu\nu}\partial_\mu
\partial_\nu = (\partial_0)^2 - (\partial_1)^2,
(\partial \cdot A) = \partial_0 A_0 - \partial_1 A_1, F_{01} = E = -
\varepsilon^{\mu\nu} \partial_\mu A_\nu = - F^{01}$. Here the 2D antisymmetric Levi-Civita tensor is chosen to satisfy $\varepsilon_{01} = + 1 = - \varepsilon^{01},
\varepsilon^{\mu\nu} \varepsilon_{\nu\lambda} = \delta^\mu_\lambda$, etc., and the Greek indices $\mu, \nu, \lambda...= 0 , 1$ correspond to the time- and space directions on the 2D flat Minkowskian spacetime manifold, respectively.
[^4]: We follow here the notations adopted in refs. \[3,35\]. In its full blaze of glory, the nilpotent $(\delta_B^2 = 0$) BRST transformations $\delta_B$ are the product of an anticommuting $(\eta C + C \eta = 0, \eta \bar C + \bar C \eta = 0)$ spacetime independent parameter $\eta$ and $s_b$ as: $\delta_B = \eta s_b$ where the nilpotency property is carried by $s_b$ (with $s_b^2 = 0$).
[^5]: In general, a set of three 1-forms can be constructed from the spacetime differential $(dx_\mu)$. These are $\alpha_1^{(1)} = dx^\mu {\cal A}^{(1)}_\mu,
\alpha_1^{(2)} = dx^\mu s^{\theta\theta} {\cal A}^{(2)}_\mu,
\alpha_1^{(3)} = dx^\mu s^{\bar\theta\bar\theta} {\cal A}^{(3)}_\mu$. A single $\star$ operation yields $\star\; (dx^\mu) = \varepsilon^{\mu\nu}
(dx_\nu \wedge d\theta \wedge d \bar\theta),
\star\; [ dx^\mu s^{\theta\theta}]
= \varepsilon^{\mu\nu} (dx_\nu \wedge d\theta \wedge d\theta),
\star\; [ dx^\mu s^{\bar\theta\bar\theta}]
= \varepsilon^{\mu\nu} (dx_\nu \wedge d\bar\theta \wedge d\bar\theta)$. For obvious reasons, such kind of a triplet of 1-forms can [*not*]{} be constructed from the 1-forms $\alpha_2$ as well as $\alpha_3$ because their Hodge dual forms are not well defined on a $(2 + 2)$-dimensional supermanifold.
[^6]: It will be noted that the non-local and non-covariant solutions to the restrictions (2.37) also exist. For the 2D Abelian case, we have $R_0 = i \bar C, R_1 = i (\partial_0 \partial_1/\nabla^2)
\bar C, \bar R_0 = i C, \bar R_1 = i
(\partial_0\partial_1/\nabla^2) C$, etc. However, for our present discussions, we avoid such kind of pathological choices. In fact, for the 4D Abelian theory, this kind of symmetries exist, too (see, e.g. \[40,41\], for details).
[^7]: We follow here the notations and conventions such that 4D Minkowskian manifold is endowed with a flat metric $\eta_{\mu\nu} =$ diag $(+1 , -1, -1, -1)$ and the totally antisymmetric 4D Levi-Civita tensor $\varepsilon_{\mu\nu\lambda\zeta}$ is chosen to satisfy $\varepsilon_{0123} = + 1 = - \varepsilon^{0123},
\varepsilon_{0ijk} = \epsilon_{ijk} = - \varepsilon^{0ijk},
\varepsilon_{\mu\nu\lambda\zeta} \varepsilon^{\mu\nu\lambda\zeta} = - 4 !,
\varepsilon_{\mu\nu\lambda\zeta} \varepsilon^{\mu\nu\lambda\rho}
= - 3! \delta^\rho_\zeta$ etc. Here the Greek indices $\mu, \nu, \lambda....= 0, 1, 2, 3$ correspond to the spacetime directions on the 4D ordinary manifold and the Latin indices $i, j, k....= 1, 2, 3$ stand for the space directions only. The 3-vectors are occasionally represented by the bold faced letters (i.e. ${\bf B} = B_i, {\bf E} = E_i, {\bf b^{(1)}} = b^{(1)}_i,
{\bf b^{(2)}} = b^{(2)}_i$, etc.)
[^8]: It will be noted that there are three 1-form differentials $ (dx_\mu), s^{\theta\theta} (dx_\mu),
s^{\bar\theta\bar\theta} (dx_\mu)$, constructed from $(dx_\mu)$, because the dual 5-form differentials on supermanifold are different for each individual of them. For instance, $\star\; (dx_\mu)
= \frac{1}{3!} \varepsilon_{\mu\nu\lambda\zeta}
(dx^\nu \wedge dx^\lambda \wedge dx^\zeta \wedge d\theta \wedge
d \bar\theta), \;
\star\; [ s^{\theta\theta} (dx_\mu) ]
= \frac{1}{3!} \varepsilon_{\mu\nu\lambda\zeta}
(dx^\nu \wedge dx^\lambda \wedge dx^\zeta \wedge d\theta \wedge d\theta),
\;\star\; [s^{\bar\theta\bar\theta} (dx_\mu) ]
= \frac{1}{3!} \varepsilon_{\mu\nu\lambda\zeta} (dx^\nu \wedge dx^\lambda
\wedge dx^\zeta \wedge d\bar\theta \wedge d\bar\theta)$. Only for the sake of brevity, a single 1-form $dx^\mu P_\mu$ (constructed from $dx^\mu$) is given in (3.7). It will be noted that such kind of a triplet of superforms cannot be associated with ${\cal O}_2$ and ${\cal O}_3$ because their Hodge duals are not well-defined on the six $(4 + 2)$-dimensional supermanifold of our present discussion.
[^9]: In principle, one can define a triplet of $\tilde \tau_6$ form in (3.16). These are $
\tilde \tau_6^{(1)}
= \frac{1}{4!} (dx^\mu \wedge dx^\nu \wedge dx^\lambda \wedge dx^\zeta)
\; \tilde T^{(1)}_{\mu\nu\lambda\zeta},
\tilde \tau_6^{(2)} = \frac{1}{4!}
(dx^\mu \wedge dx^\nu \wedge dx^\lambda \wedge dx^\zeta)\;
s^{\theta\theta}\;
\; \tilde T^{(2)}_{\mu\nu\lambda\zeta},
\tilde \tau_6^{(3)} = \frac{1}{4!}
(dx^\mu \wedge dx^\nu \wedge dx^\lambda \wedge dx^\zeta)\;
s^{\bar\theta\bar\theta}\;
\; \tilde T^{(3)}_{\mu\nu\lambda\zeta}$. It is obvious that the Hodge dual of these forms are distinct and different. For obvious reasons, no other forms in (3.16) support such kind of a triplet of superforms (as their Hodge dual forms are not well-defined on the supermanifold in the sense that they will contain more than two Grassmannian wedge products).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A new method of shower-image analysis is presented which appears very powerful as applied to those Cherenkov Imaging Telescopes with very high definition imaging capability. It provides hadron rejection on the basis of a single cut on the image shape, and simultaneously determines the energy of the electromagnetic shower and the position of the shower axis with respect to the detector. The source location is also reconstructed for each individual $\gamma$-ray shower, even with one single telescope, so for a point source the hadron rejection can be further improved. As an example, this new method is applied to data from the C[AT]{} (Cherenkov Array at Thémis) imaging telescope, which has been operational since Autumn, 1996.'
address:
- 'Centre d’Etudes Nucléaire de Bordeaux-Gradignan, France'
- 'Centre d’Etudes Spatiales des Rayonnements, Toulouse, France'
- 'Laboratoire de Physique Corpusculaire et Cosmologie, Collège de France, Paris, France'
- 'Laboratoire de Physique Nucléaire de Haute Energie, Ecole Polytechnique, Palaiseau, France'
- 'LPNHE, Lab.oratoire de Physique Nucléaire de Haute Energie, Universités de Paris VI/VII, France'
- 'Service d’Astrophysique, Centre d’Etudes de Saclay, France'
- 'Groupe de Physique Fondamentale, Université de Perpignan, France'
- 'Nuclear Center, Charles University, Prague, Czech Republic'
- 'Université Paris VII, France'
- 'Department of Physics, Purdue University, Lafayette, IN 47907, U.S.A.'
- 'Joint Laboratory of Optics Ac. Sci. and Palacky University, Olomouc, Czech Republic'
author:
- 'S. Le Bohec'
- 'B. Degrange'
- 'M. Punch'
- 'A. Barrau'
- 'R. Bazer-Bachi'
- 'H. Cabot'
- 'L.M. Chounet'
- 'G. Debiais'
- 'J.P. Dezalay'
- 'A. Djannati-Ataï'
- 'D. Dumora'
- 'P. Espigat'
- 'B. Fabre'
- 'P. Fleury'
- 'G. Fontaine'
- 'R. George'
- 'C. Ghesquière'
- 'P. Goret'
- 'C. Gouiffes'
- 'I.A. Grenier'
- 'L. Iacoucci'
- 'I. Malet'
- 'C. Meynadier'
- 'F. Munz'
- 'T.A. Palfrey'
- 'E. Paré'
- 'Y. Pons'
- 'J. Québert'
- 'K. Ragan'
- 'C. Renault'
- 'M. Rivoal'
- 'L. Rob'
- 'P. Schovanek'
- 'D. Smith'
- 'J.P. Tavernet'
- 'J. Vrana'
title: A new analysis method for very high definition Imaging Atmospheric Cherenkov Telescopes as applied to the CAT telescope
---
Gamma-Ray Astronomy, Atmospheric Cherenkov detector
Introduction
==============
Most of the recent progress in ground-based $\gamma$-ray Astronomy has been obtained from Atmospheric Cherenkov Telescopes ([ACT]{}) using the imaging technique, which permits efficient rejection of the large background of proton or nucleus-induced showers. The utility of high-definition imaging has been demonstrated by the results obtained with the Whipple (109 pixels) [@whitel], C[ANGAROO]{} (220 pixels) [@cantel], and H[EGRA]{} (271 pixels) [@hegratel] [ACT]{}’s, leading to the discovery of several firmly established very-high-energy sources; e.g., the Crab nebula [@whicrab], the pulsar [PSR]{} 1706-44 [@canpsr], and the two relatively nearby Active Galactic Nuclei, Markarian 421 [@whi421] and Markarian 501 [@whi501]. Using a camera with a large number of small pixels allows a more profound image analysis than the usual procedure based on the first and second moments of the light distribution [@scuts], with improved background rejection and energy resolution. The new image analysis described here takes advantage of the full imaging information, including the detailed longitudinal development of the shower (with a significant asymmetry between the top and the bottom of the shower) as well as its lateral extension. This method is based on an analytical model giving, for a genuine $\gamma$-ray shower, the average distribution of Cherenkov light in the focal plane as a function of the following parameters: $\gamma$-ray energy $E_{\gamma}$, distance $D$ between the shower axis and the telescope (or impact parameter), the source position (defined by the vector $\vec{\xi}$ in the focal plane), and the azimuthal position $\phi$ of the shower image about the source in the focal plane. This model is used to define a $\chi^{2}$-like function of $E_{\gamma}$, $D$, $\vec{\xi}$, and $\phi$. For each observed shower, this function is minimized with respect to the parameters: the minimized $\chi^{2}$ provides gamma-hadron discrimination; the $\gamma$-ray energy determination automatically takes account of the impact parameter which is obtained from the same fit. With this method, the origin of a genuine $\gamma$-ray shower can be reconstructed even with one single telescope. This result stems from the fact that electromagnetic showers whose energies are known have a well-defined longitudinal development with rather small fluctuations. In the focal plane of an imaging [ACT]{}, this gives a well-defined longitudinal profile for the image, which depends on the impact parameter and angular origin of the shower. Therefore, the simultaneous measurement of $E_{\gamma}$ and $D$ in the preceding fit also yields the source position $\vec{\xi}$. For the case of a well-localized point source, the energy resolution can be improved by fixing the source position during the fit since there are fewer parameters in the fit.
In this article, the preceding method is illustrated as applied to the C[AT]{} imaging telescope [@cat97], operating since Autumn 1996 in the French Pyrenees. Its very-high-definition imaging camera comprises a central region consisting of 546 phototubes in a hexagonal matrix spaced by 0.13$^\circ$; this is surrounded by 54 tubes in two “guard rings”. The angular diameter of the full field of view is $4.8^\circ$ ($3.0^\circ$ for the small pixels). The guard rings were installed in June 1997, so for the data and simulations used here, they are not included. A more complete description of the C[AT]{} imaging telescope can be found in [@cat97]. The detector response has been simulated in detail, thus allowing the optimization of the new image analysis on the basis of Monte-Carlo-generated $\gamma$-showers against the background of real off-source data.
Section 2 is devoted to the full simulation of the instrument, allowing the calculation of the variation of the equivalent detection area as a function of $E_{\gamma}$ for different zenith angles, thus defining the corresponding energy thresholds. In Section 3, the semi-analytical model giving the average Cherenkov light distribution in the focal plane for a $\gamma$-ray shower is described in detail and shown to be consistent with results from complete simulations of electromagnetic cascades. In Section 4, the performance of the analysis method is presented for different values of the zenith angle; the discussion is focused on hadron rejection by a single cut on the image shape (and for a known point source, an additional cut on direction), on the accuracy of the source location, and finally on energy resolution.
Simulation of the detector response
===================================
The development of electromagnetic or hadronic air showers in the atmosphere is simulated by a modified version of the program described in [@whimc]. For each event, the Cherenkov photons falling onto the mirror elements are followed individually according to their arrival times, initial directions, and wavelengths; reflection from the Davies-Cotton mirror is calculated, taking into account imperfections of the mirrors’ shapes and orientations. Typical aberration images from a point source are smaller than or comparable to the pixel size. A wavelength-dependent fraction of photons is kept, reproducing the effect of mirror reflectivity and Winston cones’ efficiency [@cones] as well as phototubes’ quantum efficiency. Photons from the sky-noise are also simulated, taking into account the effect of [AC]{} coupling on the read-out electronics. The contribution of typical star fields may be superposed. The time-structure and amplitude fluctuation of the pulses generated by single $\gamma$e’s have been parametrized on the basis of tests with the C[AT]{} electronics. The signal from a given pixel is obtained from the pile-up of the contributions from individual $\gamma$e’s according to their respective arrival times, taking account of the amplitude fluctuation of $\sim 0.4
\gamma$e. The comparator threshold is set at a level corresponding to three times the height of the average single-$\gamma$e signal. When a trigger occurs, the phototube signal (corrected for the effect of propagation along the $28\:{\mathrm {m}}$ cable) is integrated over the $12\:{\mathrm {ns}}$ time window provided by the fast gate, thus yielding the simulated [ADC]{} output. These [ADC]{} outputs are then used as input to the analysis procedure described in the following sections.
The preceding simulation of instrumental effects allows the calculation of the equivalent detection area for $\gamma$-rays as a function of energy for a given zenith angle $Z$. This is shown in Fig. \[fig:accept\].a for $Z$s of $0^{\circ}$, $30^{\circ}$, $45^{\circ}$, and $60^{\circ}$. Fig. \[fig:accept\].b shows the corresponding event rates of $\gamma$-rays for a reference source with the intensity and spectrum of the Crab nebula, as measured by the C[AT]{} telescope [@catcrab]:
=14.5 cm
$$\frac{d\phi}{dE} = 2.46 \times \left( \frac{E}{\mathrm{TeV}} \right)
^{-2.55} \times 10^{-11} \; \; \; {\mathrm cm^{-2} s^{-1} TeV^{-1}}$$
The energy corresponding to the maximum event rate (“mode energy”) will be considered as the nominal threshold of the telescope; it varies from 250 to $350\:{\mathrm {GeV}}$ between zenith and $Z = 30^{\circ}$ and increases to $\approx 700\:{\mathrm {GeV}}$ for $Z = 45^{\circ}$. Examples of real images from a $\gamma$-ray candidate shower and from hadron showers are shown in Fig. \[fig:images\].
$$\epsfxsize=6.2 cm
(a)\epsffile[90 210 510 630]{gamma110.ps}
\;
\epsfxsize=6.2 cm
(b)\epsffile[90 210 510 630]{hadron35.ps}$$ $$\epsfxsize=6.2 cm
(c)\epsffile[90 210 510 630]{hadron102.ps}
\;
\epsfxsize=6.2 cm
(d)\epsffile[90 210 510 630]{hadmu47.ps}$$
Whereas $\gamma$-rays produce thin elongated images (Fig. \[fig:images\].a), patterns from hadron showers are often more chaotic or patchy due to the superposition of several electromagnetic components due to $\pi^{0}$’s (Fig. \[fig:images\].b and \[fig:images\].c) and to Cherenkov rings or arcs generated by muons falling onto the mirror or close ($< 40\:{\mathrm {m}}$) to the telescope (Fig. \[fig:images\].d). For hadron showers with energies lower than $200\:{\mathrm {GeV}}$, often the only component which is seen is the Cherenkov light from a single muon; such muons are a source of background in the case where the image is reduced to a short arc, which mimics a low-energy $\gamma$-ray shower [@mu93]. It should be noted, however, that the C[AT]{} imaging telescope is surrounded by the seven detectors of the ASGAT timing array [@asgat] (whose collectors are $7\:{\mathrm {m}}$ diameter each) which should permit the rejection of most of these background events when it resumes operation after upgrade, in 1998.
Modelling of $\gamma$-shower images and image analysis method
=============================================================
For a given value of energy $E_{\gamma}$ and impact parameter, $D$, the Cherenkov image of an electromagnetic shower fluctuates about the mean image, as a result both of intrinsic fluctuations of the shower and of statistical effects in the Cherenkov light collection, as well as instrumental fluctuations. An analysis based on the comparison between individual event images and the theoretical mean image as a function of impact parameter and energy allows full advantage to be taken of the high definition of the C[AT]{} imaging telescope camera. Such an analysis requires an analytical model for $\gamma$-shower Cherenkov images.
In order to construct this model, we have used the results from a paper by Hillas [@hillas] in which the mean development of electromagnetic showers is described and parametrized on the basis of Monte-Carlo simulations. The number of charged particles ($e^{\pm}$) at a given atmospheric depth is given by the Greisen formula. Their energy spectrum and the angular distribution of their momenta with respect to the shower axis are given. Their spatial distribution around the shower axis is not explicitly stated, but parametrizations of their mean distance from this axis and of their spreads in the radial and azimuthal directions are given as a function of their energy and momentum angle. These parametrizations have been compared with our Monte-Carlo simulations, and were found to be in very good agreement, except for the mean value of the angular distribution which we have modified slightly (see Appendix 1). This description of the mean shower development permits the mean Cherenkov image to be deduced, with the additional use of the atmospheric density profile as in [@whimc], optical absorption as in [@xavier], Cherenkov emission properties, and some of the detector characteristics such as its light-collecting area, photo-tube quantum efficiency as a function of wavelength, and site altitude. The angular distance from the source to the zenith has also to be taken into account.
In practice, the shower (for a given energy and impact parameter) is divided into “slices” perpendicular to its axis at different depths, and the contribution of each slice to the image is calculated. In such a slice, the contribution of a charged particle ($e^{\pm}$) with a given energy and direction and a given lateral position with respect to the shower axis is evaluated as follows: the corresponding cone of Cherenkov emission traces a circle on the plane perpendicular to the shower axis containing the imaging telescope; rather than considering a mirror with a fixed position with respect to the shower, we allow it to rotate around the shower axis (the distance $D$ is fixed) and use the intersection of the corresponding ring with the circle of Cherenkov light to find the average contribution of these $e^{\pm}$’s to the image as well as the position of the photo-electrons in the focal plane (see Appendix 2 and Fig. \[fig:calc\]). Summing over the energy, direction, and averaging over the lateral position of all $e^{\pm}$’s in the slice gives the mean position of its image in the focal plane, its mean transverse extension, and the corresponding density of $\gamma$e’s. The sum of the contributions of all the slices gives a full mean image description in terms of the linear density of light along the image axis and the transverse extension of the image as a function of position along this axis. The form of the transverse profile is taken to be a constant shape, determined from Monte-Carlo simulations. This form is scaled to the calculated transverse extension, which varies with position along the image axis. The bi-dimensional profile of vertical $500\:{\mathrm {GeV}}$ $\gamma$-ray showers thus obtained is shown in Fig. \[fig:model\] for different values of the impact parameter $D$.
=14.5 cm
It can be seen that the longitudinal profiles of the expected images are asymmetrical and that their position and general shape depend on $D$. The consistency between this analytical calculation and the full Monte-Carlo simulation can be checked in Fig. \[fig:latlon\], which shows the lateral and longitudinal profiles of the $\gamma$-images for various impact parameters, both from full Monte Carlo simulations and as given by the model.
=14.5 cm
It can also be seen in Fig. \[fig:densite\], which shows the mean density of Cherenkov light on the ground in a $4.6^\circ$ acceptance calculated by both methods, and found to be in good agreement.
=14.5 cm
For the comparison with an individual event image, the expected distribution of $\gamma$e’s in the focal plane (for given values of $E_{\gamma}$, $D$, and $\phi$ of the shower axis projected in the image plane) is integrated over each pixel area, according to the source angular position (defined by two angular coordinates $\vec{\xi}$). A function $\chi^{2} (E_{\gamma}, D, \vec{\xi}, \phi)$ is then defined in such a way that its minimization with respect to $E_{\gamma}$, $D$, $\vec{\xi}$, and $\phi$ gives the values of those parameters which best fit the model to the image. This function is a sum of squared differences between expected and actual pixel contents divided by a quadratic error, extending over all the pixels whose expected or actual content is above a given threshold ($2\:\gamma$e). In the following expression, $Q^{\mathrm{real}}_i$ is the actual value of the charge collected in pixel $i$ (measured in equivalent number of photo-electrons), $Q^{\mathrm{mean}}_i$ is the expected mean contribution of the shower to pixel $i$ for a given set of model parameters, and $\overline{B_i}$ is the mean contribution of the noise calculated on the basis of measured randomly-gated events, which contributes to the fluctuations but not to the charge, due to the [AC]{} coupling. For the C[AT]{} detector, the electronic noise dominates the night-sky background, since the mirror and pixel size are small and the gate is short. The fluctuations in $Q^{\mathrm{real}}_i$ and $Q^{\mathrm{mean}}_i$ are supposed proportional to the square root of the charge in the pixel, giving: $$\chi^{2} = \frac{1}{k} \sum_{i}
\frac{(Q^{\mathrm{real}}_{i} - Q^{\mathrm{mean}}_{i})^{2}}
{\overline{B_i} + \frac{1}{2}
(Q^{\mathrm{real}}_{i}+Q^{\mathrm{mean}}_{i})}$$ The preceding expression is not a $\chi^{2}$ [*stricto sensu*]{} since the exact number of degrees of freedom depends somewhat on the value of the fitted parameters through the number of expected hit photo-tubes. The value of the error factor $k$ (here taken as 2.9) has no effect on the value of the fitted shower parameters. It has been adjusted on simulations in order to have an approximately flat $\chi^{2}$ probability for simulated $\gamma$-shower images well above the trigger threshold. The use of an energy or impact parameter dependent $k$-value is under study. In the case of well-localized point sources, fits in which $\vec{\xi}$ is fixed are used for improved energy resolution.
The fit is quite sensitive to the starting values, so a good estimation of these values is needed. For this purpose the usual moment-based parameters are calculated, but after a “principal cluster” image-cleaning procedure. In this procedure the pixels with a charge less than a threshold of $2\:\gamma$e are zeroed; the principal cluster consists of the largest number of contiguous pixels, all others are then zeroed. This provides moment-based parameters for $\gamma$-images which are more stable against the noise background (though these would not be useful for background rejection, as this cleaning procedure makes the hadronic background more “$\gamma$-like”). Relations between the moment-based parameters and the values to be fitted have been defined on Monte-Carlo simulations; these are then used to define the starting values for the fit.
Results of the method when applied to the CAT images
====================================================
Many of the sources in the E[GRET]{} catalogue [@egret] have well-identified radio, optical, or X-ray counterparts, which give the source position to an accuracy much better than can be achieved by [ACT]{} telescopes. Such sources with known position are usually placed at the centre of the field of an imaging telescope. However, many unidentified sources in the E[GRET]{} catalogue are located in error boxes with typical size of 1$^\circ$. Ground-based Cherenkov detectors are, in principle, able to localize such sources with a higher precision. In order to observe these sources, different methods have been developed by imaging Cherenkov telescope groups. Stereoscopy is the most direct way to find the direction of a source, but requires at least two telescopes and reduces the collection area [@hegratel]. For single-telescope experiments with sufficient background rejection on an image-shape criterion, it is possible to perform de-localized analyses assuming the source at the different points of a grid covering the field of view or by examining the distribution of the intersections of the image axes [@grid]. However, an event-by-event analysis method such as that described here is preferable since in the former methods the signal is more easily drowned-out by the background.
The method has been tested on simulated $\gamma$-images provided by the Monte-Carlo simulation program described above and a realistic simulation of the detector response, including the measured variation in collection efficiency and gain between the phototubes, and measured wavelength response of the mirrors and Winston cones. For optimization of the cut values, these simulated $\gamma$-images have been used together with the real background events from data from off-source runs with the C[AT]{} imaging telescope. Gammas from a point-like source with the Crab nebula spectrum [@catcrab] (see equation (1) above) were simulated at various elevations. The capability of the method both for source detection and for source spectrum measurement have been examined.
Source detection
----------------
As applied to the data, the method consists of minimizing the $\chi^2$ with respect to $E_\gamma$, $D$, $\vec\xi$, and $\phi$. Fig \[fig:chi2\] shows the $\chi^2$ probability distributions obtained with this fit for the simulated $\gamma$-ray events and real background events.
=14.5 cm
A cut on the $\chi^2$ probability value, $P(\chi^2)$, provides a selection of $\gamma$-like events on the basis of the image shape alone. The reconstructed angular origins obtained for simulated $\gamma$-events accumulate around the actual source position which in this case is at the centre of the field. The dispersion around the actual source position has a typical [RMS]{} spread of $0.14^\circ$. For each event, the accuracy of the angular origin determination is better by a factor two in the direction perpendicular to the image axis than in the direction of the image axis (Fig. \[fig:pointerr\]).
=14.5 cm
The [RMS]{} longitudinal error typically varies from $0.2^\circ$ to $0.1^\circ$ as the energy varies from the threshold to $2\:{\mathrm
{TeV}}$. The angular origins obtained for background events are spread over the whole field with an approximately Gaussian distribution with a $1.8^\circ$ [FWHM]{}. Since this distribution is fairly flat, 2-dimensional skymaps of the angular origins of the showers could be used for source detection, as can be seen from the reconstructed positions in data taken on Markarian 501 in Fig. \[fig:m501\].
=14.5 cm
The errors in angular reconstruction given above are for a single shower; a point source with poorly defined position could thus be localized to $\sim 1-2'$ with the combination of $\sim 100$ such events.
For the present, a conservative procedure of monitoring the background is used, based on the pointing angle $\alpha$, similar to the “orientation” of Whipple [@scuts]: $\alpha$ is the angle at the image barycentre between the actual source position and the reconstructed source position. The pointing angle does not use the full information contained in the results of the fit, but has a fairly flat distribution from $0^\circ$ to about $120^\circ$ for background events, which allows the background level to be easily monitored. The cut on $\alpha$ is more efficient than a cut on the angular distance between the source position and the reconstructed $\gamma$ origin since, as seen in Fig. \[fig:pointerr\], the position reconstructed is not symmetric about the source position. The distribution of $\alpha$ for $\gamma$-events from a Crab-like source exhibits a peak at $0^\circ$ (Fig. \[fig:alpha\]) and a small accumulation at $180^\circ$ corresponding to events which are wrongly found to point away from the source.
=14.5 cm
Around 17% of the $\gamma$-events from a Crab-like source are in this situation. The proportion of events with a wrongly reconstructed direction decreases with increasing energy, from 22% at $200\:{\mathrm {GeV}}$ to 9% at $600\:{\mathrm {GeV}}$ and 4% at $1\:{\mathrm {TeV}}$.
The significance of a signal is calculated using the usual formula: $(ON-OFF)/\sqrt{ON+OFF}$ [@lima], assuming equal time on and off-source. The significance per hour on a simulated Crab-like source at zenith has been calculated for various cut values on $\alpha$ and $P(\chi^2)$ (Fig. \[fig:signif1\].a).
$$\epsfxsize=6.5 cm
(a)\epsffile[40 30 540 490]{fig8.eps}
\;
\epsfxsize=6.5 cm
(b)\epsffile[40 30 540 490]{fig11.eps}$$
The best result in terms of both significance and efficiency for $\gamma$-events is obtained for a $P(\chi^2)>0.2$ and $\alpha<6^\circ$, which gives $5.1\sqrt{t}\ \sigma$ where $t$ is the on-source observation time in hours, retaining 34% of the $\gamma$-events while giving a rejection factor of 120 on background events. This rejection factor is smaller than for some comparable experiments as there is a large rejection factor at the trigger level, allowing a moderate background rate of $15\:{\mathrm {Hz}}$ at the zenith. At $45^\circ$ from zenith the best significance for the same cuts falls to $2.7\sqrt{t}\ \sigma$. This is essentially due to the higher energy threshold, leading to a lower event rate; on the other hand, for the same $\gamma$-ray selection efficiency the background rejection factor is comparable to that at zenith.
In order to estimate the efficiency of the $\chi^2$-method in the case of a source with a poorly-defined position, a simulated Crab-like source has been set on the edge of the trigger area ($1^\circ$ from the centre). In this case, the equivalent detection area is divided by a factor of the order of two. Even for a source with known position, when the source is not at the centre of the field it is possible to use one side of the camera as the off region for the other side [@offc]. For a Crab-like source at zenith, the same cuts in $P(\chi^2)$ and $\alpha$ as for a source in the centre of the camera give a $2.3\sqrt{t}\ \sigma$ significance, with a selection efficiency for $\gamma$’s of 36% and a rejection factor of 116 above threshold (Fig. \[fig:signif1\].b). This means that the on-source run time has to be five times larger for a source on the edge of the trigger area than for a source at the centre of the field for the same significance. The corresponding significance obtained at $45^\circ$ from zenith for an off-centre source is $1.5\sqrt{t}\ \sigma$.
$\gamma$-ray energy measurement
-------------------------------
For a source detected with a strong enough significance, the energy spectrum can be studied by a detector with good energy resolution. The fit described above in which the source position is a free parameter gives a first estimate of the energy of each event to within about 30%. However, more precise spectral studies can be carried out on point sources of $\gamma$-rays. The use of the source position as a constraint in the fit provides a higher accuracy for impact parameter measurement and, as a consequence, for energy measurement. If trigger selection effects are ignored in the Monte-Carlo program (thus accepting all events above $100\:{\mathrm {GeV}}$), the $\chi^2$ minimization with respect to $E_\gamma$, $D$, and $\phi$ provides an unbiased energy measurement within about 25% (statistical error only). Trigger selection effects are small for events well above the threshold, as can be seen for simulated $400\:{\mathrm {GeV}}$ $\gamma$-rays in Fig. \[fig:ener500\].
=14.5 cm
This figure also shows that the distribution of the fitted event energies about the true energy is Gaussian on a logarithmic scale. Consequently, the slope of a power-law spectrum can be directly estimated with this technique. Close to the threshold, however, the fitted energy $E_{\mathrm {f}}$ is overestimated as a consequence of the trigger selection. Similarly, the small remaining bias in $ \log
(E_{\mathrm {f}}/E_\gamma)$ at $400\:{\mathrm {GeV}}$ is due to showers with large impact parameters for which the trigger selection is critical at this energy since the telescope is then located at the border of the light pool (Fig. \[fig:densite\]). This effect is largely removed if only showers with a fitted impact parameter $D_{\mathrm {f}}$ lower than $125\:{\mathrm {m}}$ are included (shaded histogram in Fig. \[fig:ener500\]). The bias induced by the trigger selection at different energies is best illustrated by plotting 68% confidence intervals for $E_{\mathrm {f}}$ as a function of the true value $E_\gamma$ used in the simulation (Fig. \[fig:enerfit\]).
=14.5 cm
It can be seen that for $E_f$ below $350\:{\mathrm {GeV}}$ for vertical showers, only an upper limit can be safely derived for $E_\gamma$. Therefore, spectrum measurement is reliable only above a spectrometric threshold, that is, in the region in which $E_{\mathrm {f}}$ depends linearly on $E$. It is somewhat higher than the nominal threshold which is relevant for source discovery. Fig. \[fig:enerfit\] also shows the variation of the average values of $\log(E_{\mathrm {f}})$ on $\log(E_\gamma)$ for zenith angles $0^{\circ}$, $30^{\circ}$, $45^{\circ}$, and $60^{\circ}$, showing the increase in the spectrometric thresholds with increasing zenith angle. Restricting to events with $E_{\mathrm {f}}$ above the spectrometric threshold and $30\:{\mathrm {m}}< D_{\mathrm {f}} \cos Z < 125\:{\mathrm {m}}$, the accuracy of the preceding method is about 20% (statistical error only), independent of the zenith angle $Z$ up to $45^{\circ}$.
Conclusion
==========
The method described above, based on a realistic analytic description of electromagnetic air showers, is best suited for those Cherenkov Imaging Telescopes with a high-resolution camera. The light distribution in the focal plane is fully exploited, yielding the shower direction from the asymmetry of the longitudinal profile as well as the source position in the focal plane. Selection of $\gamma$-rays on the basis of the image shape is performed by using a single $\chi^2$-variable instead of a series of cuts on various image parameters. By combining the $\chi^2$ probability cut and a direction ($\alpha$) cut, a significance of 5$\sigma$ per hour can be achieved for a Crab-like source at the zenith. A future development of the method would be to use the distribution of selected $\gamma$-ray origins on the celestial sphere together with the known energy-dependent point spread function of the method to estimate the significance by a maximum-likelihood method. With the C[AT]{} telescope ($250\:{\mathrm {GeV}}$ threshold), sources with the intensity of the Crab nebula can be detected in one hour. This has been confirmed with the results obtained in the 96/97 observing campaign. Moreover, sources with poorly defined position can be localized with an accuracy of the order of an arc minute on the basis of about 100 showers. The accuracy on $\gamma$-ray energy is of the order of 20–25%. Biases induced by the trigger selection have been investigated; in particular, care should be taken in spectrum measurement, which is accurate only above a specific threshold, higher than that used for source detection.
Appendix 1 : Angular distribution of $e^{\pm}$’s in the analytic model
======================================================================
Let $\theta_{\mathrm {e}}$ be the angle of a charged particle from the shower axis and $E_{\mathrm {e}}$ its kinetic energy. In [@hillas], the variable of interest is $$w = \frac{2(1 - \cos \theta_{\mathrm {e}})}
{(E_{\mathrm{e}}/21\:{\mathrm{MeV}})^2} \;\;\;.$$ In order to be in good agreement with the Monte-Carlo program [@whimc], the average value of $w$ has been parametrized as $$\overline w = \frac{0.9}{1 + 120\:{\mathrm{MeV}}/E_{\mathrm {e}}}$$ instead of the corresponding formula given in [@hillas].
Appendix 2 : Calculation of the mean image longitudinal and transverse profiles
===============================================================================
The notation used in the calculation of the semi-analytical model is shown in Fig. \[fig:calc\]. The origin $O$ is taken on the shower axis $Oz$ as the point closest to the telescope mirror. We consider a charged particle ($e^{\pm}$) at an angle $\theta_{\mathrm {e}}$ from the shower axis, with energy $E_{\mathrm {e}}$. The $x$ and $y$ axes are chosen perpendicular to $Oz$ in such a way that this particle has no $y$ component of velocity. The position of the $e^{\pm}$ is then given by coordinates $(x_{\mathrm {e}},y_{\mathrm {e}},z_{\mathrm
{e}})$ with $\overline y_{\mathrm {e}}=0$ and $\overline x_{\mathrm
{e}}$ given in [@hillas]. In Fig. \[fig:calc\], point $E$ represents the $e^{\pm}$’s mean position and the circle centered on $O'$ corresponds to Cherenkov light emitted from point $E$. As explained in section 3, the telescope mirror is allowed to rotate around $O$ at distance $D$ and the corresponding circle intersects the Cherenkov light circle at $T$ and $T'$ which are the telescope locations from which the $e^{\pm}$ contributes to the image. Consider the location $T$ and denote by $\Phi$ the angle $(Ox,OT)$; $OT$ gives the direction of the image axis in the focal plane and the angular coordinates in this plane (in the longitudinal and transverse direction respectively) are given by: $$\xi_{\mathrm {l}} = \frac{D - x_{\mathrm {e}} \cos \Phi -
y_{\mathrm {e}} \sin \Phi}{z_{\mathrm {e}}-z_{\mathrm {T}}} \;\;\;\;\;\;
\xi_{\mathrm {t}} = \frac{ x_{\mathrm {e}} \sin \Phi - y_{\mathrm {e}}
\cos \Phi}{z_{\mathrm {e}}-z_{\mathrm {T}}}$$ where $z_{\mathrm {T}}$ is the $z$-coordinate of the telescope. In order to include the contribution of position fluctuations, a semi-empirical procedure has been used. If the charged particle position ($x_{\mathrm {e}}$, $y_{\mathrm {e}}$) is allowed to have a spread around $E$ of $\sigma_x$ and $\sigma_y$ (as given in [@hillas]), the Cherenkov light circle does not always intersect the circle centered on $O$ with radius $D$; this results in an apparent reduction of $\sigma_x$ and $\sigma_y$. Therefore, the image transverse spread $\sigma_{\mathrm {t}}$ due to the preceding $e^{\pm}$’s is evaluated as follows: $$\sigma_{\mathrm {t}}^2 = \alpha^2 \frac{(\overline x_{\mathrm {e}}^2+
\sigma_x^2) \sin^2 \Phi +
\sigma_y^2 \cos^2 \Phi}{(z_{\mathrm {e}}-z_{\mathrm {T}})^2}$$ where $\alpha$ has been adjusted to the value $\simeq 0.5$ in order to agree with the results of Monte-Carlo calculations. The form of this semi-empirical formula is not critical to the method, since the pixel size is much greater than $\sigma_{\mathrm {t}}$.
In the calculation of the longitudinal profile, $\xi_{\mathrm {l}}$ is calculated for $x_{\mathrm {e}} = \overline x_{\mathrm {e}}$ and $y_{\mathrm {e}} = 0$. The linear density of Cherenkov light along the image axis is given by summing over the $e^{\pm}$’s, i.e. over $\theta_{\mathrm {e}}$, $E_{\mathrm {e}}$, and $z_{\mathrm {e}}$ at fixed $\xi_{\mathrm {l}}(\theta_{\mathrm {e}},E_{\mathrm
{e}},z_{\mathrm {e}})$: $$\frac{dN}{d\xi_{\mathrm {l}}} =
\int\!\!\int\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\:
\int_{\xi_{\mathrm {l}}=\xi_{\mathrm {l}}
(\theta_{\mathrm {e}},E_{\mathrm {e}},z_{\mathrm {e}})} \,
\frac{\partial^2 N_{\mathrm {e}}}{\partial \theta_{\mathrm {e}}
\partial E_{\mathrm {e}}} \,
\frac{{\mathrm{d}}L}{{\mathrm{d}}z_{\mathrm {e}}} \,
P \, {\mathrm{d}}\theta_{\mathrm {e}} \, {\mathrm{d}}E_{\mathrm {e}}
\, {\mathrm{d}}z_{\mathrm {e}}$$ where $\partial^2 N_{\mathrm {e}}/(\partial \theta_{\mathrm {e}}
\partial E_{\mathrm {e}})$ gives the angular and energy distributions of the $e^{\pm}$’s, ${\mathrm{d}}L/{\mathrm{d}}z_{\mathrm {e}}$ is the amount of Cherenkov light emitted per unit track length and $P$ is the fraction of this light which, on average, reaches the mirror: $$P = \frac{\delta\omega}{\pi} \, \frac{S}{4 \pi D R}$$ where the first ratio is the fraction of Cherenkov light impinging on the mirror at $T$ (the angle $\delta\omega$ is defined in Fig. \[fig:calc\]) and the second the probability to find the mirror (with area $S$ and radius $R$) around $T$. Similarly, the square of the overall transverse spread of the image $\Sigma_{\mathrm {t}}$ for a given value of $\xi_{\mathrm {l}}$ is obtained by averaging $\sigma_{\mathrm {t}}^2(\theta_{\mathrm {e}},E_{\mathrm {e}},z_{\mathrm
{e}})$ over $\theta_{\mathrm {e}}$, $E_{\mathrm {e}}$, and $z_{\mathrm
{e}}$ at fixed $\xi_{\mathrm {l}}(\theta_{\mathrm {e}},E_{\mathrm
{e}},z_{\mathrm {e}})$. The light calculated in a given interval of $\xi_{\mathrm {l}}$ is considered to be distributed in the transverse direction as indicated by the following formula, scaled to the calculated spread $\Sigma_{\mathrm {t}}$: $$F(\xi_{\mathrm {t}}) = \frac{1}{\sqrt{2} \Sigma_{\mathrm {t}}}
\exp(-\frac{\sqrt{2}\mid \xi_{\mathrm {t}} \mid}{\Sigma_{\mathrm {t}}})
\;\;\;\;.$$
=10.2 cm
The authors are grateful to the French and Czech ministries of Foreign Affairs for providing grants for physicists’ travel and accommodation expenses.
[900]{}
Cawley M.F. [*et al*]{}, Exper. Astron. [**1**]{}(1990)173.
Hara T. [*et al*]{}, Nucl. Inst. and Meth. [**A332**]{}(1993)300.
Daum A., [*et al*]{}, Astroparticle Phys.,[**8**]{}(1997)1.
Vacanti G. [*et al*]{}, Ap.J. [**377**]{}(1991)467.
Kifune T. [*et al*]{}, Ap.J. [**438**]{}(1995)L91.
Punch M.[*et al*]{}, Nature [**358**]{}(1992)477.
Quinn J. [*et al*]{}, Ap.J. [**456**]{}(1996)L83.
Punch M. [*et al*]{}, Proc. $22^{nd}$ ICRC, Dublin, Ireland, [**1**]{}(1991) 464.
Barrau A. [*et al*]{}, Nucl. Inst. and Meth., accompanying paper (1998).
Kertzmann M.P. and Sembroski G., Nucl. Inst. and Meth. [**A343**]{}(1994)629.
Punch M., “Towards a Major Atmospheric Cherenkov Detector III”, Tokyo, Japan, ed. T. Kifune, Universal Academy Press, Inc. Tokyo (1994) 215.
Goret P. [*et al*]{}, “Towards a Major Atmospheric Cherenkov Detector V”, Kruger National Park, South Africa, (in press) and Astro/ph 9710260.
Degrange B., “Towards a Major Atmospheric Cherenkov Detector II”, Calgary, Canada, ed. R.C. Lamb, Iowa State University (1993) 215.
Goret P. [*et al*]{}, Astron. Astrophys, [**270**]{}(1993)401.
Hillas A.M., J. Phys.G: Nucl. Phys. [**8**]{}(1982)1461.
Sarazin X., Doctoral Thesis, Université d’Aix Marseille II, unpublished (1994).
Fichtel C.A. [*et al*]{}, Ap.J. Supp. [**94**]{}(1994)551.
Akerlof C.W. [*et al*]{}, Ap.J. [**377**]{}(1991)L97.
Li T., Ma Y., Ap.J. [**272**]{}(1994)317.
Fomin V.P. [*et al*]{}, Astroparticle Phys. [**2**]{}(1994)137.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Using Dumnicki’s approach to showing non-specialty of linear systems consisting of plane curves with prescribed multiplicities in sufficiently general points on $\mathbb{P}^2$ we develop an asymptotic method to determine lower bounds for Seshadri constants of general points on $\mathbb{P}^2$. With this method we prove the lower bound $\frac{4}{13}$ for $10$ general points on $\mathbb{P}^2$.'
address: 'Thomas Eckl, Department of Mathematical Sciences, The University of Liverpool, Mathematical Sciences Building, Liverpool, L69 7ZL, England, U.K.'
author:
- Thomas Eckl
title: 'An asymptotic version of Dumnicki’s algorithm for linear systems in $\mathbb{CP}^2$'
---
Introduction
============
A celebrated conjecture of Nagata [@Nag59] predicts that every curve in $\mathbb{P}^2 = \mathbb{CP}^2$ going through $r > 9$ very general points with multiplicity at least $m$ has degree $d \geq \sqrt{r} m$. Cast in the language of Seshadri constants, Nagata claimed in effect that $$H - \sqrt{\frac{1}{r}} \sum_{j=1}^r E_j$$ is a nef divisor on $\widetilde{X} = \mathrm{Bl}_r(\mathbb{P}^2)$, the blowup of $\mathbb{P}^2$ in the $r$ points, where $H$ is the pullback of a line in $\mathbb{P}^2$ and $E_j$ are the exceptional divisors over the blown up points.
It is well known that Nagata’s conjecture is implied by another conjecture of Harbourne and Hirschowitz about spaces $\mathcal{L}_d(m^r)$ of plane curves of given degree $d$ and multiplicity at least $m$ at $r$ general points [@Mir99; @CilMir01]. This conjecture tries to detect those of the spaces $\mathcal{L}_d(m^r)$ which do not have the expected dimension $$\max (-1, \frac{d(d+3)}{2} - r \cdot \frac{m(m+1)}{2}).$$
In [@Eck05b] the author showed that it is not necessary to know all cases of the Harbourne-Hirschowitz conjecture in order to prove Nagata’s conjecture:
\[Eckl-thm\] Let $r > 9$ be an integer and $(d_i, m_i)$ a sequence of pairs of positive integers such that $\frac{d_i^2}{m_i^2 \cdot r}
\stackrel{i \rightarrow \infty}{\longrightarrow} \frac{1}{a^2} \geq 1$ and the space $\mathcal{L}_{d_i}((m_i+1)^r)$ has expected dimension $\geq 0$. Then $$H - a \cdot \sqrt{\frac{1}{r}} \sum_{j=1}^r E_j$$ is nef on $\widetilde{X}$. In particular, Nagata’s conjecture is true for $r$ general points in $\mathbb{P}^2$, if $a = 1$.
In this paper we want to use Dumnicki’s Reduction Algorithm [@DJ05; @D06] to prove the non-specialty of linear systems $\mathcal{L}_d(m^r)$, as needed in the theorem. Dumnicki’s new idea was to consider linear systems of curves not only going through certain points with at least certain multiplicities, but also the curve equation should contain only monomials from a certain subset of all monomials of degree $\leq d$. He was able to give non-specialty criteria for such linear systems, including the following:
Let $m \in \mathbb{N}$ and let $D \subset \mathbb{N}^2$ such that $\# D = \left( \begin{array}{c}
m+1\\ 2
\end{array}\right)$. Consider the linear system $L = \mathcal{L}_D(m)$ of those curve equations $\sum_{(\alpha,\beta) \in D} c_{\alpha,\beta} x^\alpha y^\beta$, $c_{\alpha,\beta} \in \mathbb{C}$, which pass through a given point with multiplicity at least $m$. Then $L$ is non-special if and only if the points in $D$ do not lie on a curve of degree $m-1$ in $\mathbb{R}^2$.
In particular, $L$ is non-special if there are $m$ parallel lines $l_1, \ldots, l_m$ containing $1, \ldots, m$ points in $D$.
See [@D06 Prop.12]. The last statement follows from Bézout’s Theorem.
Furthermore, Dumnicki devised a recursive procedure showing the non-specialty of linear systems $\mathcal{L}_D(m_1, \ldots, m_r)$ if it terminates in the correct way:
Let $m_1, \ldots, m_{p-1}, m_p \in \mathbb{N}^\ast$, let $D \subset \mathbb{N}^2$, and let $$F: \mathbb{R}^2 \ni (a_1,a_2) \mapsto r_0 + r_1 a_1 + r_2 a_2 \in \mathbb{R},\ r_0, r_1, r_2 \in \mathbb{R},$$ be an affine function. Let $$\begin{array}{rcl}
D_1 & := & \left\{ (a_1,a_2) \in D | F(a_1, a_2) < 0 \right\}, \\
D_2 & := & \left\{ (a_1,a_2) \in D | F(a_1, a_2) > 0 \right\}.
\end{array}$$ If $D_1 \cup D_2 = D$ and $L_1 := \mathcal{L}_D(m_1, \ldots, m_{p-1})$ is non-special of dimension $\geq 0$, $L_2 := \mathcal{L}_D(m_p)$ is non-special of dimension $-1$, then $\mathcal{L}_D(m_1, \ldots, m_p)$ is non-special of dimension $\geq 0$.
See [@D06 Thm.13].
Dumnicki used this procedure to show the Harbourne-Hirschowitz conjecture up to $m=42$, but the power and simplicity of it can best be seen by some easy graphical proofs of non-specialty. For example, Dumnicki [@D06 Ex.37] used a computer to find the following proof for non-specialty of the system $L=\mathcal{L}_{21}(7^{\times 6}, 6^{\times 4},1)$:
$$\begin{array}{cc}
\begin{texdraw}
\drawdim pt
\linewd 0.5
\move(0 0)
\fcir f:0 r:0.2
\move(0 6)
\fcir f:0 r:0.2
\move(0 12)
\fcir f:0 r:0.2
\move(0 18)
\fcir f:0 r:0.2
\move(0 24)
\fcir f:0 r:0.2
\move(0 30)
\fcir f:0 r:0.2
\move(0 36)
\fcir f:0 r:0.2
\move(0 42)
\fcir f:0 r:0.2
\move(0 48)
\fcir f:0 r:0.2
\move(0 54)
\fcir f:0 r:0.2
\move(0 60)
\fcir f:0 r:0.2
\move(0 66)
\fcir f:0 r:0.2
\move(0 72)
\fcir f:0 r:0.2
\move(0 78)
\fcir f:0 r:0.2
\move(0 84)
\fcir f:0 r:0.2
\move(0 90)
\fcir f:0 r:0.2
\move(0 96)
\fcir f:0 r:0.2
\move(0 102)
\fcir f:0 r:0.2
\move(0 108)
\fcir f:0 r:0.2
\move(0 114)
\fcir f:0 r:0.2
\move(0 120)
\fcir f:0 r:0.2
\move(0 126)
\fcir f:0 r:0.2
\move(6 0)
\fcir f:0 r:0.2
\move(6 6)
\fcir f:0 r:0.2
\move(6 12)
\fcir f:0 r:0.2
\move(6 18)
\fcir f:0 r:0.2
\move(6 24)
\fcir f:0 r:0.2
\move(6 30)
\fcir f:0 r:0.2
\move(6 36)
\fcir f:0 r:0.2
\move(6 42)
\fcir f:0 r:0.2
\move(6 48)
\fcir f:0 r:0.2
\move(6 54)
\fcir f:0 r:0.2
\move(6 60)
\fcir f:0 r:0.2
\move(6 66)
\fcir f:0 r:0.2
\move(6 72)
\fcir f:0 r:0.2
\move(6 78)
\fcir f:0 r:0.2
\move(6 84)
\fcir f:0 r:0.2
\move(6 90)
\fcir f:0 r:0.2
\move(6 96)
\fcir f:0 r:0.2
\move(6 102)
\fcir f:0 r:0.2
\move(6 108)
\fcir f:0 r:0.2
\move(6 114)
\fcir f:0 r:0.2
\move(6 120)
\fcir f:0 r:0.2
\move(6 126)
\fcir f:0 r:0.2
\move(12 0)
\fcir f:0 r:0.2
\move(12 6)
\fcir f:0 r:0.2
\move(12 12)
\fcir f:0 r:0.2
\move(12 18)
\fcir f:0 r:0.2
\move(12 24)
\fcir f:0 r:0.2
\move(12 30)
\fcir f:0 r:0.2
\move(12 36)
\fcir f:0 r:0.2
\move(12 42)
\fcir f:0 r:0.2
\move(12 48)
\fcir f:0 r:0.2
\move(12 54)
\fcir f:0 r:0.2
\move(12 60)
\fcir f:0 r:0.2
\move(12 66)
\fcir f:0 r:0.2
\move(12 72)
\fcir f:0 r:0.2
\move(12 78)
\fcir f:0 r:0.2
\move(12 84)
\fcir f:0 r:0.2
\move(12 90)
\fcir f:0 r:0.2
\move(12 96)
\fcir f:0 r:0.2
\move(12 102)
\fcir f:0 r:0.2
\move(12 108)
\fcir f:0 r:0.2
\move(12 114)
\fcir f:0 r:0.2
\move(12 120)
\fcir f:0 r:0.2
\move(12 126)
\fcir f:0 r:0.2
\move(18 0)
\fcir f:0 r:0.2
\move(18 6)
\fcir f:0 r:0.2
\move(18 12)
\fcir f:0 r:0.2
\move(18 18)
\fcir f:0 r:0.2
\move(18 24)
\fcir f:0 r:0.2
\move(18 30)
\fcir f:0 r:0.2
\move(18 36)
\fcir f:0 r:0.2
\move(18 42)
\fcir f:0 r:0.2
\move(18 48)
\fcir f:0 r:0.2
\move(18 54)
\fcir f:0 r:0.2
\move(18 60)
\fcir f:0 r:0.2
\move(18 66)
\fcir f:0 r:0.2
\move(18 72)
\fcir f:0 r:0.2
\move(18 78)
\fcir f:0 r:0.2
\move(18 84)
\fcir f:0 r:0.2
\move(18 90)
\fcir f:0 r:0.2
\move(18 96)
\fcir f:0 r:0.2
\move(18 102)
\fcir f:0 r:0.2
\move(18 108)
\fcir f:0 r:0.2
\move(18 114)
\fcir f:0 r:0.2
\move(18 120)
\fcir f:0 r:0.2
\move(18 126)
\fcir f:0 r:0.2
\move(24 0)
\fcir f:0 r:0.2
\move(24 6)
\fcir f:0 r:0.2
\move(24 12)
\fcir f:0 r:0.2
\move(24 18)
\fcir f:0 r:0.2
\move(24 24)
\fcir f:0 r:0.2
\move(24 30)
\fcir f:0 r:0.2
\move(24 36)
\fcir f:0 r:0.2
\move(24 42)
\fcir f:0 r:0.2
\move(24 48)
\fcir f:0 r:0.2
\move(24 54)
\fcir f:0 r:0.2
\move(24 60)
\fcir f:0 r:0.2
\move(24 66)
\fcir f:0 r:0.2
\move(24 72)
\fcir f:0 r:0.2
\move(24 78)
\fcir f:0 r:0.2
\move(24 84)
\fcir f:0 r:0.2
\move(24 90)
\fcir f:0 r:0.2
\move(24 96)
\fcir f:0 r:0.2
\move(24 102)
\fcir f:0 r:0.2
\move(24 108)
\fcir f:0 r:0.2
\move(24 114)
\fcir f:0 r:0.2
\move(24 120)
\fcir f:0 r:0.2
\move(24 126)
\fcir f:0 r:0.2
\move(30 0)
\fcir f:0 r:0.2
\move(30 6)
\fcir f:0 r:0.2
\move(30 12)
\fcir f:0 r:0.2
\move(30 18)
\fcir f:0 r:0.2
\move(30 24)
\fcir f:0 r:0.2
\move(30 30)
\fcir f:0 r:0.2
\move(30 36)
\fcir f:0 r:0.2
\move(30 42)
\fcir f:0 r:0.2
\move(30 48)
\fcir f:0 r:0.2
\move(30 54)
\fcir f:0 r:0.2
\move(30 60)
\fcir f:0 r:0.2
\move(30 66)
\fcir f:0 r:0.2
\move(30 72)
\fcir f:0 r:0.2
\move(30 78)
\fcir f:0 r:0.2
\move(30 84)
\fcir f:0 r:0.2
\move(30 90)
\fcir f:0 r:0.2
\move(30 96)
\fcir f:0 r:0.2
\move(30 102)
\fcir f:0 r:0.2
\move(30 108)
\fcir f:0 r:0.2
\move(30 114)
\fcir f:0 r:0.2
\move(30 120)
\fcir f:0 r:0.2
\move(30 126)
\fcir f:0 r:0.2
\move(36 0)
\fcir f:0 r:0.2
\move(36 6)
\fcir f:0 r:0.2
\move(36 12)
\fcir f:0 r:0.2
\move(36 18)
\fcir f:0 r:0.2
\move(36 24)
\fcir f:0 r:0.2
\move(36 30)
\fcir f:0 r:0.2
\move(36 36)
\fcir f:0 r:0.2
\move(36 42)
\fcir f:0 r:0.2
\move(36 48)
\fcir f:0 r:0.2
\move(36 54)
\fcir f:0 r:0.2
\move(36 60)
\fcir f:0 r:0.2
\move(36 66)
\fcir f:0 r:0.2
\move(36 72)
\fcir f:0 r:0.2
\move(36 78)
\fcir f:0 r:0.2
\move(36 84)
\fcir f:0 r:0.2
\move(36 90)
\fcir f:0 r:0.2
\move(36 96)
\fcir f:0 r:0.2
\move(36 102)
\fcir f:0 r:0.2
\move(36 108)
\fcir f:0 r:0.2
\move(36 114)
\fcir f:0 r:0.2
\move(36 120)
\fcir f:0 r:0.2
\move(36 126)
\fcir f:0 r:0.2
\move(42 0)
\fcir f:0 r:0.2
\move(42 6)
\fcir f:0 r:0.2
\move(42 12)
\fcir f:0 r:0.2
\move(42 18)
\fcir f:0 r:0.2
\move(42 24)
\fcir f:0 r:0.2
\move(42 30)
\fcir f:0 r:0.2
\move(42 36)
\fcir f:0 r:0.2
\move(42 42)
\fcir f:0 r:0.2
\move(42 48)
\fcir f:0 r:0.2
\move(42 54)
\fcir f:0 r:0.2
\move(42 60)
\fcir f:0 r:0.2
\move(42 66)
\fcir f:0 r:0.2
\move(42 72)
\fcir f:0 r:0.2
\move(42 78)
\fcir f:0 r:0.2
\move(42 84)
\fcir f:0 r:0.2
\move(42 90)
\fcir f:0 r:0.2
\move(42 96)
\fcir f:0 r:0.2
\move(42 102)
\fcir f:0 r:0.2
\move(42 108)
\fcir f:0 r:0.2
\move(42 114)
\fcir f:0 r:0.2
\move(42 120)
\fcir f:0 r:0.2
\move(42 126)
\fcir f:0 r:0.2
\move(48 0)
\fcir f:0 r:0.2
\move(48 6)
\fcir f:0 r:0.2
\move(48 12)
\fcir f:0 r:0.2
\move(48 18)
\fcir f:0 r:0.2
\move(48 24)
\fcir f:0 r:0.2
\move(48 30)
\fcir f:0 r:0.2
\move(48 36)
\fcir f:0 r:0.2
\move(48 42)
\fcir f:0 r:0.2
\move(48 48)
\fcir f:0 r:0.2
\move(48 54)
\fcir f:0 r:0.2
\move(48 60)
\fcir f:0 r:0.2
\move(48 66)
\fcir f:0 r:0.2
\move(48 72)
\fcir f:0 r:0.2
\move(48 78)
\fcir f:0 r:0.2
\move(48 84)
\fcir f:0 r:0.2
\move(48 90)
\fcir f:0 r:0.2
\move(48 96)
\fcir f:0 r:0.2
\move(48 102)
\fcir f:0 r:0.2
\move(48 108)
\fcir f:0 r:0.2
\move(48 114)
\fcir f:0 r:0.2
\move(48 120)
\fcir f:0 r:0.2
\move(48 126)
\fcir f:0 r:0.2
\move(54 0)
\fcir f:0 r:0.2
\move(54 6)
\fcir f:0 r:0.2
\move(54 12)
\fcir f:0 r:0.2
\move(54 18)
\fcir f:0 r:0.2
\move(54 24)
\fcir f:0 r:0.2
\move(54 30)
\fcir f:0 r:0.2
\move(54 36)
\fcir f:0 r:0.2
\move(54 42)
\fcir f:0 r:0.2
\move(54 48)
\fcir f:0 r:0.2
\move(54 54)
\fcir f:0 r:0.2
\move(54 60)
\fcir f:0 r:0.2
\move(54 66)
\fcir f:0 r:0.2
\move(54 72)
\fcir f:0 r:0.2
\move(54 78)
\fcir f:0 r:0.2
\move(54 84)
\fcir f:0 r:0.2
\move(54 90)
\fcir f:0 r:0.2
\move(54 96)
\fcir f:0 r:0.2
\move(54 102)
\fcir f:0 r:0.2
\move(54 108)
\fcir f:0 r:0.2
\move(54 114)
\fcir f:0 r:0.2
\move(54 120)
\fcir f:0 r:0.2
\move(54 126)
\fcir f:0 r:0.2
\move(60 0)
\fcir f:0 r:0.2
\move(60 6)
\fcir f:0 r:0.2
\move(60 12)
\fcir f:0 r:0.2
\move(60 18)
\fcir f:0 r:0.2
\move(60 24)
\fcir f:0 r:0.2
\move(60 30)
\fcir f:0 r:0.2
\move(60 36)
\fcir f:0 r:0.2
\move(60 42)
\fcir f:0 r:0.2
\move(60 48)
\fcir f:0 r:0.2
\move(60 54)
\fcir f:0 r:0.2
\move(60 60)
\fcir f:0 r:0.2
\move(60 66)
\fcir f:0 r:0.2
\move(60 72)
\fcir f:0 r:0.2
\move(60 78)
\fcir f:0 r:0.2
\move(60 84)
\fcir f:0 r:0.2
\move(60 90)
\fcir f:0 r:0.2
\move(60 96)
\fcir f:0 r:0.2
\move(60 102)
\fcir f:0 r:0.2
\move(60 108)
\fcir f:0 r:0.2
\move(60 114)
\fcir f:0 r:0.2
\move(60 120)
\fcir f:0 r:0.2
\move(60 126)
\fcir f:0 r:0.2
\move(66 0)
\fcir f:0 r:0.2
\move(66 6)
\fcir f:0 r:0.2
\move(66 12)
\fcir f:0 r:0.2
\move(66 18)
\fcir f:0 r:0.2
\move(66 24)
\fcir f:0 r:0.2
\move(66 30)
\fcir f:0 r:0.2
\move(66 36)
\fcir f:0 r:0.2
\move(66 42)
\fcir f:0 r:0.2
\move(66 48)
\fcir f:0 r:0.2
\move(66 54)
\fcir f:0 r:0.2
\move(66 60)
\fcir f:0 r:0.2
\move(66 66)
\fcir f:0 r:0.2
\move(66 72)
\fcir f:0 r:0.2
\move(66 78)
\fcir f:0 r:0.2
\move(66 84)
\fcir f:0 r:0.2
\move(66 90)
\fcir f:0 r:0.2
\move(66 96)
\fcir f:0 r:0.2
\move(66 102)
\fcir f:0 r:0.2
\move(66 108)
\fcir f:0 r:0.2
\move(66 114)
\fcir f:0 r:0.2
\move(66 120)
\fcir f:0 r:0.2
\move(66 126)
\fcir f:0 r:0.2
\move(72 0)
\fcir f:0 r:0.2
\move(72 6)
\fcir f:0 r:0.2
\move(72 12)
\fcir f:0 r:0.2
\move(72 18)
\fcir f:0 r:0.2
\move(72 24)
\fcir f:0 r:0.2
\move(72 30)
\fcir f:0 r:0.2
\move(72 36)
\fcir f:0 r:0.2
\move(72 42)
\fcir f:0 r:0.2
\move(72 48)
\fcir f:0 r:0.2
\move(72 54)
\fcir f:0 r:0.2
\move(72 60)
\fcir f:0 r:0.2
\move(72 66)
\fcir f:0 r:0.2
\move(72 72)
\fcir f:0 r:0.2
\move(72 78)
\fcir f:0 r:0.2
\move(72 84)
\fcir f:0 r:0.2
\move(72 90)
\fcir f:0 r:0.2
\move(72 96)
\fcir f:0 r:0.2
\move(72 102)
\fcir f:0 r:0.2
\move(72 108)
\fcir f:0 r:0.2
\move(72 114)
\fcir f:0 r:0.2
\move(72 120)
\fcir f:0 r:0.2
\move(72 126)
\fcir f:0 r:0.2
\move(78 0)
\fcir f:0 r:0.2
\move(78 6)
\fcir f:0 r:0.2
\move(78 12)
\fcir f:0 r:0.2
\move(78 18)
\fcir f:0 r:0.2
\move(78 24)
\fcir f:0 r:0.2
\move(78 30)
\fcir f:0 r:0.2
\move(78 36)
\fcir f:0 r:0.2
\move(78 42)
\fcir f:0 r:0.2
\move(78 48)
\fcir f:0 r:0.2
\move(78 54)
\fcir f:0 r:0.2
\move(78 60)
\fcir f:0 r:0.2
\move(78 66)
\fcir f:0 r:0.2
\move(78 72)
\fcir f:0 r:0.2
\move(78 78)
\fcir f:0 r:0.2
\move(78 84)
\fcir f:0 r:0.2
\move(78 90)
\fcir f:0 r:0.2
\move(78 96)
\fcir f:0 r:0.2
\move(78 102)
\fcir f:0 r:0.2
\move(78 108)
\fcir f:0 r:0.2
\move(78 114)
\fcir f:0 r:0.2
\move(78 120)
\fcir f:0 r:0.2
\move(78 126)
\fcir f:0 r:0.2
\move(84 0)
\fcir f:0 r:0.2
\move(84 6)
\fcir f:0 r:0.2
\move(84 12)
\fcir f:0 r:0.2
\move(84 18)
\fcir f:0 r:0.2
\move(84 24)
\fcir f:0 r:0.2
\move(84 30)
\fcir f:0 r:0.2
\move(84 36)
\fcir f:0 r:0.2
\move(84 42)
\fcir f:0 r:0.2
\move(84 48)
\fcir f:0 r:0.2
\move(84 54)
\fcir f:0 r:0.2
\move(84 60)
\fcir f:0 r:0.2
\move(84 66)
\fcir f:0 r:0.2
\move(84 72)
\fcir f:0 r:0.2
\move(84 78)
\fcir f:0 r:0.2
\move(84 84)
\fcir f:0 r:0.2
\move(84 90)
\fcir f:0 r:0.2
\move(84 96)
\fcir f:0 r:0.2
\move(84 102)
\fcir f:0 r:0.2
\move(84 108)
\fcir f:0 r:0.2
\move(84 114)
\fcir f:0 r:0.2
\move(84 120)
\fcir f:0 r:0.2
\move(84 126)
\fcir f:0 r:0.2
\move(90 0)
\fcir f:0 r:0.2
\move(90 6)
\fcir f:0 r:0.2
\move(90 12)
\fcir f:0 r:0.2
\move(90 18)
\fcir f:0 r:0.2
\move(90 24)
\fcir f:0 r:0.2
\move(90 30)
\fcir f:0 r:0.2
\move(90 36)
\fcir f:0 r:0.2
\move(90 42)
\fcir f:0 r:0.2
\move(90 48)
\fcir f:0 r:0.2
\move(90 54)
\fcir f:0 r:0.2
\move(90 60)
\fcir f:0 r:0.2
\move(90 66)
\fcir f:0 r:0.2
\move(90 72)
\fcir f:0 r:0.2
\move(90 78)
\fcir f:0 r:0.2
\move(90 84)
\fcir f:0 r:0.2
\move(90 90)
\fcir f:0 r:0.2
\move(90 96)
\fcir f:0 r:0.2
\move(90 102)
\fcir f:0 r:0.2
\move(90 108)
\fcir f:0 r:0.2
\move(90 114)
\fcir f:0 r:0.2
\move(90 120)
\fcir f:0 r:0.2
\move(90 126)
\fcir f:0 r:0.2
\move(96 0)
\fcir f:0 r:0.2
\move(96 6)
\fcir f:0 r:0.2
\move(96 12)
\fcir f:0 r:0.2
\move(96 18)
\fcir f:0 r:0.2
\move(96 24)
\fcir f:0 r:0.2
\move(96 30)
\fcir f:0 r:0.2
\move(96 36)
\fcir f:0 r:0.2
\move(96 42)
\fcir f:0 r:0.2
\move(96 48)
\fcir f:0 r:0.2
\move(96 54)
\fcir f:0 r:0.2
\move(96 60)
\fcir f:0 r:0.2
\move(96 66)
\fcir f:0 r:0.2
\move(96 72)
\fcir f:0 r:0.2
\move(96 78)
\fcir f:0 r:0.2
\move(96 84)
\fcir f:0 r:0.2
\move(96 90)
\fcir f:0 r:0.2
\move(96 96)
\fcir f:0 r:0.2
\move(96 102)
\fcir f:0 r:0.2
\move(96 108)
\fcir f:0 r:0.2
\move(96 114)
\fcir f:0 r:0.2
\move(96 120)
\fcir f:0 r:0.2
\move(96 126)
\fcir f:0 r:0.2
\move(102 0)
\fcir f:0 r:0.2
\move(102 6)
\fcir f:0 r:0.2
\move(102 12)
\fcir f:0 r:0.2
\move(102 18)
\fcir f:0 r:0.2
\move(102 24)
\fcir f:0 r:0.2
\move(102 30)
\fcir f:0 r:0.2
\move(102 36)
\fcir f:0 r:0.2
\move(102 42)
\fcir f:0 r:0.2
\move(102 48)
\fcir f:0 r:0.2
\move(102 54)
\fcir f:0 r:0.2
\move(102 60)
\fcir f:0 r:0.2
\move(102 66)
\fcir f:0 r:0.2
\move(102 72)
\fcir f:0 r:0.2
\move(102 78)
\fcir f:0 r:0.2
\move(102 84)
\fcir f:0 r:0.2
\move(102 90)
\fcir f:0 r:0.2
\move(102 96)
\fcir f:0 r:0.2
\move(102 102)
\fcir f:0 r:0.2
\move(102 108)
\fcir f:0 r:0.2
\move(102 114)
\fcir f:0 r:0.2
\move(102 120)
\fcir f:0 r:0.2
\move(102 126)
\fcir f:0 r:0.2
\move(108 0)
\fcir f:0 r:0.2
\move(108 6)
\fcir f:0 r:0.2
\move(108 12)
\fcir f:0 r:0.2
\move(108 18)
\fcir f:0 r:0.2
\move(108 24)
\fcir f:0 r:0.2
\move(108 30)
\fcir f:0 r:0.2
\move(108 36)
\fcir f:0 r:0.2
\move(108 42)
\fcir f:0 r:0.2
\move(108 48)
\fcir f:0 r:0.2
\move(108 54)
\fcir f:0 r:0.2
\move(108 60)
\fcir f:0 r:0.2
\move(108 66)
\fcir f:0 r:0.2
\move(108 72)
\fcir f:0 r:0.2
\move(108 78)
\fcir f:0 r:0.2
\move(108 84)
\fcir f:0 r:0.2
\move(108 90)
\fcir f:0 r:0.2
\move(108 96)
\fcir f:0 r:0.2
\move(108 102)
\fcir f:0 r:0.2
\move(108 108)
\fcir f:0 r:0.2
\move(108 114)
\fcir f:0 r:0.2
\move(108 120)
\fcir f:0 r:0.2
\move(108 126)
\fcir f:0 r:0.2
\move(114 0)
\fcir f:0 r:0.2
\move(114 6)
\fcir f:0 r:0.2
\move(114 12)
\fcir f:0 r:0.2
\move(114 18)
\fcir f:0 r:0.2
\move(114 24)
\fcir f:0 r:0.2
\move(114 30)
\fcir f:0 r:0.2
\move(114 36)
\fcir f:0 r:0.2
\move(114 42)
\fcir f:0 r:0.2
\move(114 48)
\fcir f:0 r:0.2
\move(114 54)
\fcir f:0 r:0.2
\move(114 60)
\fcir f:0 r:0.2
\move(114 66)
\fcir f:0 r:0.2
\move(114 72)
\fcir f:0 r:0.2
\move(114 78)
\fcir f:0 r:0.2
\move(114 84)
\fcir f:0 r:0.2
\move(114 90)
\fcir f:0 r:0.2
\move(114 96)
\fcir f:0 r:0.2
\move(114 102)
\fcir f:0 r:0.2
\move(114 108)
\fcir f:0 r:0.2
\move(114 114)
\fcir f:0 r:0.2
\move(114 120)
\fcir f:0 r:0.2
\move(114 126)
\fcir f:0 r:0.2
\move(120 0)
\fcir f:0 r:0.2
\move(120 6)
\fcir f:0 r:0.2
\move(120 12)
\fcir f:0 r:0.2
\move(120 18)
\fcir f:0 r:0.2
\move(120 24)
\fcir f:0 r:0.2
\move(120 30)
\fcir f:0 r:0.2
\move(120 36)
\fcir f:0 r:0.2
\move(120 42)
\fcir f:0 r:0.2
\move(120 48)
\fcir f:0 r:0.2
\move(120 54)
\fcir f:0 r:0.2
\move(120 60)
\fcir f:0 r:0.2
\move(120 66)
\fcir f:0 r:0.2
\move(120 72)
\fcir f:0 r:0.2
\move(120 78)
\fcir f:0 r:0.2
\move(120 84)
\fcir f:0 r:0.2
\move(120 90)
\fcir f:0 r:0.2
\move(120 96)
\fcir f:0 r:0.2
\move(120 102)
\fcir f:0 r:0.2
\move(120 108)
\fcir f:0 r:0.2
\move(120 114)
\fcir f:0 r:0.2
\move(120 120)
\fcir f:0 r:0.2
\move(120 126)
\fcir f:0 r:0.2
\move(126 0)
\fcir f:0 r:0.2
\move(126 6)
\fcir f:0 r:0.2
\move(126 12)
\fcir f:0 r:0.2
\move(126 18)
\fcir f:0 r:0.2
\move(126 24)
\fcir f:0 r:0.2
\move(126 30)
\fcir f:0 r:0.2
\move(126 36)
\fcir f:0 r:0.2
\move(126 42)
\fcir f:0 r:0.2
\move(126 48)
\fcir f:0 r:0.2
\move(126 54)
\fcir f:0 r:0.2
\move(126 60)
\fcir f:0 r:0.2
\move(126 66)
\fcir f:0 r:0.2
\move(126 72)
\fcir f:0 r:0.2
\move(126 78)
\fcir f:0 r:0.2
\move(126 84)
\fcir f:0 r:0.2
\move(126 90)
\fcir f:0 r:0.2
\move(126 96)
\fcir f:0 r:0.2
\move(126 102)
\fcir f:0 r:0.2
\move(126 108)
\fcir f:0 r:0.2
\move(126 114)
\fcir f:0 r:0.2
\move(126 120)
\fcir f:0 r:0.2
\move(126 126)
\fcir f:0 r:0.2
\arrowheadtype t:V
\move(0 0)
\avec(144 0)
\move(0 0)
\avec(0 144)
\htext(148 0){$\mathbb{N}$}
\htext(-13 132){$\mathbb{N}$}
\move(0 0)
\fcir f:0 r:1
\move(0 6)
\fcir f:0 r:1
\move(0 12)
\fcir f:0 r:1
\move(0 18)
\fcir f:0 r:1
\move(0 24)
\fcir f:0 r:1
\move(0 30)
\fcir f:0 r:1
\move(0 36)
\fcir f:0 r:1
\move(0 42)
\fcir f:0 r:1
\move(0 48)
\fcir f:0 r:1
\move(0 54)
\fcir f:0 r:1
\move(0 60)
\fcir f:0 r:1
\move(0 66)
\fcir f:0 r:1
\move(0 72)
\fcir f:0 r:1
\move(0 78)
\fcir f:0 r:1
\move(0 84)
\fcir f:0 r:1
\move(0 90)
\fcir f:0 r:1
\move(0 96)
\fcir f:0 r:1
\move(0 102)
\fcir f:0 r:1
\move(0 108)
\fcir f:0 r:1
\move(0 114)
\fcir f:0 r:1
\move(0 120)
\fcir f:0 r:1
\move(0 126)
\fcir f:0 r:1
\move(6 0)
\fcir f:0 r:1
\move(6 6)
\fcir f:0 r:1
\move(6 12)
\fcir f:0 r:1
\move(6 18)
\fcir f:0 r:1
\move(6 24)
\fcir f:0 r:1
\move(6 30)
\fcir f:0 r:1
\move(6 36)
\fcir f:0 r:1
\move(6 42)
\fcir f:0 r:1
\move(6 48)
\fcir f:0 r:1
\move(6 54)
\fcir f:0 r:1
\move(6 60)
\fcir f:0 r:1
\move(6 66)
\fcir f:0 r:1
\move(6 72)
\fcir f:0 r:1
\move(6 78)
\fcir f:0 r:1
\move(6 84)
\fcir f:0 r:1
\move(6 90)
\fcir f:0 r:1
\move(6 96)
\fcir f:0 r:1
\move(6 102)
\fcir f:0 r:1
\move(6 108)
\fcir f:0 r:1
\move(6 114)
\fcir f:0 r:1
\move(6 120)
\fcir f:0 r:1
\move(12 0)
\fcir f:0 r:1
\move(12 6)
\fcir f:0 r:1
\move(12 12)
\fcir f:0 r:1
\move(12 18)
\fcir f:0 r:1
\move(12 24)
\fcir f:0 r:1
\move(12 30)
\fcir f:0 r:1
\move(12 36)
\fcir f:0 r:1
\move(12 42)
\fcir f:0 r:1
\move(12 48)
\fcir f:0 r:1
\move(12 54)
\fcir f:0 r:1
\move(12 60)
\fcir f:0 r:1
\move(12 66)
\fcir f:0 r:1
\move(12 72)
\fcir f:0 r:1
\move(12 78)
\fcir f:0 r:1
\move(12 84)
\fcir f:0 r:1
\move(12 90)
\fcir f:0 r:1
\move(12 96)
\fcir f:0 r:1
\move(12 102)
\fcir f:0 r:1
\move(12 108)
\fcir f:0 r:1
\move(12 114)
\fcir f:0 r:1
\move(18 0)
\fcir f:0 r:1
\move(18 6)
\fcir f:0 r:1
\move(18 12)
\fcir f:0 r:1
\move(18 18)
\fcir f:0 r:1
\move(18 24)
\fcir f:0 r:1
\move(18 30)
\fcir f:0 r:1
\move(18 36)
\fcir f:0 r:1
\move(18 42)
\fcir f:0 r:1
\move(18 48)
\fcir f:0 r:1
\move(18 54)
\fcir f:0 r:1
\move(18 60)
\fcir f:0 r:1
\move(18 66)
\fcir f:0 r:1
\move(18 72)
\fcir f:0 r:1
\move(18 78)
\fcir f:0 r:1
\move(18 84)
\fcir f:0 r:1
\move(18 90)
\fcir f:0 r:1
\move(18 96)
\fcir f:0 r:1
\move(18 102)
\fcir f:0 r:1
\move(18 108)
\fcir f:0 r:1
\move(24 0)
\fcir f:0 r:1
\move(24 6)
\fcir f:0 r:1
\move(24 12)
\fcir f:0 r:1
\move(24 18)
\fcir f:0 r:1
\move(24 24)
\fcir f:0 r:1
\move(24 30)
\fcir f:0 r:1
\move(24 36)
\fcir f:0 r:1
\move(24 42)
\fcir f:0 r:1
\move(24 48)
\fcir f:0 r:1
\move(24 54)
\fcir f:0 r:1
\move(24 60)
\fcir f:0 r:1
\move(24 66)
\fcir f:0 r:1
\move(24 72)
\fcir f:0 r:1
\move(24 78)
\fcir f:0 r:1
\move(24 84)
\fcir f:0 r:1
\move(24 90)
\fcir f:0 r:1
\move(24 96)
\fcir f:0 r:1
\move(24 102)
\fcir f:0 r:1
\move(30 0)
\fcir f:0 r:1
\move(30 6)
\fcir f:0 r:1
\move(30 12)
\fcir f:0 r:1
\move(30 18)
\fcir f:0 r:1
\move(30 24)
\fcir f:0 r:1
\move(30 30)
\fcir f:0 r:1
\move(30 36)
\fcir f:0 r:1
\move(30 42)
\fcir f:0 r:1
\move(30 48)
\fcir f:0 r:1
\move(30 54)
\fcir f:0 r:1
\move(30 60)
\fcir f:0 r:1
\move(30 66)
\fcir f:0 r:1
\move(30 72)
\fcir f:0 r:1
\move(30 78)
\fcir f:0 r:1
\move(30 84)
\fcir f:0 r:1
\move(30 90)
\fcir f:0 r:1
\move(30 96)
\fcir f:0 r:1
\move(36 0)
\fcir f:0 r:1
\move(36 6)
\fcir f:0 r:1
\move(36 12)
\fcir f:0 r:1
\move(36 18)
\fcir f:0 r:1
\move(36 24)
\fcir f:0 r:1
\move(36 30)
\fcir f:0 r:1
\move(36 36)
\fcir f:0 r:1
\move(36 42)
\fcir f:0 r:1
\move(36 48)
\fcir f:0 r:1
\move(36 54)
\fcir f:0 r:1
\move(36 60)
\fcir f:0 r:1
\move(36 66)
\fcir f:0 r:1
\move(36 72)
\fcir f:0 r:1
\move(36 78)
\fcir f:0 r:1
\move(36 84)
\fcir f:0 r:1
\move(36 90)
\fcir f:0 r:1
\move(42 0)
\fcir f:0 r:1
\move(42 6)
\fcir f:0 r:1
\move(42 12)
\fcir f:0 r:1
\move(42 18)
\fcir f:0 r:1
\move(42 24)
\fcir f:0 r:1
\move(42 30)
\fcir f:0 r:1
\move(42 36)
\fcir f:0 r:1
\move(42 42)
\fcir f:0 r:1
\move(42 48)
\fcir f:0 r:1
\move(42 54)
\fcir f:0 r:1
\move(42 60)
\fcir f:0 r:1
\move(42 66)
\fcir f:0 r:1
\move(42 72)
\fcir f:0 r:1
\move(42 78)
\fcir f:0 r:1
\move(42 84)
\fcir f:0 r:1
\move(48 0)
\fcir f:0 r:1
\move(48 6)
\fcir f:0 r:1
\move(48 12)
\fcir f:0 r:1
\move(48 18)
\fcir f:0 r:1
\move(48 24)
\fcir f:0 r:1
\move(48 30)
\fcir f:0 r:1
\move(48 36)
\fcir f:0 r:1
\move(48 42)
\fcir f:0 r:1
\move(48 48)
\fcir f:0 r:1
\move(48 54)
\fcir f:0 r:1
\move(48 60)
\fcir f:0 r:1
\move(48 66)
\fcir f:0 r:1
\move(48 72)
\fcir f:0 r:1
\move(48 78)
\fcir f:0 r:1
\move(54 0)
\fcir f:0 r:1
\move(54 6)
\fcir f:0 r:1
\move(54 12)
\fcir f:0 r:1
\move(54 18)
\fcir f:0 r:1
\move(54 24)
\fcir f:0 r:1
\move(54 30)
\fcir f:0 r:1
\move(54 36)
\fcir f:0 r:1
\move(54 42)
\fcir f:0 r:1
\move(54 48)
\fcir f:0 r:1
\move(54 54)
\fcir f:0 r:1
\move(54 60)
\fcir f:0 r:1
\move(54 66)
\fcir f:0 r:1
\move(54 72)
\fcir f:0 r:1
\move(60 0)
\fcir f:0 r:1
\move(60 6)
\fcir f:0 r:1
\move(60 12)
\fcir f:0 r:1
\move(60 18)
\fcir f:0 r:1
\move(60 24)
\fcir f:0 r:1
\move(60 30)
\fcir f:0 r:1
\move(60 36)
\fcir f:0 r:1
\move(60 42)
\fcir f:0 r:1
\move(60 48)
\fcir f:0 r:1
\move(60 54)
\fcir f:0 r:1
\move(60 60)
\fcir f:0 r:1
\move(60 66)
\fcir f:0 r:1
\move(66 0)
\fcir f:0 r:1
\move(66 6)
\fcir f:0 r:1
\move(66 12)
\fcir f:0 r:1
\move(66 18)
\fcir f:0 r:1
\move(66 24)
\fcir f:0 r:1
\move(66 30)
\fcir f:0 r:1
\move(66 36)
\fcir f:0 r:1
\move(66 42)
\fcir f:0 r:1
\move(66 48)
\fcir f:0 r:1
\move(66 54)
\fcir f:0 r:1
\move(66 60)
\fcir f:0 r:1
\move(72 0)
\fcir f:0 r:1
\move(72 6)
\fcir f:0 r:1
\move(72 12)
\fcir f:0 r:1
\move(72 18)
\fcir f:0 r:1
\move(72 24)
\fcir f:0 r:1
\move(72 30)
\fcir f:0 r:1
\move(72 36)
\fcir f:0 r:1
\move(72 42)
\fcir f:0 r:1
\move(72 48)
\fcir f:0 r:1
\move(72 54)
\fcir f:0 r:1
\move(78 0)
\fcir f:0 r:1
\move(78 6)
\fcir f:0 r:1
\move(78 12)
\fcir f:0 r:1
\move(78 18)
\fcir f:0 r:1
\move(78 24)
\fcir f:0 r:1
\move(78 30)
\fcir f:0 r:1
\move(78 36)
\fcir f:0 r:1
\move(78 42)
\fcir f:0 r:1
\move(78 48)
\fcir f:0 r:1
\move(84 0)
\fcir f:0 r:1
\move(84 6)
\fcir f:0 r:1
\move(84 12)
\fcir f:0 r:1
\move(84 18)
\fcir f:0 r:1
\move(84 24)
\fcir f:0 r:1
\move(84 30)
\fcir f:0 r:1
\move(84 36)
\fcir f:0 r:1
\move(84 42)
\fcir f:0 r:1
\move(90 0)
\fcir f:0 r:1
\move(90 6)
\fcir f:0 r:1
\move(90 12)
\fcir f:0 r:1
\move(90 18)
\fcir f:0 r:1
\move(90 24)
\fcir f:0 r:1
\move(90 30)
\fcir f:0 r:1
\move(90 36)
\fcir f:0 r:1
\move(96 0)
\fcir f:0 r:1
\move(96 6)
\fcir f:0 r:1
\move(96 12)
\fcir f:0 r:1
\move(96 18)
\fcir f:0 r:1
\move(96 24)
\fcir f:0 r:1
\move(96 30)
\fcir f:0 r:1
\move(102 0)
\fcir f:0 r:1
\move(102 6)
\fcir f:0 r:1
\move(102 12)
\fcir f:0 r:1
\move(102 18)
\fcir f:0 r:1
\move(102 24)
\fcir f:0 r:1
\move(108 0)
\fcir f:0 r:1
\move(108 6)
\fcir f:0 r:1
\move(108 12)
\fcir f:0 r:1
\move(108 18)
\fcir f:0 r:1
\move(114 0)
\fcir f:0 r:1
\move(114 6)
\fcir f:0 r:1
\move(114 12)
\fcir f:0 r:1
\move(120 0)
\fcir f:0 r:1
\move(120 6)
\fcir f:0 r:1
\move(126 0)
\fcir f:0 r:1
\move(0 90)
\move(-2 88)
\lvec(2 92)
\move(2 88)
\lvec(-2 92)
\move(0 96)
\move(-2 94)
\lvec(2 98)
\move(2 94)
\lvec(-2 98)
\move(0 102)
\move(-2 100)
\lvec(2 104)
\move(2 100)
\lvec(-2 104)
\move(0 108)
\move(-2 106)
\lvec(2 110)
\move(2 106)
\lvec(-2 110)
\move(0 114)
\move(-2 112)
\lvec(2 116)
\move(2 112)
\lvec(-2 116)
\move(0 120)
\move(-2 118)
\lvec(2 122)
\move(2 118)
\lvec(-2 122)
\move(0 126)
\move(-2 124)
\lvec(2 128)
\move(2 124)
\lvec(-2 128)
\move(6 90)
\move(4 88)
\lvec(8 92)
\move(8 88)
\lvec(4 92)
\move(6 96)
\move(4 94)
\lvec(8 98)
\move(8 94)
\lvec(4 98)
\move(6 102)
\move(4 100)
\lvec(8 104)
\move(8 100)
\lvec(4 104)
\move(6 108)
\move(4 106)
\lvec(8 110)
\move(8 106)
\lvec(4 110)
\move(6 114)
\move(4 112)
\lvec(8 116)
\move(8 112)
\lvec(4 116)
\move(6 120)
\move(4 118)
\lvec(8 122)
\move(8 118)
\lvec(4 122)
\move(12 90)
\move(10 88)
\lvec(14 92)
\move(14 88)
\lvec(10 92)
\move(12 96)
\move(10 94)
\lvec(14 98)
\move(14 94)
\lvec(10 98)
\move(12 102)
\move(10 100)
\lvec(14 104)
\move(14 100)
\lvec(10 104)
\move(12 108)
\move(10 106)
\lvec(14 110)
\move(14 106)
\lvec(10 110)
\move(12 114)
\move(10 112)
\lvec(14 116)
\move(14 112)
\lvec(10 116)
\move(18 90)
\move(16 88)
\lvec(20 92)
\move(20 88)
\lvec(16 92)
\move(18 96)
\move(16 94)
\lvec(20 98)
\move(20 94)
\lvec(16 98)
\move(18 102)
\move(16 100)
\lvec(20 104)
\move(20 100)
\lvec(16 104)
\move(18 108)
\move(16 106)
\lvec(20 110)
\move(20 106)
\lvec(16 110)
\move(24 90)
\move(22 88)
\lvec(26 92)
\move(26 88)
\lvec(22 92)
\move(24 96)
\move(22 94)
\lvec(26 98)
\move(26 94)
\lvec(22 98)
\move(24 102)
\move(22 100)
\lvec(26 104)
\move(26 100)
\lvec(22 104)
\move(30 90)
\move(28 88)
\lvec(32 92)
\move(32 88)
\lvec(28 92)
\move(30 96)
\move(28 94)
\lvec(32 98)
\move(32 94)
\lvec(28 98)
\move(36 90)
\move(34 88)
\lvec(38 92)
\move(38 88)
\lvec(34 92)
\move(90 0)
\move(88 -2)
\lvec(92 2)
\move(92 -2)
\lvec(88 2)
\move(90 6)
\move(88 4)
\lvec(92 8)
\move(92 4)
\lvec(88 8)
\move(90 12)
\move(88 10)
\lvec(92 14)
\move(92 10)
\lvec(88 14)
\move(90 18)
\move(88 16)
\lvec(92 20)
\move(92 16)
\lvec(88 20)
\move(90 24)
\move(88 22)
\lvec(92 26)
\move(92 22)
\lvec(88 26)
\move(90 30)
\move(88 28)
\lvec(92 32)
\move(92 28)
\lvec(88 32)
\move(90 36)
\move(88 34)
\lvec(92 38)
\move(92 34)
\lvec(88 38)
\move(96 0)
\move(94 -2)
\lvec(98 2)
\move(98 -2)
\lvec(94 2)
\move(96 6)
\move(94 4)
\lvec(98 8)
\move(98 4)
\lvec(94 8)
\move(96 12)
\move(94 10)
\lvec(98 14)
\move(98 10)
\lvec(94 14)
\move(96 18)
\move(94 16)
\lvec(98 20)
\move(98 16)
\lvec(94 20)
\move(96 24)
\move(94 22)
\lvec(98 26)
\move(98 22)
\lvec(94 26)
\move(96 30)
\move(94 28)
\lvec(98 32)
\move(98 28)
\lvec(94 32)
\move(102 0)
\move(100 -2)
\lvec(104 2)
\move(104 -2)
\lvec(100 2)
\move(102 6)
\move(100 4)
\lvec(104 8)
\move(104 4)
\lvec(100 8)
\move(102 12)
\move(100 10)
\lvec(104 14)
\move(104 10)
\lvec(100 14)
\move(102 18)
\move(100 16)
\lvec(104 20)
\move(104 16)
\lvec(100 20)
\move(102 24)
\move(100 22)
\lvec(104 26)
\move(104 22)
\lvec(100 26)
\move(108 0)
\move(106 -2)
\lvec(110 2)
\move(110 -2)
\lvec(106 2)
\move(108 6)
\move(106 4)
\lvec(110 8)
\move(110 4)
\lvec(106 8)
\move(108 12)
\move(106 10)
\lvec(110 14)
\move(110 10)
\lvec(106 14)
\move(108 18)
\move(106 16)
\lvec(110 20)
\move(110 16)
\lvec(106 20)
\move(114 0)
\move(112 -2)
\lvec(116 2)
\move(116 -2)
\lvec(112 2)
\move(114 6)
\move(112 4)
\lvec(116 8)
\move(116 4)
\lvec(112 8)
\move(114 12)
\move(112 10)
\lvec(116 14)
\move(116 10)
\lvec(112 14)
\move(120 0)
\move(118 -2)
\lvec(122 2)
\move(122 -2)
\lvec(118 2)
\move(120 6)
\move(118 4)
\lvec(122 8)
\move(122 4)
\lvec(118 8)
\move(126 0)
\move(124 -2)
\lvec(128 2)
\move(128 -2)
\lvec(124 2)
\move(0 0)
\move(-2 -2)
\lvec(2 2)
\move(2 -2)
\lvec(-2 2)
\move(0 6)
\move(-2 4)
\lvec(2 8)
\move(2 4)
\lvec(-2 8)
\move(0 12)
\move(-2 10)
\lvec(2 14)
\move(2 10)
\lvec(-2 14)
\move(0 18)
\move(-2 16)
\lvec(2 20)
\move(2 16)
\lvec(-2 20)
\move(0 24)
\move(-2 22)
\lvec(2 26)
\move(2 22)
\lvec(-2 26)
\move(0 30)
\move(-2 28)
\lvec(2 32)
\move(2 28)
\lvec(-2 32)
\move(0 36)
\move(-2 34)
\lvec(2 38)
\move(2 34)
\lvec(-2 38)
\move(6 0)
\move(4 -2)
\lvec(8 2)
\move(8 -2)
\lvec(4 2)
\move(6 6)
\move(4 4)
\lvec(8 8)
\move(8 4)
\lvec(4 8)
\move(6 12)
\move(4 10)
\lvec(8 14)
\move(8 10)
\lvec(4 14)
\move(6 18)
\move(4 16)
\lvec(8 20)
\move(8 16)
\lvec(4 20)
\move(6 24)
\move(4 22)
\lvec(8 26)
\move(8 22)
\lvec(4 26)
\move(6 30)
\move(4 28)
\lvec(8 32)
\move(8 28)
\lvec(4 32)
\move(12 0)
\move(10 -2)
\lvec(14 2)
\move(14 -2)
\lvec(10 2)
\move(12 6)
\move(10 4)
\lvec(14 8)
\move(14 4)
\lvec(10 8)
\move(12 12)
\move(10 10)
\lvec(14 14)
\move(14 10)
\lvec(10 14)
\move(12 18)
\move(10 16)
\lvec(14 20)
\move(14 16)
\lvec(10 20)
\move(12 24)
\move(10 22)
\lvec(14 26)
\move(14 22)
\lvec(10 26)
\move(18 0)
\move(16 -2)
\lvec(20 2)
\move(20 -2)
\lvec(16 2)
\move(18 6)
\move(16 4)
\lvec(20 8)
\move(20 4)
\lvec(16 8)
\move(18 12)
\move(16 10)
\lvec(20 14)
\move(20 10)
\lvec(16 14)
\move(18 18)
\move(16 16)
\lvec(20 20)
\move(20 16)
\lvec(16 20)
\move(24 0)
\move(22 -2)
\lvec(26 2)
\move(26 -2)
\lvec(22 2)
\move(24 6)
\move(22 4)
\lvec(26 8)
\move(26 4)
\lvec(22 8)
\move(24 12)
\move(22 10)
\lvec(26 14)
\move(26 10)
\lvec(22 14)
\move(30 0)
\move(28 -2)
\lvec(32 2)
\move(32 -2)
\lvec(28 2)
\move(30 6)
\move(28 4)
\lvec(32 8)
\move(32 4)
\lvec(28 8)
\move(36 0)
\move(34 -2)
\lvec(38 2)
\move(38 -2)
\lvec(34 2)
\linewd 0.5
\move(45 -6)
\lvec(-6 45)
\move(-6 87)
\lvec(48 87)
\move(87 -6)
\lvec(87 48)
\end{texdraw}
&
\begin{texdraw}
\drawdim pt
\linewd 0.5
\move(0 0)
\fcir f:0 r:0.2
\move(0 6)
\fcir f:0 r:0.2
\move(0 12)
\fcir f:0 r:0.2
\move(0 18)
\fcir f:0 r:0.2
\move(0 24)
\fcir f:0 r:0.2
\move(0 30)
\fcir f:0 r:0.2
\move(0 36)
\fcir f:0 r:0.2
\move(0 42)
\fcir f:0 r:0.2
\move(0 48)
\fcir f:0 r:0.2
\move(0 54)
\fcir f:0 r:0.2
\move(0 60)
\fcir f:0 r:0.2
\move(0 66)
\fcir f:0 r:0.2
\move(0 72)
\fcir f:0 r:0.2
\move(0 78)
\fcir f:0 r:0.2
\move(0 84)
\fcir f:0 r:0.2
\move(6 0)
\fcir f:0 r:0.2
\move(6 6)
\fcir f:0 r:0.2
\move(6 12)
\fcir f:0 r:0.2
\move(6 18)
\fcir f:0 r:0.2
\move(6 24)
\fcir f:0 r:0.2
\move(6 30)
\fcir f:0 r:0.2
\move(6 36)
\fcir f:0 r:0.2
\move(6 42)
\fcir f:0 r:0.2
\move(6 48)
\fcir f:0 r:0.2
\move(6 54)
\fcir f:0 r:0.2
\move(6 60)
\fcir f:0 r:0.2
\move(6 66)
\fcir f:0 r:0.2
\move(6 72)
\fcir f:0 r:0.2
\move(6 78)
\fcir f:0 r:0.2
\move(6 84)
\fcir f:0 r:0.2
\move(12 0)
\fcir f:0 r:0.2
\move(12 6)
\fcir f:0 r:0.2
\move(12 12)
\fcir f:0 r:0.2
\move(12 18)
\fcir f:0 r:0.2
\move(12 24)
\fcir f:0 r:0.2
\move(12 30)
\fcir f:0 r:0.2
\move(12 36)
\fcir f:0 r:0.2
\move(12 42)
\fcir f:0 r:0.2
\move(12 48)
\fcir f:0 r:0.2
\move(12 54)
\fcir f:0 r:0.2
\move(12 60)
\fcir f:0 r:0.2
\move(12 66)
\fcir f:0 r:0.2
\move(12 72)
\fcir f:0 r:0.2
\move(12 78)
\fcir f:0 r:0.2
\move(12 84)
\fcir f:0 r:0.2
\move(18 0)
\fcir f:0 r:0.2
\move(18 6)
\fcir f:0 r:0.2
\move(18 12)
\fcir f:0 r:0.2
\move(18 18)
\fcir f:0 r:0.2
\move(18 24)
\fcir f:0 r:0.2
\move(18 30)
\fcir f:0 r:0.2
\move(18 36)
\fcir f:0 r:0.2
\move(18 42)
\fcir f:0 r:0.2
\move(18 48)
\fcir f:0 r:0.2
\move(18 54)
\fcir f:0 r:0.2
\move(18 60)
\fcir f:0 r:0.2
\move(18 66)
\fcir f:0 r:0.2
\move(18 72)
\fcir f:0 r:0.2
\move(18 78)
\fcir f:0 r:0.2
\move(18 84)
\fcir f:0 r:0.2
\move(24 0)
\fcir f:0 r:0.2
\move(24 6)
\fcir f:0 r:0.2
\move(24 12)
\fcir f:0 r:0.2
\move(24 18)
\fcir f:0 r:0.2
\move(24 24)
\fcir f:0 r:0.2
\move(24 30)
\fcir f:0 r:0.2
\move(24 36)
\fcir f:0 r:0.2
\move(24 42)
\fcir f:0 r:0.2
\move(24 48)
\fcir f:0 r:0.2
\move(24 54)
\fcir f:0 r:0.2
\move(24 60)
\fcir f:0 r:0.2
\move(24 66)
\fcir f:0 r:0.2
\move(24 72)
\fcir f:0 r:0.2
\move(24 78)
\fcir f:0 r:0.2
\move(24 84)
\fcir f:0 r:0.2
\move(30 0)
\fcir f:0 r:0.2
\move(30 6)
\fcir f:0 r:0.2
\move(30 12)
\fcir f:0 r:0.2
\move(30 18)
\fcir f:0 r:0.2
\move(30 24)
\fcir f:0 r:0.2
\move(30 30)
\fcir f:0 r:0.2
\move(30 36)
\fcir f:0 r:0.2
\move(30 42)
\fcir f:0 r:0.2
\move(30 48)
\fcir f:0 r:0.2
\move(30 54)
\fcir f:0 r:0.2
\move(30 60)
\fcir f:0 r:0.2
\move(30 66)
\fcir f:0 r:0.2
\move(30 72)
\fcir f:0 r:0.2
\move(30 78)
\fcir f:0 r:0.2
\move(30 84)
\fcir f:0 r:0.2
\move(36 0)
\fcir f:0 r:0.2
\move(36 6)
\fcir f:0 r:0.2
\move(36 12)
\fcir f:0 r:0.2
\move(36 18)
\fcir f:0 r:0.2
\move(36 24)
\fcir f:0 r:0.2
\move(36 30)
\fcir f:0 r:0.2
\move(36 36)
\fcir f:0 r:0.2
\move(36 42)
\fcir f:0 r:0.2
\move(36 48)
\fcir f:0 r:0.2
\move(36 54)
\fcir f:0 r:0.2
\move(36 60)
\fcir f:0 r:0.2
\move(36 66)
\fcir f:0 r:0.2
\move(36 72)
\fcir f:0 r:0.2
\move(36 78)
\fcir f:0 r:0.2
\move(36 84)
\fcir f:0 r:0.2
\move(42 0)
\fcir f:0 r:0.2
\move(42 6)
\fcir f:0 r:0.2
\move(42 12)
\fcir f:0 r:0.2
\move(42 18)
\fcir f:0 r:0.2
\move(42 24)
\fcir f:0 r:0.2
\move(42 30)
\fcir f:0 r:0.2
\move(42 36)
\fcir f:0 r:0.2
\move(42 42)
\fcir f:0 r:0.2
\move(42 48)
\fcir f:0 r:0.2
\move(42 54)
\fcir f:0 r:0.2
\move(42 60)
\fcir f:0 r:0.2
\move(42 66)
\fcir f:0 r:0.2
\move(42 72)
\fcir f:0 r:0.2
\move(42 78)
\fcir f:0 r:0.2
\move(42 84)
\fcir f:0 r:0.2
\move(48 0)
\fcir f:0 r:0.2
\move(48 6)
\fcir f:0 r:0.2
\move(48 12)
\fcir f:0 r:0.2
\move(48 18)
\fcir f:0 r:0.2
\move(48 24)
\fcir f:0 r:0.2
\move(48 30)
\fcir f:0 r:0.2
\move(48 36)
\fcir f:0 r:0.2
\move(48 42)
\fcir f:0 r:0.2
\move(48 48)
\fcir f:0 r:0.2
\move(48 54)
\fcir f:0 r:0.2
\move(48 60)
\fcir f:0 r:0.2
\move(48 66)
\fcir f:0 r:0.2
\move(48 72)
\fcir f:0 r:0.2
\move(48 78)
\fcir f:0 r:0.2
\move(48 84)
\fcir f:0 r:0.2
\move(54 0)
\fcir f:0 r:0.2
\move(54 6)
\fcir f:0 r:0.2
\move(54 12)
\fcir f:0 r:0.2
\move(54 18)
\fcir f:0 r:0.2
\move(54 24)
\fcir f:0 r:0.2
\move(54 30)
\fcir f:0 r:0.2
\move(54 36)
\fcir f:0 r:0.2
\move(54 42)
\fcir f:0 r:0.2
\move(54 48)
\fcir f:0 r:0.2
\move(54 54)
\fcir f:0 r:0.2
\move(54 60)
\fcir f:0 r:0.2
\move(54 66)
\fcir f:0 r:0.2
\move(54 72)
\fcir f:0 r:0.2
\move(54 78)
\fcir f:0 r:0.2
\move(54 84)
\fcir f:0 r:0.2
\move(60 0)
\fcir f:0 r:0.2
\move(60 6)
\fcir f:0 r:0.2
\move(60 12)
\fcir f:0 r:0.2
\move(60 18)
\fcir f:0 r:0.2
\move(60 24)
\fcir f:0 r:0.2
\move(60 30)
\fcir f:0 r:0.2
\move(60 36)
\fcir f:0 r:0.2
\move(60 42)
\fcir f:0 r:0.2
\move(60 48)
\fcir f:0 r:0.2
\move(60 54)
\fcir f:0 r:0.2
\move(60 60)
\fcir f:0 r:0.2
\move(60 66)
\fcir f:0 r:0.2
\move(60 72)
\fcir f:0 r:0.2
\move(60 78)
\fcir f:0 r:0.2
\move(60 84)
\fcir f:0 r:0.2
\move(66 0)
\fcir f:0 r:0.2
\move(66 6)
\fcir f:0 r:0.2
\move(66 12)
\fcir f:0 r:0.2
\move(66 18)
\fcir f:0 r:0.2
\move(66 24)
\fcir f:0 r:0.2
\move(66 30)
\fcir f:0 r:0.2
\move(66 36)
\fcir f:0 r:0.2
\move(66 42)
\fcir f:0 r:0.2
\move(66 48)
\fcir f:0 r:0.2
\move(66 54)
\fcir f:0 r:0.2
\move(66 60)
\fcir f:0 r:0.2
\move(66 66)
\fcir f:0 r:0.2
\move(66 72)
\fcir f:0 r:0.2
\move(66 78)
\fcir f:0 r:0.2
\move(66 84)
\fcir f:0 r:0.2
\move(72 0)
\fcir f:0 r:0.2
\move(72 6)
\fcir f:0 r:0.2
\move(72 12)
\fcir f:0 r:0.2
\move(72 18)
\fcir f:0 r:0.2
\move(72 24)
\fcir f:0 r:0.2
\move(72 30)
\fcir f:0 r:0.2
\move(72 36)
\fcir f:0 r:0.2
\move(72 42)
\fcir f:0 r:0.2
\move(72 48)
\fcir f:0 r:0.2
\move(72 54)
\fcir f:0 r:0.2
\move(72 60)
\fcir f:0 r:0.2
\move(72 66)
\fcir f:0 r:0.2
\move(72 72)
\fcir f:0 r:0.2
\move(72 78)
\fcir f:0 r:0.2
\move(72 84)
\fcir f:0 r:0.2
\move(78 0)
\fcir f:0 r:0.2
\move(78 6)
\fcir f:0 r:0.2
\move(78 12)
\fcir f:0 r:0.2
\move(78 18)
\fcir f:0 r:0.2
\move(78 24)
\fcir f:0 r:0.2
\move(78 30)
\fcir f:0 r:0.2
\move(78 36)
\fcir f:0 r:0.2
\move(78 42)
\fcir f:0 r:0.2
\move(78 48)
\fcir f:0 r:0.2
\move(78 54)
\fcir f:0 r:0.2
\move(78 60)
\fcir f:0 r:0.2
\move(78 66)
\fcir f:0 r:0.2
\move(78 72)
\fcir f:0 r:0.2
\move(78 78)
\fcir f:0 r:0.2
\move(78 84)
\fcir f:0 r:0.2
\move(84 0)
\fcir f:0 r:0.2
\move(84 6)
\fcir f:0 r:0.2
\move(84 12)
\fcir f:0 r:0.2
\move(84 18)
\fcir f:0 r:0.2
\move(84 24)
\fcir f:0 r:0.2
\move(84 30)
\fcir f:0 r:0.2
\move(84 36)
\fcir f:0 r:0.2
\move(84 42)
\fcir f:0 r:0.2
\move(84 48)
\fcir f:0 r:0.2
\move(84 54)
\fcir f:0 r:0.2
\move(84 60)
\fcir f:0 r:0.2
\move(84 66)
\fcir f:0 r:0.2
\move(84 72)
\fcir f:0 r:0.2
\move(84 78)
\fcir f:0 r:0.2
\move(84 84)
\fcir f:0 r:0.2
\arrowheadtype t:V
\move(0 0)
\avec(102 0)
\move(0 0)
\avec(0 102)
\htext(106 0){$\mathbb{N}$}
\htext(-13 90){$\mathbb{N}$}
\move(0 42)
\fcir f:0 r:1
\move(0 48)
\fcir f:0 r:1
\move(0 54)
\fcir f:0 r:1
\move(0 60)
\fcir f:0 r:1
\move(0 66)
\fcir f:0 r:1
\move(0 72)
\fcir f:0 r:1
\move(0 78)
\fcir f:0 r:1
\move(0 84)
\fcir f:0 r:1
\move(6 36)
\fcir f:0 r:1
\move(6 42)
\fcir f:0 r:1
\move(6 48)
\fcir f:0 r:1
\move(6 54)
\fcir f:0 r:1
\move(6 60)
\fcir f:0 r:1
\move(6 66)
\fcir f:0 r:1
\move(6 72)
\fcir f:0 r:1
\move(6 78)
\fcir f:0 r:1
\move(6 84)
\fcir f:0 r:1
\move(12 30)
\fcir f:0 r:1
\move(12 36)
\fcir f:0 r:1
\move(12 42)
\fcir f:0 r:1
\move(12 48)
\fcir f:0 r:1
\move(12 54)
\fcir f:0 r:1
\move(12 60)
\fcir f:0 r:1
\move(12 66)
\fcir f:0 r:1
\move(12 72)
\fcir f:0 r:1
\move(12 78)
\fcir f:0 r:1
\move(12 84)
\fcir f:0 r:1
\move(18 24)
\fcir f:0 r:1
\move(18 30)
\fcir f:0 r:1
\move(18 36)
\fcir f:0 r:1
\move(18 42)
\fcir f:0 r:1
\move(18 48)
\fcir f:0 r:1
\move(18 54)
\fcir f:0 r:1
\move(18 60)
\fcir f:0 r:1
\move(18 66)
\fcir f:0 r:1
\move(18 72)
\fcir f:0 r:1
\move(18 78)
\fcir f:0 r:1
\move(18 84)
\fcir f:0 r:1
\move(24 18)
\fcir f:0 r:1
\move(24 24)
\fcir f:0 r:1
\move(24 30)
\fcir f:0 r:1
\move(24 36)
\fcir f:0 r:1
\move(24 42)
\fcir f:0 r:1
\move(24 48)
\fcir f:0 r:1
\move(24 54)
\fcir f:0 r:1
\move(24 60)
\fcir f:0 r:1
\move(24 66)
\fcir f:0 r:1
\move(24 72)
\fcir f:0 r:1
\move(24 78)
\fcir f:0 r:1
\move(24 84)
\fcir f:0 r:1
\move(30 12)
\fcir f:0 r:1
\move(30 18)
\fcir f:0 r:1
\move(30 24)
\fcir f:0 r:1
\move(30 30)
\fcir f:0 r:1
\move(30 36)
\fcir f:0 r:1
\move(30 42)
\fcir f:0 r:1
\move(30 48)
\fcir f:0 r:1
\move(30 54)
\fcir f:0 r:1
\move(30 60)
\fcir f:0 r:1
\move(30 66)
\fcir f:0 r:1
\move(30 72)
\fcir f:0 r:1
\move(30 78)
\fcir f:0 r:1
\move(30 84)
\fcir f:0 r:1
\move(36 6)
\fcir f:0 r:1
\move(36 12)
\fcir f:0 r:1
\move(36 18)
\fcir f:0 r:1
\move(36 24)
\fcir f:0 r:1
\move(36 30)
\fcir f:0 r:1
\move(36 36)
\fcir f:0 r:1
\move(36 42)
\fcir f:0 r:1
\move(36 48)
\fcir f:0 r:1
\move(36 54)
\fcir f:0 r:1
\move(36 60)
\fcir f:0 r:1
\move(36 66)
\fcir f:0 r:1
\move(36 72)
\fcir f:0 r:1
\move(36 78)
\fcir f:0 r:1
\move(36 84)
\fcir f:0 r:1
\move(42 0)
\fcir f:0 r:1
\move(42 6)
\fcir f:0 r:1
\move(42 12)
\fcir f:0 r:1
\move(42 18)
\fcir f:0 r:1
\move(42 24)
\fcir f:0 r:1
\move(42 30)
\fcir f:0 r:1
\move(42 36)
\fcir f:0 r:1
\move(42 42)
\fcir f:0 r:1
\move(42 48)
\fcir f:0 r:1
\move(42 54)
\fcir f:0 r:1
\move(42 60)
\fcir f:0 r:1
\move(42 66)
\fcir f:0 r:1
\move(42 72)
\fcir f:0 r:1
\move(42 78)
\fcir f:0 r:1
\move(42 84)
\fcir f:0 r:1
\move(48 0)
\fcir f:0 r:1
\move(48 6)
\fcir f:0 r:1
\move(48 12)
\fcir f:0 r:1
\move(48 18)
\fcir f:0 r:1
\move(48 24)
\fcir f:0 r:1
\move(48 30)
\fcir f:0 r:1
\move(48 36)
\fcir f:0 r:1
\move(48 42)
\fcir f:0 r:1
\move(48 48)
\fcir f:0 r:1
\move(48 54)
\fcir f:0 r:1
\move(48 60)
\fcir f:0 r:1
\move(48 66)
\fcir f:0 r:1
\move(48 72)
\fcir f:0 r:1
\move(48 78)
\fcir f:0 r:1
\move(54 0)
\fcir f:0 r:1
\move(54 6)
\fcir f:0 r:1
\move(54 12)
\fcir f:0 r:1
\move(54 18)
\fcir f:0 r:1
\move(54 24)
\fcir f:0 r:1
\move(54 30)
\fcir f:0 r:1
\move(54 36)
\fcir f:0 r:1
\move(54 42)
\fcir f:0 r:1
\move(54 48)
\fcir f:0 r:1
\move(54 54)
\fcir f:0 r:1
\move(54 60)
\fcir f:0 r:1
\move(54 66)
\fcir f:0 r:1
\move(54 72)
\fcir f:0 r:1
\move(60 0)
\fcir f:0 r:1
\move(60 6)
\fcir f:0 r:1
\move(60 12)
\fcir f:0 r:1
\move(60 18)
\fcir f:0 r:1
\move(60 24)
\fcir f:0 r:1
\move(60 30)
\fcir f:0 r:1
\move(60 36)
\fcir f:0 r:1
\move(60 42)
\fcir f:0 r:1
\move(60 48)
\fcir f:0 r:1
\move(60 54)
\fcir f:0 r:1
\move(60 60)
\fcir f:0 r:1
\move(60 66)
\fcir f:0 r:1
\move(66 0)
\fcir f:0 r:1
\move(66 6)
\fcir f:0 r:1
\move(66 12)
\fcir f:0 r:1
\move(66 18)
\fcir f:0 r:1
\move(66 24)
\fcir f:0 r:1
\move(66 30)
\fcir f:0 r:1
\move(66 36)
\fcir f:0 r:1
\move(66 42)
\fcir f:0 r:1
\move(66 48)
\fcir f:0 r:1
\move(66 54)
\fcir f:0 r:1
\move(66 60)
\fcir f:0 r:1
\move(72 0)
\fcir f:0 r:1
\move(72 6)
\fcir f:0 r:1
\move(72 12)
\fcir f:0 r:1
\move(72 18)
\fcir f:0 r:1
\move(72 24)
\fcir f:0 r:1
\move(72 30)
\fcir f:0 r:1
\move(72 36)
\fcir f:0 r:1
\move(72 42)
\fcir f:0 r:1
\move(72 48)
\fcir f:0 r:1
\move(72 54)
\fcir f:0 r:1
\move(78 0)
\fcir f:0 r:1
\move(78 6)
\fcir f:0 r:1
\move(78 12)
\fcir f:0 r:1
\move(78 18)
\fcir f:0 r:1
\move(78 24)
\fcir f:0 r:1
\move(78 30)
\fcir f:0 r:1
\move(78 36)
\fcir f:0 r:1
\move(78 42)
\fcir f:0 r:1
\move(78 48)
\fcir f:0 r:1
\move(84 0)
\fcir f:0 r:1
\move(84 6)
\fcir f:0 r:1
\move(84 12)
\fcir f:0 r:1
\move(84 18)
\fcir f:0 r:1
\move(84 24)
\fcir f:0 r:1
\move(84 30)
\fcir f:0 r:1
\move(84 36)
\fcir f:0 r:1
\move(84 42)
\fcir f:0 r:1
\move(0 48)
\move(-2 46)
\lvec(2 50)
\move(2 46)
\lvec(-2 50)
\move(0 54)
\move(-2 52)
\lvec(2 56)
\move(2 52)
\lvec(-2 56)
\move(0 60)
\move(-2 58)
\lvec(2 62)
\move(2 58)
\lvec(-2 62)
\move(0 66)
\move(-2 64)
\lvec(2 68)
\move(2 64)
\lvec(-2 68)
\move(0 72)
\move(-2 70)
\lvec(2 74)
\move(2 70)
\lvec(-2 74)
\move(0 78)
\move(-2 76)
\lvec(2 80)
\move(2 76)
\lvec(-2 80)
\move(0 84)
\move(-2 82)
\lvec(2 86)
\move(2 82)
\lvec(-2 86)
\move(6 54)
\move(4 52)
\lvec(8 56)
\move(8 52)
\lvec(4 56)
\move(6 60)
\move(4 58)
\lvec(8 62)
\move(8 58)
\lvec(4 62)
\move(6 66)
\move(4 64)
\lvec(8 68)
\move(8 64)
\lvec(4 68)
\move(6 72)
\move(4 70)
\lvec(8 74)
\move(8 70)
\lvec(4 74)
\move(6 78)
\move(4 76)
\lvec(8 80)
\move(8 76)
\lvec(4 80)
\move(6 84)
\move(4 82)
\lvec(8 86)
\move(8 82)
\lvec(4 86)
\move(12 60)
\move(10 58)
\lvec(14 62)
\move(14 58)
\lvec(10 62)
\move(12 66)
\move(10 64)
\lvec(14 68)
\move(14 64)
\lvec(10 68)
\move(12 72)
\move(10 70)
\lvec(14 74)
\move(14 70)
\lvec(10 74)
\move(12 78)
\move(10 76)
\lvec(14 80)
\move(14 76)
\lvec(10 80)
\move(12 84)
\move(10 82)
\lvec(14 86)
\move(14 82)
\lvec(10 86)
\move(18 66)
\move(16 64)
\lvec(20 68)
\move(20 64)
\lvec(16 68)
\move(18 72)
\move(16 70)
\lvec(20 74)
\move(20 70)
\lvec(16 74)
\move(18 78)
\move(16 76)
\lvec(20 80)
\move(20 76)
\lvec(16 80)
\move(18 84)
\move(16 82)
\lvec(20 86)
\move(20 82)
\lvec(16 86)
\move(24 72)
\move(22 70)
\lvec(26 74)
\move(26 70)
\lvec(22 74)
\move(24 78)
\move(22 76)
\lvec(26 80)
\move(26 76)
\lvec(22 80)
\move(24 84)
\move(22 82)
\lvec(26 86)
\move(26 82)
\lvec(22 86)
\move(30 78)
\move(28 76)
\lvec(32 80)
\move(32 76)
\lvec(28 80)
\move(30 84)
\move(28 82)
\lvec(32 86)
\move(32 82)
\lvec(28 86)
\move(36 84)
\move(34 82)
\lvec(38 86)
\move(38 82)
\lvec(34 86)
\move(48 78)
\move(46 76)
\lvec(50 80)
\move(50 76)
\lvec(46 80)
\move(54 66)
\move(52 64)
\lvec(56 68)
\move(56 64)
\lvec(52 68)
\move(54 72)
\move(52 70)
\lvec(56 74)
\move(56 70)
\lvec(52 74)
\move(60 54)
\move(58 52)
\lvec(62 56)
\move(62 52)
\lvec(58 56)
\move(60 60)
\move(58 58)
\lvec(62 62)
\move(62 58)
\lvec(58 62)
\move(60 66)
\move(58 64)
\lvec(62 68)
\move(62 64)
\lvec(58 68)
\move(66 42)
\move(64 40)
\lvec(68 44)
\move(68 40)
\lvec(64 44)
\move(66 48)
\move(64 46)
\lvec(68 50)
\move(68 46)
\lvec(64 50)
\move(66 54)
\move(64 52)
\lvec(68 56)
\move(68 52)
\lvec(64 56)
\move(66 60)
\move(64 58)
\lvec(68 62)
\move(68 58)
\lvec(64 62)
\move(72 30)
\move(70 28)
\lvec(74 32)
\move(74 28)
\lvec(70 32)
\move(72 36)
\move(70 34)
\lvec(74 38)
\move(74 34)
\lvec(70 38)
\move(72 42)
\move(70 40)
\lvec(74 44)
\move(74 40)
\lvec(70 44)
\move(72 48)
\move(70 46)
\lvec(74 50)
\move(74 46)
\lvec(70 50)
\move(72 54)
\move(70 52)
\lvec(74 56)
\move(74 52)
\lvec(70 56)
\move(78 18)
\move(76 16)
\lvec(80 20)
\move(80 16)
\lvec(76 20)
\move(78 24)
\move(76 22)
\lvec(80 26)
\move(80 22)
\lvec(76 26)
\move(78 30)
\move(76 28)
\lvec(80 32)
\move(80 28)
\lvec(76 32)
\move(78 36)
\move(76 34)
\lvec(80 38)
\move(80 34)
\lvec(76 38)
\move(78 42)
\move(76 40)
\lvec(80 44)
\move(80 40)
\lvec(76 44)
\move(78 48)
\move(76 46)
\lvec(80 50)
\move(80 46)
\lvec(76 50)
\move(84 6)
\move(82 4)
\lvec(86 8)
\move(86 4)
\lvec(82 8)
\move(84 12)
\move(82 10)
\lvec(86 14)
\move(86 10)
\lvec(82 14)
\move(84 18)
\move(82 16)
\lvec(86 20)
\move(86 16)
\lvec(82 20)
\move(84 24)
\move(82 22)
\lvec(86 26)
\move(86 22)
\lvec(82 26)
\move(84 30)
\move(82 28)
\lvec(86 32)
\move(86 28)
\lvec(82 32)
\move(84 36)
\move(82 34)
\lvec(86 38)
\move(86 34)
\lvec(82 38)
\move(84 42)
\move(82 40)
\lvec(86 44)
\move(86 40)
\lvec(82 44)
\move(6 36)
\move(4 34)
\lvec(8 38)
\move(8 34)
\lvec(4 38)
\move(12 30)
\move(10 28)
\lvec(14 32)
\move(14 28)
\lvec(10 32)
\move(18 24)
\move(16 22)
\lvec(20 26)
\move(20 22)
\lvec(16 26)
\move(18 30)
\move(16 28)
\lvec(20 32)
\move(20 28)
\lvec(16 32)
\move(24 18)
\move(22 16)
\lvec(26 20)
\move(26 16)
\lvec(22 20)
\move(24 24)
\move(22 22)
\lvec(26 26)
\move(26 22)
\lvec(22 26)
\move(30 12)
\move(28 10)
\lvec(32 14)
\move(32 10)
\lvec(28 14)
\move(30 18)
\move(28 16)
\lvec(32 20)
\move(32 16)
\lvec(28 20)
\move(30 24)
\move(28 22)
\lvec(32 26)
\move(32 22)
\lvec(28 26)
\move(36 6)
\move(34 4)
\lvec(38 8)
\move(38 4)
\lvec(34 8)
\move(36 12)
\move(34 10)
\lvec(38 14)
\move(38 10)
\lvec(34 14)
\move(36 18)
\move(34 16)
\lvec(38 20)
\move(38 16)
\lvec(34 20)
\move(42 0)
\move(40 -2)
\lvec(44 2)
\move(44 -2)
\lvec(40 2)
\move(42 6)
\move(40 4)
\lvec(44 8)
\move(44 4)
\lvec(40 8)
\move(42 12)
\move(40 10)
\lvec(44 14)
\move(44 10)
\lvec(40 14)
\move(42 18)
\move(40 16)
\lvec(44 20)
\move(44 16)
\lvec(40 20)
\move(48 0)
\move(46 -2)
\lvec(50 2)
\move(50 -2)
\lvec(46 2)
\move(48 6)
\move(46 4)
\lvec(50 8)
\move(50 4)
\lvec(46 8)
\move(48 12)
\move(46 10)
\lvec(50 14)
\move(50 10)
\lvec(46 14)
\move(54 0)
\move(52 -2)
\lvec(56 2)
\move(56 -2)
\lvec(52 2)
\move(54 6)
\move(52 4)
\lvec(56 8)
\move(56 4)
\lvec(52 8)
\move(54 12)
\move(52 10)
\lvec(56 14)
\move(56 10)
\lvec(52 14)
\move(60 0)
\move(58 -2)
\lvec(62 2)
\move(62 -2)
\lvec(58 2)
\move(60 6)
\move(58 4)
\lvec(62 8)
\move(62 4)
\lvec(58 8)
\move(66 0)
\move(64 -2)
\lvec(68 2)
\move(68 -2)
\lvec(64 2)
\move(66 6)
\move(64 4)
\lvec(68 8)
\move(68 4)
\lvec(64 8)
\move(72 0)
\move(70 -2)
\lvec(74 2)
\move(74 -2)
\lvec(70 2)
\move(78 0)
\move(76 -2)
\lvec(80 2)
\move(80 -2)
\lvec(76 2)
\linewd 0.5
\move(81 0)
\lvec(0 40)
\move(0 45)
\lvec(42 87)
\move(85 0)
\lvec(42 88)
\end{texdraw}
\end{array}$$
$$\begin{array}{cc}
\begin{texdraw}
\drawdim pt
\linewd 0.5
\move(0 0)
\fcir f:0 r:0.2
\move(0 6)
\fcir f:0 r:0.2
\move(0 12)
\fcir f:0 r:0.2
\move(0 18)
\fcir f:0 r:0.2
\move(0 24)
\fcir f:0 r:0.2
\move(0 30)
\fcir f:0 r:0.2
\move(0 36)
\fcir f:0 r:0.2
\move(0 42)
\fcir f:0 r:0.2
\move(0 48)
\fcir f:0 r:0.2
\move(0 54)
\fcir f:0 r:0.2
\move(0 60)
\fcir f:0 r:0.2
\move(0 66)
\fcir f:0 r:0.2
\move(0 72)
\fcir f:0 r:0.2
\move(0 78)
\fcir f:0 r:0.2
\move(0 84)
\fcir f:0 r:0.2
\move(6 0)
\fcir f:0 r:0.2
\move(6 6)
\fcir f:0 r:0.2
\move(6 12)
\fcir f:0 r:0.2
\move(6 18)
\fcir f:0 r:0.2
\move(6 24)
\fcir f:0 r:0.2
\move(6 30)
\fcir f:0 r:0.2
\move(6 36)
\fcir f:0 r:0.2
\move(6 42)
\fcir f:0 r:0.2
\move(6 48)
\fcir f:0 r:0.2
\move(6 54)
\fcir f:0 r:0.2
\move(6 60)
\fcir f:0 r:0.2
\move(6 66)
\fcir f:0 r:0.2
\move(6 72)
\fcir f:0 r:0.2
\move(6 78)
\fcir f:0 r:0.2
\move(6 84)
\fcir f:0 r:0.2
\move(12 0)
\fcir f:0 r:0.2
\move(12 6)
\fcir f:0 r:0.2
\move(12 12)
\fcir f:0 r:0.2
\move(12 18)
\fcir f:0 r:0.2
\move(12 24)
\fcir f:0 r:0.2
\move(12 30)
\fcir f:0 r:0.2
\move(12 36)
\fcir f:0 r:0.2
\move(12 42)
\fcir f:0 r:0.2
\move(12 48)
\fcir f:0 r:0.2
\move(12 54)
\fcir f:0 r:0.2
\move(12 60)
\fcir f:0 r:0.2
\move(12 66)
\fcir f:0 r:0.2
\move(12 72)
\fcir f:0 r:0.2
\move(12 78)
\fcir f:0 r:0.2
\move(12 84)
\fcir f:0 r:0.2
\move(18 0)
\fcir f:0 r:0.2
\move(18 6)
\fcir f:0 r:0.2
\move(18 12)
\fcir f:0 r:0.2
\move(18 18)
\fcir f:0 r:0.2
\move(18 24)
\fcir f:0 r:0.2
\move(18 30)
\fcir f:0 r:0.2
\move(18 36)
\fcir f:0 r:0.2
\move(18 42)
\fcir f:0 r:0.2
\move(18 48)
\fcir f:0 r:0.2
\move(18 54)
\fcir f:0 r:0.2
\move(18 60)
\fcir f:0 r:0.2
\move(18 66)
\fcir f:0 r:0.2
\move(18 72)
\fcir f:0 r:0.2
\move(18 78)
\fcir f:0 r:0.2
\move(18 84)
\fcir f:0 r:0.2
\move(24 0)
\fcir f:0 r:0.2
\move(24 6)
\fcir f:0 r:0.2
\move(24 12)
\fcir f:0 r:0.2
\move(24 18)
\fcir f:0 r:0.2
\move(24 24)
\fcir f:0 r:0.2
\move(24 30)
\fcir f:0 r:0.2
\move(24 36)
\fcir f:0 r:0.2
\move(24 42)
\fcir f:0 r:0.2
\move(24 48)
\fcir f:0 r:0.2
\move(24 54)
\fcir f:0 r:0.2
\move(24 60)
\fcir f:0 r:0.2
\move(24 66)
\fcir f:0 r:0.2
\move(24 72)
\fcir f:0 r:0.2
\move(24 78)
\fcir f:0 r:0.2
\move(24 84)
\fcir f:0 r:0.2
\move(30 0)
\fcir f:0 r:0.2
\move(30 6)
\fcir f:0 r:0.2
\move(30 12)
\fcir f:0 r:0.2
\move(30 18)
\fcir f:0 r:0.2
\move(30 24)
\fcir f:0 r:0.2
\move(30 30)
\fcir f:0 r:0.2
\move(30 36)
\fcir f:0 r:0.2
\move(30 42)
\fcir f:0 r:0.2
\move(30 48)
\fcir f:0 r:0.2
\move(30 54)
\fcir f:0 r:0.2
\move(30 60)
\fcir f:0 r:0.2
\move(30 66)
\fcir f:0 r:0.2
\move(30 72)
\fcir f:0 r:0.2
\move(30 78)
\fcir f:0 r:0.2
\move(30 84)
\fcir f:0 r:0.2
\move(36 0)
\fcir f:0 r:0.2
\move(36 6)
\fcir f:0 r:0.2
\move(36 12)
\fcir f:0 r:0.2
\move(36 18)
\fcir f:0 r:0.2
\move(36 24)
\fcir f:0 r:0.2
\move(36 30)
\fcir f:0 r:0.2
\move(36 36)
\fcir f:0 r:0.2
\move(36 42)
\fcir f:0 r:0.2
\move(36 48)
\fcir f:0 r:0.2
\move(36 54)
\fcir f:0 r:0.2
\move(36 60)
\fcir f:0 r:0.2
\move(36 66)
\fcir f:0 r:0.2
\move(36 72)
\fcir f:0 r:0.2
\move(36 78)
\fcir f:0 r:0.2
\move(36 84)
\fcir f:0 r:0.2
\move(42 0)
\fcir f:0 r:0.2
\move(42 6)
\fcir f:0 r:0.2
\move(42 12)
\fcir f:0 r:0.2
\move(42 18)
\fcir f:0 r:0.2
\move(42 24)
\fcir f:0 r:0.2
\move(42 30)
\fcir f:0 r:0.2
\move(42 36)
\fcir f:0 r:0.2
\move(42 42)
\fcir f:0 r:0.2
\move(42 48)
\fcir f:0 r:0.2
\move(42 54)
\fcir f:0 r:0.2
\move(42 60)
\fcir f:0 r:0.2
\move(42 66)
\fcir f:0 r:0.2
\move(42 72)
\fcir f:0 r:0.2
\move(42 78)
\fcir f:0 r:0.2
\move(42 84)
\fcir f:0 r:0.2
\move(48 0)
\fcir f:0 r:0.2
\move(48 6)
\fcir f:0 r:0.2
\move(48 12)
\fcir f:0 r:0.2
\move(48 18)
\fcir f:0 r:0.2
\move(48 24)
\fcir f:0 r:0.2
\move(48 30)
\fcir f:0 r:0.2
\move(48 36)
\fcir f:0 r:0.2
\move(48 42)
\fcir f:0 r:0.2
\move(48 48)
\fcir f:0 r:0.2
\move(48 54)
\fcir f:0 r:0.2
\move(48 60)
\fcir f:0 r:0.2
\move(48 66)
\fcir f:0 r:0.2
\move(48 72)
\fcir f:0 r:0.2
\move(48 78)
\fcir f:0 r:0.2
\move(48 84)
\fcir f:0 r:0.2
\move(54 0)
\fcir f:0 r:0.2
\move(54 6)
\fcir f:0 r:0.2
\move(54 12)
\fcir f:0 r:0.2
\move(54 18)
\fcir f:0 r:0.2
\move(54 24)
\fcir f:0 r:0.2
\move(54 30)
\fcir f:0 r:0.2
\move(54 36)
\fcir f:0 r:0.2
\move(54 42)
\fcir f:0 r:0.2
\move(54 48)
\fcir f:0 r:0.2
\move(54 54)
\fcir f:0 r:0.2
\move(54 60)
\fcir f:0 r:0.2
\move(54 66)
\fcir f:0 r:0.2
\move(54 72)
\fcir f:0 r:0.2
\move(54 78)
\fcir f:0 r:0.2
\move(54 84)
\fcir f:0 r:0.2
\move(60 0)
\fcir f:0 r:0.2
\move(60 6)
\fcir f:0 r:0.2
\move(60 12)
\fcir f:0 r:0.2
\move(60 18)
\fcir f:0 r:0.2
\move(60 24)
\fcir f:0 r:0.2
\move(60 30)
\fcir f:0 r:0.2
\move(60 36)
\fcir f:0 r:0.2
\move(60 42)
\fcir f:0 r:0.2
\move(60 48)
\fcir f:0 r:0.2
\move(60 54)
\fcir f:0 r:0.2
\move(60 60)
\fcir f:0 r:0.2
\move(60 66)
\fcir f:0 r:0.2
\move(60 72)
\fcir f:0 r:0.2
\move(60 78)
\fcir f:0 r:0.2
\move(60 84)
\fcir f:0 r:0.2
\move(66 0)
\fcir f:0 r:0.2
\move(66 6)
\fcir f:0 r:0.2
\move(66 12)
\fcir f:0 r:0.2
\move(66 18)
\fcir f:0 r:0.2
\move(66 24)
\fcir f:0 r:0.2
\move(66 30)
\fcir f:0 r:0.2
\move(66 36)
\fcir f:0 r:0.2
\move(66 42)
\fcir f:0 r:0.2
\move(66 48)
\fcir f:0 r:0.2
\move(66 54)
\fcir f:0 r:0.2
\move(66 60)
\fcir f:0 r:0.2
\move(66 66)
\fcir f:0 r:0.2
\move(66 72)
\fcir f:0 r:0.2
\move(66 78)
\fcir f:0 r:0.2
\move(66 84)
\fcir f:0 r:0.2
\move(72 0)
\fcir f:0 r:0.2
\move(72 6)
\fcir f:0 r:0.2
\move(72 12)
\fcir f:0 r:0.2
\move(72 18)
\fcir f:0 r:0.2
\move(72 24)
\fcir f:0 r:0.2
\move(72 30)
\fcir f:0 r:0.2
\move(72 36)
\fcir f:0 r:0.2
\move(72 42)
\fcir f:0 r:0.2
\move(72 48)
\fcir f:0 r:0.2
\move(72 54)
\fcir f:0 r:0.2
\move(72 60)
\fcir f:0 r:0.2
\move(72 66)
\fcir f:0 r:0.2
\move(72 72)
\fcir f:0 r:0.2
\move(72 78)
\fcir f:0 r:0.2
\move(72 84)
\fcir f:0 r:0.2
\move(78 0)
\fcir f:0 r:0.2
\move(78 6)
\fcir f:0 r:0.2
\move(78 12)
\fcir f:0 r:0.2
\move(78 18)
\fcir f:0 r:0.2
\move(78 24)
\fcir f:0 r:0.2
\move(78 30)
\fcir f:0 r:0.2
\move(78 36)
\fcir f:0 r:0.2
\move(78 42)
\fcir f:0 r:0.2
\move(78 48)
\fcir f:0 r:0.2
\move(78 54)
\fcir f:0 r:0.2
\move(78 60)
\fcir f:0 r:0.2
\move(78 66)
\fcir f:0 r:0.2
\move(78 72)
\fcir f:0 r:0.2
\move(78 78)
\fcir f:0 r:0.2
\move(78 84)
\fcir f:0 r:0.2
\move(84 0)
\fcir f:0 r:0.2
\move(84 6)
\fcir f:0 r:0.2
\move(84 12)
\fcir f:0 r:0.2
\move(84 18)
\fcir f:0 r:0.2
\move(84 24)
\fcir f:0 r:0.2
\move(84 30)
\fcir f:0 r:0.2
\move(84 36)
\fcir f:0 r:0.2
\move(84 42)
\fcir f:0 r:0.2
\move(84 48)
\fcir f:0 r:0.2
\move(84 54)
\fcir f:0 r:0.2
\move(84 60)
\fcir f:0 r:0.2
\move(84 66)
\fcir f:0 r:0.2
\move(84 72)
\fcir f:0 r:0.2
\move(84 78)
\fcir f:0 r:0.2
\move(84 84)
\fcir f:0 r:0.2
\arrowheadtype t:V
\move(0 0)
\avec(102 0)
\move(0 0)
\avec(0 102)
\htext(106 0){$\mathbb{N}$}
\htext(-13 90){$\mathbb{N}$}
\move(0 42)
\fcir f:0 r:1
\move(6 42)
\fcir f:0 r:1
\move(6 48)
\fcir f:0 r:1
\move(12 36)
\fcir f:0 r:1
\move(12 42)
\fcir f:0 r:1
\move(12 48)
\fcir f:0 r:1
\move(12 54)
\fcir f:0 r:1
\move(18 36)
\fcir f:0 r:1
\move(18 42)
\fcir f:0 r:1
\move(18 48)
\fcir f:0 r:1
\move(18 54)
\fcir f:0 r:1
\move(18 60)
\fcir f:0 r:1
\move(24 30)
\fcir f:0 r:1
\move(24 36)
\fcir f:0 r:1
\move(24 42)
\fcir f:0 r:1
\move(24 48)
\fcir f:0 r:1
\move(24 54)
\fcir f:0 r:1
\move(24 60)
\fcir f:0 r:1
\move(24 66)
\fcir f:0 r:1
\move(30 30)
\fcir f:0 r:1
\move(30 36)
\fcir f:0 r:1
\move(30 42)
\fcir f:0 r:1
\move(30 48)
\fcir f:0 r:1
\move(30 54)
\fcir f:0 r:1
\move(30 60)
\fcir f:0 r:1
\move(30 66)
\fcir f:0 r:1
\move(30 72)
\fcir f:0 r:1
\move(36 24)
\fcir f:0 r:1
\move(36 30)
\fcir f:0 r:1
\move(36 36)
\fcir f:0 r:1
\move(36 42)
\fcir f:0 r:1
\move(36 48)
\fcir f:0 r:1
\move(36 54)
\fcir f:0 r:1
\move(36 60)
\fcir f:0 r:1
\move(36 66)
\fcir f:0 r:1
\move(36 72)
\fcir f:0 r:1
\move(36 78)
\fcir f:0 r:1
\move(42 24)
\fcir f:0 r:1
\move(42 30)
\fcir f:0 r:1
\move(42 36)
\fcir f:0 r:1
\move(42 42)
\fcir f:0 r:1
\move(42 48)
\fcir f:0 r:1
\move(42 54)
\fcir f:0 r:1
\move(42 60)
\fcir f:0 r:1
\move(42 66)
\fcir f:0 r:1
\move(42 72)
\fcir f:0 r:1
\move(42 78)
\fcir f:0 r:1
\move(42 84)
\fcir f:0 r:1
\move(48 18)
\fcir f:0 r:1
\move(48 24)
\fcir f:0 r:1
\move(48 30)
\fcir f:0 r:1
\move(48 36)
\fcir f:0 r:1
\move(48 42)
\fcir f:0 r:1
\move(48 48)
\fcir f:0 r:1
\move(48 54)
\fcir f:0 r:1
\move(48 60)
\fcir f:0 r:1
\move(48 66)
\fcir f:0 r:1
\move(48 72)
\fcir f:0 r:1
\move(54 18)
\fcir f:0 r:1
\move(54 24)
\fcir f:0 r:1
\move(54 30)
\fcir f:0 r:1
\move(54 36)
\fcir f:0 r:1
\move(54 42)
\fcir f:0 r:1
\move(54 48)
\fcir f:0 r:1
\move(54 54)
\fcir f:0 r:1
\move(54 60)
\fcir f:0 r:1
\move(60 12)
\fcir f:0 r:1
\move(60 18)
\fcir f:0 r:1
\move(60 24)
\fcir f:0 r:1
\move(60 30)
\fcir f:0 r:1
\move(60 36)
\fcir f:0 r:1
\move(60 42)
\fcir f:0 r:1
\move(60 48)
\fcir f:0 r:1
\move(66 12)
\fcir f:0 r:1
\move(66 18)
\fcir f:0 r:1
\move(66 24)
\fcir f:0 r:1
\move(66 30)
\fcir f:0 r:1
\move(66 36)
\fcir f:0 r:1
\move(72 6)
\fcir f:0 r:1
\move(72 12)
\fcir f:0 r:1
\move(72 18)
\fcir f:0 r:1
\move(72 24)
\fcir f:0 r:1
\move(78 6)
\fcir f:0 r:1
\move(78 12)
\fcir f:0 r:1
\move(84 0)
\fcir f:0 r:1
\move(12 54)
\move(10 52)
\lvec(14 56)
\move(14 52)
\lvec(10 56)
\move(18 54)
\move(16 52)
\lvec(20 56)
\move(20 52)
\lvec(16 56)
\move(18 60)
\move(16 58)
\lvec(20 62)
\move(20 58)
\lvec(16 62)
\move(24 54)
\move(22 52)
\lvec(26 56)
\move(26 52)
\lvec(22 56)
\move(24 60)
\move(22 58)
\lvec(26 62)
\move(26 58)
\lvec(22 62)
\move(24 66)
\move(22 64)
\lvec(26 68)
\move(26 64)
\lvec(22 68)
\move(30 60)
\move(28 58)
\lvec(32 62)
\move(32 58)
\lvec(28 62)
\move(30 66)
\move(28 64)
\lvec(32 68)
\move(32 64)
\lvec(28 68)
\move(30 72)
\move(28 70)
\lvec(32 74)
\move(32 70)
\lvec(28 74)
\move(36 60)
\move(34 58)
\lvec(38 62)
\move(38 58)
\lvec(34 62)
\move(36 66)
\move(34 64)
\lvec(38 68)
\move(38 64)
\lvec(34 68)
\move(36 72)
\move(34 70)
\lvec(38 74)
\move(38 70)
\lvec(34 74)
\move(36 78)
\move(34 76)
\lvec(38 80)
\move(38 76)
\lvec(34 80)
\move(42 60)
\move(40 58)
\lvec(44 62)
\move(44 58)
\lvec(40 62)
\move(42 66)
\move(40 64)
\lvec(44 68)
\move(44 64)
\lvec(40 68)
\move(42 72)
\move(40 70)
\lvec(44 74)
\move(44 70)
\lvec(40 74)
\move(42 78)
\move(40 76)
\lvec(44 80)
\move(44 76)
\lvec(40 80)
\move(42 84)
\move(40 82)
\lvec(44 86)
\move(44 82)
\lvec(40 86)
\move(48 60)
\move(46 58)
\lvec(50 62)
\move(50 58)
\lvec(46 62)
\move(48 66)
\move(46 64)
\lvec(50 68)
\move(50 64)
\lvec(46 68)
\move(48 72)
\move(46 70)
\lvec(50 74)
\move(50 70)
\lvec(46 74)
\linewd 0.5
\move(-1 47)
\lvec(60 62)
\end{texdraw}
&
\begin{texdraw}
\drawdim pt
\linewd 0.5
\move(0 0)
\fcir f:0 r:0.2
\move(0 7)
\fcir f:0 r:0.2
\move(0 14)
\fcir f:0 r:0.2
\move(0 21)
\fcir f:0 r:0.2
\move(0 28)
\fcir f:0 r:0.2
\move(0 35)
\fcir f:0 r:0.2
\move(0 42)
\fcir f:0 r:0.2
\move(0 49)
\fcir f:0 r:0.2
\move(0 56)
\fcir f:0 r:0.2
\move(0 63)
\fcir f:0 r:0.2
\move(0 70)
\fcir f:0 r:0.2
\move(7 0)
\fcir f:0 r:0.2
\move(7 7)
\fcir f:0 r:0.2
\move(7 14)
\fcir f:0 r:0.2
\move(7 21)
\fcir f:0 r:0.2
\move(7 28)
\fcir f:0 r:0.2
\move(7 35)
\fcir f:0 r:0.2
\move(7 42)
\fcir f:0 r:0.2
\move(7 49)
\fcir f:0 r:0.2
\move(7 56)
\fcir f:0 r:0.2
\move(7 63)
\fcir f:0 r:0.2
\move(7 70)
\fcir f:0 r:0.2
\move(14 0)
\fcir f:0 r:0.2
\move(14 7)
\fcir f:0 r:0.2
\move(14 14)
\fcir f:0 r:0.2
\move(14 21)
\fcir f:0 r:0.2
\move(14 28)
\fcir f:0 r:0.2
\move(14 35)
\fcir f:0 r:0.2
\move(14 42)
\fcir f:0 r:0.2
\move(14 49)
\fcir f:0 r:0.2
\move(14 56)
\fcir f:0 r:0.2
\move(14 63)
\fcir f:0 r:0.2
\move(14 70)
\fcir f:0 r:0.2
\move(21 0)
\fcir f:0 r:0.2
\move(21 7)
\fcir f:0 r:0.2
\move(21 14)
\fcir f:0 r:0.2
\move(21 21)
\fcir f:0 r:0.2
\move(21 28)
\fcir f:0 r:0.2
\move(21 35)
\fcir f:0 r:0.2
\move(21 42)
\fcir f:0 r:0.2
\move(21 49)
\fcir f:0 r:0.2
\move(21 56)
\fcir f:0 r:0.2
\move(21 63)
\fcir f:0 r:0.2
\move(21 70)
\fcir f:0 r:0.2
\move(28 0)
\fcir f:0 r:0.2
\move(28 7)
\fcir f:0 r:0.2
\move(28 14)
\fcir f:0 r:0.2
\move(28 21)
\fcir f:0 r:0.2
\move(28 28)
\fcir f:0 r:0.2
\move(28 35)
\fcir f:0 r:0.2
\move(28 42)
\fcir f:0 r:0.2
\move(28 49)
\fcir f:0 r:0.2
\move(28 56)
\fcir f:0 r:0.2
\move(28 63)
\fcir f:0 r:0.2
\move(28 70)
\fcir f:0 r:0.2
\move(35 0)
\fcir f:0 r:0.2
\move(35 7)
\fcir f:0 r:0.2
\move(35 14)
\fcir f:0 r:0.2
\move(35 21)
\fcir f:0 r:0.2
\move(35 28)
\fcir f:0 r:0.2
\move(35 35)
\fcir f:0 r:0.2
\move(35 42)
\fcir f:0 r:0.2
\move(35 49)
\fcir f:0 r:0.2
\move(35 56)
\fcir f:0 r:0.2
\move(35 63)
\fcir f:0 r:0.2
\move(35 70)
\fcir f:0 r:0.2
\move(42 0)
\fcir f:0 r:0.2
\move(42 7)
\fcir f:0 r:0.2
\move(42 14)
\fcir f:0 r:0.2
\move(42 21)
\fcir f:0 r:0.2
\move(42 28)
\fcir f:0 r:0.2
\move(42 35)
\fcir f:0 r:0.2
\move(42 42)
\fcir f:0 r:0.2
\move(42 49)
\fcir f:0 r:0.2
\move(42 56)
\fcir f:0 r:0.2
\move(42 63)
\fcir f:0 r:0.2
\move(42 70)
\fcir f:0 r:0.2
\move(49 0)
\fcir f:0 r:0.2
\move(49 7)
\fcir f:0 r:0.2
\move(49 14)
\fcir f:0 r:0.2
\move(49 21)
\fcir f:0 r:0.2
\move(49 28)
\fcir f:0 r:0.2
\move(49 35)
\fcir f:0 r:0.2
\move(49 42)
\fcir f:0 r:0.2
\move(49 49)
\fcir f:0 r:0.2
\move(49 56)
\fcir f:0 r:0.2
\move(49 63)
\fcir f:0 r:0.2
\move(49 70)
\fcir f:0 r:0.2
\move(56 0)
\fcir f:0 r:0.2
\move(56 7)
\fcir f:0 r:0.2
\move(56 14)
\fcir f:0 r:0.2
\move(56 21)
\fcir f:0 r:0.2
\move(56 28)
\fcir f:0 r:0.2
\move(56 35)
\fcir f:0 r:0.2
\move(56 42)
\fcir f:0 r:0.2
\move(56 49)
\fcir f:0 r:0.2
\move(56 56)
\fcir f:0 r:0.2
\move(56 63)
\fcir f:0 r:0.2
\move(56 70)
\fcir f:0 r:0.2
\move(63 0)
\fcir f:0 r:0.2
\move(63 7)
\fcir f:0 r:0.2
\move(63 14)
\fcir f:0 r:0.2
\move(63 21)
\fcir f:0 r:0.2
\move(63 28)
\fcir f:0 r:0.2
\move(63 35)
\fcir f:0 r:0.2
\move(63 42)
\fcir f:0 r:0.2
\move(63 49)
\fcir f:0 r:0.2
\move(63 56)
\fcir f:0 r:0.2
\move(63 63)
\fcir f:0 r:0.2
\move(63 70)
\fcir f:0 r:0.2
\move(70 0)
\fcir f:0 r:0.2
\move(70 7)
\fcir f:0 r:0.2
\move(70 14)
\fcir f:0 r:0.2
\move(70 21)
\fcir f:0 r:0.2
\move(70 28)
\fcir f:0 r:0.2
\move(70 35)
\fcir f:0 r:0.2
\move(70 42)
\fcir f:0 r:0.2
\move(70 49)
\fcir f:0 r:0.2
\move(70 56)
\fcir f:0 r:0.2
\move(70 63)
\fcir f:0 r:0.2
\move(70 70)
\fcir f:0 r:0.2
\move(77 0)
\fcir f:0 r:0.2
\move(77 7)
\fcir f:0 r:0.2
\move(77 14)
\fcir f:0 r:0.2
\move(77 21)
\fcir f:0 r:0.2
\move(77 28)
\fcir f:0 r:0.2
\move(77 35)
\fcir f:0 r:0.2
\move(77 42)
\fcir f:0 r:0.2
\move(77 49)
\fcir f:0 r:0.2
\move(77 56)
\fcir f:0 r:0.2
\move(77 63)
\fcir f:0 r:0.2
\move(77 70)
\fcir f:0 r:0.2
\move(84 0)
\fcir f:0 r:0.2
\move(84 7)
\fcir f:0 r:0.2
\move(84 14)
\fcir f:0 r:0.2
\move(84 21)
\fcir f:0 r:0.2
\move(84 28)
\fcir f:0 r:0.2
\move(84 35)
\fcir f:0 r:0.2
\move(84 42)
\fcir f:0 r:0.2
\move(84 49)
\fcir f:0 r:0.2
\move(84 56)
\fcir f:0 r:0.2
\move(84 63)
\fcir f:0 r:0.2
\move(84 70)
\fcir f:0 r:0.2
\move(91 0)
\fcir f:0 r:0.2
\move(91 7)
\fcir f:0 r:0.2
\move(91 14)
\fcir f:0 r:0.2
\move(91 21)
\fcir f:0 r:0.2
\move(91 28)
\fcir f:0 r:0.2
\move(91 35)
\fcir f:0 r:0.2
\move(91 42)
\fcir f:0 r:0.2
\move(91 49)
\fcir f:0 r:0.2
\move(91 56)
\fcir f:0 r:0.2
\move(91 63)
\fcir f:0 r:0.2
\move(91 70)
\fcir f:0 r:0.2
\move(98 0)
\fcir f:0 r:0.2
\move(98 7)
\fcir f:0 r:0.2
\move(98 14)
\fcir f:0 r:0.2
\move(98 21)
\fcir f:0 r:0.2
\move(98 28)
\fcir f:0 r:0.2
\move(98 35)
\fcir f:0 r:0.2
\move(98 42)
\fcir f:0 r:0.2
\move(98 49)
\fcir f:0 r:0.2
\move(98 56)
\fcir f:0 r:0.2
\move(98 63)
\fcir f:0 r:0.2
\move(98 70)
\fcir f:0 r:0.2
\arrowheadtype t:V
\move(0 0)
\avec(119 0)
\move(0 0)
\avec(0 91)
\htext(123 0){$\mathbb{N}$}
\htext(-13 77){$\mathbb{N}$}
\move(0 49)
\fcir f:0 r:1
\move(7 49)
\fcir f:0 r:1
\move(7 56)
\fcir f:0 r:1
\move(14 42)
\fcir f:0 r:1
\move(14 49)
\fcir f:0 r:1
\move(14 56)
\fcir f:0 r:1
\move(21 42)
\fcir f:0 r:1
\move(21 49)
\fcir f:0 r:1
\move(21 56)
\fcir f:0 r:1
\move(28 35)
\fcir f:0 r:1
\move(28 42)
\fcir f:0 r:1
\move(28 49)
\fcir f:0 r:1
\move(28 56)
\fcir f:0 r:1
\move(35 35)
\fcir f:0 r:1
\move(35 42)
\fcir f:0 r:1
\move(35 49)
\fcir f:0 r:1
\move(35 56)
\fcir f:0 r:1
\move(35 63)
\fcir f:0 r:1
\move(42 28)
\fcir f:0 r:1
\move(42 35)
\fcir f:0 r:1
\move(42 42)
\fcir f:0 r:1
\move(42 49)
\fcir f:0 r:1
\move(42 56)
\fcir f:0 r:1
\move(42 63)
\fcir f:0 r:1
\move(49 28)
\fcir f:0 r:1
\move(49 35)
\fcir f:0 r:1
\move(49 42)
\fcir f:0 r:1
\move(49 49)
\fcir f:0 r:1
\move(49 56)
\fcir f:0 r:1
\move(49 63)
\fcir f:0 r:1
\move(56 21)
\fcir f:0 r:1
\move(56 28)
\fcir f:0 r:1
\move(56 35)
\fcir f:0 r:1
\move(56 42)
\fcir f:0 r:1
\move(56 49)
\fcir f:0 r:1
\move(56 56)
\fcir f:0 r:1
\move(56 63)
\fcir f:0 r:1
\move(63 21)
\fcir f:0 r:1
\move(63 28)
\fcir f:0 r:1
\move(63 35)
\fcir f:0 r:1
\move(63 42)
\fcir f:0 r:1
\move(63 49)
\fcir f:0 r:1
\move(63 56)
\fcir f:0 r:1
\move(63 63)
\fcir f:0 r:1
\move(63 70)
\fcir f:0 r:1
\move(70 14)
\fcir f:0 r:1
\move(70 21)
\fcir f:0 r:1
\move(70 28)
\fcir f:0 r:1
\move(70 35)
\fcir f:0 r:1
\move(70 42)
\fcir f:0 r:1
\move(70 49)
\fcir f:0 r:1
\move(70 56)
\fcir f:0 r:1
\move(77 14)
\fcir f:0 r:1
\move(77 21)
\fcir f:0 r:1
\move(77 28)
\fcir f:0 r:1
\move(77 35)
\fcir f:0 r:1
\move(77 42)
\fcir f:0 r:1
\move(84 7)
\fcir f:0 r:1
\move(84 14)
\fcir f:0 r:1
\move(84 21)
\fcir f:0 r:1
\move(84 28)
\fcir f:0 r:1
\move(91 7)
\fcir f:0 r:1
\move(91 14)
\fcir f:0 r:1
\move(98 0)
\fcir f:0 r:1
\move(0 49)
\move(-2 47)
\lvec(2 51)
\move(2 47)
\lvec(-2 51)
\move(7 49)
\move(5 47)
\lvec(9 51)
\move(9 47)
\lvec(5 51)
\move(7 56)
\move(5 54)
\lvec(9 58)
\move(9 54)
\lvec(5 58)
\move(14 42)
\move(12 40)
\lvec(16 44)
\move(16 40)
\lvec(12 44)
\move(14 49)
\move(12 47)
\lvec(16 51)
\move(16 47)
\lvec(12 51)
\move(14 56)
\move(12 54)
\lvec(16 58)
\move(16 54)
\lvec(12 58)
\move(21 42)
\move(19 40)
\lvec(23 44)
\move(23 40)
\lvec(19 44)
\move(21 49)
\move(19 47)
\lvec(23 51)
\move(23 47)
\lvec(19 51)
\move(21 56)
\move(19 54)
\lvec(23 58)
\move(23 54)
\lvec(19 58)
\move(28 35)
\move(26 33)
\lvec(30 37)
\move(30 33)
\lvec(26 37)
\move(28 42)
\move(26 40)
\lvec(30 44)
\move(30 40)
\lvec(26 44)
\move(28 49)
\move(26 47)
\lvec(30 51)
\move(30 47)
\lvec(26 51)
\move(35 35)
\move(33 33)
\lvec(37 37)
\move(37 33)
\lvec(33 37)
\move(35 42)
\move(33 40)
\lvec(37 44)
\move(37 40)
\lvec(33 44)
\move(42 28)
\move(40 26)
\lvec(44 30)
\move(44 26)
\lvec(40 30)
\move(42 35)
\move(40 33)
\lvec(44 37)
\move(44 33)
\lvec(40 37)
\move(49 28)
\move(47 26)
\lvec(51 30)
\move(51 26)
\lvec(47 30)
\move(56 21)
\move(54 19)
\lvec(58 23)
\move(58 19)
\lvec(54 23)
\move(56 28)
\move(54 26)
\lvec(58 30)
\move(58 26)
\lvec(54 30)
\move(63 21)
\move(61 19)
\lvec(65 23)
\move(65 19)
\lvec(61 23)
\move(70 14)
\move(68 12)
\lvec(72 16)
\move(72 12)
\lvec(68 16)
\linewd 0.5
\move(17 60)
\lvec(97 -4)
\end{texdraw}
\\
\begin{texdraw}
\drawdim pt
\linewd 0.5
\move(0 0)
\fcir f:0 r:0.2
\move(0 7)
\fcir f:0 r:0.2
\move(0 14)
\fcir f:0 r:0.2
\move(0 21)
\fcir f:0 r:0.2
\move(0 28)
\fcir f:0 r:0.2
\move(0 35)
\fcir f:0 r:0.2
\move(0 42)
\fcir f:0 r:0.2
\move(0 49)
\fcir f:0 r:0.2
\move(0 56)
\fcir f:0 r:0.2
\move(0 63)
\fcir f:0 r:0.2
\move(0 70)
\fcir f:0 r:0.2
\move(7 0)
\fcir f:0 r:0.2
\move(7 7)
\fcir f:0 r:0.2
\move(7 14)
\fcir f:0 r:0.2
\move(7 21)
\fcir f:0 r:0.2
\move(7 28)
\fcir f:0 r:0.2
\move(7 35)
\fcir f:0 r:0.2
\move(7 42)
\fcir f:0 r:0.2
\move(7 49)
\fcir f:0 r:0.2
\move(7 56)
\fcir f:0 r:0.2
\move(7 63)
\fcir f:0 r:0.2
\move(7 70)
\fcir f:0 r:0.2
\move(14 0)
\fcir f:0 r:0.2
\move(14 7)
\fcir f:0 r:0.2
\move(14 14)
\fcir f:0 r:0.2
\move(14 21)
\fcir f:0 r:0.2
\move(14 28)
\fcir f:0 r:0.2
\move(14 35)
\fcir f:0 r:0.2
\move(14 42)
\fcir f:0 r:0.2
\move(14 49)
\fcir f:0 r:0.2
\move(14 56)
\fcir f:0 r:0.2
\move(14 63)
\fcir f:0 r:0.2
\move(14 70)
\fcir f:0 r:0.2
\move(21 0)
\fcir f:0 r:0.2
\move(21 7)
\fcir f:0 r:0.2
\move(21 14)
\fcir f:0 r:0.2
\move(21 21)
\fcir f:0 r:0.2
\move(21 28)
\fcir f:0 r:0.2
\move(21 35)
\fcir f:0 r:0.2
\move(21 42)
\fcir f:0 r:0.2
\move(21 49)
\fcir f:0 r:0.2
\move(21 56)
\fcir f:0 r:0.2
\move(21 63)
\fcir f:0 r:0.2
\move(21 70)
\fcir f:0 r:0.2
\move(28 0)
\fcir f:0 r:0.2
\move(28 7)
\fcir f:0 r:0.2
\move(28 14)
\fcir f:0 r:0.2
\move(28 21)
\fcir f:0 r:0.2
\move(28 28)
\fcir f:0 r:0.2
\move(28 35)
\fcir f:0 r:0.2
\move(28 42)
\fcir f:0 r:0.2
\move(28 49)
\fcir f:0 r:0.2
\move(28 56)
\fcir f:0 r:0.2
\move(28 63)
\fcir f:0 r:0.2
\move(28 70)
\fcir f:0 r:0.2
\move(35 0)
\fcir f:0 r:0.2
\move(35 7)
\fcir f:0 r:0.2
\move(35 14)
\fcir f:0 r:0.2
\move(35 21)
\fcir f:0 r:0.2
\move(35 28)
\fcir f:0 r:0.2
\move(35 35)
\fcir f:0 r:0.2
\move(35 42)
\fcir f:0 r:0.2
\move(35 49)
\fcir f:0 r:0.2
\move(35 56)
\fcir f:0 r:0.2
\move(35 63)
\fcir f:0 r:0.2
\move(35 70)
\fcir f:0 r:0.2
\move(42 0)
\fcir f:0 r:0.2
\move(42 7)
\fcir f:0 r:0.2
\move(42 14)
\fcir f:0 r:0.2
\move(42 21)
\fcir f:0 r:0.2
\move(42 28)
\fcir f:0 r:0.2
\move(42 35)
\fcir f:0 r:0.2
\move(42 42)
\fcir f:0 r:0.2
\move(42 49)
\fcir f:0 r:0.2
\move(42 56)
\fcir f:0 r:0.2
\move(42 63)
\fcir f:0 r:0.2
\move(42 70)
\fcir f:0 r:0.2
\move(49 0)
\fcir f:0 r:0.2
\move(49 7)
\fcir f:0 r:0.2
\move(49 14)
\fcir f:0 r:0.2
\move(49 21)
\fcir f:0 r:0.2
\move(49 28)
\fcir f:0 r:0.2
\move(49 35)
\fcir f:0 r:0.2
\move(49 42)
\fcir f:0 r:0.2
\move(49 49)
\fcir f:0 r:0.2
\move(49 56)
\fcir f:0 r:0.2
\move(49 63)
\fcir f:0 r:0.2
\move(49 70)
\fcir f:0 r:0.2
\move(56 0)
\fcir f:0 r:0.2
\move(56 7)
\fcir f:0 r:0.2
\move(56 14)
\fcir f:0 r:0.2
\move(56 21)
\fcir f:0 r:0.2
\move(56 28)
\fcir f:0 r:0.2
\move(56 35)
\fcir f:0 r:0.2
\move(56 42)
\fcir f:0 r:0.2
\move(56 49)
\fcir f:0 r:0.2
\move(56 56)
\fcir f:0 r:0.2
\move(56 63)
\fcir f:0 r:0.2
\move(56 70)
\fcir f:0 r:0.2
\move(63 0)
\fcir f:0 r:0.2
\move(63 7)
\fcir f:0 r:0.2
\move(63 14)
\fcir f:0 r:0.2
\move(63 21)
\fcir f:0 r:0.2
\move(63 28)
\fcir f:0 r:0.2
\move(63 35)
\fcir f:0 r:0.2
\move(63 42)
\fcir f:0 r:0.2
\move(63 49)
\fcir f:0 r:0.2
\move(63 56)
\fcir f:0 r:0.2
\move(63 63)
\fcir f:0 r:0.2
\move(63 70)
\fcir f:0 r:0.2
\move(70 0)
\fcir f:0 r:0.2
\move(70 7)
\fcir f:0 r:0.2
\move(70 14)
\fcir f:0 r:0.2
\move(70 21)
\fcir f:0 r:0.2
\move(70 28)
\fcir f:0 r:0.2
\move(70 35)
\fcir f:0 r:0.2
\move(70 42)
\fcir f:0 r:0.2
\move(70 49)
\fcir f:0 r:0.2
\move(70 56)
\fcir f:0 r:0.2
\move(70 63)
\fcir f:0 r:0.2
\move(70 70)
\fcir f:0 r:0.2
\move(77 0)
\fcir f:0 r:0.2
\move(77 7)
\fcir f:0 r:0.2
\move(77 14)
\fcir f:0 r:0.2
\move(77 21)
\fcir f:0 r:0.2
\move(77 28)
\fcir f:0 r:0.2
\move(77 35)
\fcir f:0 r:0.2
\move(77 42)
\fcir f:0 r:0.2
\move(77 49)
\fcir f:0 r:0.2
\move(77 56)
\fcir f:0 r:0.2
\move(77 63)
\fcir f:0 r:0.2
\move(77 70)
\fcir f:0 r:0.2
\move(84 0)
\fcir f:0 r:0.2
\move(84 7)
\fcir f:0 r:0.2
\move(84 14)
\fcir f:0 r:0.2
\move(84 21)
\fcir f:0 r:0.2
\move(84 28)
\fcir f:0 r:0.2
\move(84 35)
\fcir f:0 r:0.2
\move(84 42)
\fcir f:0 r:0.2
\move(84 49)
\fcir f:0 r:0.2
\move(84 56)
\fcir f:0 r:0.2
\move(84 63)
\fcir f:0 r:0.2
\move(84 70)
\fcir f:0 r:0.2
\move(91 0)
\fcir f:0 r:0.2
\move(91 7)
\fcir f:0 r:0.2
\move(91 14)
\fcir f:0 r:0.2
\move(91 21)
\fcir f:0 r:0.2
\move(91 28)
\fcir f:0 r:0.2
\move(91 35)
\fcir f:0 r:0.2
\move(91 42)
\fcir f:0 r:0.2
\move(91 49)
\fcir f:0 r:0.2
\move(91 56)
\fcir f:0 r:0.2
\move(91 63)
\fcir f:0 r:0.2
\move(91 70)
\fcir f:0 r:0.2
\move(98 0)
\fcir f:0 r:0.2
\move(98 7)
\fcir f:0 r:0.2
\move(98 14)
\fcir f:0 r:0.2
\move(98 21)
\fcir f:0 r:0.2
\move(98 28)
\fcir f:0 r:0.2
\move(98 35)
\fcir f:0 r:0.2
\move(98 42)
\fcir f:0 r:0.2
\move(98 49)
\fcir f:0 r:0.2
\move(98 56)
\fcir f:0 r:0.2
\move(98 63)
\fcir f:0 r:0.2
\move(98 70)
\fcir f:0 r:0.2
\arrowheadtype t:V
\move(0 0)
\avec(119 0)
\move(0 0)
\avec(0 91)
\htext(123 0){$\mathbb{N}$}
\htext(-13 77){$\mathbb{N}$}
\move(28 56)
\fcir f:0 r:1
\move(35 49)
\fcir f:0 r:1
\move(35 56)
\fcir f:0 r:1
\move(35 63)
\fcir f:0 r:1
\move(42 42)
\fcir f:0 r:1
\move(42 49)
\fcir f:0 r:1
\move(42 56)
\fcir f:0 r:1
\move(42 63)
\fcir f:0 r:1
\move(49 35)
\fcir f:0 r:1
\move(49 42)
\fcir f:0 r:1
\move(49 49)
\fcir f:0 r:1
\move(49 56)
\fcir f:0 r:1
\move(49 63)
\fcir f:0 r:1
\move(56 35)
\fcir f:0 r:1
\move(56 42)
\fcir f:0 r:1
\move(56 49)
\fcir f:0 r:1
\move(56 56)
\fcir f:0 r:1
\move(56 63)
\fcir f:0 r:1
\move(63 28)
\fcir f:0 r:1
\move(63 35)
\fcir f:0 r:1
\move(63 42)
\fcir f:0 r:1
\move(63 49)
\fcir f:0 r:1
\move(63 56)
\fcir f:0 r:1
\move(63 63)
\fcir f:0 r:1
\move(63 70)
\fcir f:0 r:1
\move(70 21)
\fcir f:0 r:1
\move(70 28)
\fcir f:0 r:1
\move(70 35)
\fcir f:0 r:1
\move(70 42)
\fcir f:0 r:1
\move(70 49)
\fcir f:0 r:1
\move(70 56)
\fcir f:0 r:1
\move(77 14)
\fcir f:0 r:1
\move(77 21)
\fcir f:0 r:1
\move(77 28)
\fcir f:0 r:1
\move(77 35)
\fcir f:0 r:1
\move(77 42)
\fcir f:0 r:1
\move(84 7)
\fcir f:0 r:1
\move(84 14)
\fcir f:0 r:1
\move(84 21)
\fcir f:0 r:1
\move(84 28)
\fcir f:0 r:1
\move(91 7)
\fcir f:0 r:1
\move(91 14)
\fcir f:0 r:1
\move(98 0)
\fcir f:0 r:1
\move(28 56)
\move(26 54)
\lvec(30 58)
\move(30 54)
\lvec(26 58)
\move(35 49)
\move(33 47)
\lvec(37 51)
\move(37 47)
\lvec(33 51)
\move(35 56)
\move(33 54)
\lvec(37 58)
\move(37 54)
\lvec(33 58)
\move(35 63)
\move(33 61)
\lvec(37 65)
\move(37 61)
\lvec(33 65)
\move(42 42)
\move(40 40)
\lvec(44 44)
\move(44 40)
\lvec(40 44)
\move(42 49)
\move(40 47)
\lvec(44 51)
\move(44 47)
\lvec(40 51)
\move(42 56)
\move(40 54)
\lvec(44 58)
\move(44 54)
\lvec(40 58)
\move(42 63)
\move(40 61)
\lvec(44 65)
\move(44 61)
\lvec(40 65)
\move(49 35)
\move(47 33)
\lvec(51 37)
\move(51 33)
\lvec(47 37)
\move(49 42)
\move(47 40)
\lvec(51 44)
\move(51 40)
\lvec(47 44)
\move(49 49)
\move(47 47)
\lvec(51 51)
\move(51 47)
\lvec(47 51)
\move(49 56)
\move(47 54)
\lvec(51 58)
\move(51 54)
\lvec(47 58)
\move(49 63)
\move(47 61)
\lvec(51 65)
\move(51 61)
\lvec(47 65)
\move(56 35)
\move(54 33)
\lvec(58 37)
\move(58 33)
\lvec(54 37)
\move(56 42)
\move(54 40)
\lvec(58 44)
\move(58 40)
\lvec(54 44)
\move(56 49)
\move(54 47)
\lvec(58 51)
\move(58 47)
\lvec(54 51)
\move(56 56)
\move(54 54)
\lvec(58 58)
\move(58 54)
\lvec(54 58)
\move(56 63)
\move(54 61)
\lvec(58 65)
\move(58 61)
\lvec(54 65)
\move(63 56)
\move(61 54)
\lvec(65 58)
\move(65 54)
\lvec(61 58)
\move(63 63)
\move(61 61)
\lvec(65 65)
\move(65 61)
\lvec(61 65)
\move(63 70)
\move(61 68)
\lvec(65 72)
\move(65 68)
\lvec(61 72)
\linewd 0.5
\move(53 24)
\lvec(69 70)
\end{texdraw}
&
\begin{texdraw}
\drawdim pt
\linewd 0.5
\move(0 0)
\fcir f:0 r:0.2
\move(0 7)
\fcir f:0 r:0.2
\move(0 14)
\fcir f:0 r:0.2
\move(0 21)
\fcir f:0 r:0.2
\move(0 28)
\fcir f:0 r:0.2
\move(0 35)
\fcir f:0 r:0.2
\move(0 42)
\fcir f:0 r:0.2
\move(0 49)
\fcir f:0 r:0.2
\move(0 56)
\fcir f:0 r:0.2
\move(7 0)
\fcir f:0 r:0.2
\move(7 7)
\fcir f:0 r:0.2
\move(7 14)
\fcir f:0 r:0.2
\move(7 21)
\fcir f:0 r:0.2
\move(7 28)
\fcir f:0 r:0.2
\move(7 35)
\fcir f:0 r:0.2
\move(7 42)
\fcir f:0 r:0.2
\move(7 49)
\fcir f:0 r:0.2
\move(7 56)
\fcir f:0 r:0.2
\move(14 0)
\fcir f:0 r:0.2
\move(14 7)
\fcir f:0 r:0.2
\move(14 14)
\fcir f:0 r:0.2
\move(14 21)
\fcir f:0 r:0.2
\move(14 28)
\fcir f:0 r:0.2
\move(14 35)
\fcir f:0 r:0.2
\move(14 42)
\fcir f:0 r:0.2
\move(14 49)
\fcir f:0 r:0.2
\move(14 56)
\fcir f:0 r:0.2
\move(21 0)
\fcir f:0 r:0.2
\move(21 7)
\fcir f:0 r:0.2
\move(21 14)
\fcir f:0 r:0.2
\move(21 21)
\fcir f:0 r:0.2
\move(21 28)
\fcir f:0 r:0.2
\move(21 35)
\fcir f:0 r:0.2
\move(21 42)
\fcir f:0 r:0.2
\move(21 49)
\fcir f:0 r:0.2
\move(21 56)
\fcir f:0 r:0.2
\move(28 0)
\fcir f:0 r:0.2
\move(28 7)
\fcir f:0 r:0.2
\move(28 14)
\fcir f:0 r:0.2
\move(28 21)
\fcir f:0 r:0.2
\move(28 28)
\fcir f:0 r:0.2
\move(28 35)
\fcir f:0 r:0.2
\move(28 42)
\fcir f:0 r:0.2
\move(28 49)
\fcir f:0 r:0.2
\move(28 56)
\fcir f:0 r:0.2
\move(35 0)
\fcir f:0 r:0.2
\move(35 7)
\fcir f:0 r:0.2
\move(35 14)
\fcir f:0 r:0.2
\move(35 21)
\fcir f:0 r:0.2
\move(35 28)
\fcir f:0 r:0.2
\move(35 35)
\fcir f:0 r:0.2
\move(35 42)
\fcir f:0 r:0.2
\move(35 49)
\fcir f:0 r:0.2
\move(35 56)
\fcir f:0 r:0.2
\move(42 0)
\fcir f:0 r:0.2
\move(42 7)
\fcir f:0 r:0.2
\move(42 14)
\fcir f:0 r:0.2
\move(42 21)
\fcir f:0 r:0.2
\move(42 28)
\fcir f:0 r:0.2
\move(42 35)
\fcir f:0 r:0.2
\move(42 42)
\fcir f:0 r:0.2
\move(42 49)
\fcir f:0 r:0.2
\move(42 56)
\fcir f:0 r:0.2
\move(49 0)
\fcir f:0 r:0.2
\move(49 7)
\fcir f:0 r:0.2
\move(49 14)
\fcir f:0 r:0.2
\move(49 21)
\fcir f:0 r:0.2
\move(49 28)
\fcir f:0 r:0.2
\move(49 35)
\fcir f:0 r:0.2
\move(49 42)
\fcir f:0 r:0.2
\move(49 49)
\fcir f:0 r:0.2
\move(49 56)
\fcir f:0 r:0.2
\move(56 0)
\fcir f:0 r:0.2
\move(56 7)
\fcir f:0 r:0.2
\move(56 14)
\fcir f:0 r:0.2
\move(56 21)
\fcir f:0 r:0.2
\move(56 28)
\fcir f:0 r:0.2
\move(56 35)
\fcir f:0 r:0.2
\move(56 42)
\fcir f:0 r:0.2
\move(56 49)
\fcir f:0 r:0.2
\move(56 56)
\fcir f:0 r:0.2
\move(63 0)
\fcir f:0 r:0.2
\move(63 7)
\fcir f:0 r:0.2
\move(63 14)
\fcir f:0 r:0.2
\move(63 21)
\fcir f:0 r:0.2
\move(63 28)
\fcir f:0 r:0.2
\move(63 35)
\fcir f:0 r:0.2
\move(63 42)
\fcir f:0 r:0.2
\move(63 49)
\fcir f:0 r:0.2
\move(63 56)
\fcir f:0 r:0.2
\move(70 0)
\fcir f:0 r:0.2
\move(70 7)
\fcir f:0 r:0.2
\move(70 14)
\fcir f:0 r:0.2
\move(70 21)
\fcir f:0 r:0.2
\move(70 28)
\fcir f:0 r:0.2
\move(70 35)
\fcir f:0 r:0.2
\move(70 42)
\fcir f:0 r:0.2
\move(70 49)
\fcir f:0 r:0.2
\move(70 56)
\fcir f:0 r:0.2
\move(77 0)
\fcir f:0 r:0.2
\move(77 7)
\fcir f:0 r:0.2
\move(77 14)
\fcir f:0 r:0.2
\move(77 21)
\fcir f:0 r:0.2
\move(77 28)
\fcir f:0 r:0.2
\move(77 35)
\fcir f:0 r:0.2
\move(77 42)
\fcir f:0 r:0.2
\move(77 49)
\fcir f:0 r:0.2
\move(77 56)
\fcir f:0 r:0.2
\move(84 0)
\fcir f:0 r:0.2
\move(84 7)
\fcir f:0 r:0.2
\move(84 14)
\fcir f:0 r:0.2
\move(84 21)
\fcir f:0 r:0.2
\move(84 28)
\fcir f:0 r:0.2
\move(84 35)
\fcir f:0 r:0.2
\move(84 42)
\fcir f:0 r:0.2
\move(84 49)
\fcir f:0 r:0.2
\move(84 56)
\fcir f:0 r:0.2
\move(91 0)
\fcir f:0 r:0.2
\move(91 7)
\fcir f:0 r:0.2
\move(91 14)
\fcir f:0 r:0.2
\move(91 21)
\fcir f:0 r:0.2
\move(91 28)
\fcir f:0 r:0.2
\move(91 35)
\fcir f:0 r:0.2
\move(91 42)
\fcir f:0 r:0.2
\move(91 49)
\fcir f:0 r:0.2
\move(91 56)
\fcir f:0 r:0.2
\move(98 0)
\fcir f:0 r:0.2
\move(98 7)
\fcir f:0 r:0.2
\move(98 14)
\fcir f:0 r:0.2
\move(98 21)
\fcir f:0 r:0.2
\move(98 28)
\fcir f:0 r:0.2
\move(98 35)
\fcir f:0 r:0.2
\move(98 42)
\fcir f:0 r:0.2
\move(98 49)
\fcir f:0 r:0.2
\move(98 56)
\fcir f:0 r:0.2
\arrowheadtype t:V
\move(0 0)
\avec(119 0)
\move(0 0)
\avec(0 77)
\htext(123 0){$\mathbb{N}$}
\htext(-13 63){$\mathbb{N}$}
\move(63 28)
\fcir f:0 r:1
\move(63 35)
\fcir f:0 r:1
\move(63 42)
\fcir f:0 r:1
\move(63 49)
\fcir f:0 r:1
\move(70 21)
\fcir f:0 r:1
\move(70 28)
\fcir f:0 r:1
\move(70 35)
\fcir f:0 r:1
\move(70 42)
\fcir f:0 r:1
\move(70 49)
\fcir f:0 r:1
\move(70 56)
\fcir f:0 r:1
\move(77 14)
\fcir f:0 r:1
\move(77 21)
\fcir f:0 r:1
\move(77 28)
\fcir f:0 r:1
\move(77 35)
\fcir f:0 r:1
\move(77 42)
\fcir f:0 r:1
\move(84 7)
\fcir f:0 r:1
\move(84 14)
\fcir f:0 r:1
\move(84 21)
\fcir f:0 r:1
\move(84 28)
\fcir f:0 r:1
\move(91 7)
\fcir f:0 r:1
\move(91 14)
\fcir f:0 r:1
\move(98 0)
\fcir f:0 r:1
\move(84 7)
\move(82 5)
\lvec(86 9)
\move(86 5)
\lvec(82 9)
\end{texdraw}
\\
\end{array}$$
Since the Nagata conjecture for square-free integers $r > 9$ involves irrational square roots, it seems appropriate to look for an asymptotic version of Dumnicki’s reduction algorithm. To this purpose we introduce the following notion:
\[asymp-sys-Def\] Let $m_1, \ldots, m_r \in \mathbb{R}_{>0}$. A subset $P \subset \mathbb{R}^2_{\geq 0}$ contains asymptotically $(m_1, \ldots, m_r)$-non-special systems (of dimension $\geq d$) iff for all $\delta > 0$ and all $k >> 0$ there exist $m_1^{(k)}, \ldots, m_r^{(k)} \in \mathbb{N}$ and a $D_k \subset k \cdot P \cap \mathbb{N}^2_{\geq 0}$ such that
- $\mathcal{L}_{D_k}(m_1^{(k)}, \ldots, m_r^{(k)})$ is non-special (of dimension $\geq d$) and
- $\left| \frac{m_i^{(k)}-km}{km_i} \right| < \delta$, $i=1, \ldots , r$.
With this notion we prove the following method of obtaining bounds on Seshadri constants on $\mathbb{P}^2$ (see Section \[LowerBound-sec\] for the proof):
\[nef-crit\] If the set $P := \{ x + y \leq 1 \} \cap \mathbb{R}^2_{\geq 0}$ contains asymptotically $(m^r)$-non-special systems of dimension $\geq 0$ then $$H - m \sum_{j=1}^r E_j$$ is nef on $X$, where $X$ is the blow-up of $\mathbb{P}^2$ in $r$ very general points, $H$ is the pullback of a line in $\mathbb{P}^2$ to $X$, and the $E_i$ are the exceptional divisors on $X$.
To show the existence of $(m^r)$-non-special sytems we develop an asymptotic version of Dumnicki’s reduction algorithm (see Thm. \[As-red-alg-Thm\]), and together with a criterion for asymptotic $(m)$-non-specialty (see Thm. \[as-m-non-spec-crit\]), we are able to give the following bound on the Seshadri constant of $10$ very general points on $\mathbb{P}^2$ (see again Section \[LowerBound-sec\] for the proof):
\[Seshadri-bd-thm\] Let $X$ be the blow-up of $\mathbb{P}^2$ in 10 very general points, let $E_1, \ldots, E_{10}$ be the exceptional divisors on $X$, and let $H$ be the pull back of a line in $\mathbb{P}^2$. Then the divisor $$H - \frac{4}{13} \sum_{i=1}^{10} E_i$$ is nef on $X$.
In recent years many authors tried to give lower bounds for the Seshadri constants of a fixed number of general points on algebraic surfaces, and especially on $\mathbb{P}^2$ [@Xu95; @ST02; @Harb03; @HR04; @HR05]. Some of these bounds are even better than $\frac{4}{13} \approx 0.307$, which is still not really close to $\frac{1}{\sqrt{10}} \approx 0.3162$: Tutaj-Gasińska [@Tut03] achieved $\frac{2}{11}\sqrt{3} \approx 0.314$, Biran [@Bir99] $\frac{6}{19} \approx 0.3158$, and Harbourne-Roé [@HR03] even $\frac{177}{560} \approx 0.31607$. At least, our bound is better than what can be achieved by using the non-specialty of all non-empty linear systems $\mathcal{L}_d(m^{10})$ up to $m \leq 42$, shown by Dumnicki [@D06]: The expected dimension of $\mathcal{L}_d(41^{10})$ is $\geq 0$ iff $d \geq 132$, hence Thm. \[Eckl-thm\] applied to the constant sequence $(132,40)$ gives the bound $\frac{40}{132} \approx 0.303$. For all other multiplicities $m \leq 42$ the bound gets smaller.
In any case the true interest in this bound lies in the fact that it was shown with an asymptotic method.
*Acknowledgement.* The author would like to thank Felix Schüller who explained Dumnicki’s techniques in his diploma thesis [@Sch07].
Monotone Reordering
===================
In this section we collect elementary, but useful facts about the *monotone reordering* of functions:
Let $f: [a,b] \rightarrow \mathbb{R}$ be a measurable function on a closed interval $[a,b] \subset \mathbb{R}$. Then the monotone reordering $f^\#: (0,b-a] \rightarrow \mathbb{R}$ of $f$ is defined by $$t \mapsto \inf \left\{ s: t \leq \mathrm{length\ of\ } \{t^\prime \in [a,b]: f(t^\prime) \leq s\} \right\}.$$
This notion will be used to state the criterion of $(m)$-non-specialty (see Thm. \[as-m-non-spec-crit\]).
Let $f: [a,b] \rightarrow \mathbb{R}$ be a step function. Then $f^\#: (0, b-a] \rightarrow \mathbb{R}$ reorders its steps such that they increase monotonely. Another example is given in the following diagram which shows the monotone reordering of a piecewise-linear function:
(150,60)(-8,-8) (-5,0)[(1,0)[55]{}]{} (0,-5)[(0,1)[55]{}]{}
(0,0)[(1,2)[20]{}]{} (20,40)[(1,-2)[20]{}]{}
(-1,40)[(1,0)[2]{}]{} (20,-1)[(0,1)[2]{}]{} (-8,37)[h]{} (17,-8)[t]{} (37,-8)[2t]{}
(55,20)[(1,0)[10]{}]{}
(75,0)[(1,0)[55]{}]{} (80,-5)[(0,1)[55]{}]{}
(80,0)[(1,1)[40]{}]{}
(79,40)[(1,0)[2]{}]{} (120,-1)[(0,1)[2]{}]{} (72,37)[h]{} (117,-8)[2t]{}
\[mon-re-inc-prop\] The monotone reordering $f^\#$ of a function $f: [a,b] \rightarrow \mathbb{R}$ is monotonely increasing and lower semi-continuous.
If $t_1 < t_2$ then $t_2 \leq \mathrm{length\ of\ } \{ f(t^\prime) \leq s\}$ implies $t_1 \leq \mathrm{length\ of\ } \{ f(t^\prime) \leq s\}$, hence $$\begin{array}{rcccl}
f^\#(t_1) & = & \inf \left\{ s: t_1 \leq \mathrm{length\ of\ } \{ f(t^\prime) \leq s\} \right\} & & \\
& \leq & \inf \left\{ s: t_2 \leq \mathrm{length\ of\ } \{ f(t^\prime) \leq s\} \right\} & = & f^\#(t_2).
\end{array}$$ Furthermore, set $s := f^\#(t)$ and assume for given $\epsilon > 0$ that $$\widetilde{t} \leq \mathrm{length\ of\ } \{ f(t^\prime) \leq s - \epsilon \}$$ for all $\widetilde{t} < t$. This implies $t \leq \mathrm{length\ of\ } \{ f(t^\prime) \leq s - \epsilon \}$, a contradiction to $f^\#(t)=s$. Hence there exists a $\overline{t} < t$ such that $f^\#((\overline{t}, b-a)) > s - \epsilon$, and $f^\#$ is lower semi-continuous.
\[monre-ineq-prop\] Let $f_1, f_2: [a,b] \rightarrow \mathbb{R}$ be two measurable functions such that $f_1 \leq f_2$. Then $f_1^\# \leq f_2^\#$.
If $f_1 \leq f_2$ then for fixed $s$, $$\mathrm{length\ of\ } \{ f_1 \leq s \} \geq \mathrm{length\ of\ } \{ f_2 \leq s \}.$$ This implies for fixed $t$ that $$\left\{ s: t \leq \mathrm{length\ of\ } \{ f_1 \leq s\} \right\} \supset
\left\{ s: t \leq \mathrm{length\ of\ } \{ f_2 \leq s\} \right\},$$ hence $f_1^\#(t) \leq f_2^\#(t)$.
Let $f: [a,b] \rightarrow \mathbb{R}$ be a continuous function. Then the monotone reordering $f^\#:(0, b-a] \rightarrow \mathbb{R}$ is also continuous.
We already know from Prop. \[mon-re-inc-prop\] that $f^\#$ is lower semi-continuous. Let $t \in (0, b-a]$ and set $s := f^\#(t)$. For $t = b-a$ or $s = \max\{ f(t^\prime): t^\prime \in [a,b]\}$ nothing is to prove. Since $f$ is continuous the set $\{t^\prime: s < f(t^\prime) < s+\epsilon \}$ is open and non-empty for all $\epsilon > 0$ and has consequently a positive length $\delta_\epsilon$. Then $$\mathrm{length\ of\ } \{ f \leq s+\epsilon \} \geq
\mathrm{length\ of\ } \{ f \leq s\} + \mathrm{length\ of\ } \{ s < f < s+\epsilon \} \geq t+\delta_\epsilon,$$ hence $f^\#(t^\prime) < s + 2\epsilon$ for all $t^\prime < t+\delta_\epsilon$, and $f^\#$ is upper semi-continuous in $t$.
\[cutoff-thm\] Let $f: [a,b] \rightarrow \mathbb{R}$ be a continuous function on the closed interval $[a,b] \subset \mathbb{R}$. Then for all $\epsilon > 0$ there exists a $\delta > 0$ such that for all closed intervals $[a^\prime, b^\prime] \subset [a,b]$ with $(b-a) - (b^\prime-a^\prime) < \delta$ the monotone reorderings $f^\#: (0, b-a] \rightarrow \mathbb{R}$ and $(f_{|[a^\prime,b^\prime]})^\#: (0, b^\prime-a^\prime] \rightarrow \mathbb{R}$ satisfy $$\parallel \!\! (f^\#)_{|(0, b^\prime-a^\prime]} - (f_{|[a^\prime,b^\prime]})^\# \!\! \parallel_{\mathrm{max}} <
\epsilon.$$
Since $f$ is continuous on the compact interval $[a,b]$ the function is uniformly continuous on $[a,b]$. Consequently, for every $\epsilon > 0$ there exists a $\delta > 0$ such that $| t - t^\prime | < \delta$ implies $| f(t) - f(t^\prime) | < \epsilon$.
*Claim 1*. $| t - t^\prime | < \delta$ also implies $| f^\#(t) - f^\#(t^\prime) | < 2\epsilon$.
Since $f$ is continuous $f$ has a minimum $s_{\min}$ on $[a,b]$ which also must be a lower bound for $f^\#$ on $(0,b-a]$. By construction, $\mathrm{length\ of\ } \{ f \leq s_{\min}+\epsilon\} \geq \delta$, hence $$| f^\#(t) - f^\#(t^\prime) | \leq \epsilon < 2\epsilon\
\mathrm{for\ all\ } 0 < t \leq t^\prime < \delta.$$
Now let $t^\prime \geq \delta$. If $f^\#(t^\prime) - 2\epsilon < s_{\min}$, we have $$| f^\#(t^\prime) - f^\#(t) | \leq f^\#(t^\prime) - s_{\min} < 2\epsilon\ \mathrm{for\ } t < t^\prime,$$ since $f^\#$ is monotonely increasing by Prop. \[mon-re-inc-prop\].
Otherwise $f^\#(t^\prime) - 2\epsilon \geq s_{\min}$, and we show two claims:
*Claim 1.1*. $f: [a,b] \rightarrow \mathbb{R}$ continuous $\Rightarrow$ $\mathrm{length\ of\ } \{ f < f^\#(t^\prime) \} \leq t^\prime$.
The characteristic function of the sets $\{ s-\epsilon < f < s \}$ tend pointwise to $0$ for $\epsilon \rightarrow 0$, and they are dominated by the integrable characteristic function of $[a,b]$. Hence, by Lebesgue’s dominated convergence, $$\mathrm{length\ of\ } \{ s-\epsilon < f \leq s \} \rightarrow 0,\ \mathrm{for\ } \epsilon \rightarrow 0.$$ If $\mathrm{length\ of\ } \{ f < f^\#(t^\prime) \} > t^\prime$, this limit would imply the existence of an $\epsilon>0$ such that $$\mathrm{length\ of\ } \{ f < f^\#(t^\prime)-\epsilon \} > t^\prime,$$ a contradiction to the definition of $f^\#(t^\prime)$.
*Claim 1.2*. $\mathrm{Length\ of\ } \{ f^\#(t^\prime) - 2\epsilon < f < f^\#(t^\prime) \} \geq \delta$.
$f^\#(t^\prime)$ is a value of $f$ on $[a,b]$: Otherwise $f^\#(t^\prime)$ would be bigger than the maximum $s_{\max}$ of $f$ on $[a,b]$, hence $$\mathrm{length\ of\ } \{ f \leq f^\#(t^\prime) \} = \mathrm{length\ of\ } \{ f \leq s_{\max} \},$$ contradicting the definition of $f^\#(t^\prime)$. Furthermore, $f^\#(t^\prime) - 2\epsilon > s_{\min}$, hence by continuity, there exists a $\overline{t}$ such that $f(\overline{t}) = f^\#(t^\prime) - \epsilon$. But then the construction of $\delta$ shows that $$f((\overline{t}-\delta, \overline{t}+\delta)) \subset (f^\#(t^\prime) - 2\epsilon, f^\#(t^\prime)).$$ Since w.l.o.g. we can assume that $\delta < b-a$, the interval $(\overline{t}-\delta, \overline{t}+\delta) \cap [a,b]$ has length $\geq \delta$, hence the claim.
From these two claims we deduce $$\begin{aligned}
\lefteqn{\mathrm{length\ of\ } \{ f \leq f^\#(t^\prime) - 2\epsilon \} = } \\
& = & \mathrm{length\ of\ } \{ f < f^\#(t^\prime) \} -
\mathrm{length\ of\ } \{ f^\#(t^\prime) - 2\epsilon < f < f^\#(t^\prime) \} \\
& \leq & t^\prime - \delta,\end{aligned}$$ hence we have $f^\#(t^\prime) \geq f^\#(t) > f^\#(t^\prime) - 2\epsilon$ for all $t^\prime - \delta < t \leq t^\prime$. This proves Claim 1.
Now choose $a^\prime, b^\prime \in [a,b]$ such that $d := (b-a) - (b^\prime - a^\prime) < \delta$. Since $f$ is continuous it has a maximum $M$ and a minimum $m$ on the compact set $[a^\prime, b^\prime]$. Define $$\underline{f}(t) :=
\left\{ \begin{array}{ll}
m-\epsilon & \mathrm{for\ } t \in [a,a^\prime) \cup (b^\prime,b] \\
f(t) & \mathrm{else}
\end{array} \right. ,\ \
\overline{f}(t) :=
\left\{ \begin{array}{ll}
M+\epsilon & \mathrm{for\ } t \in [a,a^\prime) \cup (b^\prime,b] \\
f(t) & \mathrm{else.}
\end{array} \right.$$ Then $\underline{f} \leq f \leq \overline{f}$, hence $\underline{f}^\# \leq f^\# \leq \overline{f}^\#$ by Prop. \[monre-ineq-prop\], and furthermore $$\underline{f}^\#(t) =
\left\{ \begin{array}{ll}
m-\epsilon & \mathrm{for\ } t \leq d \\
(f_{|[a^\prime,b^\prime]})^\#(t-d) & \mathrm{else}
\end{array} \right. ,\ \
\overline{f}^\#(t) =
\left\{ \begin{array}{ll}
(f_{|[a^\prime,b^\prime]})^\#(t) & \mathrm{for\ } t \leq b^\prime - a^\prime \\
M+\epsilon & \mathrm{else.}
\end{array} \right.$$ Consequently, for $t \leq b^\prime - a^\prime$, $$\begin{aligned}
| (f_{|[a^\prime,b^\prime]})^\#(t) - f^\#(t) | & \leq & | \overline{f}^\#(t) - \underline{f}^\#(t) | \\
& = & \left\{ \begin{array}{ll}
(f_{|[a^\prime,b^\prime]})^\#(t) - (m-\epsilon) & \mathrm{for\ } t \leq d \\
(f_{|[a^\prime,b^\prime]})^\#(t) - (f_{|[a^\prime,b^\prime]})^\#(t-d) & \mathrm{else.}
\end{array} \right. \\
& < & \left\{ \begin{array}{c}
3\epsilon\\
2\epsilon
\end{array} \right.\end{aligned}$$ using $d < \delta$, $m \leq (f_{|[a^\prime,b^\prime]})^\#(t)$ for all $t \in (0,b^\prime - a^\prime]$ and Claim 1 applied to $f_{|[a^\prime,b^\prime]}$.
\[max-norm-bd-prop\] Let $f,g: [a,b] \rightarrow \mathbb{R}$ be two continuous function on $[a,b]$. Then for all $\epsilon > 0$, $$\parallel\!\! f - g \!\!\parallel_{\max} < \epsilon\ \ \Longrightarrow \ \
\parallel\!\! f^\# - g^\# \!\!\parallel_{\max} < 2\epsilon.$$
$\parallel\!\! f - g \!\!\parallel_{\max} < \epsilon$ implies $f-\epsilon < g < f+\epsilon$. Since $(f \pm \epsilon)^\# = f^\# \pm \epsilon$, we obtain from Prop. \[monre-ineq-prop\] $$f^\# - \epsilon \leq g^\# \leq f^\# + \epsilon,$$ hence the claim.
\[mon-re-id-prop\] Let $f: [a,b] \rightarrow \mathbb{R}_{\geq 0}$ be a continuous concave function, $f^\#: (0,b-a] \rightarrow \mathbb{R}_{\geq 0}$ its monotone reordering and $M := \max \{ f(t): t \in [a,b] \}$. If $M \geq b-a$ then $f^\#(t) \geq t$ for all $t \in (0, b-a]$.
Since $f$ is continuous, it achieves its maximum in some point $c \in [a,b]$. From now on suppose $c \in (a,b)$. If $c=a$ or $c=b$ the arguments are similar but easier. Set $$g: [a,b] \rightarrow \mathbb{R}_{\geq 0},
t \mapsto \left\{ \begin{array}{ll}
\frac{t-a}{c-a} \cdot (b-a)\ \mathrm{for\ all\ } a \leq t \leq c \\
\frac{b-t}{b-c} \cdot (b-a)\ \mathrm{for\ all\ } c \leq t \leq b. \rule{0cm}{0.4cm}
\end{array} \right.$$ Since $\mathrm{length\ of\ } \{ g \leq s \} = s$, we have $g^\#(t) = t$ for all $t \in (0,b-a]$. On the other hand, $g \leq f$ because $M \geq b-a$ and $f$ is concave. Consequently, $$g^\# \leq f^\#$$ by Prop. \[monre-ineq-prop\], which proves the claim.
The asymptotic version of Dumnicki’s algorithm
==============================================
The asymptotic version of Dumnicki’s reduction algorithm now reads as follows:
\[As-red-alg-Thm\] Let $m_1, \ldots. m_{p-1}, m_p \in \mathbb{R}_{> 0}$ and $P \subset \mathbb{R}^2_{\geq 0}$. For $$F: \mathbb{R}^2_{\geq 0} \ni (\alpha_1, \alpha_2) \mapsto r_0 + r_1 \alpha_1 + r_2 \alpha_2,\
r_0, r_1, r_2 \in \mathbb{R},$$ an affine function, define $$\begin{aligned}
P_1 & := & P \cap \{ (\alpha_1, \alpha_2) : F(\alpha_1, \alpha_2) < 0\} \\
P_2 & := & P \cap \{ (\alpha_1, \alpha_2) : F(\alpha_1, \alpha_2) > 0\}. \end{aligned}$$ If $P_1$ contains asymptotically $(m_p)$-non-special systems of dimension $-1$ and $P_2$ contains asymptotically $(m_1, \ldots, m_{p-1})$-non-special systems of dimension $\geq 0$, then $P$ contains asymptotically $(m_1, \ldots , m_p)$-non-special systems of dimension $\geq 0$.
By assumption, the set $n \cdot P_1 \cap \mathbb{N}^2_{\geq 0}$ contains an $m_p^{(n)}$-non-special system of dimension $-1$, and $n \cdot P_2 \cap \mathbb{N}^2_{\geq 0}$ contains an $(m_1^{(n)}, \ldots, m_{p-1}^{(n)})$-non-special system of dimension $\geq 0$, such that the $m_i^{(n)}$ satisfy the inequalities $(ii)$ of Def. \[asymp-sys-Def\] for a given $\delta$, for all $n \gg 0$.
Consequently, Dumnicki’s reduction algorithm applied to $n \cdot P \cap \mathbb{N}^2_{\geq 0}$ and $$F_n: \mathbb{R}^2_{\geq 0} \ni (\alpha_1, \alpha_2) \mapsto nr_0 + r_1 \alpha_1 + r_2 \alpha_2,$$ shows that $n \cdot P \cap \mathbb{N}^2_{\geq 0}$ contains an $(m_1^{(n)}, \ldots, m_p^{(n)})$-non-special system of dimension $\geq 0$. Since the $m_i^{(n)}$ still satisfy the inequalities $(ii)$ of Def. \[asymp-sys-Def\], for given $\delta$, the algorithm is justified.
The facts shown in the last section can be used to prove a criterion for asymptotic $(m)$-non-specialty:
\[as-m-non-spec-crit\] Let $P$ be a convex open subset of $\mathbb{R}^2_{\geq 0}$, such that its closure $\overline{P}$ is compact. Set $[a,b] := p_x(\overline{P})$ where $p_x: \mathbb{R}^2_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ is the projection of $\mathbb{R}^2_{\geq 0}$ onto the positive $x$-axis. Define $$f: [a,b] \rightarrow \mathbb{R}_{\leq 0}, t \mapsto f(t) := \mathrm{length\ of\ } p_x^{-1}(t) \cap \overline{P},$$ and let $f^\#: (0, b-a] \rightarrow \mathbb{R}_{\leq 0}$ be the monotone reordering of $f$.
If $m < b-a$ and $f^\#(t) \geq t$, for all $t \in (0,m]$, then $P$ contains asymptotically $(m)$-non-special systems of dimension $-1$.
The following diagram illustrates how we obtain the height function$f$ from a convex polygon $P \subset \mathbb{R}^2_{\geq 0}$:
(180,80)(-5,-5) (-5,0)[(1,0)[85]{}]{} (0,-5)[(0,1)[65]{}]{}
(0,30)[(2,1)[40]{}]{} (40,50)[(1,-1)[20]{}]{} (60,30)[(-1,-2)[10]{}]{} (50,10)[(-1,0)[30]{}]{} (20,10)[(-1,1)[20]{}]{}
(30,25)[P]{}
(20,-1)[(0,1)[2]{}]{} (40,-1)[(0,1)[2]{}]{} (50,-1)[(0,1)[2]{}]{} (60,-1)[(0,1)[2]{}]{}
(8,-8)[$t_1$]{} (28,-8)[$t_2$]{} (43,-8)[$t_3$]{} (53,-8)[$t_4$]{}
(90,30)[(1,0)[10]{}]{}
(110,0)[(1,0)[85]{}]{} (115,-5)[(0,1)[65]{}]{}
(115,0)[(2,3)[20]{}]{} (135,30)[(2,1)[20]{}]{} (155,40)[(1,-1)[10]{}]{} (165,30)[(1,-3)[10]{}]{}
(135,40)[f]{}
(135,-1)[(0,1)[2]{}]{} (155,-1)[(0,1)[2]{}]{} (165,-1)[(0,1)[2]{}]{} (175,-1)[(0,1)[2]{}]{}
(133,-8)[$t_1$]{} (153,-8)[$t_2$]{} (163,-8)[$t_3$]{} (173,-8)[$t_4$]{}
For the proof of the theorem we need some further properties of the function $f$:
\[conc-cont-prop\] Let $P$ be a convex open subset of $\mathbb{R}^2_{\geq 0}$, $\overline{P}$ compact. Define $f: [a,b] \rightarrow \mathbb{R}_{\leq 0}$ as in Thm. \[as-m-non-spec-crit\]. Then $f$ is concave and continuous on $[a,b]$.
Let $f^+$ resp. $f^-$ be the functions assigning to each $t \in [a,b]$ the upper resp. lower bound of the interval $p_x^{-1}(t) \cap P$. Then $f(t) = f^+(t) - f^-(t)$.
$f^+$ is concave: Choose $t_1, t_2 \in [a,b]$. Then the convexity of $\overline{P}$ implies $$\left( \lambda t_1 + (1-\lambda) t_2, \lambda f^+(t_1) + (1-\lambda)f^+(t_2) \right) \in \overline{P},$$ hence $$\lambda f^+(t_1) + (1-\lambda)f^+(t_2) \leq f^+(\lambda t_1 + (1-\lambda) t_2).$$ Similarly $f^-$ is convex, hence $f$ as the difference of a concave and a convex function is concave.
The following lemma shows that $f$ is also continuous on $(a,b)$ and that $\lim_{t \rightarrow a} f^\pm(t)$ and $\lim_{t \rightarrow b} f^\pm(t)$ exist.
We still have to prove that $\lim_{t \rightarrow a} f^\pm(t) = f^\pm(a)$ resp. $\lim_{t \rightarrow b} f^\pm(t) = f^\pm(b)$: Closedness implies $(a, \lim_{t \rightarrow a} f^+(t)) \in \overline{P}$, hence $\lim_{t \rightarrow a} f^+(t) \leq f^+(a)$. If $\lim_{t \rightarrow a} f^+(t) < f^+(a)$ then $$\lambda f^+(a) + (1-\lambda) f^+(t) > (1-\lambda) f^+(t)$$ for $\lambda$ sufficiently close to $1$, contradicting the concavity of $f^+$. The same types of arguments hold for the other limits.
Let $f: [a,b] \rightarrow \mathbb{R}$ be a concave function. Then $f$ is continuous on $(a,b)$, and the limits $\lim_{t \rightarrow a} f(t)$ and $\lim_{t \rightarrow b} f(t)$ exist.
For any triple $a \leq t^\prime < t_0 < t^{\prime\prime} \leq b$, we have $t_0 = \frac{t^{\prime\prime} - t_0}{t^{\prime\prime} - t^\prime} t^\prime +
\frac{t_0 - t^\prime}{t^{\prime\prime} - t^\prime} t^{\prime\prime}$. The concavity of $f$ implies $$f(t_0) \geq \frac{t^{\prime\prime} - t_0}{t^{\prime\prime} - t^\prime} f(t^\prime) +
\frac{t_0 - t^\prime}{t^{\prime\prime} - t^\prime} f(t^{\prime\prime}).$$ Subtracting $f(t^\prime)$ from both sides leads to the left hand inequality of $$\frac{f(t_0)-f(t^\prime)}{t_0-t^\prime} \geq \frac{f(t^{\prime\prime})-f(t^\prime)}{t^{\prime\prime}-t^\prime} \geq
\frac{f(t^{\prime\prime})-f(t_0)}{t^{\prime\prime}-t_0},$$ subtracting $f(t^{\prime\prime})$ and multiplying with $-1$ to the right hand inequality. Renaming $t_0,t^{\prime}, t^{\prime\prime}$ the left hand inequality implies that the difference quotients $\frac{f(t^{\prime\prime})-f(t_0)}{t^{\prime\prime}-t_0}$ are increasing for $t^{\prime\prime} \stackrel{>}{\rightarrow} t_0$, so they are bounded from below for $t^{\prime\prime}$ close to $t_0$. Since $t_0 \in (a,b)$ there is always a $t^\prime < t_0$ in $(a,b)$, hence the inequality chain shows that the difference quotients are also bounded from above. Consequently, $$| f(t^{\prime\prime})-f(t_0) | < M \cdot | t^{\prime\prime}-t_0 |$$ for appropriate $M > 0$ and $t^{\prime\prime} > t_0$ close to $t_0$. The same can be shown for $t^\prime < t_0$ close to $t_0$, hence $f$ is continuous in $t_0$.
The situation is more complicated for the boundary points $a$ and $b$. If $\lim_{t \stackrel{<}{\rightarrow} b} f(t)$ does not exist there are $3$ possibilities:
- There are at least $2$ accumulation points, for different sequences $(t_n^\pm) \stackrel{<}{\rightarrow} b$, say $-\infty \leq y^- < y^+ \leq +\infty$. Then it is possible to choose a triple $t_0 < t_n^- < t_m^+$, $n,m \gg 0$, which contradicts the concavity of $f$ (see the end of the proof in the proposition before).
- $\lim_{t \stackrel{<}{\rightarrow} b} f(t) = -\infty$: Then it is possible to choose a triple $t_0 < t < b$ contradicting the concavity of $f$ as before.
- $\lim_{t \stackrel{<}{\rightarrow} b} f(t) = +\infty$: Then it is possible to choose $t_0 < t^\prime < t^{\prime\prime} < b$, such that the point $(t^{\prime\prime}, f(t^{\prime\prime}))$ lies over the line connecting $(t_0, f(t_0))$ and $(t^\prime, f(t^\prime))$. But then the point $(t^\prime, f(t^\prime))$ lies under the line segment connecting $(t_0, f(t_0))$ and $(t^{\prime\prime}, f(t^{\prime\prime}))$. This contradicts the concavity of $f$.
(200,180)(-20,-20) (0,20)(140,20)(140,160) (0,12)[(2,1)[140]{}]{} (0,17)[(4,1)[140]{}]{}
(-10,0)[(1,0)[160]{}]{}
(15,-2)[(0,1)[4]{}]{} (80,-2)[(0,1)[4]{}]{} (118,-2)[(0,1)[4]{}]{} (142,-2)[(0,1)[4]{}]{}
(12,-10)[$t_0$]{} (77,-10)[$t^\prime$]{} (115,-10)[$t^{\prime\prime}$]{} (139,-10)[$b$]{}
The same type of arguments hold for $\lim_{t \stackrel{>}{\rightarrow} a} f(t)$.
*Proof of Thm. \[as-m-non-spec-crit\].* We start with a construction: Take any number $n \in \mathbb{N}$. Set $$S_{(m_x,m_y)} := \left[ \frac{m_x}{n} - \frac{1}{2n}, \frac{m_x}{n} + \frac{1}{2n} \right] \times
\left[ \frac{m_y}{n} - \frac{1}{2n}, \frac{m_x}{n} + \frac{1}{2n} \right],
(m_x, m_y) \in \mathbb{N}^2.$$ The $S_{(m_x,m_y)}$ are closed squares of sidelength $\frac{1}{n}$ such that the coordinates of their centers are multiples of $\frac{1}{n}$. Then set $P_n := \bigcup_{S_{(m_x,m_y)} \subset \overline{P}} S_{(m_x,m_y)}$, the union of all such squares in $\overline{P}$.
Since $P_n \subset \overline{P}$ we have $p_x(P_n) \subset [a,b]$. Hence we can define the analog of $f$ for $P_n$: $$f_n: [a,b] \rightarrow \mathbb{R}_{\geq 0},\ t \mapsto \mathrm{length\ of\ } p_x^{-1}(t) \cap P_n.$$ By construction, $f_n \leq f$. Furthermore the following holds:
*Claim 1.* For all $a < a_0 < b_0 < b$, the functions $f_n$ converge uniformly against $f$ on $[a_0,b_0]$, for $n \rightarrow \infty$.
Let $f^+$ and $f^-$ be defined as in the proof of Prop. \[conc-cont-prop\]. This proof shows that $f^+$ and $f^-$ are continuous functions, hence they are uniformly continuous on the compact interval $[a,b]$. Consequently, for every $\epsilon > 0$ there exists a $\delta > 0$ such that $$|x-y| < \delta \Rightarrow |f^\pm(x) - f^\pm(y)| < \epsilon\ \mathrm{for\ all\ } x,y \in [a,b].$$ Next, choose $a < a^\prime < a_0 < b_0 < b^\prime < b$. Since $P$ is a convex set and the projection $p_x$ is an open and continuous map, $p_x(P) = (a,b)$, and $p_x^{-1}(t) \cap P$ is a non-empty open interval, for all $t \in (a,b)$. Consequently, $f > 0$ on $(a,b)$, and $f$ achieves a strictly positive minimum on $[a^\prime, b^\prime]$.
Let $\epsilon > 0$ be any real number such that $4\epsilon$ is smaller than this minimum. Let $n \in \mathbb{N}$ be an integer such that $\frac{1}{n} < \epsilon$, $\frac{1}{n} < \delta$, $\frac{1}{n} < \min \{a_0-a^\prime, b^\prime-b_0, b_0-a_0\}$. Let $\frac{k}{n} \in [a_0,b_0]$. By assumption, $$f^+(k/n) - f^-(k/n) > 4\epsilon > \epsilon + \frac{1}{2n} + \frac{1}{n} + \frac{1}{2n} + \epsilon.$$ Hence the interval $[f^-(k/n) + \epsilon + \frac{1}{2n}, f^+(k/n) - \epsilon - \frac{1}{2n}]$ has length $> \frac{1}{n}$, consequently it contains at least one number of the form $\frac{m}{n}$.
*Claim 1.1.* $[\frac{k}{n}-\frac{1}{2n}, \frac{k}{n}+\frac{1}{2n}] \times [\frac{m}{n}-\frac{1}{2n}, \frac{m}{n}+\frac{1}{2n}]
\subset \overline{P}$.
Let $t \in [\frac{k}{n}-\frac{1}{2n}, \frac{k}{n}+\frac{1}{2n}] \subset [a^\prime, b^\prime]$. Then $|t-\frac{k}{n}|<\delta$, hence $|f^\pm(t)-f^\pm(k/n)|<\epsilon$. This implies $f^+(t) > f^+(k/n)-\epsilon \geq \frac{m}{n}+\frac{1}{2n}$ and $\frac{m}{n}-\frac{1}{2n} \geq f^-(k/n)+\epsilon > f^-(t)$.
If $\frac{L}{n}$ is maximal among all $\frac{m}{n} \in [f^-(k/n) + \epsilon + \frac{1}{2n}, f^+(k/n) - \epsilon - \frac{1}{2n}]$, then $\frac{L}{n} \geq f^+(k/n) - \epsilon - \frac{1}{2n} - \frac{1}{n}$, and if $\frac{l}{n}$ is minimal, then $\frac{l}{n} \leq f^-(k/n) + \epsilon + \frac{1}{2n} + \frac{1}{n}$. Consequently $$f^+(t) - (\frac{L}{n} + \frac{1}{2n}) < f^+(t) - (f^+(k/n) - \epsilon - \frac{1}{n}) \leq
|f^+(t) - f^+(k/n)| + \epsilon + \frac{1}{n} \leq 2\epsilon + \frac{1}{n}.$$ Similarly we deduce $\frac{l}{n} - \frac{1}{2n} - f^-(t) < 2\epsilon + \frac{1}{n}$.
Now, $f(t) = f^+(t) - f^-(t) \geq f_n(t) > \frac{L}{n} - \frac{l}{n}$ for $t \in [\frac{k}{n}-\frac{1}{2n}, \frac{k}{n}+\frac{1}{2n}]$, hence $$f(t) - f_n(t) \leq f^+(t) - \frac{L}{n} + \frac{l}{n} - f^-(t) < 4\epsilon + \frac{3}{n} < 7\epsilon.$$ Claim 1 is proven.
*Claim 2.* For every $\epsilon > 0$ and $n \gg 0$, $$\parallel\!\! (f^\#)_{|(0,m]} - (f_n^\#)_{|(0,m]} \!\!\parallel_{\max} < \epsilon.$$
Given $\epsilon > 0$, there is a $\delta > 0$ such that $$\parallel\!\! (f^\#)_{|(0,b_0-a_0]} - (f_{|[a_0,b_0]})^\# \!\!\parallel_{\max} < \frac{\epsilon}{3}$$ as long as $(a_0-a)+(b-b_0) < \delta$, by means of Thm. \[cutoff-thm\]. Furthermore, there is an $n \gg 0$ such that $\parallel\!\! f - f_n \!\!\parallel_{\max} < \frac{\epsilon}{6}$, by Claim 1, hence also $\parallel\!\! f_{|[a_0,b_0]} - f_{n|[a_0,b_0]} \!\!\parallel_{\max} < \frac{\epsilon}{6}$. Then Prop. \[max-norm-bd-prop\] implies $$\parallel\!\! (f_{|[a_0,b_0]})^\# - (f_{n|[a_0,b_0]})^\# \!\!\parallel_{\max} < \frac{\epsilon}{3}.$$ Next we choose $\delta$ small enough to ensure $\parallel\!\! (f_{n|[a_0,b_0]})^\# - (f_n^\#)_{|(0,b_0-a_0]} \!\!\parallel_{\max} < \frac{\epsilon}{3}$; this is possible again by Thm. \[cutoff-thm\].
Finally we choose $a_0, b_0$ such that in addition to $(a_0-a)+(b-b_0) < \delta$, we also have $m < b_0-a_0 < b-a$. By expanding $(f^\#)_{|(0,m]} - (f_n^\#)_{|(0,m]}$ to $$(f^\#)_{|(0,m]} - (f_{|[a_0,b_0]})^\#_{|(0,m]} + (f_{|[a_0,b_0]})^\#_{|(0,m]} - (f_{n|[a_0,b_0]})^\#_{|(0,m]} +
(f_{n|[a_0,b_0]})^\#_{|(0,m]} -(f_n^\#)_{|(0,m]}$$ we get the claim.
For all $\epsilon > 0$, Claim 2 implies $f_n^\#(t) > t-\epsilon$ for all $t \in [0,m]$ if $n \gg 0$.
Now for $\epsilon$ small enough, consider all integers $e$ with $0 \leq e \leq \lfloor mn \rfloor - \lceil n\epsilon \rceil - 1$, and set $e^\prime := \lfloor mn \rfloor - \lceil n\epsilon \rceil - e - 1$. Then $e + 1 + \lceil n\epsilon \rceil \leq \lfloor mn \rfloor$, hence $\frac{e + 1 + \lceil n\epsilon \rceil}{n} \leq m$, and we can apply $f_n^\#$ on $\frac{e + 1 + \lceil n\epsilon \rceil}{n}$: $$f_n^\#(\frac{e + 1 + \lceil n\epsilon \rceil}{n}) = f_n^\#(\frac{\lfloor mn \rfloor - e^\prime}{n}) >
\frac{\lfloor mn \rfloor - e^\prime}{n} - \epsilon \geq
\frac{\lfloor mn \rfloor - e^\prime - \lceil n\epsilon \rceil}{n} = \frac{e+1}{n},$$ for $n \gg 0$. Consequently for each such e the step function $f_n^\#$ has a step of height at least $\frac{e+1}{n}$, and these steps can be chosen pairwise distinct.
On the other hand, $f_n^\#$ is just a reordering of the steps in $f_n$, so $f_n$ has the same property. By construction the height of the step of $f_n$ over $\frac{k}{n}$ counts the number of points $(\frac{k}{n}, \frac{l}{n})$ inside $P$. But this means that $n \cdot P \cap \mathbb{N}^2_{\geq 0}$ contains a non-special linear system $\mathcal{L}_{D_n}(\lfloor mn \rfloor - \lceil n\epsilon \rceil)$ of dimension $-1$, by Dumnicki’s non-specialty criterion. Since $$\left| \frac{\lfloor mn \rfloor - \lceil n\epsilon \rceil - nm}{nm} \right| \leq
\left| \frac{\lfloor mn \rfloor - nm}{nm} \right| + \frac{\lceil n\epsilon \rceil}{nm} \leq
\frac{1}{mn} + \frac{\epsilon}{m} + \frac{1}{nm} \rightarrow 0$$ for $\epsilon \rightarrow 0$, $n \rightarrow \infty$, the theorem is proven. $\Box$
A lower bound for the Seshadri constant of 10 points in $\mathbb{CP}^2$ {#LowerBound-sec}
=======================================================================
The following proposition allows to prove the nefness criterion Prop. \[nef-crit\]:
\[inc-non-spc-prop\] Let $m_1, \ldots, m_r \in \mathbb{N}_{>0}$, let $D_1 \subset D_2 \subset \mathbb{N}^2$ be finite subsets, and let $p_1, \ldots, p_r \in \mathbb{P}^2$ be points. If $\mathcal{L}_{D_1}(m_1p_1, \ldots, m_rp_r)$ has expected dimension $\geq 0$, then also $\mathcal{L}_{D_2}(m_1p_1, \ldots, m_rp_r)$.
The subspace $\mathcal{L}_{D_1}(m_1p_1, \ldots, m_rp_r) \subset \mathcal{L}_{D_2}(m_1p_1, \ldots, m_rp_r)$ is described by the intersection of $\mathcal{L}_{D_2}(m_1p_1, \ldots, m_rp_r) \subset \mathbb{P}(\sum_{i+j \leq d} a_{ij}x^iy^j)$ with the linear subspace $$\{ a_{ij} = 0: (i,j) \in D_2 \setminus D_1 \}.$$ Here, $d$ is chosen such that $D_1 \subset D_2 \subset \{ (i,j) : i+j \leq d \}$. Consequently, $$\dim \mathcal{L}_{D_1}(m_1p_1, \ldots, m_rp_r) \geq \mathcal{L}_{D_2}(m_1p_1, \ldots, m_rp_r) -
\#(D_2 \setminus D_1).$$ In particular, $\mathcal{L}_{D_1}(m_1p_1, \ldots, m_rp_r)$ cannot have expected dimension if $\mathcal{L}_{D_2}(m_1p_1, \ldots, m_rp_r)$ has not.
*Proof of Prop. \[nef-crit\].* The assumption on $P$ implies that there exists a sequence $\delta_n \stackrel{>}{\rightarrow}0$, natural numbers $d_n, m_1^{(n)}, \ldots, m_r^{(n)}$ for all $n \in \mathbb{N}$, $d_n \rightarrow \infty$ for $n \rightarrow \infty$, and subsets $D_n \subset d_n \cdot P \cap \mathbb{N}^2_{\geq 0}$ such that $\mathcal{L}_{D_n}(m_1^{(n)}, \ldots, m_r^{(n)})$ is non-special of dimension $\geq 0$ and $$\left| \frac{m_i^{(n)}}{d_n \cdot m} - 1 \right| = \left| \frac{m_i^{(n)}- d_n m}{d_n m} \right| < \delta_n,\ \
i = 1, \ldots, r.$$ Let $m_n := \min \{ m_1^{(n)}, \ldots, m_r^{(n)} \}$. Then Prop. \[inc-non-spc-prop\] implies that $\mathcal{L}_{d_n}(m_n^r)$ is non-special of dimension $\geq 0$.
Since $\frac{d_n}{m_n-1}$ and $\frac{d_n}{m_n}$ have the same limit, and for some $i_n \in \{ 1, \ldots, r\}$, $$\frac{d_n}{m_n} = \frac{d_n}{m_{i_n}^{(n)}} = \frac{d_n m}{m_{i_n}^{(n)}} \cdot \frac{1}{m} \rightarrow \frac{1}{m},$$ the proposition follows. $\Box$
*Proof of Thm. \[Seshadri-bd-thm\].* In the following diagram, let the points $O,A,B, \ldots , R, S$ be given by the coordinates $$\begin{array}{lllll}
O = (0,0), & A = (1,0), & C = (\frac{9}{13}, 0), & E = (\frac{9}{13}, \frac{4}{13}), & G = (\frac{5}{13}, 0), \\
& & & & \\
& B = (0,1), & D = (0, \frac{9}{13}), & F = (\frac{4}{13}, \frac{9}{13}), & H = (0, \frac{5}{13}), \\
& & & & \\
& I = (\frac{4}{13}, 0), & K = (\frac{7}{13}, \frac{6}{13}), & M = (\frac{6}{13}, \frac{3}{13}), & \\
& & & & \\
& J = (0, \frac{4}{13}), & L = (\frac{6}{13}, \frac{7}{13}), & N = (\frac{3}{13}, \frac{6}{13}), & \\
& & & & \\
& P = (\frac{9}{26}, \frac{9}{26}), & Q = (\frac{2}{13}, \frac{2}{13}), &
R = (\frac{7}{13}, \frac{2}{13}), & S = (\frac{9}{26}, 0).
\end{array}$$
(280,280)(-10,-10) (0,0)[(1,0)[270]{}]{} (0,0)[(0,1)[270]{}]{} (0,260)[(1,-1)[260]{}]{}
(0,80)[(1,-1)[80]{}]{} (180,0)[(0,1)[80]{}]{} (0,180)[(1,0)[80]{}]{} (100,0)[(1,1)[80]{}]{} (0,100)[(1,1)[80]{}]{} (100,0)[(1,3)[40]{}]{} (0,100)[(3,1)[120]{}]{} (60,120)[(1,-1)[60]{}]{} (40,40)[(1,1)[50]{}]{} (140,120)(140,80)(140,40) (90,90)(90,45)(90,0)
(-2,-8)[O]{} (258,-8)[A]{} (178,-8)[C]{} (98,-8)[G]{} (88,-8)[S]{} (78,-8)[I]{} (-8,256)[B]{} (-8,176)[D]{} (-8,96)[H]{} (-8,76)[J]{} (178,84)[E]{} (118,144)[L]{} (138,124)[K]{} (78,184)[F]{} (66,116)[N]{} (92,92)[P]{} (114,66)[M]{} (32,36)[Q]{} (132,40)[R]{}
(20,20)[$P_{1}$]{} (198,20)[$P_{2}$]{} (20,208)[$P_{3}$]{} (148,20)[$P_{4}$]{} (20,148)[$P_{5}$]{} (148,78)[$P_{6}$]{} (80,138)[$P_{7}$]{} (110,110)[$P_{8}$]{} (95,40)[$P_{9}$]{} (40,78)[$P_{10}$]{}
The diagram is possible since $E, K, L, F$ lie on the line $AB$, the points $I, G, C$ on the line $OA$, the points $J, H, D$ on the line $OB$, the point $M$ on the line $GK$, the point $N$ on the line $HL$, the point $P$ on the line $NM$ and $Q$ on the line $IJ$. Furthermore, $S$ lies between $I$ and $G$ on $OA$, and $R$ lies on the line $GE$.
The diagram shows the dissection of the $2$-dimensional simplex $OAB$ into 10 polygons $P_1, \ldots, P_{10}$ by straight lines. The indices of the polygons denote the sequence of dissections. To prove the theorem we apply the asymptotic version of Dumnicki’s reduction algorithm to this sequence of dissections. That is, we have to show that for every $m < \frac{4}{13}$ each of the polygons $P_i$, $i = 1, \ldots, 9$, contains asymptotically $(m)$-non-special systems of dimension $-1$, and that $P_{10}$ contains asymptotically $(m)$-non-special systems of dimension $\geq 0$.
By construction the polygons are convex. Hence Thm. \[as-m-non-spec-crit\] together with Prop. \[mon-re-id-prop\] and Prop. \[conc-cont-prop\] imply that it is enough to show the following, for every polygon $P_i$: The projection of $P_i$ onto the $x$-axis is an interval of length $\geq \frac{4}{13} > m$, and there is a vertical section of $P_i$ of length $\geq \frac{4}{13} > m$. By symmetry, it is also possible to show these inequalities for the projection onto the $y$-axis and a horizontal section. Furthermore, since the lengths are $> m$, it will be always possible to add some monomials to $D_{10}^{(n)}$ in $P_{10}^{(n)}$. Prop. \[inc-non-spc-prop\] shows that this produces $m_{10}^{(n)}$-non-special systems of dimension $\geq 0$ in $P_{10}^{(n)}$, for $n \gg 0$.
For the polygons $P_1, \ldots, P_5$ the inequalities are obvious. For the polygon $P_6$ the projection to the $x$-axis is the interval $GC$ which has length $\frac{4}{13}$. The vertical section $KR$ has also length $\frac{4}{13}$. By symmetry, $P_7$ also satisfies the inequalities.
The projection of $P_8$ onto the $x$-axis is the interval $[\frac{3}{13}, \frac{7}{13}]$ (these are the $x$-coordinates of $N$ and $M$), and the vertical section $LM$ has length $\frac{4}{13}$. The projection of $P_9$ onto the $x$-axis is $[\frac{2}{13}, \frac{6}{13}]$, and the vertical section $PS$ has length $\frac{4}{13}$. By symmetry, $P_{10}$ also satisfies the necessary inequalities. $\Box$
Since the bound $\frac{13}{4}$ is rational there might be a pair $(d,m)$ with $\mathcal{L}_d(10^{m+1})$ non-special of non-negative dimension and $\frac{d}{m} = \frac{13}{4}$. From such a pair the theorem would follow by Thm. \[Eckl-thm\] without any limit process. But it is difficult to find such a pair: $\mathcal{L}_d(10^{m+1})$ has expected dimension $-1$ up to $m = 92$, and then it is still not clear how to prove non-specialty. For example, the cutting proposed in the proof of the theorem above might require an even bigger $m$.
[Dum06]{}
Paul Biran. Constructing new ample divisors out of old ones. , 98(1):113–135, 1999.
C. Ciliberto and R. Miranda. . In [*[Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001)]{}*]{}, volume 36 of [*NATO Sci. Ser. II Math. Phys. Chem.*]{}, pages [37–51]{}. Kluwer Acad. Publ., Dordrecht, 2001.
M. Dumnicki and W. Jarnicki. . arXiv:math/0505183, 2005.
M. Dumnicki. . arXiv:math/0606716, 2006.
Thomas Eckl. Seshadri constants via lelong numbers. preprint math.AG/0508561, to be published in Math. Nachr., August 2005.
Brian Harbourne. Seshadri constants and very ample divisors on algebraic surfaces. , 559:115–122, 2003.
B. Harbourne and J. Ro[é]{}. . preprint, arXiv:math/0309064v3, 2003.
Brian Harbourne and Joaquim Ro[é]{}. Linear systems with multiple base points in [$\Bbb P\sp 2$]{}. , 4(1):41–59, 2004.
B. Harbourne and J. Ro[é]{}. Multipoint [S]{}eshadri constants on [$\Bbb P\sp 2$]{}. , 63(1):99–102, 2005.
R. Miranda. . , 46:192–201, 1999.
M. Nagata. . , 81:766–772, 1959.
F. Schüller. Ein neuer ansatz zur harbourne-hirschowitz-vermutung. diploma thesis, Universität zu Köln, 2007. http://www.mi.uni-koeln.de/\~kebekus/teaching/diplomarbeiten.html.
Tomasz Szemberg and Halszka Tutaj-Gasi[ń]{}ska. General blow-ups of the projective plane. , 130(9):2515–2524 (electronic), 2002.
Halszka Tutaj-Gasi[ń]{}ska. A bound for [S]{}eshadri constants on [${\Bbb P}\sp 2$]{}. , 257:108–116, 2003.
Geng Xu. Curves in [${\bf P}\sp 2$]{} and symplectic packings. , 299(4):609–613, 1994.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study bosonic atoms in optical honeycomb lattices with anisotropic tunneling and find dimerized Mott insulator phases with fractional filling. These incompressible insulating phases are characterized by an interaction-driven localization of particles in respect to the individual dimers and large local particle-number fluctuations within the dimers. We calculate the ground-state phase diagrams and the excitation spectra using an accurate cluster mean-field method. The cluster treatment enables us to probe the fundamental excitations of the dimerized Mott insulator where the excitation gap is dominated by the intra-dimer tunneling amplitude. This allows the distinction from normal Mott insulating phases gapped by the on-site interaction. In addition, we present analytical results for the phase diagram derived by a higher-order strong-coupling perturbative expansion approach. By computing finite lattices with large diameters the influence of a harmonic confinement is discussed in detail. It is shown that a large fraction of atoms forms the dimerized Mott insulator under experimental conditions. The necessary anisotropic tunneling can be realized either by periodic driving of the optical lattice or by engineering directly a dimerized lattice potential. The dimers can be mapped to to their antisymmetric states creating a lattice with coupled $p$-orbitals.'
author:
- 'Ole Jürgensen,$^1$ and Dirk-Sören Lühmann,$^1$'
title: Dimerized Mott insulators in hexagonal optical lattices
---
In recent years, experiments with ultracold atoms in optical lattices have attracted a lot of attention due to their unique possibilities to simulate condensed matter systems in a highly controllable environment. The pioneering experiments in this field have been realized in cubic lattices demonstrating the observability of the transition between the superfluid and the Mott insulating phase [@Jaksch1998; @Greiner2002]. Lately, more sophisticated setups allow experiments with a variety of non-cubic optical lattice geometries including superlattices [@Guidoni1997; @Peil2003; @Santos2004; @Anderlini2007; @Trotzky2008], checkerboard [@Sebby-Strabley2006; @Wirth2011], Kagomé [@Jo2012], and honeycomb lattices [@Becker2010; @SoltanPanahi2011; @SoltanPanahi2012; @Tarruell2012; @Greif2013; @Uehlinger2013; @Luhmann2014]. The latter is of particular interest due to its analogy to graphene [@Zhu2007; @Wu2008; @Chen2011; @Zhang2012; @Polini2013]. Furthermore, the periodic modulation of the phase of the lattice beams offers an additional tool to engineer the tunneling matrix elements [@Eckardt2005; @Lignier2007; @Struck2011; @Struck2012; @Hauke2012].
![ (a) A honeycomb lattice with anisotropic tunneling matrix elements $J_1$ and $J_2$. The sites within a dimer are coupled by the tunneling matrix element $J_1$, while the coupling between neighboring dimers is associated with the smaller tunneling $J_2$. For $J_2\ll J_1$ a quasi-quadratic lattice structure of dimers is formed. (b) In the Mott insulator phase with integer filling $\rho=1$ each atom is localized at a lattice site. (c) In the *dimerized* Mott insulator phase with $\rho=1/2$, $3/2$, $...$ the atoms are still delocalized within individual dimers, while the superfluid order parameter of the lattice vanishes. (d) By lattice shaking the tunneling matrix elements along the dimers can be tuned to negative values, where each dimer wave function resembles a $p$-orbital state. In the superfluid state a stripe order is formed.[]{data-label="fig:lattice"}](Figure1.pdf "fig:"){width="\linewidth"}\
Superlattices with different site offsets and barrier heights can be created by superposing lattices with different wavelengths [@Roth2003; @Peil2003; @Rabl2003; @Santos2004; @Buonsante2004a; @Sebby-Strabley2006; @Rousseau2006; @Anderlini2007; @Trotzky2008; @Tarruell2012; @Uehlinger2013; @Greif2013] or by employing polarization-dependent light potentials [@Becker2010; @SoltanPanahi2011; @SoltanPanahi2012; @Luhmann2014]. The unit cells of these lattices consist of multiple sites, adding a new degree of complexity to the system. If the sites within the unit cell have different energy offsets, normal Mott insulators with a population imbalance emerge [@Roth2003; @Rabl2003; @Buonsante2004a; @Chen2010; @Luhmann2014], where the atom are localized on individual sites. For the case without site-offsets, anisotropic tunneling couplings lead to non-trivial insulator phases with fractional filling in between the conventional Mott insulator phases [@Buonsante2004b; @Buonsante2005a; @Buonsante2005b; @Danshita2007; @Muth2008]. Here, the particles localize on individual unit cells with a vanishing superfluid order parameter but are still delocalized within each unit cell. Originally, respective experiments have been proposed for one-dimensional superlattices, but so far, these phases with fractional filling have not been observed. For this the reason is two-fold. First, the phase diagram suggests a very small fraction of atoms to occupy the dimerized phase in a confined system. Second, a clear signature for discriminating the fractionally filled phase from the conventional Mott insulator was missing.
In this work, we theoretically study bosonic atoms in honeycomb optical lattices with adjustable tunneling matrix elements. The latter is achieved either by a periodic driving of a honeycomb lattice or by engineering directly a dimerized potential [@Tarruell2012; @Greif2013; @Uehlinger2013]. In addition to normal Mott insulating phases with integer filling, we find insulating phases with half-integer filling where the particles are delocalized on dimers. The dimers are naturally defined by the biatomic unit cell of the honeycomb lattice, allowing for non-trivial fractional-filling phases [@Buonsante2004b; @Buonsante2005a; @Buonsante2005b; @Danshita2007; @Muth2008]. The phase diagram is studied by means of the strong-coupling expansion approach similar to [Ref. [@Buonsante2005a]]{} as well as by the cluster Gutzwiller mean-field method giving accurate results for honeycomb lattices [@Luhmann2013]. This method grants the great advantage of the access to the excitation spectrum. We show that the characteristic local excitations allow distinguishing experimentally between the conventional and this *dimerized Mott insulator* state. The excitation gap gives an estimate of the required temperatures for observing this quantum phase. Furthermore, we simulate two-dimensional lattice planes with harmonic confinement using realistic experimental parameters. We find that the dimerized state is formed by a large fraction of atoms ($> 70\%$) and thus is well observable in the proposed experiment.
Phase diagram
=============
There are two perspectives to realize the dimerized Mott insulator phase in experiments with hexagonal lattices. First, we propose to use an optical honeycomb lattice generated by three running laser beams as in Refs. [@Becker2010; @SoltanPanahi2011; @SoltanPanahi2012]. A honeycomb lattice features a two-atomic unit cell, where the intra-cell bond and inter-cell bonds have different orientations, which allows addressing them independently by lattice shaking techniques. By modulating the relative phases of the beams, the lattice is periodically accelerated on an elliptical orbit [@Eckardt2005; @Struck2011]. Employing Floquet theory, one can obtain an effective time-averaged Hamiltonian, where the tunneling matrix elements are modified by a Bessel function depending on the driving parameters [@Eckardt2005; @Lignier2007; @Struck2011; @Struck2012; @Hauke2012]. This allows engineering two different and tunable tunneling matrix elements $J_1$ and $J_2$ in the vertical and horizontal direction (see [Fig. \[fig:lattice\]]{}a). When $J_1$ is larger than $J_2$, the honeycomb lattice separates into a dimerized square lattice of double-wells coupled by the reduced matrix element $J_2$. Second, a dimerized lattice can also be obtained in the setup [@Tarruell2012; @Greif2013; @Uehlinger2013], where the dimerized honeycomb lattice sketched in [Fig. \[fig:lattice\]]{} is created due to the interference of two collinear phase-shifted laser beams. Here, an additional shaking is not required but the control of the tunneling parameters is not independent. In the first case it is, e.g., possible to achieve negative tunneling matrix elements.
![Phase diagram of the driven honeycomb lattice with $J_1 = 10 J_2$. For shallow lattices, i.e., large tunneling $J_1$ and $J_2$ the superfluid (SF) is favorable. For deep lattices the common Mott phases with integer fillings (MI) appear accompanied by dimerized Mott insulator phases with fractional fillings (DMI) in between.[]{data-label="fig:PD"}](Figure2.pdf){width="\linewidth"}
The phase diagram for this setup is shown in [Fig. \[fig:PD\]]{} for $J_1=10 J_2$ as a function of the chemical potential $\mu$ and the tunneling energy $J_1$ in units of the repulsive on-site interaction $U$. Both the Mott insulator phases (MI) and the dimerized Mott insulator phases (DMI) are characterized by a vanishing superfluid order parameter $\psi=0$. In the remaining regions of the phase diagram with $\psi\neq0$ the particles are in a superfluid state. In the Mott insulator phase, the particles are localized at individual lattice sites as depicted in [Fig. \[fig:lattice\]]{}b. In contrast, the dimerized Mott insulator phase with fractional filling $\rho=1/2$, $3/2$, $...$ is characterized by a delocalization of particles within the dimers ([Fig. \[fig:lattice\]]{}c). In the limit of fully separated dimers ($J_2\to 0$), the local ground state on a dimer for filling $\rho=1/2$ reads $\ket{s}=\frac{1}{\sqrt{2}}(\ket{1,0}+\ket{0,1})$, where $\ket{n_\text{L},n_\text{R}}$ denotes the occupation of left and right dimer sites. In the dimerized insulator phase the local particle-number fluctuations $(\Delta n)^2$ are large, whereas they are strongly suppressed in a conventional Mott insulator, which can be used to distinguish between the phases. Thus, the single-site resolution available in a quantum-gas microscope experiment would reveal a random distribution on individual lattice sites but a fixed integer occupation when summing up both dimer sites. When the tunneling matrix element $J_1$ is negative, which can be achieved with the lattice shaking technique, the antisymmetric state $\ket{a} = \frac{1}{\sqrt{2}} (\ket{1,0} - \ket{0,1})$ becomes the dimer ground state resembling a $p$-orbital. Hence, we can understand the setup as a square lattice of $p$-orbitals. In the superfluid phase, a negative tunneling matrix element leads to an alternating sign of the superfluid order parameter, which can be mapped onto a classical spin model and is therefore referred to as antiferromagnetic coupling. The phases align due to the positive dimer coupling $J_2 > 0$ such that the sign of the dimer wave function is the same along the respective bonds ([Fig. \[fig:lattice\]]{}d), which minimizes the tunneling energy. This alignment leads to a stripe order of the superfluid order parameter. In the Mott state where the superfluid order parameter vanishes, this alignment persists in the nearest-neighbor correlations. A suitable gauge transformation maps the symmetric (ferromagnetic) onto the antisymmetric ground state which is not frustrated. Therefore, we restrict ourselves to the symmetric case for $J_1>0$ in the following. We should stress however that the aforementioned equivalence means that the dimerized lattice resembles $p$-orbital-like physics even without antiferromagnetic driving of the lattice.
{width=".8\linewidth"}
The solid lines in the phase diagram in [Fig. \[fig:PD\]]{} are computed using a cluster mean-field approach with a cluster size of 12 sites, as described in Sec. \[sec:CMF\]. Due to the structure of the dimerized state, conventional mean-field theory is not able to capture the dimerized Mott insulator phases. As a second approach, we apply strong-coupling perturbative expansion to third order (dashed lines) which allows analytical results for the phase diagram. This approach is detailed below in Sec. \[sec:PT\], where we also give the explicit expression for the second-order perturbation (dotted lines) as a reasonable approximation. Both methods use the tight-binding Bose-Hubbard Hamiltonian $$\begin{split}\label{eq:H_full}
\hat H = \frac{U}{2}\sum_i \hat n_i (\hat n_i - 1)- J_1 \sum_{\langle i,j \rangle} \hat b_i^\dagger \hat b_{j} - J_2 \sum_{\langle\!\langle i,j \rangle\!\rangle} \hat b_i^\dagger \hat b_{j}. \end{split}$$ Here, the brackets $\langle i,j \rangle$ denote sites $i$ and $j$ on the same dimer connected via $J_1$, while sites $\langle\!\langle i,j \rangle\!\rangle$ are nearest-neighbors sites connected via $J_2$ on different dimers. The repulsive on-site interaction is denoted by $U$ with the particle number operator $\hat n_i=\hat b_i^\dagger \hat b_i$.
Excitation spectrum {#sec:spectrum}
===================
Due to the internal structure of the dimerized Mott insulator state, its fundamental excitations differ strongly from the normal Mott insulator phases. Therefore the excitation spectrum allows distinguishing between the two insulating phases. In the Mott insulator, the lowest excitation is the creation of a particle-hole pair resulting in an empty and a doubly occupied site. The corresponding excitation energy is the additional on-site interaction $U$. The excitation spectrum of the fractional insulator with $\rho=1/2$ is not $U$-gaped since each dimer offers an empty site, where the particle from a neighboring dimer can tunnel to. Even for dimerized Mott insulators with higher fillings the $U$-gap vanishes.
In [Fig. \[fig:spectrum\]]{}c the ground-state configuration of the dimerized Mott insulator is depicted for two unit cells, each being populated by the symmetric state $\ket{s}$. There are two different fundamental excitations both on the order of $2J_1$. First, the particle-hole excitation $E_{\mathrm{ph}}$, where one particle is excited by hopping to a neighboring dimer as depicted in [Fig. \[fig:spectrum\]]{}b. The excitation energy corresponds to the loss of delocalization energy $J_1$ within the empty dimer reduced by the interaction energy on the doubly occupied dimer. Second, a particle can be excited within the same dimer from the symmetric ground state $\ket{s}$ to the antisymmetric state $\ket{a}$ associated with the energy $E_\text{as}=2J_1$ (see [Fig. \[fig:spectrum\]]{}a).
Harmonic confinement {#sec:inhomog}
====================
{width="0.9\linewidth"}
The excitation spectrum as a function of $J_2/J_1$ is depicted in [Fig. \[fig:spectrum\]]{}d in units of the tunneling energy $J_1$ for $J_1=0.2U$. The black markers represent the numerical data calculated with the cluster mean-field approach using 16 sites within the insulating phase with $\psi=0$. The shaded areas indicate the results of first-order strong-coupling perturbation theory. The antisymmetric excitation $E_\text{as}$ is not broadened within first-order perturbation and is only affected by higher-order processes. In the limit $J_2 \rightarrow 0$, a particle-hole excitation for an insulating phase with $n$ particles per dimer has the energy $E_{\mathrm{ph}}=E_{n+1}+E_{n-1} - 2E_n$. Thus, the particle-hole energy for the fractional insulator with $n=1$ is $E_{\mathrm{ph}}= E_2-2 E_1\approx 2J_1-4J_1^2/U$. For finite $J_2$, first order perturbation (blue shaded area) leads to a delocalization of the particle and the hole. The minimum and maximum of the emerging band are given by $$E_{\mathrm{ph}}= E_2-2 E_1 \pm 4 J_2 (j_1^2+j_{2}^2).$$ where $j_n^2+j_{n+1}^2\approx 1$ for $U\gg J_1$ (see Sec. \[sec:PT\] for details).
The excitation gap $E_\mathrm{ph} \approx J_1$ gives an estimate of the required temperature for the dimerized phase $T \approx E_\mathrm{ph}/k_\mathrm{B}$, where $k_\mathrm{B}$ is the Boltzmann constant. Assuming the experimental parameters given in section \[sec:inhomog\] and $V=6.5\, E_\mathrm{rec}$ we obtain $T \approx 20\, \mathrm{nK}$. In general, this value increases with $J_1$ and decreases with the ratio $J_2/J_1$. Alternatively, a larger value of the scattering length $a_s/a$, where $a$ is the lattice constant, allows for a larger value of $J_1$, increasing the excitation gap.
The higher excitations in the energy spectrum are combinations of the two fundamental excitations described above, i.e. two particle-hole excitations $2E_{\mathrm{ph}}$, two asymmetric excitations $2E_\text{as}$, and a combination of both $E_{\mathrm{ph}}+E_\text{as}$. The numerical spectrum is distorted by higher-order tunneling processes and the interaction between individual excitations. Due to the finite size of the cluster the band width is reduced, which is in particular noticeable for two particle-hole excitations due to the limited possibilities to delocalize. The $U$-excitation at $E\approx 5J_1$ corresponding to a doubly occupied site lies in the continuous part of the spectrum for most parameters.
In an experiment, the lattice modulation technique proposed in [Ref. [@Konabe2006]]{} can be applied for the direct observation of the dimerized Mott insulator phase by addressing these fundamental excitations. The excitation gap on the order of $2 J_1$ is therefore the characteristic signature of the fractional insulator and could serve as prove for its experimental realization.
In this section, the question is addressed whether the dimerized Mott insulator in the honeycomb lattice can be observed in an experimental setup, where the optical lattice is superimposed by a harmonic confinement, leading to a spatially varying chemical potential $\mu(\mathbf{r})$. The relatively small extend of the dimerized Mott insulator phase in the phase diagram might suggest that a dominating part of the atoms are in the superfluid or in the Mott insulator with $\rho=1$ coexisting with the dimerized phase. The cluster mean-field approach (see Sec. \[sec:CMF\]) allows the simulation of a two-dimensional lattice of realistic size by iteratively moving the cluster through the lattice [@Pisarski2011; @Luhmann2013]. This introduces a site-dependent mean-field, where at every iteration the *local* order parameter is updated until the results converge. If the ratio $J_1/U$ is considerably smaller than the tip position of the dimerized insulator phase, the results are influenced only to a minor degree by the cluster size. Using six-site clusters, we can determine the extent of the phases in a lattice with harmonic confinement accurately. In [Fig. \[fig:inhomog\]]{} the results are shown for $J_1=0.1 U$, $J_2=0.1 J_1$ and a chemical potential $\mu_c$ in the center of the trap. As an example, for $^{87}\mathrm{Rb}$ with a scattering length of $a_s \approx 100\, a_0$, this corresponds to a lattice depth of $V=9.5\, E_\mathrm{rec}$ [@SoltanPanahi2011; @Luhmann2014], where $a_0$ is the Bohr radius, $E_\mathrm{rec}=\frac{\hbar^2 k^2}{2m}$ is the recoil energy, $k=\frac{2\pi}{\lambda}$ is the wave vector of the lattice beams with a wavelength of $\lambda=830\, \mathrm{nm}$ and $m$ is the atomic mass of $^{87}\mathrm{Rb}$. For each lattice site we apply the local density-approximation $\mu(\mathbf{r}_i)=\mu_\text{c}-V_\text{trap}(\mathbf{r}_i)$ with a harmonic trapping potential $V_\text{trap}$ and trap frequencies of $32\, \mathrm{Hz}$ (a-c) to $35\, \mathrm{Hz}$ (d-f).
For $\mu_\text{c}=0.02 U$, only the dimerized insulator persists ([Fig. \[fig:inhomog\]]{}a-c), whereas for $\mu_\text{c}=0.1U$ we observe the coexistence of normal and dimerized Mott insulator ([Fig. \[fig:inhomog\]]{}d-f). The superfluid order parameter $\psi$ shown in [Fig. \[fig:inhomog\]]{}b,e vanishes in the insulating phases and increases to a finite value in between forming superfluid rings. For a threshold of $\Psi<0.1$, we find that the dimerized Mott insulator phase is occupied by 490 of a total of 680 atoms (a-c) and 400 of 1160 atoms (d-f), respectively. The density in [Fig. \[fig:inhomog\]]{}a,d shows the typical wedding cake structure but with half-integer steps. In the dimerized Mott insulator phase with $\rho=1/2$, a periodic density modulation along the dimer axis appears. While the average density on a dimer is fixed to $\rho=1/2$, the density on the two dimer sites adjust itself according to the gradient of the chemical potential along the dimers. The persistence of the insulating phase despite this strong impact of the gradient illustrates its robustness.
A cut through the trap along the dimer axes is shown in [Fig. \[fig:inhomog\]]{}c, f. The density profile (red lines) and the superfluid order parameter (blue lines) clearly indicate the Mott insulator and dimerized Mott insulator regime. This agrees well with the expectation from the phase diagram in [Fig. \[fig:PD\]]{} for infinite size (for $\mu<\mu_c$). The dimerized Mott insulator plateaus in the density profile show an oscillating behavior along the dimer axis which is caused by the trap as discussed above. The green lines represent the total particle number and indicate a comparatively large occupation of the dimerized phase in the case of [Fig. \[fig:inhomog\]]{}f.
The local particle-number fluctuations $(\Delta n)^2$ are shown as black lines and demonstrate the expected large value of $(\Delta n)^2 = 0.25$ in the dimerized insulator phase. The large particle-number fluctuations in combination with the vanishing order parameter characterize the dimerized Mott insulator phase, whereas in the conventional Mott insulator all fluctuations are suppressed.
Perturbation theory {#sec:PT}
===================
The phase diagram [Fig. \[fig:PD\]]{} as well as the excitation spectrum [Fig. \[fig:spectrum\]]{}d can be approximated with the strong-coupling perturbative expansion technique [@Freericks1994; @Freericks1996; @Buonsante2004b; @Buonsante2004c; @Buonsante2005a]. For this we recast the full Hamiltonian to an effective model with dimer unit cells. As a first step we find the eigenstates $|n\rangle$ of $n$ particles in a dimer with energies $E_n$. In contrast to [Ref. [@Buonsante2005a]]{}, we can restrict the calculations to the respective symmetric ground states, due to the large ratio $J_1/J_2$. This simplifies the approach significantly and allows us to include perturbations up to third order. The energies for the lowest values of $n$ read $E_0=0$, $E_1=-J_1$, $E_2=\frac{1}{2}(U-\sqrt{U^2+16 J_1^2})$. This approximation is valid as long as the perturbation $z J_2$ with the coordination number $z=4$ is much smaller than the energy $E_n^{(1)}- E_n^{(0)}$, i.e., $z J_2\ll U, J_1 $. The coupling between neighboring dimers is given by the operator $$\hat J_2 = - J_2 \sum_{\langle i,j \rangle} \hat d_i^\dagger \hat d_{j},$$ where the brackets $\langle i,j \rangle$ label neighboring dimers $i, j$ instead of sites and the operators $\hat d_i (\hat d_i^\dagger)$ annihilate (create) a particle on a dimer. More precisely, for the annihilation of a particle on a dimer the operator is given by the projection on the $n$-particle ground states $$\hat d = \sum_n |n-1\rangle\langle n-1 | \hat b |n\rangle\langle n |,$$ where $\hat b$ acts on one of the two equivalent sites of the dimer. As a compact notation we define the coupling parameter $j_n = \langle n-1 | \hat b |n\rangle$. For a conventional single-site lattice model it is $j_n=\sqrt{n}$, whereas here $j_n$ depends on the exact form of the ground states that are functions of $J_1/U$. With the above restriction we obtain an effective lattice model $$\label{eq:H_eff}
\hat H_\mathrm{eff} = \sum_i (E_{\hat n_i} - \mu \hat n_i) - J_2 \sum_{\langle i,j \rangle} \hat d_i^\dagger \hat d_{j}.$$ Within this dimerized model, all insulating phases are treated on the same footing as product states of dimer ground states $|n\rangle$. Odd fillings $n$ correspond to dimerized insulators and even $n$ to conventional Mott insulators.
We now discuss the first lowest three orders of perturbation by inter-dimer tunneling $J_2$. In the unperturbed case of $J_2=0$, the creation of a particle (hole) excitation is associated with the energy $$\begin{aligned}
E_\mathrm{p}^{(0)} &= E_{n + 1} - E_n - \mu \\
E_\mathrm{h}^{(0)} &= E_{n - 1} - E_n + \mu,\end{aligned}$$ with the chemical potential $\mu$. In this case, only insulating phases exist and their phase boundaries $\mu_{p|h}^{(0)}$ can be obtained from the condition $E_{p|h}=0$, where the chemical potential compensates for the energy of one additional or missing particle per dimer.
In first order perturbation, the delocalization of a particle or a hole over the lattice results in lower and upper energy bounds $$\begin{aligned}
E_{\mathrm{p}\pm}^{(1)} &= E_\mathrm{p}^{(0)} \pm z J_2 j_{n+1}^2 \\
E_{\mathrm{h}\pm}^{(1)} &= E_\mathrm{h}^{(0)} \pm z J_2 j_n^2.\end{aligned}$$ Thus, the energy band for a particle-hole excitations lies between $E_{{\mathrm{ph}}\pm} = E_{\mathrm{p}\pm}^{(1)}+ E_{\mathrm{h}\pm}^{(1)}$ for one and $2 E_{{\mathrm{ph}}\pm}$ for two excitations indicated in figure [Fig. \[fig:spectrum\]]{}d (colored areas). In addition, the asymmetric excitation that goes beyond the effective model is indicated as unbroadened line.
The coupling between the dimers gives rise to superfluid phases surrounding the insulating phases. The phase boundaries shown in [Fig. \[fig:PD\]]{} are obtained from the minimum energy of the excitations. In first order perturbation they read $$\begin{aligned}
\mu_\mathrm{p}^{(1)} &= E_{n+1} - E_n - z J_2 j_{n+1}^2 \\
\mu_\mathrm{h}^{(1)} &= E_n - E_{n-1} + z J_2 j_n^2.\end{aligned}$$
For the determination of second- and third-order energy, one has to include processes with amplitudes on the order of $J_2^2$ and $J_2^3$. The phase boundaries in second-order perturbation read $$\begin{aligned}
\begin{split} \label{eq_mup2}
\mu_\mathrm{p}^{(2)} = \mu_\mathrm{p}^{(1)} &+ 8 \frac{(J_2 j_n j_{n+1})^2}{E_{n+1}+E_{n-1}-2E_n} \\
&- 4 \frac{(J_2 j_n j_{n+2})^2}{E_{n+2}+E_{n-1}-E_n-E_{n+1}}
\end{split} \\
\begin{split}
\mu_\mathrm{h}^{(2)} = \mu_\mathrm{h}^{(1)} &+ 4 \frac{(J_2 j_n j_{n+1})^2}{E_{n+1}+E_{n-1}-2E_n} \\
&+ 4 \frac{(J_2 j_{n-1} j_{n+1})^2}{E_{n-2}+E_{n+1}-E_n-E_{n-1}}
\end{split}.\end{aligned}$$ The first additional term in accounts for the second-order energy of the insulator that is obtained by processes via a virtual particle-hole excitations. When a particle excitation is present, these bidirectional processes are inhibited on neighboring bonds leading to a factor of $2 z = 8$. They are partly substituted by processes via the intermediate state where a particle tunnels onto the excitation, which is captured by the second additional term. Analogously, the energy of a hole-excitation relative to the insulator accounts for all possible second-order processes.
We compute the phase diagram up to third order and find a good agreement with the cluster mean-field approach detailed below. The pointy tip of the insulator lobes is due to the vanishing energy of particle-hole excitations at the crossing point of the phase boundaries. Here, the perturbation series cannot be limited to finite-order processes.
Cluster Gutzwiller theory {#sec:CMF}
=========================
Our numerical simulations are performed using a cluster mean-field approach [@Buonsante2004b; @Jain2004; @Hen2009; @Pisarski2011; @McIntosh2012; @Yamamoto2012b; @Luhmann2013]. The cluster approach allows capturing phases with strong short-range correlations such as fractional insulators formed on unit cells covering more than one site. Conventional single-site mean-field approaches such as the Gutzwiller approach are not capable of finding the dimerized insulator phases. Furthermore the cluster mean-field approach is well suited to obtain results for inhomogeneous systems as discussed in [Sec. \[sec:inhomog\]]{} and gives detailed information on the local excitation spectrum presented in [Sec. \[sec:spectrum\]]{}.
The cluster Gutzwiller approach decouples a cluster of sites from the rest of the lattice and couples it to a mean-field at its borders. The latter is determined from the exact diagonalization of the cluster and is updated in an iterative process. This allows taking into account local correlations exactly and thereby gives far more precise results than conventional mean-field methods. The improvement is especially pronounced for lattices with a small number of nearest neighbors, such as hexagonal lattices. In the many-particle cluster basis $\ket{N}$ the Hamiltonian matrix $$\hat H_{MN}=\bra{M} \hat H_\text{cluster} + \hat H_\text{boundary} \ket{N}
\label{eq:HamiltionanMatrix}$$ decomposes in two parts describing the cluster according to Eq. and its boundary. The Hamiltonian $\hat H_\text{boundary} $ describes the coupling of sites at the boundary $\sigma$ of the cluster to sites outside the cluster and reads $$\hat H_\mathrm{boundary}=-J_2 \sum_{i \in \sigma} \nu_i \hat b^\dagger_i \braket{\hat b} + c.c.,$$ where $\nu_i$ is the number of mean-field bonds at site $i$. The superfluid order parameter $\psi = \braket{\hat b}$ at the boundary is obtained from the innermost site in the cluster. When at least one of the two tunneling matrix elements is negative, the order parameter $\psi$ shows an alternating sign as depicted in [Fig. \[fig:lattice\]]{}d. Note that we apply periodic boundary conditions along one direction of the clusters, which increases the ratio of inner cluster bonds to mean-field bonds. We use clusters of up to 16 sites and restrict the basis $\ket{N}$ further using cut-offs on the fluctuations and the number of particles per site (for further details see [Ref. [@Luhmann2013]]{}). We carefully checked for convergence.
The excitation spectrum obtained from the cluster mean-field approach agrees well with the perturbation theory. The reduced band width of the band of two particle-hole excitations is due to the finite size of the cluster, as the two excitations already spread over four dimers on a eight-dimer cluster, which limits the possibilities for delocalization.
The phase diagram matches the predictions of the perturbation theory well for all phases ($0 \leq n \leq 4$). At the tips of the lobes, the perturbation theory is not expected to give precise results due to the vanishing energy of particle-hole excitations. We expect further deviations due to the restriction to the symmetric ground states on each dimer in the perturbation theory approach. The cluster mean-field approach on the other hand is restricted to a finite cluster size. The good agreement between the two methods indicates that the approximations are justified and the true phase boundary is well approximated.
Conclusions
===========
In conclusion, we have shown that when anisotropic tunneling is introduced to optical honeycomb lattices, dimerized Mott insulator phases with fractional fillings appear. In these phases, the superfluid order parameter vanishes but large particle number fluctuations persist on the individual lattice sites. We have calculated the phase diagram using two different approaches, namely a cluster Gutzwiller approach and the strong-coupling perturbation expansion technique, and found excellent agreement. The former method allows us to study the excitation spectrum, which allows the distinction of normal and dimerized Mott insulator. In a possible experiment with a harmonic confinement the dimerized insulator is formed by a large fraction of the atoms and should therefore be observable. Therefore, optical honeycomb lattice experiments should be well suited to realize and probe the proposed dimerized Mott insulator phase, which to our best knowledge has not been measured experimentally so far. When driving an optical honeycomb lattice the intra-dimer tunneling coupling can be tuned negative to realize $p$-orbital like physics.
After submission, we became aware of [Ref. [@Gawryluk2013]]{} discussing a similar setup.
Acknowledgments
===============
We thank K. Sengstock, J.Struck and M. Weinberg for helpful discussions. We acknowledge funding by the Deutsche Forschungsgemeinschaft (grants SFB 925 and GRK 1355).
[48]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, , , , , ****, ().
, , , , , ****, ().
, , , , ****, ().
, , , , , , , , ****, ().
, , , , , , ****, ().
, , , , , , ****, ().
, , , , , , , , , , ****, ().
, , , , ****, ().
, , , ****, ().
, , , , , , ****, ().
, , , , , , ****, ().
, , , , , , , , , , ****, ().
, , , , , ****, ().
, , , , , ****, ().
, , , , , ****, ().
, , , , , , , ****, ().
, , , , , , ().
, , , ****, ().
, ****, ().
, ****, ().
, , , ****, ().
, , , , , ****, ().
, , , ****, ().
, , , , , , , ****, ().
, , , , , , , , ****, ().
, , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, ****, ().
, , , , , ****, ().
, ****, ().
, , , , , , ****, ().
, , , , ****, ().
, , , ****, ().
, ****, ().
, , , ****, ().
, , , , ****, ().
, , , ****, ().
, ****, ().
, , , ****, ().
, , , ****, ().
, ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, ****, ().
, , , , ****, ().
, , , ****, ().
, , , ().
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We apply generalisations of the Swendson-Wang and Wolff cluster algorithms, which are based on the construction of Fortuin-Kasteleyn clusters, to the three-dimensional $\pm 1$ random-bond Ising model. The behaviour of the model is determined by the temperature $T$ and the concentration $p$ of negative (anti-ferromagnetic) bonds. The ground state is ferromagnetic for $0 \le p<p_c$, and a spin glass for $p_c < p \le 0.5$ where $p_c \simeq 0.222$. We investigate the percolation transition of the Fortuin-Kasteleyn clusters as function of temperature. Except for $p=0$ the Fortuin-Kasteleyn percolation transition occurs at a higher temperature than the magnetic ordering temperature. This was known before for $p=1/2$ but here we provide evidence for a difference in transition temperatures even for $p$ arbitrarily small. Furthermore, for all values of $p>0$, our data suggest that the percolation transition is universal, irrespective of whether the ground state exhibits ferromagnetic or spin-glass order, and is in the universality class of standard percolation. This shows that correlations in the bond occupancy of the Fortuin-Kasteleyn clusters are irrelevant, except for $p=0$ where the clusters are tied to Ising correlations so the percolation transition is in the Ising universality class.'
author:
- Hauke Fajen
- 'Alexander K. Hartmann'
- 'A. Peter Young'
bibliography:
- 'references.bib'
title: 'Percolation of Fortuin-Kasteleyn clusters for the random-bond Ising model'
---
Introduction
============
Magnetic systems with quenched disorder, such as spin glasses (SGs) [@binder1986; @mezard1987; @young1998; @nishimori2001] and random field systems, exhibit phase transitions between low-temperature ordered and high-temperature disordered (paramagnetic) phases in high enough dimensions. This is similar to the case of pure systems like ferromagnets [@ising1925] but spin glasses in particular exhibit a much richer behaviour and many aspects of the low-temperature phase are still not well understood. Since most disordered models cannot be solved analytically, one has to resort to computer simulations.[@practical_guide2015] For the special case of zero temperature, there are often efficient algorithms[@opt-phys2001]. However, for systems coupled to a heat bath at finite temperature, Monte Carlo simulations [@landau2000; @newman1999] are generally used. For the *pure* Ising model, efficient cluster Monte Carlo (MC) approaches exist,[@swendsen1987; @wolff1989] which are based on the construction of Fortuin-Kasteleyn (FK) [@fortuin1972] clusters of spins. This gives fast equilibration even close to the phase transition point. The reason is that the FK clusters percolate [@stauffer1994introduction] *precisely* at the phase transition.[@coniglio1980]
It is also possible to implement cluster MC algorithms like the Wolff algorithm for spin glasses, but unfortunately these are not efficient because, in the vicinity of the spin glass phase transition, each update flips almost all the spins.[@kessler1990] The reason is that percolation of the FT clusters happens at much higher temperatures than the magnetic-ordering phase transition temperature.[@deArcangelis1991] Other approaches for cluster algorithms for spin glasses have been tried, [@niedermayer1988; @liang1992; @houdayer2001; @joerg2005; @zhu2015] but in the end none turned out to be efficient for three-dimensional spin glasses and related models. Thus, single-spin flip algorithms are still used for studying spin glasses numerically. Some improvement is obtained by using parallel tempering,[@geyer1991; @hukushima1996] and by running parallel tempering on a special-purpose high-performance computer “JANUS” [@janus2009] it has been possible to simulate an $N=48^3$ spin glass model near the transition temperature.
To obtain a better understanding of the nature of FK clusters and their percolation transitions, as well as algorithmic efficiency, we study here the $\pm 1$ random-bond Ising model,[@kirkpatrick1977] which is a generalisation of the standard spin glass. It consist of $N$ Ising spins $\sigma_i=\pm 1$ placed on a $d$-dimensional hyper-cubic lattice of linear size $L$, i.e. $N=L^d$. The Hamiltonian is given by $$H =
- \sum_{\langle i,j\rangle} J_{ij} \sigma_i \sigma_j \,.$$ Each spin $i$ interacts with its nearest neighbours $j$ via an interaction which is a quenched random variable $J_{ij}$. Here we use a bimodal distribution so each bond is anti-ferromagnetic ($J_{ij}=-1$) with probability $p$ and ferromagnetic ($J_{ij}=+1$) with probability $1-p$. As usual for quenched disorder, the result of any measurement will depend on the realisation of the disorder, so one has to perform an average over many realizations of disorder in addition to doing the thermal average.
We consider here the case of a simple cubic lattice for which the low temperature phase is ferromagnetic for a small concentration ($p$) of anti-ferromagnetic bonds and a spin glass for a larger concentration. We denote the paramagnet to ferromagnet transition temperature by $T_c(p)$ and the paramagnet to spin glass transition by $T_{\rm SG}(p)$. The phase diagram in the $p$–$T$ plane has been determined by Monte Carlo simulations, [@reger1986; @hasenbusch2007] see Fig. \[fig:phasediagram\]. For $T=0$ the transition point between the ferromagnetic and spin-glass phases was found [@art_threshold1999] to be approximately $p_c=0.222(5)$.
![\[fig:phasediagram\] Phase diagram showing the line of percolation transitions of the FK clusters, and the lines of phase transitions between ferromagnetic (FM), paramagnetic (PM) and spin glass (SG) phases. Lines are guides for the eyes only. The data for the ferromagnetic transition temperature $T_c(p)$ is from Ref. \[\], the data for the spin glass transition temperature $T_{\rm SG}$ is from Ref. \[\], and the value of $p$ where the spin glass and ferromagnetic phases meet at $T=0$, $p_{\rm
c}$, is from Ref. \[\]. The data for the percolation transition temperature $T_{FK}(p)$ is from this work. ](T_c.eps){width="\columnwidth"}
In the present study, we investigate the behaviour of FK clusters and, related to this, the performance of the Wolff algorithm, in the $p$-$T$ plane. We know that for the pure ($p=0$) ferromagnet the FK percolation transition coincides with the ferromagnet-paramagnet, and here we investigate whether this is true for any other values of $p$. Results of some test simulations performed previously [@hasenbusch2007] suggest this is not the case at least for some values of $p$. Also, close to the ferromagnetic-spin glass boundary, where frustration is lower than for the standard spin glass model which has $p=1/2$, we investigate whether the Wolff algorithm performs better than for the standard spin glass model. If this were the case, one might be able to study low-temperature spin-glass behaviour for larger samples, by working in this range of $p$.
We present an extensive study of the FK percolation transition in the full range of interest $0\le p\le 1/2$, which indicates that this transition happens above the phase transition line *for all* $p>0$. Only for the pure ferromagnet, $p=0$, does it coincides with the FM-PM transition. In addition, the critical exponents seem to be those of the (uncorrelated) percolation problem everywhere along the FK transition percolation transition line, including both the ferromagnetic and spin-glass regions, see Fig. \[fig:phasediagram\]. The only exception is for $p$ *precisely* equal to 0, the pure ferromagnet, for which the critical exponents are those of the Ising model. Finally, our result indicate that in the spin glass region close to $p_c$ the Wolff algorithm does not perform notably better than for the standard ($p=1/2$) spin-glass case.
Our paper is organised as follows. In Sec. \[sec:methods\], we review the algorithms we have used. Next, in Sec. \[sec:results\], we present our results, and finally in Sec. \[sec:summary\] we give a summary and discussion.
\[sec:method\]Methods {#sec:methods}
=====================
To study the FK percolation transition and to investigate the efficiency of the Wolff algorithm we construct FK clusters at each step as follows:
- Bonds where $J_{i,j}\sigma_i \sigma_j>0$ are said to be *satisfied*, and we *activate* them with probability $p_{\mathrm{act}}=1-e^{-2\beta
|J_{ij}|}$. Unsatisfied bonds are never activated.
- We determine all clusters of spins connected by activated bonds, as in bond percolation.
A cluster is said to be *wrapping* or *percolating* if it spans the lattice across between the periodic boundaries and so is connected back to itself. For each step, we record whether a cluster is wrapping (this is typically the largest one), and we also monitor the sizes of all clusters to investigate the distribution of cluster sizes. Finally, we generate the next configuration according the Wolff algorithm by selecting a spin at random and flipping the the spins (i.e. with “acceptance probability” one) in the cluster which contains it.
Averages are done both over the spin configurations for a given realization and a disorder average over a large number of different realizations. The quantities that we measure are:
- The average wrapping probability, $p_{\rm wrap}$.
- The fraction of sites in the largest cluster, $P$.
- The number $n_s$ of clusters of size $s$.
- The average size $S$ of the clusters excluding the largest one (this would be the percolating cluster in the percolating phase). The average is done with respect to all sites, i.e. $S = \sum_s s^2n_s/\sum_s sn_s$.
- The average size of the flipped clusters, $n_{\rm Wolff}$.
For high temperatures the activation probability $p_{\mathrm{act}}$ is small, leading to many small clusters which do not wrap. On the other hand, for low temperatures, $p_{\mathrm{act}}$ will be large leading to few clusters and typically one big wrapping cluster. Thus, in between, there exists a percolation transition of the FK clusters at some temperature $T_{\rm FK }$, such that, in the thermodynamic limit, $N\to\infty$, one finds $p_{\rm wrap}\to 0$ for $T>T_{\rm FK}$ and $p_{\rm wrap}\to 1$ for $T<T_{\rm FK}$.
We analyse our data using finite-size scaling (FSS), as is standard in percolation transitions.[@stauffer1994introduction] According to FSS, at a second order percolation transition near the critical point, the wrapping probability should exhibit a scaling behaviour $$p_{\rm wrap}(L,T) = f_{\rm wrap}((T-T_{\rm FK})L^{1/\nu})\,,
\label{eq:fss:wrapping}$$ where $\nu$ is the critical exponent which describes the divergence of the correlation length of the FK clusters. Thus, the parameters $T_{\rm FK}$ and $\nu$ can be determined by varying them until the data for different system sizes collapse on to the same universal curve $f_{\rm wrap}(\tilde x)$.
Furthermore, in the percolating phase, the fraction of sites in the largest (i.e. percolating) cluster in an infinite system goes to zero like $P \sim (T_{\rm FK} -T)^{\beta}$ as $T$ approaches $T_{\rm FK}$ from below. For a finite system, this becomes, according to FSS, $$P(L,T) = L^{-\beta/\nu}f_{P}((T-T_{\rm FK})L^{1/\nu})\,,
\label{eq:clustersize}$$ allowing us to obtain the critical exponent $\beta$. The average cluster size behaves in a similar way, as described by the finite-size scaling relation $$S (L,T) = L^{\gamma/\nu}f_{S}((T-T_{\rm FK})L^{1/\nu})\, .$$ Note that in computing $S$ we neglect the largest cluster, so $S$ has a maximum near the percolation transition, because in the non-percolating phase there are only many small clusters, while in the percolating phase most sites belong to the percolating cluster which is neglected. Thus, the scaling function $f_{\rm S}$ exhibits a peak at some value $x_{\rm peak}$, corresponding to a temperature $T_{\rm peak}=T_{\rm FK}+x_{\rm peak}L^{-1/\nu}$, which means that the height of $S^\star$ at the peak scales with a power-law $$S^\star \sim L^{\gamma/\nu}\, ,
\label{Sstar}$$ allowing us to obtain the critical exponent $\gamma$. Finally, at the critical point $T_{\rm FK}$, the distribution $n_s$ of cluster sizes for an infinite system is expected to follow a power-law $$n_s(T_{\rm FK})\sim s^{-\tau}\, ,$$ defining another critical exponent $\tau$.
The critical exponents are not independent of each other. Instead, they are connected through scaling relations, such that there are only two independent exponents. The scaling relations for the standard percolation problem are often expressed [@stauffer1994introduction] as functions of exponents describing the shape of $n_s$, i.e. for an infinite system $$n_s = s^{-\tau} f_n\left( s^\sigma\, (T-T_{\rm FK}) \right) \, ,$$ which defines another exponent $\sigma$. In terms of $\tau$ and $\sigma$ the standard scaling relations are [@stauffer1994introduction] $$\label{eq:beta}
\nu=\frac{\tau-1}{\sigma d}\,,\;
\gamma=\frac{3-\tau}{\sigma}\,,\;
\beta=\frac{\tau-2}{\sigma}\,.$$
We don’t measure $\sigma$, since this would require additional numerical effort, but we can remove $\sigma$ from the equations by solving the first equation with respect to $\sigma$ and inserting the solution into the other two, resulting in: $$\gamma=\frac{3-\tau}{\tau-1}\nu d\,,\;
\beta=\frac{\tau-2}{\tau-1}\nu d\,.
\label{eq:scaling:relations}$$
We will verify that our computed values for $\nu,\tau,\gamma$ and $\beta$ obey these relations.
Results {#sec:results}
=======
We perform simulations for various values of $p\in [0,0.5]$. For each value of $p$ we treated different system sizes $L \in [10,100]$,, and for a few values of $p$ we also did simulations for $L=200$, see below. All results are disorder averages over typically 1000 realisations. For each realisation we perform Monte Carlo simulations using the Wolff algorithm for 72 temperatures equally spaced in $[3.615,4.68]$, i.e. with spacing $\Delta T = 0.015$. For the selected cases of $p=0.1$, $0.3$ and $0.5$ (and also for $p=0$ as a comparison with other work and a check on our code), we studied 20 additional temperatures spaced by $\Delta T=0.003$ very close to $T_{\rm FK}$, in order to determine the critical properties precisely.
To check for equilibration we average over intervals $[t/2,t]$ for a logarithmically increasing set of times $t$, and require that there is no systematic trend for the last several values of $t$. Typically, due to the high temperatures, equilibration is achieved within a few steps. For small systems, $L\le 30$, we perform 2$\times 10^5$ Wolff steps per realisation, while for the larger systems, which run slower but still need only a few steps to equilibrate, we do $5\times 10^3$ steps.
To determine the position of the FK percolation transitions, we monitor the wrapping probability of the FK clusters. An example is shown for $p=0.1$ in the inset of Fig. \[fig:kollaps\_nu\]. A clear decrease of the wrapping probability beyond $T\approx 4$ is visible. We performed a data collapse according to Eq. (\[eq:fss:wrapping\]), see main plot of Fig. \[fig:kollaps\_nu\], to determine $T_{\rm FK}$ and the critical exponent $\nu$ of the percolation length, resulting in $T_{\rm FK}= 4.059(3)$ and $\nu=0.89(8)$. The best fit parameters were determined from the method discussed in the appendix of Ref. \[\] and in Ref. \[\].
![\[fig:kollaps\_nu\] Wrapping probability as function of temperature $T$ for $p=0.1$, for various system sizes $L$. The inset shows the raw data, while in the main plot a data collapse to determine $T_{\rm FK}$ and $\nu$ gives $1/\nu=1.10(5)$ and $T_{\rm FK}=4.059(3)$.](scaling_p0_1_kollaps.eps){width="\columnwidth"}
In a similar way, we analysed the data for other values of $p$. The resulting values of $T_{\rm FK}$ as a function of $p$ are shown in the phase diagram in Fig. \[fig:phasediagram\], along with the values for the FM-PM and SG-PM phase transitions obtained from the literature [@reger1986; @hasenbusch2007], and the critical concentration $p_c$ for the zero-temperature FM-SG transition.[@art_threshold1999] Interestingly, the FK percolation transition seems to coincide with magnetic-ordering transition only for the pure ferromagnetic system ($p=0$). For all other values of $p$, $T_c<T_{\rm FK}$ even close to the pure ferromagnet. Hence, even if the ground state is ferromagnetic, i.e. for $0<p<p_{\rm c}$, the FM-PM phase transition cannot be understood as a percolation transition of the FK clusters.
The resulting values of $\nu$ as a function of $p$ are shown in Fig. \[fig:nu\]. For $p=0$, we recover the literature value for the pure Ising ferromagnet,[@baillie1992] but with larger error bars (which is natural, because our main numerical effort goes into the necessary disorder average and considering several values of $p$). For all other values of $p$, including both ferromagnetic and spin glass regions, we find that $\nu$ is compatible with the previously found[@deArcangelis1991] value of $\nu=0.88(5)$. This is also compatible with the value[@wang2013] for the standard percolation problem, in which there are no correlations between the occupancies of the bonds. By contrast, in FT clusters there *are* correlations for all $p$ but interestingly they do not seem to affect the critical behavior, except for $p=0$ where the bond occupancies are *rigorously constrained* to follow Ising correlations.
![\[fig:nu\] The critical exponent $\nu$ as a function of $p$. The value of $\nu$ for $p=0$ is from Ref. \[\] and the value for $p=0.5$ indicated by a triangle is from Ref. \[\]. ](nu.eps){width="\columnwidth"}
To investigate universality more carefully we have evaluated the other critical exponents with additional data near $T_{\rm FK}$ for the values $p=0$ (for a consistency check), $p=0.1$ (a ferromagnetic case), $p=0.3$ and $p=0.5$ (SG cases; for the latter value the critical behavior is already partially known[@deArcangelis1991]).
![\[fig:max\_cluster\_size\] (color online) The fraction of sites in the infinite cluster, $P$, as a function of the temperature $T$ in the vicinity of $T_{\rm FK}$, for $p=0.3$ and various system sizes $L$. The inset shows the raw data, while the main plot shows the data rescaled according to Eq. (\[eq:clustersize\]), with best fitting values $\beta=0.48(4)$, $\nu=0.87(8)$, and $T_{\rm FK}=3.939(3)$.](scaling_beta_p0_3_kollaps.eps){width="\columnwidth"}
For the fraction of sites in the infinite cluster (the order parameter), we show data for $p=0.3$ in Fig. \[fig:max\_cluster\_size\]. From a finite-size scaling collapse of the data we obtain the best fitting parameters $\beta=0.48(4)$, $\nu=0.87(8)$ and $T_{\rm FK}=3.939(3)$. The result for the other intensively studied cases are shown in Table \[tab:exponents\]. Note that for $T_{\rm FK}$ and $\nu$, we usually have several independent estimates available and the stated values and their error bars are chosen such that they are compatible with all results.
$p$ $T_{\rm FK}$ $\nu$ $\tau$ $\gamma$ $\beta$ $\gamma=\frac{3-\tau}{\tau-1}\nu d$ $\beta=\frac{\tau-2}{\tau-1}\nu d$
----- -------------- --------- ---------- ---------- --------- ------------------------------------- ------------------------------------
0.0 4.5116(5) 0.65(4) 2.27(3) 1.18(6) 0.31(4) 1.1(2) 0.4(1)
0.1 4.059(3) 0.89(8) 2.196(8) 1.82(8) 0.48(4) 1.79(12) 0.44(4)
0.3 3.941(3) 0.89(8) 2.23(5) 1.84(6) 0.41(4) 1.7(2) 0.49(16)
0.5 3.934(3) 0.88(9) 2.26(1) 1.8(1) 0.41(5) 1.6(2) 0.54(9)
In addition to the order parameter, we have also analysed the data for the average cluster size $S$. As example, we show the result for $p=0.1$ and size $L=50$ as a function of temperature $T$ in Fig. \[fig:mean\_cluster\_size\]. The data exhibits a peak at some point ($T^\star, S^\star$). One can read off the critical exponent $\gamma$ from the $L^{\gamma/\nu}$ scaling, see Eq. (\[Sstar\]), of the peak height as a function of $L$. The data is shown in the inset of Fig. \[fig:mean\_cluster\_size\]. For different values of $p$, the resulting values of $\gamma$ are also shown in Table \[tab:exponents\]. Again, we observe that for $p>0$ the results seem to agree with each other.
![\[fig:mean\_cluster\_size\] Mean cluster size $S$ at $L=50$ and $p=0.1$ as a function of temperature $T$, near the FK percolation transition $T_{\rm FK}$. The data exhibits a peak with peak height $S^\star$. The inset shows the peak height as function of $L$.](mean_cluster_size_max_p0_1.eps){width="\columnwidth"}
To obtain the critical exponent $\tau$, we analyse the distribution of cluster sizes, excluding the largest cluster, at the critical point for a rather large system size, $L=200$. As an example, we present our results for $p=0.3$ in Fig. \[fig:tau\]. The data exhibits a high quality which allows us to observe a power law over about 10 decades in probability. A fit resulted in a value $\tau=2.23(5)$. This value, and the results for the three other selected cases, are also shown in Table \[tab:exponents\]. The values we have found for all values of $p>0$ are compatible with the values for standard percolation in three dimensions.
![\[fig:tau\]Cluster size distribution at $p=0.3$ with a system size of $N=200^3$.](cluster_size_T3_93808_L200_p0_3_10E5.eps){width="\columnwidth"}
The exponents should obey the scaling relations in Eqs. (\[eq:scaling:relations\]). The values obtained when inserting the measured values for $\nu$ and $\tau$ from Table \[tab:exponents\] to estimate $\gamma$ and $\beta$ from the scaling relations are shown in the last two columns in Table \[tab:exponents\]. All these values are compatible with the directly measured values within error bars. Note that the error bars from the scaling relations are larger than the error bars of the directly measured exponents due to error propagation.
Finally we consider the question of whether the Wolff algorithm might be more efficient in the spin-glass phase near the FM-SG transition, i.e. for $p$ just slightly greater than $p_c$, rather than for $p=1/2$. In Fig. \[fig:gzb\] we show the average effective size of the flipped cluster (which is not always the largest one), as a function of the temperature $T$ for $p=0.25\, (>p_{\rm c})$. By “effective” we mean that if the cluster of flipped spins is larger than half of the system size, then the spins which are not flipped are counted. We see that the clusters which are flipped near the SG phase transition are very small. One could already expect this from the phase diagram in Fig. \[fig:phasediagram\], which shows that for $p=0.25$ the FK percolation transition is considerably above the critical temperature $T_{\rm SG}$. Thus, applying the Wolff algorithm in the spin glass phase but near the spin glass-ferromagnet phase boundary, does not lead to any benefit relative to studying the standard spin glass model which has $p=1/2$.
![\[fig:gzb\] Average size of the clusters flipped (or not flipped if this is smaller) by the Wolff algorithm as a function of temperature $T$ for $L=10$ and $p=0.25$.](gzb.eps){width="\columnwidth"}
Summary {#sec:summary}
=======
We have studied the percolation transitions of Fortuin-Kasteleyn clusters for the three-dimensional random-bond Ising model. Near the cluster percolation transition the Wolff algorithm can be used to efficiently sample equilibrium configurations. However, except for the pure Ising case ($p=0$), the temperature of the percolation transition is higher that of the ferromagnet-paramagnet and spin glass-paramagnet transitions, and for most values of $p$ it is *much* higher, see Fig. \[fig:phasediagram\]. This renders the Wolff algorithm inefficient for the magnetic transitions except for $p=0$. Indications of this behaviour were already found in some test simulations of a previous study,[@hasenbusch2007] where, for the FM-PM phase boundary at one value of $p>0$, cluster algorithms were tried but turned out to be inefficient.
We have determined the critical exponents at the FK cluster percolation transition. For $p=0$, the pure Ising case, we obtain the known values, which are those of the Ising model since the FK clusters are controlled by Ising correlations in this limit. For all other values, $0<p\le 1/2$, our results are compatible with the universal behaviour of standard percolation, irrespective of whether the ground state exhibits ferromagnetic or spin-glass order. Since standard percolation has no correlations between the occupancy of the bonds, whereas bonds in the FK clusters *are* correlated, this implies that the correlations are irrelevant for universal properties, and so presumably are of short range for $p > 0$.
For future studies, it would be interesting to investigate other types of cluster algorithms [@niedermayer1988; @joerg2005; @zhu2015] for the three-dimensional random-bond Ising model. So far, from the literature studies known to us, none of them turned out to be efficient enough to study the pure spin-glass case ($p=1/2$) for large enough systems, but it could be that some will work well close to the FM-SG phase boundary or perhaps at least for ferromagnetic ordering of the random ($p>0$) case.
Acknowledgements
================
APY thanks the Alexander von Humboldt Foundation for financial support through a Research Award. The simulations were performed at the HPC facilities GWDG Göttingen, HERO and CARL. HERO and CARL are both located at the University of Oldenburg (Germany) and funded by the DFG through its Major Research Instrumentation Programme (INST 184/108-1 FUGG and INST 184/157-1 FUGG) and the Ministry of Science and Culture (MWK) of the Lower Saxony State.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The Hardy-Littlewood maximal function $\mathcal{M}$ and the trigonometric function $\sin{x}$ are two central objects in harmonic analysis. We prove that $\mathcal{M}$ characterizes $\sin{x}$ in the following way: let $f \in C^{\alpha}(\mathbb{R}, \mathbb{R})$ be a periodic function and $\alpha > 1/2$. If there exists a real number $0 < \gamma < \infty$ such that the averaging operator $$(A_xf)(r) = \frac{1}{2r}\int_{x-r}^{x+r}{f(z)dz}$$ has a critical point in $r = \gamma$ for every $x \in \mathbb{R}$, then $$f(x) = a+b\sin{(cx + d)} \qquad \mbox{for some}~a,b,c,d \in \mathbb{R}.$$ This statement can be used to derive a characterization of trigonometric functions as those nonconstant functions for which the computation of the maximal function $\mathcal{M}$ is as simple as possible. The proof uses the Lindemann-Weierstrass theorem from transcendental number theory.'
address: 'Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, CT 06511, USA'
author:
- Stefan Steinerberger
title: |
A Rigidity Phenomenon for the\
Hardy-Littlewood Maximal Function
---
Introduction and main results
=============================
Introduction
------------
Maximal functions are a central object in harmonic analysis; conversely, harmonic analysis is built up from trigonometric functions. We were motivated by the simple question whether a maximal function is able to ’recognize’ a trigonometric function in any particular way. We focus on the centered Hardy-Littlewood maximal function on the real line $$(\mathcal{M}f)(x) = \sup_{r > 0}\frac{1}{2r}\int_{x-r}^{x+r}{|f(z)|dz}.$$ Classical results are the embedding $\mathcal{M}:L^1 \rightarrow L^{1, \infty}$, where the sharp constant is known [@mel], as well as the embedding $\mathcal{M}:L^p \rightarrow L^p$ for $1 < p \leq \infty$. The wealth of theory developed around maximal functions can no longer be succinctly summarized: we refer to the book of Stein [@stein] for the classical theory and a survey of Wolff [@wo] on the Kakeya problem.
The interval length function $r_f(x)$.
--------------------------------------
Usual questions around maximal functions are concerned with their size: since $(\mathcal{M}f)(x) \geq |f(x)|$ in all Lebesgue points, it is of interest to understand mapping properties in $L^p$ spaces. Our question goes in an orthogonal direction: how complicated is the dynamical behavior of the ’maximal’ intervals? The question is not well-posed because there might be more than one interval centered at $x$ over which the average value of the function coincides with the maximal function: in these cases, we opt for taking the smallest such interval. Formally, we define the length function $r_f(x):\mathbb{R} \rightarrow \mathbb{R}_{\geq 0}$ associated to a periodic function $f$ by $$r_f(x) = \inf \left\{r \geq 0: \frac{1}{2r}\int_{x-r}^{x+r}{f(z)dz} = \sup_{s > 0}{\frac{1}{2s}\int_{x-s}^{x+s}{f(z)dz}}\right\},$$ where the integral is to be understood as the point evaluation if $r = 0$. It is easy to see that $r_f$ is well-defined and finite for periodic functions.
Main results.
-------------
The purpose of this paper is to study the situation, where for all $x \in \mathbb{R}$ both $r_f(x)$ and $r_{-f}(x)$ are either 0 or a fixed positive real number and to show that this characterizes the trigonometric function. This theorem may be understood as a characterization of trigonometric functions by means of a dynamical aspect of the Hardy-Littlewood maximal function. It seems to have surprisingly little to do with traditional characterizations involving geometry, power series, differential equations or spectral theory. Indeed, we failed to find a slick reduction to any of the classical characterizations and ended up needing tools from transcendental number theory.
Let $f \in C^{\alpha}(\mathbb{R}, \mathbb{R})$ be a periodic function and $\alpha > 1/2$. There exists a positive number $\gamma > 0$ such that the averaging operator $$(A_xf)(r) = \frac{1}{2r}\int_{x-r}^{x+r}{f(z)dz}$$ has a critical point in $r = \gamma$ for every $x$ if and only if $$f(x) = a+b\sin{(cx + d)} \qquad \mbox{for some}~a,b,c,d \in \mathbb{R}.$$
Theorem 1 is the strongest statement in this paper; it is relatively easy to deduce the following statement, which formulates everything in terms of the complexity of the maximal intervals.
Let $f \in C^{\alpha}(\mathbb{R}, \mathbb{R})$ be a periodic function and $\alpha > 1/2$. Then $$\left| \bigcup_{x \in \mathbb{R}}{\left\{r_f(x), r_{-f}(x)\right\}} \right| \leq 2$$ if and only if $$f(x) = a+b\sin{(cx + d)} \qquad \mbox{for some}~a,b,c,d \in \mathbb{R}.$$
We emphasize that we do not even know whether the statement remains true if the constant 2 is replaced by a larger positive integer (but conjecture that it does). Another way of stating Theorem 2 is as follows: suppose the periodic function $f(x)$ does not change sign and that both $\mathcal{M}f$ and $\mathcal{M}(-f)$ can be computed by checking the average over an interval of fixed interval and comparing it with point evaluation, i.e. suppose there exists a fixed number $0 < \gamma < \infty$ such that $f$ satisfies the equation $$(\mathcal{M}f)(x) = \max\left(|f(x)|, \frac{1}{2\gamma}\int_{x-\gamma}^{x+\gamma}{|f(z)|dz}\right) \qquad \mbox{for all}~x \in \mathbb{R}$$ and the same condition (with the same value $\gamma$) holds for $\mathcal{M}(-f)$, then $$f(x) = a+b\sin{(cx + d)} \qquad \mbox{for some}~a,b,c,d \in \mathbb{R}.$$
A delay differential equation.
------------------------------
Perhaps the most natural first step after seeing Theorem 1 would be to try a combination of differentiation and algebraic manipulations to obtain an ordinary differential equation (with the hope of it being $y'' + y =0$). As it turns out, this does not work and produces much more interesting results instead. Differentiation in $r$ implies that $$0 = \partial_r \frac{1}{2r}\int_{x-r}^{x+r}{f(z)dz}\big|_{r = \gamma} = -\frac{1}{2\gamma^2}\int_{x-\gamma}^{x+\gamma}{f(z)dz} + \frac{1}{2\gamma}(f(x+\gamma) + f(x-\gamma)).$$ Assuming $f \in C^1$, this equation can now be differentiated in $x$ and yields $$f'(x+\gamma) - \frac{1}{\gamma}f(x+\gamma) = -f'(x-\gamma) - \frac{1}{\gamma}f(x-\gamma).$$ The qualitative theory of delay differential equations is a lot more complicated than the theory of ordinary differential equations because the space of solutions is *much* larger (uncountable): any $C^1$ function $g:[0, 2\gamma] \rightarrow \mathbb{R}$ with correct boundary conditions can always be extended to a solution of the delay differential equation. However, as an easy consequence of Theorem 2, we can show that there are few periodic solutions.
Let $\gamma > 0$ be fixed and let $f \in C^1(\mathbb{R}, \mathbb{R})$ be a solution of the delay differential equation $$f'(x+\gamma) - \frac{1}{\gamma}f(x+\gamma) = -f'(x-\gamma) - \frac{1}{\gamma}f(x-\gamma).$$ If $f$ is periodic, then $$f(x) = a+b\sin{(cx + d)} \qquad \mbox{for some}~a,b,c,d \in \mathbb{R}.$$
Considering the large (uncountable) number of solutions, it is utterly remarkable that there are so few periodic solutions. We have not been able to locate any type of argument in the literature that would allow to prove a result of this type.
Open questions.
---------------
We believe that many of the assumptions can be weakened. Periodicity of the functions is necessary to allow the use of Fourier series on which our argument is based, however, it seems natural to assume that the properties discussed could not hold for a nonperiodic function anyway. The assumption $f \in C^{\alpha}(\mathbb{R}, \mathbb{R})$ with $\alpha > 1/2$ is required at one point in the proof to enforce uniform convergence of the Fourier series; again, it seems to be an artefact of the method. We note that the condition $f \in C^{\alpha}(\mathbb{R}, \mathbb{R})$ with $\alpha > 1/2$ in our statements could always be replaced with the condition of $f$ having an absolutely convergent Fourier series. The strongest statement we believe could to true is the following.
> *Conjecture.* If $f \in L^{\infty}(\mathbb{R})$ is real-valued and $r_f(x)$ assumes only finitely many different values, then $$f(x) = a+b\sin{(cx + d)} \qquad \mbox{for some}~a,b,c,d \in \mathbb{R}.$$
A more daring conjecture would be that it suffices to assume that $$\bigcup_{x \in \mathbb{R}}{\left\{r_f(x)\right\}} \subset \mathbb{R}
\qquad \mbox{is a Lebesgue-null set.}$$ If $r_f(x)$ is contained in a set of ’small’ non-zero Lebesgue measure, what does that imply for the function? It seems to indicate, in some weak sense, that Fourier frequencies interact weakly (perhaps in the sense of a $\Lambda(p)-$property?). Furthermore, it seems that if $f$ is given by a lacunary Fourier series, then $$\bigcup_{x \in \mathbb{R}}{\left\{r_f(x)\right\}} \subset \mathbb{R} \qquad \mbox{can have 'fractal' structure}$$ again in a vague sense (small measure and a very large number of connected components): it could be of interest to try to understand quantitative versions of this basic intuition. One could also ask to which extent this persists in higher dimensions: in $\mathbb{R}^d$, if we consider the maximal function associated to axis-parallel rectangles and the natural analogue $r_f(x):\mathbb{R}^d \rightarrow \mathbb{R}^d$, then setting $$f(x_1, \dots, x_d) = \prod_{i=1}^{d}{(a_i + b_i\sin{x_i} + c_i\cos{x_i})} \quad \mbox{implies} \quad \left| \bigcup_{x \in \mathbb{R}}{\left\{r_f(x), r_{-f}(x)\right\}} \right| \leq 2^d.$$ Already in two dimensions, there are many natural maximal functions and it is not clear to us whether similar statements hold for any of them. We recall the Pompeiu conjecture [@pomp]: if $K \subset \mathbb{R}^n$ is a simply connected Lipschitz domain and $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is a nonzero continuous function such that the integral of f vanishes over every congruent copy of $K$ – does this imply that $K$ is a ball? Is there a connection between the Pompeiu conjecture and the maximal problem for disks?\
*The discrete setting.* In light of the recent results [@bob; @car; @car1] concerning the behavior of the maximal function on the lattice, this might be another interesting direction to investigate. For a function $f:\mathbb{Z} \rightarrow \mathbb{R}$, we define the maximal function as $$(\mathcal{M}f)(n) = \sup_{r \in \mathbb{N}_{\geq 0}}{\frac{1}{2r +1}{\sum_{k=n-r}^{n+r}{f(k)}}}.$$ The length function $r_{f}:\mathbb{Z} \rightarrow \mathbb{N}$ is defined as above. Numerical experiments show that the continuous case translates into the discrete setting: for generic parameters functions of the form $$f(n) = a + b \sin{(c n + d)} \qquad a,b,c,d \in \mathbb{R}$$ seem to give rise to two-valued $r_f$. We do not have a formal proof of this statement, it should be equivalent to a series of trigonometric inequalities that might actually be in the literature. The property is stable under small perturbations. However, there also exist completely different functions with a two-valued $r_f$: the following example was found by Xiuyuan Cheng (personal communication). Taking $$f(n) = \frac{(-1)^n}{\left(|n|+\frac{1}{2}\right)^{\alpha}} \qquad \mbox{for certain} \quad 0 < \alpha < \frac{1}{2},$$ introducing a cut-off and making it periodic can yield functions with $r_f:\mathbb{Z} \rightarrow \left\{0,2\right\}$.
This example is structurally totally different from the sine: having oscillations at scale 2 seems crucial. We do not know whether there are any solutions of other types and consider it to be a fascinating problem. A natural conjecture would be that if a periodic function $f:\mathbb{Z} \rightarrow \mathbb{R}$ satisfies $$|f(n+1) - f(n)| \leq \varepsilon \|f\|_{\ell^{\infty}}$$ and has a two-valued $r_f$, then $$\inf_{a,b,c,d \in \mathbb{R}} ~~ \sup_{n \in \mathbb{Z}} ~~\left| f(n) - (a + b \sin{(cn + d)}) \right| \leq c(\varepsilon)\|f\|_{\ell^{\infty}}$$ for some function $c:\mathbb{R}_{+} \rightarrow \mathbb{R}_{+}$ tending to 0 as $\varepsilon$ tends to 0.
Related work.
-------------
We are not aware of any related work in this direction. Our interest in the Hardy-Littlewood maximal function itself, however, stems from a series of recent interesting results studying fine properties of $\mathcal{M}f$: since $\mathcal{M}f$ tries to maximize local averages, there is every reason to believe that it should decrease total variation – this turns out to be a surprisingly intricate problem. Motivated by a question of Kinnunen [@ki], Tanaka [@tan] showed for the *uncentered* Hardy-Littlewood maximal function $ \mathcal{ M^*}$ that $$\| (\mathcal{M}^* f)'\| \leq 2 \| f'\|_{L^1},$$ where the constant 2 was then improved to 1 by J. M. Aldaz & J. Pérez Lázaro [@al]. Kurka [@k] has recently proven the same inequality for the centered Hardy-Littlewood maximal function for a large universal constant. Carneiro & Svaiter [@car2] give corresponding results for the maximal heat flow and the maximal Poisson operator. The discrete question on the lattice $\mathbb{Z}$ has been investigated by Bober, Carneiro, Hughes & Pierce [@bob] and Carneiro & Hughes [@car] in higher dimensions. The result of Kurka in the discrete setting has been proven by Temur [@tem]. These results are well in line of what one would expect from a maximal function, however, it is quite interesting that all of them seem quite difficult to prove; indeed, the sharp constant 1 for the centered maximal function on the real line is still merely conjectural.\
*Stokes wave.* Equations of the type appearing in our proof seems to have previously surfaced in a completely different context: in a 1987 paper on on the behaviour of the Stokes wave of extreme form near its crest, Amick & Fraenkel [@amick] encounter the equation $$\sqrt{3}(1+z) = \tan{\left( \frac{\pi}{2} z\right)} \qquad z \in \mathbb{C}$$ and require statements about the linear independence of the solutions of such an equation. All solutions $z_1, z_2, \dots$ with $Re(z) > -1$ are simple and the Amick-Fraenkel conjecture says that $$\left\{ 1, z_1, z_2, \dots \right\} \qquad \mbox{is linearly independent over the rationals}.$$ Shargorodsky [@shar] showed that this is implied by the Schanuel conjecture. Our proof encounters similar issues but can be unconditionally resolved using the Lindemann-Weierstrass theorem.
Proofs
======
Outline.
--------
This section contains all the proofs. We first prove Theorem 1 and then show how it implies Theorem 2 and Theorem 3. The proof of Theorem 1 uses an expansion into Fourier series and the fact that averaging over intervals acts diagonally on the Fourier basis. This implies that a particular Fourier series has to vanish identically which implies that all Fourier coefficients have to vanish identically – this can be reduced to a system of ’diophantine’ equations (over $\mathbb{N} \times \mathbb{N} \times \mathbb{R}_{+}$). Using arguments from transcendental number theory, we can show that this system only has the trivial solution, which implies that the function has to be localized at one point in the frequency spectrum. The latter part of the argument is exploiting the arising structure in a very particular way and seems to only work in the very special case we are considering.
Proof of Theorem 1.
-------------------
Suppose that $f \in C^{\alpha}(\mathbb{R}, \mathbb{R})$ for some $\alpha > 1/2$ is periodic. Without loss of generality, we can use the symmetries of the statement to assume that the function has mean value 0 and the smallest period is $2\pi$ and that it can be written as $$f(x) = \sum_{k =1}^{\infty}{a_k \sin{k x} + b_k \cos{k x}}.$$ We assume now that $$\left( \partial_r \frac{1}{2r}\int_{x-r}^{x+r}{f(z)dz} \right) \big|_{r = \gamma} = 0 \qquad \mbox{for all}~x \in \mathbb{R}.$$ The trigonometric identities $$\begin{aligned}
\sin{(x+r)} - \sin{(x-r)} &= 2\sin{r}\cos{x} \\
\cos{(x+r)} - \cos{(x-r)} &= -2 \sin{r} \sin{x}\end{aligned}$$ yield that $$\frac{1}{2r}\int_{x-r}^{x+r}{f(z)dz} = \sum_{k=1}^{\infty}{\frac{\sin{rk}}{rk}\left(a_k\sin{kx} + b_k\cos{kx}\right)}.$$ Here we invoke the classical theorem of Bernstein (see, e.g. [@katz]) stating that periodic functions in $f \in C^{\alpha}(\mathbb{R}, \mathbb{R})$ for some $\alpha > 1/2$ have an absolutely convergent Fourier series. Furthermore, this allows us to interchange the sum and derivative with respect to $r$ $$0 = \partial_r \frac{1}{2r}\int_{x-r}^{x+r}{f(z)dz} \big|_{r = \gamma} = \sum_{k=1}^{\infty}{\frac{\gamma k \cos{\gamma k} - \sin{\gamma k}}{\gamma^2 k}\left(a_k\sin{kx} + b_k\cos{kx}\right)}$$ because $$\left| \frac{\gamma k \cos{\gamma k} - \sin{\gamma k}}{\gamma^2 k} \right| \leq \frac{\gamma k + 1}{\gamma^2 k} \leq \frac{\gamma+1}{\gamma^2}$$ and therefore $$\sum_{k=1}^{\infty}{\left|\frac{\gamma k \cos{\gamma k} - \sin{\gamma k}}{\gamma^2 k}\left(a_k\sin{kx} + b_k\cos{kx}\right)\right|} \leq \frac{\gamma+1}{\gamma^2}\sum_{k=1}^{\infty}{\left|a_k\sin{kx} + b_k\cos{kx}\right|} < \infty.$$ The only way for a Fourier series to vanish everywhere is for all the coefficients to vanish. Note that $\gamma > 0$ and therefore $$\frac{\gamma k \cos{\gamma k} - \sin{\gamma k}}{\gamma^2 k} = 0 \Leftrightarrow \gamma k = \tan{\gamma k}.$$ For any fixed $k$, it is certainly possible to choose $\gamma$ in such a way that the equation is satisfied. It remains to show that no two such equations can be satisfied at the same time. We prove this by contradiction and assume now that $$a_k^2 + b_k^2 > 0 \qquad \mbox{for at least two different values of}~k \in \mathbb{N}.$$ This would imply the existence of a solution $(\gamma, m, n) \in \mathbb{R} \times \mathbb{N} \times \mathbb{N}$ $$\begin{aligned}
\tan{\gamma m} &= \gamma m \\
\tan{\gamma n} &= \gamma n\end{aligned}$$ with $\gamma > 0$ and $m \neq n$. If we could derive a contradiction from this assumption, it would imply that, independently of the value $\gamma$, $$a_k^2 + b_k^2 > 0 \qquad \mbox{can hold for at most one value of}~k \in \mathbb{N}$$ from which the statement follows since then $$f(x) = a_k\sin{kx} + b_k\cos{kx}.$$ It is not surprising that number theory enters here: one way of rephrasing the problem is that any two elements in the set $$\left\{ z \in \mathbb{R}_{+}: \tan(z) = z \right\} \qquad \mbox{are linearly independent over}~\mathbb{Q}.$$ The rest of the argument can be summarized as follows: the tangent has the powerful property of sending nonzero algebraic numbers to transcendental numbers. Any nonzero solution $\gamma \in \mathbb{R}$ of the equation $\tan{(\gamma m)} = \gamma m$ must therefore be transcendental, which means that it is never the root of a polynomial with rational coefficients. Using multiple angle formulas for the tangent, the assumption of any nontrivial solution $\gamma$ satisfying two of these equations at the same time allows to construct an explicit polynomial for which $\gamma$ is a root – this contradiction will conclude the proof. We start with the cornerstone of the argument.
> **Claim** (taken from [@mor])**.** *If $x \neq 0$ is algebraic over ${\mathbb{Q}}$, then $\tan{x}$ is transcendental over ${\mathbb{Q}}$.*
>
> Suppose $\tan{x}$ is algebraic over $\mathbb{Q}$, then we would have that for some $n \in \mathbb{N}$ and some $r_k \in \mathbb{Q}$ $$\sum_{k = 0}^{n}{r_k (\tan{x})^k} = 0.$$ We rewrite $x$ using the exponential function $$\tan{x} = \frac{1}{i}\frac{e^{ix} - e^{-ix}}{e^{ix} + e^{-ix}}.$$ Inserting this expression and multiplying by $(e^{ix} + e^{-ix})^n$ on both sides allows us deduce that $$\sum_{k = 0}^{n}{r_k \left( \frac{e^{ix} - e^{-ix}}{i}\right)^k(e^{ix} + e^{-ix})^{n-k} } = 0.$$ Expanding all brackets, we may deduce that $$\sum_{k = -n}^{n}{r_k^*e^{i k x} } = 0$$ for some $r_k^* \in \mathbb{Q}[[i]]$ not all of which are 0. We invoke the Lindemann-Weierstrass theorem in the formulation of Baker [@bak]: if $ b_0, b_1, \dots, b_m$ are non-zero algebraic numbers and $\beta_0, \beta_1, \dots, \beta_m$ are distinct algebraic numbers, then $$b_0 e^{\beta_0} + b_1 e^{\beta_1} + \dots + b_m e^{\beta_m} \neq 0$$ and this contradiction completes our proof.
We now prove a little statement showing that integer multiples of fixed points $\tan{x} = x$ have a well-defined tangent. Equivalently, we want to guarantee that if $\tan{\gamma} = \gamma$, then $n \gamma = (m+1/2)\pi$ has no solutions $(n,m) \in \mathbb{N}^2$.
> **Claim.** *If $\gamma > 0$ satisfies $\tan{\gamma n} = \gamma n$ for some $n \in \mathbb{N}_{>0}$, then $\gamma$ and $\pi$ are linearly independent over $\mathbb{Q}$.*
>
> Suppose that the statement fails and $$\gamma n =(p/q)\pi.$$ Note that, by definition, $$\tan\left(\tan\left( \pi p/q \right) \right) = \tan\left(\tan\left(\gamma n \right) \right) = \tan{\gamma n} = \tan\left( \pi p/q \right),$$ It is known that $ \tan\left( \pi p/q \right)$ is an algebraic number (even the degree of the minimal polynomial is known, see [@ca]). This would be an instance of the tangent mapping the nonzero algebraic number $\tan\left( \pi p/q \right)$ to an algebraic number, which is a contradiction to the statement proven above.
Suppose now that $(\gamma, m, n) \in \mathbb{R} \times \mathbb{N} \times \mathbb{N}$ is a nontrivial solution of $$\begin{aligned}
\tan{\gamma m} &= \gamma m \\
\tan{\gamma n} &= \gamma n.\end{aligned}$$ Then $\gamma$ has to be transcendental: if $\gamma$ were algebraic, then $\gamma m$ would be algebraic from which we could deduce that $\tan{\gamma m}$ is transcendental, which contradicts $\tan{\gamma m} = \gamma m$. Now in order to derive a final contradiction exploiting the fact that $\gamma$ is transcendental we use an addition theorem for the tangent: $$\tan{((n+1) x)} = \tan{(n x + x)} = \frac{\tan{n x} + \tan{x}}{1-\tan{nx}\tan{x}}.$$ Iterating this multiple-angle formula, we have $$\tan{n x} = \frac{p_n(\tan(x))}{q_n(\tan(x))}$$ for two sequences of polynomials with integer coefficients satisfying the initial conditions $p_1(x) = x$ and $q_1(x) = 1$ and the recursion formulas $$\begin{aligned}
p_{n+1}(x) &= p_n(x) + x q_n(x) \\
q_{n+1}(x) &= q_n(x) - xp_n(x).\end{aligned}$$ We know that $(\gamma, m, n) \in \mathbb{R}_{>0} \times \mathbb{N} \times \mathbb{N}$ solves $$0 = n\tan{\gamma m} - m\tan{\gamma n} = n\frac{p_m(\tan{\gamma})}{q_m(\tan{\gamma})} - m\frac{p_n(\tan{\gamma})}{q_n(\tan{\gamma})}$$ and therefore $$0 = n q_n(\tan{\gamma}) p_m(\tan{\gamma}) - m q_m(\tan{\gamma}) p_n(\tan{\gamma}).$$ It is easy to see that the polynomial on the right-hand side does not vanish identically by checking that $$\frac{d^3}{dx^3}n \tan{x m} - m \tan{x n}\big|_{x=0} = \frac{1}{3}(nm^3 - mn^3) \neq 0.$$ This implies that $\tan{\gamma}$ satisfies a polynomial equation with integer coefficients and thus $\tan{\gamma}$ is algebraic, which is a contradiction.
Theorem 1 implies Theorem 2.
----------------------------
Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a periodic function of regularity $C^{\alpha}$ with $\alpha > 1/2$ such that $$\left| \bigcup_{x \in \mathbb{R}}{ \left\{ r_f(x), r_{-f}(x) \right\} } \right| \leq 2.$$ Using the symmetries of the maximal function, we may assume without loss of generality that $f$ is periodic with period $2\pi$ and has vanishing mean value. Let us first assume that $$\left| \bigcup_{x \in \mathbb{R}}{ \left\{ r_f(x), r_{-f}(x) \right\} } \right| = 1.$$ Since $f$ is periodic, it assumes a global maximum from which it follows that if $r_f$ were to be constant, it would have to be 0 from which it follows that $f$ is constant and the statement holds. Thus we can focus on the remaining case of $r_f$ and $r_{-f}$ being two-valued (by the same reasoning 0 always has to be one of the two values): $$\bigcup_{x \in \mathbb{R}}{\left\{r_f(x), r_{-f}(x)\right\}} = \left\{0, \gamma\right\} \qquad \mbox{for some real number} ~\gamma > 0.$$ We are now in the case where $f$ is continuous, non-constant and has vanishing mean: this allows to partition $\mathbb{R}$ into three nonempty sets $$\begin{aligned}
I_1 &= \left\{x \in \mathbb{R}: f(x) < 0\right\} \\
I_2 &= \left\{x \in \mathbb{R}: f(x) = 0\right\} \\
I_3 &= \left\{x \in \mathbb{R}: f(x) > 0\right\}.\end{aligned}$$ We will now prove that $$\left( \partial_r \frac{1}{2r}\int_{x-r}^{x+r}{f(z)dz} \right) \big|_{r = \gamma} = 0 \qquad \mbox{for all}~x \in \mathbb{R}.$$ This is easy to see on $I_1$: if $x \in I_1$ and $r_{f}(x) = 0$, then the maximal function would have the value $f(x) < 0$. However, by taking the maximal interval of length $2\pi$, we can at least get an average value of 0, which exceeds $f(x)$. This implies that $r_{f}(x) = \gamma,$ which implies the statement. A similar argument works for $x \in I_3$, where the same reasoning implies $r_{-f}(x) = \gamma$, which implies $$\left( \partial_r \frac{1}{2r}\int_{x-r}^{x+r}{-f(z)dz} \right) \big|_{r = \gamma} = 0$$ and gives the statement after multiplication with $-1$. For $x \in I_2$ we have to argue a bit differently: suppose $r_f(x) = 0$. Then let us consider the function $h:[0, \pi] \rightarrow \mathbb{R}$ $$h(r) = \frac{1}{2r}\int_{x-r}^{x+r}{f(z)dz}.$$ By assumption, we have that $h(0) = h(\pi) = 0$ and $h(r) \leq 0$. If $h$ vanishes identically, the derivative vanishes everywhere and in particular also in $\gamma$. Suppose now that $h$ does not vanish identically, then it assumes a global minimum on that interval. By definition, this implies that $r_{-f}(x) > 0$ and thus by assumption $r_{-f}(x) = \gamma$ and this implies the statement as before – this completes the reduction of Theorem 1 to Theorem 2.\
Theorem 1 implies the Theorem 3.
--------------------------------
Let $\gamma > 0$ be fixed and let $f \in C^1(\mathbb{R}, \mathbb{R})$ be a periodic solution of the delay differential equation $$f'(x+\gamma) - \frac{1}{\gamma}f(x+\gamma) = -f'(x-\gamma) - \frac{1}{\gamma}f(x-\gamma).$$ This can be rephrased as $$f'(x+\gamma) + f'(x-\gamma) = \frac{1}{\gamma}\left(f(x+\gamma) - f(x-\gamma)\right).$$ Integrating with respect to $x$ on both sides yields $$f(x+\gamma) + f(x-\gamma) = \frac{1}{\gamma}\int_{x-\gamma}^{x+\gamma}{f(z)dz} + c,$$ where $c \in \mathbb{R}$ is some undetermined constant. Since $f$ is periodic with some period $P$, we can deduce that the average value of the left-hand side of the equation is precisely $$\lim_{y \rightarrow \infty}{\frac{1}{y} \int_{0}^{y}{f(x+\gamma) + f(x-\gamma) dx}} = \frac{2}{P}\int_{0}^{P}{f(x)dx}.$$ On the other hand $$\lim_{y \rightarrow \infty}{\frac{1}{y} \int_{0}^{y}{ \left( \frac{1}{\gamma}\int_{x-\gamma}^{x+\gamma}{f(z)dz} \right)+ c~dx}} = c + \frac{2}{P}\int_{0}^{P}{f(x)dx}$$ and thus $c=0$. Thus, multiplying the equation with $(2\gamma)^{-1}$, we get $$0 = \frac{1}{2\gamma}(f(x+\gamma) + f(x-\gamma)) - \frac{1}{2\gamma^2}\int_{x-\gamma}^{x+\gamma}{f(z)dz} = \left( \partial_r \frac{1}{2r}\int_{x-r}^{x+r}{f(z)dz} \right) \big|_{r = \gamma}.$$ This is precisely the condition in Theorem 2 (with slightly higher regularity on $f$) and implies the result.
Concluding remarks
==================
A conjectured stronger statement.
---------------------------------
It seems reasonable to assume that for a periodic $C^1-$function $f:\mathbb{R} \rightarrow \mathbb{R}$ already $$\left| \bigcup_{x \in \mathbb{R}}{\left\{r_f(x)\right\}} \right| \leq 2$$ implies that $f$ has to be a trigonometric function, i.e. that it suffices to demand that the computation of $\mathcal{M}f$ is ’simple’ (in the sense described above) and not additionally that the computation of $\mathcal{M}(-f)$ be simple as well. After adding a suitable constant we can assume w.l.o.g. that $f$ has vanishing mean and, by using the dilation symmetry, that $r_{f}(x) \in \left\{0, 1\right\}$. Then we know that $r_f(x) = 1$ whenever $f(x) < 0$ and that therefore $$\partial_x \left( \partial_r \frac{1}{2r} \int_{x-r}^{x+r}{f(y) dy} \big|_{r=1} \right) = 0 \qquad \mbox{whenever}~f(x) < 0.$$ The same explicit computation as before implies that one could derive the following statement.
> *Conjecture.* Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is $C^1$ and satisfies $$f'(x+1) - f(x+1) = -f'(x-1)-f(x-1) \qquad \mbox{whenever}~f(x) < 0.$$ Then $$f(x) = a+b\sin{(cx + d)} \qquad \mbox{for some}~a,b,c,d \in \mathbb{R}.$$
This statement would be a *quite* curious strengthening of Theorem 3.
A Poincaré inequality.
----------------------
The purpose of this short section is to note a basic observation for the uncentered maximal function (which fails for the centered maximal function): if the uncentered maximal intervals are all rather short, then this should imply the presence of strong oscillation in the function. We give a very simple form of that statement. Let $f \in C^1([0,1])$ and consider the uncentered maximal function $\mathcal{M}^*$ defined via $$(\mathcal{M}^*f)(x) = \sup_{J \ni x}{\frac{1}{|J|}\int_{J}{f(x)dx}},$$ where $J$ ranges over all intervals $J \subset [0,1]$ containing $x$. We define analogously $r^*_f(x)$ as the length of the shortest interval necessary to achieve the maximal possible value.
(0.7,0) circle \[radius=0.02\]; (0.7,-0.5) circle \[radius=0.015\]; (0.2,-0.5) circle \[radius=0.015\]; (0.2,-0.5) – (0.7,-0.5); (0,0) – (2,0); (0,0) – (0,1); plot (,[0.4\*sin(400\*)]{}); at (0,-0.4) [$J$]{}; at (0.7,0) [$x$]{};
(3.4,0) circle \[radius=0.02\]; (3.4,-0.5) circle \[radius=0.015\]; (3.2,-0.5) circle \[radius=0.015\]; (3.15,-0.5) – (3.4,-0.5); (3,0) – (5,0); (3,0) – (3,1); plot (,[0.4\*sin(600\*-1800)]{}); (3.6,0) – (5,0); at (3,-0.4) [$J$]{}; at (3.4,0) [$x$]{};
It is clear from examples that, in a loose sense, strong oscillation implies that $r^*_f(x)$ is small: it will be optimal to choose the interval to be either a point evaluation or in such a way that one captures the one or two adjacent large amplitudes. An inverse result can be quantified as follows.
Assume $f \in C^1([0,1])$ has mean value $\overline{f}$. Then we have the Poincaré inequality $$\int_{0}^{1}{\left| f(x) - \overline{f} \right| dx} \leq 4\| r^*_f \|_{L^{\infty}(\mathbb{R})} \int_{0}^{1}{|f'(x)|dx}.$$
If we were to replace $4\|r^*_f\|_{L^{\infty}}$ by the constant $1/2$, this would be the classical $L^1-$Poincaré inequality on $[0,1]$. Put differently, for a function $f:[0,1] \rightarrow \mathbb{R}$ with vanishing mean, we have an uncertainty relation between the total variation and $ \|r^*_f\|_{L^{\infty}}$ $$\mbox{var}(f) \|r^*_f\|_{L^{\infty}} \geq \frac{1}{4}\int_{0}^{1}{|f(x)|dx}.$$ It is easy to see that the statement has the sharp scaling: consider $$f(x) = \sin{N \pi x} \quad \mbox{where} \qquad \|r^*_f \|_{L^{\infty}} \sim N^{-1} ~ \mbox{and} ~\| f'\|_{L^1} \sim N.$$ Another example is given by taking a positive bump function $\phi \in C^{\infty}_{c}(0,1)$ and consider the rescaled function $$f(x) = a \phi'(b x) \quad \mbox{where} \qquad \int_{0}^{1}{\left| f(x) - \overline{f} \right| dx} \sim a b^{-1}, \|r^* \|_{L^{\infty}} \sim b^{-1} ~ \mbox{and} ~\| f' \|_{L^1} \sim a.$$ We note that the result is not true for the centered maximal function: the function $f(x) = 1 - (x-0.5)^2$ (or, more generally, any strictly concave function) satisfies $r_f \equiv 0$.
We may suppose w.l.o.g. that $f$ has vanishing mean $\overline{f} = 0$. Now we write $$\int_{0}^{1}{\left| f(x) \right| dx} = 2\int_{0}^{1}{\chi_{f < 0}(x)|f(x)| dx}$$ and estimate the number on the right. It is easy to see that the set $$\left\{x \in [0,1]: f(x) < 0\right\}$$ cannot contain an interval of length larger than $2\|r^{*}_f\|_{L^{\infty}}$ because, since $\overline{f} = 0$, the maximal function is always nonnegative (one can always simply choose the entire interval). Clearly, for any $g \in C^1$, $$\int_{s}^{t}{|g(x)|dx} \leq (t-s)\int_{s}^{t}{|g'(x)| dx} \qquad \mbox{if}~g(s)=0.$$ We can now use that inequality on every connected component $ \left\{x \in [0,1]: f(x) < 0\right\}$: by the reasoning above, we will always have $t-s \leq 2\|r^{*}_f\|_{L^{\infty}}$ and therefore $$\int_{0}^{1}{\chi_{f < 0}|f| dx} \leq 2\|r^{*}_f\|_{L^{\infty}} \int_{0}^{1}{| f'(x)|dx},$$ which implies the statement.
It could be quite interesting to understand under which conditions and to which extent such improved Poincaré inequalities are true in higher dimensions and how they depend on the maximal function involved.\
**Acknowledgement.** I am grateful for discussions with Raphy Coifman and indebted to Lillian Pierce for a number helpful remarks, which greatly improved the manuscript. My interest in non-standard aspects of the maximal function traces back to a series of very enjoyable conversations with Emanuel Carneiro at the Oberwolfach Workshop 1340 – I am grateful to both him and the organizers of the workshop.
[4]{}
J. M. Aldaz and J. Pérez Lázaro, Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities. Trans. Amer. Math. Soc. 359 (2007), no. 5, 2443-2461.
C. J. Amick and L.E. Fraenkel, On the behavior near the crest of waves of extreme form. Trans. Amer. Math. Soc. 299 (1987), no. 1, 273-298.
A. Baker, Transcendental number theory. Cambridge University Press, London-New York, 1975.
J. Bober, E. Carneiro, K. Hughes and L. Pierce, On a discrete version of Tanaka’s theorem for maximal functions. Proc. Amer. Math. Soc. 140 (2012), no. 5, 1669-1680.
J. Calcut, Gaussian integers and arctangent identities for π. Amer. Math. Monthly 116 (2009), no. 6, 515-530.
E. Carneiro and K. Hughes, On the endpoint regularity of discrete maximal operators. Math. Res. Lett. 19 (2012), no. 6, 1245-1262.
E. Carneiro and J. Madrid, Derivative bounds for fractional maximal functions, to appear in Transactions of the American Mathematical Society, arXiv:1510.02965
E. Carneiro and B. Svaiter, On the variation of maximal operators of convolution type. J. Funct. Anal. 265 (2013), no. 5, 837-865.
Y. Katznelson, An introduction to harmonic analysis. Dover Publications, Inc., New York, 1976. xiv+264 pp.
J. Kinnunen, The Hardy-Littlewood maximal function of a Sobolev function. Israel J. Math. 100 (1997), 117-124.
O. Kurka, On the variation of the Hardy-Littlewood maximal function, arxiv:1210.0496.
A. Melas, The best constant for the centered Hardy-Littlewood maximal inequality. Ann. of Math. (2) 157 (2003), no. 2, 647-688.
P. Morandi, Field and Galois theory. Graduate Texts in Mathematics, 167. Springer-Verlag, New York, 1996.
D. Pompeiu, Sur certains systemes d’équations linéaires et sur une propriété intégrale des fonctions de plusieurs variables, Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 188 (1929): 1138–1139
E. Shargorodsky, On the Amick-Fraenkel conjecture. Q. J. Math. 65 (2014), no. 1, 267-278.
E. Stein Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970
H. Tanaka, A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function. Bull. Austral. Math. Soc. 65 (2002), no. 2, 253-258.
F. Temur, On regularity of the discrete Hardy-Littlewood maximal function, arXiv:1303:3993.
T. Wolff, Recent work connected with the Kakeya problem. Prospects in mathematics (Princeton, NJ, 1996), 129-162, Amer. Math. Soc., Providence, RI, 1999.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We review the recent developments in the field of cuprate superconductors with the special focus on the recently observed charge order in the underdoped compounds. We introduce new theoretical developments following the study of the antiferromagnetic (AF) quantum critical point (QCP) in two dimensions, in which preemptive orders in the charge and superconducting (SC) sectors emerged, that are in turn related by an SU(2) symmetry. We consider the implications of this proliferation of orders in the underdoped region, and provide a study of the type of fluctuations which characterize the SU(2) symmetry. We identify an intermediate energy scale where the SU(2) pairing fluctuations are dominant and argue that they are unstable towards the formation of a Resonant Peierls Excitonic (RPE) state at the pseudogap (PG) temperature $T^{*}$. We discuss the implications of this scenario for a few key experiments.'
author:
- 'T. Kloss$^{1}$, X. Montiel$^{1}$, V. S. de Carvalho$^{2}$, H. Freire$^{2}$, C. Pépin$^{1}$'
bibliography:
- 'Cuprates.bib'
date: 'December 11, 2015'
title: 'Charge orders, magnetism and pairings in the cuprate superconductors'
---
= 1
Introduction
============
The last decade has seen a strong revival of interest in cuprate superconductors, with the observation of charge orders in the underdoped regime of these materials. Maybe the starting point of this intense period of investigation was the observation by STM of checkerboard-type patterns inside the vortices in Bi-2212 [@Hoffman02; @Fujita14]. Subsequent studies with Fermi surface reconstruction showed that this feature was generic [@Vershinin:2004gk; @daSilvaNeto:2014vy] (also verified in Bi-2201 [@He14; @Wise08]) and that the charge patterns corresponded to two axial wave vectors $\left(0,Q_{y}\right)$ and $\left(Q_{x},0\right)$, incommensurate with the lattice periodicity, and the magnitude of the wave vectors decreases with oxygen-doping. The charge excitation was also found to be non-dispersive in temperature, and correlated with the “hot-spots” - the points of the Fermi surface where the AF zone boundary is intersected. The picture refined itself a bit later, and we now believe the charge order emerges at the tip of the Fermi arcs [@Hamidian15; @Hamidian15a]. The study of charge order in underdoped cuprates stayed in a *status quo* until the observation of quantum oscillations (QO) under a strong magnetic field in YBCO [@Doiron-Leyraud07; @LeBoeuf07]. This result pointed directly to a reconstruction of the Fermi surface induced by magnetic field and received several explanations in terms of stripe and charge patterns until the link was made with the bi-axial charge order observed by STM [@Doiron-Leyraud13; @Sebastian10; @Sebastian12; @Cyr15a]. In particular, models for the reconstruction of the Fermi surface involved charge ordering with bi-axial wave vectors similar to those unveiled by STM. A subsequent Nuclear Magnetic Resonance (NMR) study finally found some charge splitting under a magnetic field $B\geq17$T, which brought the final confirmation that charge order under a finite magnetic field is coherent, static and long-ranged [@Wu11; @Wu13a; @Wu:2015bt]. The field versus temperature phase diagram was later completed by ultrasound experiments, which showed evidence for a flat transition line at $B_{c}=17$T [@LeBoeuf13]. For $B\leq B_{c}$, YBCO is a $d$-wave superconductor. The increase of the magnetic field then creates vortices whose cores show the typical charge ordering [@Wu13a]. For $B\geq B_{c}$ YBCO shows long range charge order with a typical ordering temperature remarkably similar in magnitude with the SC $T_{c}$. In the PG regime at $B=0,$ both hard x-ray [@Chang12; @Blackburn13a] and soft x-rays [@Ghiringhelli12; @LeTacon11; @Blanco-Canosa13; @Blanco-Canosa14] study showed the presence of a sizable short range excitation at the bi-axial wave vectors. A softening of the phonon spectrum has been observed in the PG phase, while a softening at the charge order wave vectors occurs below $T_{c}$ [@Blackburn13b; @LeTacon14]. Note that a recent state-of-the-art x-ray experiment at $B=17$T showed that the charge order becomes uni-axial and tri-dimensional at high field [@Chang16]. One preliminary conclusion that one can infer from these experiments is that the charge and superconducting sectors are of the same order of magnitude in cuprate superconductors. We will use this observation later when we introduces the emergent SU(2) rotations that appear between these two sectors.
We turn now to one of the most enduring mysteries of cuprate superconductors, the pseudogap (PG) regime. The PG phase was observed in 1989 by NMR experiments [@Alloul89; @Warren89], where a gradual drop in the Knight-shift was observed at a crossover temperature $T^{*}$. This gap was attributed to a loss of density in the electronic carriers, and it was shown to decrease when the oxygen-doping increases but no obvious symmetry breaking was associated with this phase transition. We focus here on a few properties of the PG phase which we will use later in the SU(2)-interpretation of the experiments. The first remark that one can make is that the PG is an extremely robust feature of the phase diagram. It seems insensitive to disorder [@Alloul:2010ko; @RullierAlbenque:2000fl] and magnetic field [@RullierAlbenque:2007bm] and is closely associated to a regime of linear-in-$T$ resistivity on its right hand side [@Hussey:2008tw; @Hussey:2011kp; @husseycupratescriticality]. One other very striking observation in the PG regime detected by angle-resolved photoemission (ARPES), is the formation of Fermi arcs, instead of a closed-contour Fermi surface [@Campuzano98; @Campuzano99; @Chatterjee06; @Kanigel07; @Shen:2004ek; @Shen:2005ir]. Recently, a momentum scale of similar magnitude as the one observed in the CDW was associated to the opening of the PG in the anti-nodal region of the Brillouin zone (BZ) [@Shen05; @He11; @Yoshida:2012kh], and led to an interpretation in terms of a pair-density-wave (PDW) [@Lee14; @Wang15b]- or a finite momentum superconducting state Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) [@Fulde:1964dq; @Larkin65]. Coherent neutron scattering showed a $\mathbf{Q}=0$ signal [@Fauque06; @Li08; @Bourges11; @Sidis13; @Mangin_Thro14; @ManginThro:2015fg], which was interpreted in terms of intra-unit-cell loop currents [@Varma06; @AjiVarma07; @AjiVarma07a]. Although a $\mathbf{Q}=0$ phase is unable to open a gap in the electronic density of states, the loop-current line surprisingly follows the $T^{*}$-line. Note that NMR [@Roos2011; @Mounce2013] and $\mu$SR [@MacDougall08; @Sonier2009] techniques were not able to detect such loop current. An explanation could be the longer time scale of local probes ($\approx 10^{-8}-10^{-6}s$) compared with the INS time scale ($\approx 10^{-11}s$). At lower temperature, a Kerr effect signal has been reported, hinting at a breaking of time-reversal (TR) symmetry inside the PG [@Xia08]. This last observation is widely discussed by the community, but it is necessarily related to the $\mathbf{Q}=0$ loop currents [@Aji:2013eo; @Yakovenko:2015fd]. The inelastic neutron scattering (INS) is also interesting for revealing collective modes of the system. A resonance at $41$meV was found in YBCO the early days of cuprate superconductivity [@Rossat] and at similar energies in other compounds [@Bourges2005; @Sidis2001; @Fong99; @He02]. It was first believed that this collective excitation existed only in the SC phase, where it has a typical “hour-glass” shape centered around $41$ meV at $\mathbf{Q}=\left(\pi,\pi\right)$, as a function of energy and wave-vector. It was later shown that the resonance exists as well in the PG phase above $T_{c}$, where it is still centered around $41$meV, but shows a typical “Y”-shape with a long energy-extension at $\mathbf{Q}=\left(\pi,\pi\right)$[@Bourges19052000; @Hinkov04; @Hinkov07; @Hayden04]. Many theoretical approaches have been invoked to describe the resonance below the SC transition [@Norman07; @Eschrig:2000bf; @Eremin:2005ba; @Demler95]. This observation of the resonance around similar typical energies in the SC and PG phases, however, has never received a theoretical description, and constrains theories of the PG to keep some reminiscence of the SC phase. The neutron resonance was also observed in mono-layer tetragonal compounds (Hg-1201), where the long energy extension at $\mathbf{Q}=\left(\pi,\pi\right)$ persists below $T_{c}$ [@Tabis14].
Collective modes of a material give useful insights to probe symmetries of an effective model. One example is a resonance observed in the Raman $A_{1g}$ channel, that appears at energies very similar to the ones where a collective mode was observed by INS [@Gallais04; @Blanc:2009vo; @Sacuto2015TS]. Raman scattering typically probes the symmetries of the Fermi surface and the presence of “two gaps” in the underdoped regime of the cuprates was observed below $T_{c}$ [@LeTacon06; @Civelli:2008vr; @Blanc:2010tm]. This fact was corroborated in a series of ARPES experiments on BSCO from which the gap velocity $v_{\Delta}$ at the nodes was extracted and shown to differ from the Fermi velocity. Three regions in the phase diagram were identified [@Vishik12]. Starting from the over-doped region and decreasing the doping, $v_{\Delta}$ is shown to first increase then to reach a plateau in the underdoped region -down to dopings of the order of 5 %, and after that it drops at lower dopings when the system gets close to the insulating Mott-transition. The key question associated with the PG phase is whether it is a “strong-coupling” phenomenon, emerging as a direct consequence of the Mott transition [@Anderson87; @Georges96; @Katanin04; @Sordi:2012jc; @Gull:2013hh], or whether it is a a very unusual collective phenomenon which is sensitive to other peculiarities of the physics of the cuprates, like its low dimensionality, the antiferromagnetic fluctuations or its fermiology [@LeHur:2009iw; @Rice12; @Chowdhury:2015gx; @Kloss15a]. In this work, we argue that the key to explain the mystery of the PG phase resides in an underlying emergent SU(2) symmetry, which produces SU(2) pairing fluctuations at intermediate energy scales. These fluctuations are in turn unstable toward the formation of a new kind of excitonic state, the (RPE) state, which is responsible for gapping out the Fermi surface in the anti-nodal region of the BZ [@Kloss15a].
The paper is organized as follows: In section \[sectionSU2\], we present the basics of the emergent symmetry model with SU(2) symmetry. Section \[sectionfluc\] discusses the competition between the U(1) and SU(2) paring fluctuations in the framework of the non linear $\sigma$ model. In particular, we propose to explain the PG state as a new type of charge order: the Resonant Peierls Excitonic (RPE) state coming from the SU(2) fluctuations. We also demonstrate that the CDW state is a secondary instability produced by U(1) fluctuations mediated by a Leggett mode. In section \[sectionDiscussion\], we discuss the possible experimental evidence of this phase before to conclude in section \[conclusion\].
The emergent SU(2) symmetry {#sectionSU2}
===========================
The concept of emergent symmetry in the context of the cuprate superconductors can be traced back to the work of Yang and later Zhang [@Yang89; @Yang:1990cf] where a representation with pseudo-spin operators was introduced which rotated the $d$-wave SC state onto a $d$-wave bi-partite charge order. The lowering and raising pseudo-spin operators $\eta^{+},$ $\eta^{-}=\left(\eta^{+}\right)^{\dagger}$, and $\eta_{z}$, which follow from the definition
\[eq:1\] $$\begin{aligned}
\eta^{+} & =\sum_{\mathbf{k}}c_{\mathbf{k}\uparrow}^{\dagger}c_{-\mathbf{\mathbf{k}+Q}\downarrow}^{\dagger}\\
\eta_{z} & =\sum_{\mathbf{k}}\left(c_{\mathbf{k}\uparrow}^{\dagger}c_{\mathbf{k}\uparrow}+c_{\mathbf{k+Q}\downarrow}^{\dagger}c_{\mathbf{k+Q}\downarrow}-1\right).\end{aligned}$$
The operators (\[eq:1\]) form an SU(2) Lie algebra. Noticeably, the $\eta$-pairing stat – which is equivalent to finite center of mass pairing of vector $\mathbf{Q}=\left(\pi,\pi\right)$, or FFLO state – is an eigenstate of the Hubbard Hamiltonian, both for positive and negative U. The simplest irreducible representation for the pseudo-spin is the triplet vector $\Delta_{m},$ with $m=\left\{ -1,0,1\right\} $ defined as
\[eq:1-1\] $$\begin{aligned}
\Delta_{1} & =-\frac{1}{\sqrt{2}}\sum_{\mathbf{k}}c_{\mathbf{k}\uparrow}^{\dagger}c_{-\mathbf{k}\downarrow}^{\dagger},\\
\Delta_{0} & =\frac{1}{2}\sum_{\mathbf{k},\sigma}c_{\mathbf{k}\sigma}^{\dagger}c_{\mathbf{k+Q}\sigma},\\
\Delta_{-1} & =-\Delta_{1}^{\dagger},\end{aligned}$$
which represents the two conjugated s-wave SC states $\left( \Delta_{-1} , \Delta_{1} \right)$ and the charge ordering state $ \Delta_{0}$. The SU(2) pseudo-spin operators (\[eq:1\]) rotate each component of the multiplet (\[eq:1-1\]) into one another in the standard way ($l$ is the rank of the irreducible representation Eq. (\[eq:1-1\]), here $l=1$)
\[eq:3\] $$\begin{aligned}
\left[\eta^{\pm},\Delta_{m}\right] & =\sqrt{l\left(l+1\right)-m\left(m\pm1\right)}\Delta_{m\pm1},\\
\left[\eta_{z},\Delta_{m}\right] & =m\Delta_{m}.\end{aligned}$$
The effective theory describing the pseudo-spin symmetry is the $SO\left(4\right)$ [\[]{}$SO\left(4\right)=(SU\left(2\right)\times SU\left(2\right))/Z_{2}$[\]]{} non-linear $\sigma$-model which excites thermally from the SC state to the ordered state. This model describes transitions from one state to the other within the generic framework of “spin-flop” transitions. In the case above one has a pseudo spin-flop from the s-wave SC to the CDW states, whereas the standard spin-flop transition from easy axis to easy plane belongs to the SO(3) group [@Zhang:1997ew]. The concept of SU(2)-symmetry was used later on in an effective theory of the PG leading to a rotation from the $d$-wave superconductor to the d-density wave state [@Nayak00]. Here the generators of the symmetry are simply $i\eta^{+}$, $i\eta^{-}$ and $\eta^{z}$ and the effective theory is the $O\left(4\right)$ non-linear $\sigma$-model. Let us mention a similar rotation between the nematic $d$-wave bond order $ \Delta_{nem} =\frac{1}{2}$ $\sum_{\mathbf{k}\sigma}d_{\mathbf{k}} c_{\mathbf{k}\sigma}^{\dagger}c_{\mathbf{k}\sigma} $ and $d$-wave states $ \Delta_{dsc}^{+} =-\frac{1}{\sqrt{2}}\sum_{\mathbf{k}}d_{\mathbf{k}} c_{\mathbf{k}\uparrow}^{\dagger}c_{\mathbf{k}\downarrow}^{\dagger} $, and $ \Delta_{dsc}^{-} =\frac{1}{\sqrt{2}}\sum_{\mathbf{k}}d_{\mathbf{k}} c_{\mathbf{k}\downarrow}c_{\mathbf{k}\uparrow} $ where $d_{\mathbf{k}}=\cos k_{x}-\cos k_{y}$ [@Kee:2008gw]. The pseudo-spin generators in this case take the form $L^{+}$, $L^{-}=\left(L^{+}\right)^{\dagger}$, $L_{0}$ with $$L^{+}=\sum_{\mathbf{k}}c_{\mathbf{k}\uparrow}^{\dagger}c_{-\mathbf{k}\downarrow}^{\dagger},\qquad L_{0}=\sum_{\mathbf{k}\sigma}\left(c_{\mathbf{k}\sigma}^{\dagger}c_{\mathbf{k}\sigma}-1\right).\label{eq:1-2}$$ Note that the chemical potential couples to the generator $\eta^{z}$ (or $L_{0}$) and thus a finite chemical potential breaks the SU(2) symmetry in favor of the SC state.
![\[fig:so5\] (Color online) Schematic phase diagram of the SO(5) model [@zhang97]. Four types of scenarios are discussed in Ref. [@zhang97]: 1) a direct first order transition with a bi-critical point, 2) two second order transitions with an intermediate coexistence regime, 3) one single second order transition terminating at a QCP at zero temperature and 4) two second order transitions with a quantum disordered phase. Although the SO(5) symmetry is broken in scenario 1), 2) and 3) at zero temperature, thermal fluctuations lead to a restoration below the mean-field critical temperature $T_{MF}$. Adapted from Ref. [@zhang97].](fig1){width="70mm"}
Another rotation, this time from the SC state towards the AF state, was introduced early on and became famous as the SO(5) theory [@Zhang:1997ew; @Zhang:1999fe; @Demler04] . The SO(5) theory is the one of a non-linear $\sigma$-model which operates on a five state “superspin” $\left(\begin{array}{ccccc}
n_{1}, & n_{2}, & n_{3}, & n_{4}, & n_{5}\end{array}\right)$- two SC states $\left(\begin{array}{cc}
n_{1}=\Delta_{s}, & n_{5}=\Delta_{s}^{\dagger}\end{array}\right)$ and three AF vectors $\left(\begin{array}{ccc}
n_{2}=s^{+}, & n_{3}=s^{-}, & n_{4}=s^{z}\end{array}\right)$ [@Zhang:1997ew]. The superspin $n_{a}$ is a vector representation of the SO(5) algebra. The SO(5) theory was based on the idea that both the SC and the AF states are key players of the physics of these compounds and are close enough in energy so that in between their respective phase transition an SO(5)-symmetric state is found where SC and AFM are undistinguishable. This phase was naturally associated with the PG of the cuprates. A typical SO(5) non-linear $\sigma$-model was introduced to describe the effective physics of the system, and four typical phase diagrams were derived which are depicted in Fig. \[fig:so5\]. The mechanism favoring one of the states in the non-linear $\sigma$-model can be understood as a spin-flop transition- also called “super spin flop” transition for the SO(5) symmetries. As mentioned above, one gets a very accurate picture by thinking of the spin-flop transition of the antiferromagnetic state in a uniform magnetic field $\mathbf{B}$ along the easy$z$-axis [@Chakravarty:1988uu; @Chakravarty89]. The magnetic field creates an easy plane $xy$, so that at a critical value of the field, the Néel wave vector changes its orientation abruptly from the $z$-axis to the $xy$-plane. Hence although in each of the above cases the symmetries are different, the underlying physics is as simple as the one on a spin-flop transition. The four typical phase diagrams show the various phases as a function of temperature and an external parameter which breaks the symmetry and are depicted in Fig. \[fig:so5\]. They correspond to the cases where: 1) a first order transition between the two states terminates at a bi-critical point; 2) there is a coexistence phase between the two orders; 3) the transition between the two orders terminates at $T=0$ at a quantum critical point (QCP) or 4) the two orders are disconnected. In cases 1), 2) and 3), although the symmetry is broken at zero temperature, thermal fluctuations restore the SO(5)- or SU(2)- symmetry, leading to an invariant phase under the mean field transition $T_{MF}$. In the case of the SO(5)-symmetry, the typical phase diagram of the cuprates has the shape depicted in case 4), where the two orders are disconnected from one another, and this situation leads to a splitting of the big group to the $SO(3)\times U(1)$ subgroups describing fluctuations associated to each order separately. Furthermore, the SO(5) symmetry led to the prediction of a giant proximity effect in a SC-AF-SC junction [@Demler:1998iw; @denHertog:1999dk]. Unfortunately, this proximity effect was never verified experimentally [@Bozovic:2003hx]. One reason invoked here was that the SC state is by nature itinerant while the AF state is an insulator in those compounds. Hence the typical energy difference between those two states is big, of the order of the Coulomb U.
In the present work, we revive the concept of emergent symmetry, with an the SU(2) symmetry which rotates from a $d$-wave SC state to an incommensurate $d$-wave charge order. The pseudo-spin generators have the same form as depicted in Eqn. (\[eq:1\]), with the exception that the wave vector $\mathbf{Q=Q_{0}}$ is now the charge order wave vector and is not necessarily commensurate with the lattice. The $l=1$ irreducible representation is given by the $d$-wave version of Eqn. (\[eq:1-1\]) with namely $\Delta_{0}=\chi_{CDW}$, $\Delta_{1}=\Delta_{dsc}^{\dagger}$and $\Delta_{-1}=-\Delta_{1}^{\dagger},$ namely
\[eq:5\] $$\begin{aligned}
\Delta_{1} & =-\frac{1}{\sqrt{2}}\sum_{\mathbf{k}}d_{\mathbf{k}}c_{\mathbf{k}\uparrow}^{\dagger}c_{-\mathbf{k}\downarrow}^{\dagger},\\
\Delta_{0} & =\frac{1}{2}\sum_{\mathbf{k},\sigma}d_{\mathbf{k}}c_{\mathbf{k}\sigma}^{\dagger}c_{\mathbf{k+Q}\sigma},\\
\Delta_{-1} & =\frac{1}{\sqrt{2}}\sum_{\mathbf{k}}d_{\mathbf{k}}c_{\mathbf{k}\downarrow}c_{-\mathbf{k}\uparrow}.\end{aligned}$$
The CDW ordering wave vector could be the axial CDW wave vector observed through many recent experiments ( STM, Quantum oscillations, X-rays, ARPES) or it could be another wave vector carefully chosen so that the SU(2) symmetry is fully respected. As it turns out, the Eight Hot Spots (EHS) model depicted in Fig. \[fig:ehsmodel\] provides an exact realization of such a symmetry, as was first mentioned in Ref.[@Metlitski10; @Metlitski10a; @Efetov13].
![\[fig:ehsmodel\] (Color online) Schematic representation of the first Brillouin zone with the Fermi surface of cuprate superconductors. Hotspots are located at the intersection points of the AFM zone boundary with the Fermi surface. The diagonal coupling vector $\mathbf{Q}_{diag}$ between two hotspots is shown in blue and the AFM ordering vector $\bf{Q}$ in magenta.](fig2){width="50mm"}
This model is a simplified version of the spin-fermion model, which describes the vicinity of an AF QCP within a metallic substrate [@Abanov00]. At this point it is useful to recall that the spin-fermion model played an important role at the beginning of the theoretical investigation on the cuprates [@Barzykin:1995fr; @Scalapino:2006uw; @Monthoux:2007ha]. For a recent link to strongly correlated systems note also Ref. [@Ferraz:2015voa]. Two different views were (and are still) competing for the understanding of the phase diagram of these compounds. Observing that the SC phase is close to a Mott insulator, a first group of theoreticians consider that the system is fundamentally strongly correlated, namely that the Coulomb energy $U=1$eV is affecting the qualitative behavior down to very low temperatures [@Anderson87; @Anderson04; @Gull:2013hh]. This viewpoint has been extensively developed around the resonating valence bond (RVB) suggestion made by Anderson as early as 1987 [@Anderson87], and now explored via extensive numerical calculations which can capture the strongly interacting behavior [@Gull:2013hh; @Sordi2012]. Another part of the physics community defends the viewpoint that the Mott transition has a strong qualitative influence up to 6-7% doping, beyond which the physics of the system is mainly driven by the presence of AF fluctuations [@Pines2002]. Proponents of this viewpoint hence start the theoretical investigation with the spin-fermion (SF) model which couples conduction electrons to AF paramagnon modes $\Phi=\left(\Phi^{x},\Phi^{y},\Phi^{z}\right)$ on the brink of criticality with the propagator $$\begin{aligned}
\left\langle \Phi_{\omega,\mathbf{q}}^{\alpha}\Phi_{-\omega,-\mathbf{q}}^{\beta}\right\rangle & =\frac{\delta_{\alpha\beta}}{c^{-2}\omega^{2}+\left(\mathbf{q-Q}\right)^{2}+\xi_{AF}^{-2}},\end{aligned}$$ where **Q** is the AF wave vector and $\xi_{AF}^{-1}$ is the effective mass of the paramagnons which defines the distance to the QCP. Note, however, that in the present theory the SF model can be considered as an effective theory for which the SU(2) symmetry is approximately verified -in the case of hot regions [@Kloss15], or exactly verified in the case of the EHS model [@Metlitski10; @Efetov13]. Although the proximity to the AF QCP may not be verified in the cuprates, we believe that the concept of emergent $SU(2)$ symmetry is robust and will remain when dust settles down. The conduction electrons have the kinetic energy $H_{K}=\sum_{k,\sigma}c_{k,\sigma}^{\dagger}\epsilon_{k}c_{k,\sigma}$ and interact with the paramagnons through a simple spin-spin interaction term $H_{int}=J\sum_{i}\mathbf{\Phi}_{i}\cdot c_{i}^{\dagger}\mathbf{\sigma}c_{i}$. In the EHS model, a further simplification is implemented with the reduction of the Fermi surface to eight “hot spots” which are the points at $T=0$ where the electrons scatter through the AF $\Phi$-modes. When the electron dispersion $\epsilon_{k}\simeq v_{hs}k$ is linearized at the hot spots, the model possesses an exact $SU(2)$ symmetry defined by the operators of Eqn. (\[eq:1\]) but with $Q=Q_{diag}$, being the diagonal wave vector depicted in Fig. \[fig:ehsmodel\]. This model was further studied in Ref.[@Efetov13] and an $SU(2)$ precursor of the AF state was found, where quadrupolar density wave (QDW) with diagonal wave vector, which is equivalent to a $d$-wave CDW with diagonal wave vector, is degenerate with the $d$-wave SC state. This new state can be understood as a kind of non-Abelian superconductor with order parameter $\hat{b}$ that, instead of having a U(1) phase, has an $SU(2)$ unitary matrix fluctuating between the charge and SC sector [@Efetov13; @Kloss15] $$\begin{aligned}
\hat{b} & =b\left(\begin{array}{cc}
\Delta_{CDW} & \Delta_{SC}\\
-\Delta_{SC}^{*} & \Delta_{CDW}^{*}
\end{array}\right)_{SU\left(2\right)}\end{aligned}$$ subject to the constraint $\left|\Delta_{CDW}\right|^{2}+\left|\Delta_{SC}\right|^{2}=1$. Within the framework of the EHS model, and the related $O\left(4\right)$ non linear $\sigma$-model, several experimental findings were successfully addressed [@Meier13; @Einenkel14; @Hayward14]. The general picture follows closely the ideas expressed in the $SO(5)$ theory, which are valid for all theories of emergent symmetries. A small curvature term in the electron dispersion breaks the symmetry in favor of the SC state. Hence at $T=0$ the system is a superconductor. Once the temperature is raised, thermal fluctuations then excite the system between the two pseudo-spin states, restoring the $SU(2)$ invariance below the PG dome. Conversely, an applied magnetic field breaks the $SU(2)$ symmetry in favor of the CDW state and beyond a certain critical field $B_{c}$, a “pseudo spin-flop” is observed where the ground state “flips” from the SC state to CDW order. This “pseudo spin-flop” was precisely observed in experiments performed under magnetic field, with a critical field $B_{c}\sim17T$ [@Doiron-Leyraud07; @Sebastian12; @Wu13]. In particular, the ultra-sound experiment [@LeBoeuf13] shows that the typical **B** versus T phase diagram in Fig. \[fig:tb\] is very similar to Fig.\[fig:so5\]-2). Within the EHS model, this experiment was addressed in Ref. [@Einenkel14]. Note that a co-existence phase is present in this phase diagram, which accentuates the similarity with the phase diagram 2) in Fig. \[fig:so5\] of the $SO(5)$ theory. Notice as well that the CDW and SC temperatures are of the same order of magnitude, which was never the case for the AF and SC states. It is another indication that the $SU(2)$ symmetry is more likely verified in the underdoped cuprates than the $SO(5)$ symmetry.
![\[fig:tb\] (Color online) Experimental $B$-$T$ phase diagram from sound velocity measurements in YBCO from Ref. [@LeBoeuf13]. The “pseudo spin-flop” is visible from the SC to CDW transition beyond a critical field $B_{c}\simeq18\,$T.](fig3){width="70mm"}
Of course, a question can be raised at this point, which is that the exact realization of the $SU(2)$ symmetry within the EHS model gives a charge wave vector on the diagonal, while only axial charge order was experimentally observed [@Hoffman02; @Wise08; @Fujita14; @Ghiringhelli12; @Shen05; @He11; @Achkar13; @Blanco-Canosa14; @Blackburn13a]. It is an important question in the $SU(2)$ theory and we will address it in details in the next section. For the moment let us notice that similar rotations as in Eqn.(\[eq:1\]) can be generated for the axial wave vector $\mathbf{Q}=\{\mathbf{Q}_{x},\mathbf{Q}_{y}\}$ observed experimentally, which rotates similar multiplets as in Eqn.(\[eq:5\]) but for the axial wave vector. This idea of a rotation between the $d$-wave SC state and the axial charge order [@Hayward14] was used to explain that the CDW signal is peaked at $T_{c}$ [@Ghiringhelli12; @Blackburn13a]. It was also used in explaining the $A_{1g}$ mode observed in Raman scattering as a collective mode associated to this specific rotation [@Montiel15a; @Sacuto13; @SacutoSidis02].
The notion an emergent symmetry is more general than any of its specific representations. It is indeed very nice to have a model, although very simplified, where the $SU(2)$ symmetry is exactly realized (at all energy scales), but the main concern is whether this symmetry is approximately realized at finite temperatures in the underdoped region of the phase diagram. That is the interest of the concept of emergent symmetry: although it can be exactly realized in only a few effective models, if the splitting between the two pseudo-spin states is smaller than the typical energy of each state, it can also be approximately realized at low energies in the more realistic 2D $t-t'$ Hubbard model (this was verified explicitly in Refs. [@deCarvalho:2014tj; @Freire:2015kg] using two-loop RG techniques).
Another remark that can be made at this stage, is that another type of $SU(2)$ symmetry was identified early on, which consists of performing a particle-hole transformation on each site $c_{i\sigma}^{\dagger}\rightarrow c_{i-\sigma}$, which translates in the reciprocal space as $c_{k\sigma}^{\dagger}\rightarrow c_{k-\sigma}$ for all k vector. This symmetry is interesting for the phase diagram of the cuprates because it is exact at half-filling and will be gradually broken with doping [@Kotliar88a; @Lee06]. The operators for this symmetry group take the form
\[eq:1-3\] $$\begin{aligned}
\eta_{ph}^{+} & =\sum_{\mathbf{k}}c_{\mathbf{k}\uparrow}^{\dagger}c_{\mathbf{k}\downarrow}^{\dagger}\\
\eta_{z} & =\sum_{\mathbf{k}\sigma}\left(c_{\mathbf{k}\sigma}^{\dagger}c_{\mathbf{k}\sigma}-1\right),\end{aligned}$$
while one irreducible $l=1$ representation associated to it can be taken as [^1]
\[eq:5-1\] $$\begin{aligned}
\Delta_{1} & =-\frac{1}{\sqrt{2}}\sum_{\mathbf{k}}d_{\mathbf{k}} c_{\mathbf{k}\uparrow}^{\dagger}c_{-\mathbf{k}\downarrow}^{\dagger} ,\\
\Delta_{0} & =\frac{1}{2}\sum_{\mathbf{k},\sigma}d_{\mathbf{k}} c_{\mathbf{k}\sigma}^{\dagger}c_{\mathbf{-k}\sigma} ,\\
\Delta_{-1} & =\frac{1}{\sqrt{2}}\sum_{\mathbf{k}}d_{\mathbf{k}} c_{\mathbf{k}\downarrow}c_{-\mathbf{k}\uparrow} .\end{aligned}$$
Interestingly, within the EHS model, the operators (\[eq:1-3\]) and (\[eq:5-1\]) are, respectively, the same as (\[eq:1\]) and (\[eq:5\]) with a diagonal wave vector, since there the summation over **k** is reduced to the eight hot spots. The EHS model is also an exact realization of the $SU\left(2\right)$ symmetry associated with particle-hole transformation.
Using (\[eq:1-3\]) one can also ask oneself what is the $SU(2)$ partner of the observed axial CDW. To fix the ideas let us take a uni-axial CDW order with ordering wave vector $\mathbf{Q}_{x}$ relating two hot spots. One can then construct the $l=1$ irreducible representation using the particle- hole transformation. This gives
\[eq:5-1-1\] $$\begin{aligned}
\Delta_{1} & =-\frac{1}{\sqrt{2}}\sum_{\mathbf{k}}d_{\mathbf{k}} c_{\mathbf{k}\uparrow}^{\dagger}c_{-\mathbf{k+Q}_{y}\downarrow}^{\dagger} ,\\
\Delta_{0} & =\frac{1}{2}\sum_{\mathbf{k},\sigma}d_{\mathbf{k}} c_{\mathbf{k}\sigma}^{\dagger}c_{\mathbf{k+Q}_{x}\sigma} ,\\
\Delta_{-1} & =\frac{1}{\sqrt{2}}\sum_{\mathbf{k}}d_{\mathbf{k}} c_{-\mathbf{k+Q}_{y}\downarrow}c_{\mathbf{k}\uparrow} ,\end{aligned}$$
which means that the $SU(2)$ partner of the $\mathbf{Q}_{x}$ CDW is the pair density wave (PDW), namely a non zero center of mass SC state, with $\mathbf{Q}_{y}$ wave vector. This notion of PDW state was introduced recently to explain the very unusual ARPES data tracing the formation of the PG in Bi-2201[@Lee14; @Agterberg:2014wf]. In this theory, the formation of the PDW is suggested as the primary mechanism for the formation of the PG state, which means that the observed CDW is a secondary order. As such, it should be observed at a wave vector twice as big as the PDW wave vector $\mathbf{Q}_{CDW}=2\mathbf{Q}_{PDW}$. In contrast, if the mechanism governing the underdoped region is a hidden pseudospin $SU(2)$ symmetry, then the partner of a bi-axial CDW is a bi-axial PDW with the same wave vectors $\mathbf{Q}_{CDW}=\mathbf{Q}_{PDW}$ (see the Refs. [@Pepin14; @Freire:2015kg; @Wang15b; @Carvalho15b]). The latter scenario has recently been verified experimentally [@Hamidian16].
Non linear $\sigma$-model, and $SU(2)$ vs. $U(1)$ pairing fluctuations {#sectionfluc}
======================================================================
The idea of emergent symmetries received a recent critique, that when the group of symmetry is large enough, the symmetric phase is unstable to smaller subgroups [@Fradkin15]. For example, the symmetric phase associated to the $SO(5)$ symmetry which was intended to describe the PG shall decompose into the $SU(2)\times U(1)$ group describing fluctuations around the AF and SC phases respectively. Similarly, the $SU(2)$ symmetry which rotates between the CDW and SC channel shall decompose into the $U(1)\times U(1)$ groups. In this section, we consider seriously the criticism that the symmetric phase of large non-abelian groups is unstable, but wonder more particularly about the fate of SC fluctuations.
The role of SC fluctuations in the physics of cuprates is indeed very mysterious. We know that they are a few orders of magnitude more intense that in standard metals like $Al$, or $Cu$ [@EmeryVJ:1995dr], but experiments detecting pure the Josephson effect were observed only a few tens of degrees above $T_{c}$ [@Bergeal:2008gf; @Alloul:2010ko; @RullierAlbenque:2011ji]. In the deeply underdoped phase, $U(1)$ SC fluctuations form a dome shape that we will discuss further in this section [@RullierAlbenque:2011ji; @Li:2011cs; @Li:2013ed; @Wang:2002ke; @Wang:2006fa] . Direct observation of pre-formed pairs in the PG phase was always negative, but a giant proximity effect was observed in the Lanthanum-compounds induced in the PG phase when it is surrounded by optimal SC phases [@Decca:2000hc; @Bozovic:2004cp; @Morenzoni11]. The very easy injection of pairs from optimally doped into the PG phase suggests that the PG phase is related to the SC phase through a hidden symmetry. Such proximity effects were predicted in the case of the $SO(5)$ symmetry and never observed [@Demler:1998iw], but specific predictions in the case of the $SU(2)$ symmetry were never discussed in detail.
The $SU(2)$ SC fluctuations
---------------------------
In this section we assume that at an intermediate energy scale SC fluctuations are present, protected by an $SU(2)$ symmetry between the CDW and SC channel. The microscopic derivation of the non linear $\sigma$-model describing the $SU(2)$ fluctuations can be found, in the context of the EHS model in Ref.[@Efetov13], and in the context where regions of the Brillouin zone instead of points are “hot” -or hot anti-nodal regions-, in Ref. [@Kloss15a]. The massless $O(4)$ effective free energy has the following form $$\begin{aligned}
F_{SU(2)} & =\frac{T^{2}}{2}\sum_{\varepsilon,\omega}\int_{\mathbf{k},\mathbf{q}}tr\delta\hat{u}_{-k,q}^{\dagger}\left[J_{0,k}\omega^{2}+J_{1,k}q^{2}\right]\delta\hat{u}_{k,q}\label{eq:nlsm2-1-1}\end{aligned}$$ where $\hat{u}_{\mathbf{k},\mathbf{q}}=\left(\begin{array}{cc}
\Delta_{CDW} & \Delta_{SC}\\
-\Delta_{SC}^{*} & \Delta_{CDW}^{*}
\end{array}\right)$ is the $SU(2)$ matrix associated with the condition $\left|\Delta_{CDW}\right|^{2}+\left|\Delta_{SC}\right|^{2}=1$, and the $tr$ runs on the $SU(2)$ structure. The coefficients write $J_{0,k}=\left|M_{\mathbf{k}}\right|^{2}/\left|G^{-1}\right|^{2}$, and $J_{1,k}=J_{0,k}v_{\mathbf{k}}^{2}$, where $M_{\mathbf{k}}$ is the magnitude of the mean-field $SU(2)$ order parameter $\hat{M}_{\mathbf{k},\mathbf{q}}=\left(\begin{array}{cc}
& \hat{m}_{\mathbf{k,}q}\\
\hat{m}_{\mathbf{k,}q}^{\dagger}
\end{array}\right)_{\Lambda}$ , with $\hat{m}_{\mathbf{k},\mathbf{q}}=M_{\mathbf{k}}\hat{u}_{\mathbf{k},\mathbf{q}}$, which has a $4\times4$ structure in the $\tau\times\Lambda$-$SU(2)$ spaces where $\tau$ is the particle-hole transformation and $\Lambda$ is the $\mathbf{Q}$-translation. The Green’s function writes $\hat{G}^{-1}=\hat{G}_{0,k}^{-1}+\hat{M}_{k,0}$, with $\hat{G}_{0,k}^{-1}=i\omega-\left(\tau_{3}\xi_{\mathbf{k}}^{s}-\xi_{\mathbf{k}}^{a}\right)\Lambda_{3}$, with $\xi_{\mathbf{k}}^{s,a}=\left(\epsilon_{\mathbf{k}}\pm\epsilon_{-\mathbf{k}-\mathbf{Q}}\right)/2$, and $\epsilon_{\mathbf{k}}$ being the electron dispersion. Note that no information was given on the value of the $\mathbf{Q}$-wave vector for the CDW sector. It corresponds in all generality to the $SU(2)$ operators (\[eq:1\]) and (\[eq:1-1\]). The exact $SU(2)$ symmetry is verified when $\xi_{\mathbf{k}}^{s}=0$ which effectively kills the $\tau_{3}$-term in the equation for $\hat{G}_{0,k}^{-1}$. $\xi_{\mathbf{k}}^{s}$ hence models the symmetry breaking term associated with this specific wave vector and contributing to the free energy as $$F_{SB}=\frac{T^{2}}{2}\sum_{\varepsilon,\omega}\int_{\mathbf{k},\mathbf{q}}J_{3,k}\,tr\left[\delta\hat{u}_{-k,q}^{\dagger}\tau_{3}\delta\hat{u}_{k,q}\tau_{3}\right],\label{eq:10a}$$ with $$\begin{aligned}
J_{3,k}= & \frac{1}{4}\frac{\left|m_{0,k}\right|^{2}\left(\xi_{\mathbf{k}}^{s}\right)^{2}}{\left|G^{-1}\right|^{2}}.\label{eq:10}\end{aligned}$$ The shape of the symmetry-breaking term Eqn.(\[eq:10\]) is visualized in Fig. \[fig:symbreak\]. One can observe the anisotropy of the mass in various directions in the Brillouin Zone : the mass is much bigger in the nodal direction than in the anti-nodal one.
a)![\[fig:symbreak\] (Color online) a) Visualization of the SU(2) symmetry breaking contribution $(\xi_{{\bf {k}}}^{s})^{2}$ in the positive region of the first Brillouin zone. It is small in the blue region and vanishes for the two black lines crossing the hotspot, but grows in the nodal line to the upper edge. b) Variation of $(\xi_{{\bf {k}}}^{s})^{2}$ along the Fermi surface (shown as a gray line in panel a)) parametrized as a function of $k_{x}$. While $(\xi_{{\bf {k}}}^{s})^{2}$ vanishes at the hotspot and stays small at the antinodes at $k_{x}=\pi$, it grows at the left side when approaching the nodal position.](fig4a "fig:"){width="27mm"}
b)![\[fig:symbreak\] (Color online) a) Visualization of the SU(2) symmetry breaking contribution $(\xi_{{\bf {k}}}^{s})^{2}$ in the positive region of the first Brillouin zone. It is small in the blue region and vanishes for the two black lines crossing the hotspot, but grows in the nodal line to the upper edge. b) Variation of $(\xi_{{\bf {k}}}^{s})^{2}$ along the Fermi surface (shown as a gray line in panel a)) parametrized as a function of $k_{x}$. While $(\xi_{{\bf {k}}}^{s})^{2}$ vanishes at the hotspot and stays small at the antinodes at $k_{x}=\pi$, it grows at the left side when approaching the nodal position.](fig4b "fig:"){width="40mm"}
Resonant Peierls Excitonic (RPE) state
--------------------------------------
We now integrate the $SU(2)$ SC fluctuations out of the partition function, and evaluate the consequences of them in the charge channel. We get the following effective action $$S_{\text{eff}}[c]=\sum_{\mathbf{kk'q}}\pi_{k,k',q}c_{\uparrow\mathbf{k}}^{\dagger}c_{\uparrow\mathbf{k'}}c_{\downarrow-\mathbf{k+}q}^{\dagger}c_{\downarrow-\mathbf{k'}+\mathbf{q}},\label{eq:sfinfull}$$ with $$\pi_{k,k',q}=\langle\bar{\Delta}_{k,q}\Delta_{k',q}\rangle=\frac{\pi_{0}\left(\delta_{k,-k'}+\delta_{k,k'}\right)}{\left(J_{0}\omega_{n}^{2}+J_{1}({\bf {v}}\cdot{\bf {q}})^{2}+a_{0,k}\right)},$$ where the form of the SC fluctuations comes from Eqns.(\[eq:nlsm2-1-1\],\[eq:10a\]). The self-consistent Dyson equation ( or “gap equation”) writes $$\chi_{k,k'}=\sum_{q}\pi_{k,k',q}[\hat{G}(q-k,q-k')]_{12},\label{eq:chieq3}$$ with $\chi_{k,k'}=\sum_{q}\pi_{k,k',q}\langle c_{\sigma\mathbf{k}}^{\dagger}c_{\sigma\mathbf{k'}}\rangle$ and with $$\begin{aligned}
[\hat{G}_{k,k'}]_{12} & =-\langle c_{\sigma}^{\dagger}(k)c_{\sigma}(k')\rangle\nonumber \\
& =-\frac{\chi_{k,k'}}{(i\epsilon_{n}-\xi_{{\bf {k}}})(i\epsilon_{n}'-\xi_{k'})-\chi_{k,k'}^{2}}.\label{eq:chieq2}\end{aligned}$$
a\) ![\[fig:1bz\] (Color online) left) Density of the charge order parameter $|\chi_{k,-k}|$ in the first Brillouin zone from the RPE state. The charge density follows the Fermi surface, but due to a SU(2) dependent mass contribution, does not stabilize in the nodal region. right) Charge order parameter around the hotspot position for a constant ${2{\bf {p}}_{\text{F}}}$ ordering vector. From Ref. [@Kloss15a].](fig5a "fig:"){width="40mm"}
b\) ![\[fig:1bz\] (Color online) left) Density of the charge order parameter $|\chi_{k,-k}|$ in the first Brillouin zone from the RPE state. The charge density follows the Fermi surface, but due to a SU(2) dependent mass contribution, does not stabilize in the nodal region. right) Charge order parameter around the hotspot position for a constant ${2{\bf {p}}_{\text{F}}}$ ordering vector. From Ref. [@Kloss15a].](fig5b "fig:"){width="30mm"}
![\[fig:2pf\] (Color online) Set of degenerate ${2{\bf {p}}_{\text{F}}}$ couplings between electrons on opposed Fermi surfaces in the antinodal region due to the RPE state. From Ref. [@Kloss15a].](fig6){width="26mm"}
To get some idea about the nonlocal nature the order parameter $\chi_{\bf{k,k'}}$, it is illustrative to consider one of the labels having a constant shift $\bf{k'}=\bf{k+P}$. The order parameter can then be decomposed as $\chi_{\bf{P,k}} = \chi_{\bf{P}} F_{\bf{P}}$ [@Kloss15a], where $\chi_{\bf{P}}$ is a plain wave and $F_{\bf{k}}$ a formfactor for the electron-hole pair with a finite extent, see Fig. \[fig:1bz\]b). The total order $\chi_{\bf{k,k'}}$ is thus a superposition of all local $\chi_{\bf{P,k}}$ orders, where $\bf{P}$ is running over the whole set of degenerate ${2{\bf {p}}_{\text{F}}}$ couplings as visualized in Fig. \[fig:2pf\]. The gap in the antinodal region is $\Delta^{PG}_{\bf{k}} = \sum_{\{ \bf{P}\}} \chi_{\bf{P},k}$. Note that in Eqns. (\[eq:chieq3\],\[eq:chieq2\]) the external wave vectors $\mathbf{k},\mathbf{k'}$ are *a priori* not defined, but are let free to find self-consistently the most favorable solution. We studied numerically the possible excitonic solutions of the gap equations and the result is depicted in Fig. \[fig:1bz\]. We obtained an excitonic state in which a large number of wave vectors are degenerate with a typically $\mathbf{k}-\mathbf{k'}=2\mathbf{k}_{F}$ which are spread out in the anti-nodal region of the Brillouin zone producing a depletion of the density of states in this region (see Fig. \[fig:2pf\]). Due to the angular dependence of the fluctuation mass ($a_{0,node}\gg a_{0,anti-node}$), we obtain a preferential gapping out of the anti-nodal region, which is characteristic of the $SU(2)$ SC fluctuations compared to the original $U(1)$.
Long range charge order
-----------------------
At this stage we have proposed a theory for the PG phase of cuprates superconductors. In the following we will give some more arguments that this theory is a promising candidate to describe high T$_{c}$ superconductors. We know from a body of experimental evidence, though, that the PG phase is distinct from the observed uni-axial CDW. At zero field, the CDW dome decreases when the oxygen doping decreases, which is at variance with the PG $T^{*}$-line, which increases with the chemical potential (or oxygen doping) [@Achkar12; @Achkar13; @Bakr13; @Ghiringhelli12; @CyrChoiniere:2015wu; @Blanco-Canosa14]. Moreover, recent studies of the Fermi surface reconstruction under a magnetic field of $17$T infer that the PG is formed *before* the Fermi surface is reconstructed by CDW order -recall that the CDW becomes range and three dimensional beyond $B=17$T. There are many proposals for the PG phase [@LeHur:2009iw; @Rice12; @Lee14; @Fradkin15; @Senthil03; @Wang15b; @Hayward14; @Bulut2015; @Nie2015; @Gull:2013hh; @Sordi2012; @Sordi2012B], but since ours consists of a special type of excitonic liquid, it is important to shed light on the relationship between the excitonic RPE phase and the observed axial CDW which is stabilized under magnetic field.
Within the EHS model, or within all sorts of simple weak coupling RPA evaluation, we find that the axial charge order is a secondary instability, weaker than CDW with a wave vector on the diagonal. The question that is then raised, is why nothing at all is observed on the diagonal, whether it is by STM [@Wise08; @He14; @Fujita14; @Hamidian15a] or by X-rays [@Blackburn13a; @Ghiringhelli12; @Achkar12; @Blanco-Canosa14] The simplest explanation is that the pseudogap is forming, gapping out the anti-nodal region of the Brillouin Zone, and wiping out the CDW instability on the diagonal. When the mechanism of formation of the pseudogap instability has operated, then the secondary instability can be visible, at the tip of the Fermi arcs. Many suggestions have been made for the formation of axial CDW order. The fact that this wave vector is present as a secondary instability in any weak coupling theory, and stabilized for example, in the presence of additional effect like Coulomb interactions [@Pepin14; @Wang14; @Allais14c], within both one-loop and two-loop RG [@deCarvalho2013738; @deCarvalho:2014tj; @Whitsitt:2014eq; @Carvalho15; @Freire:2015kg] or starting from a three band model [@Bulut2015], or invoking the proximity to the van Hove singularity [@Volkov:2015vh] has been outlined in many works, including ours. All these studies are based on the observation that axial CDW is distinct from the formation of the PG state and starts to get formed at the tip of the arcs. In all mentioned scenarios though, it is quit unclear why the CDW dome is increasing with doping, in contrast with the PG line $T^{*}$. The $SU(2)$-paradigm which related the axial CDW to SC fluctuations that gives a good explanation for the maximum of amplitude of the CDW order at $T_{c}$ [@Hayward14] seems to have been lost in the attempts to rotate the leading wave vector from the diagonal to the axes.
![\[fig:nernst\] (Color online) upper panel) Temperature-hole doping phase diagram deduced from Nernst effect experiments (from Ref.[@Wang:2006fa]). The gray area corresponds to CDW phase where Nernst coefficient is not zero. lower panel) Temperature-hole doping phase diagram with the superconducting critical temperature (black dots) and the onset temperature of the CDW axial order (red dots) deduced from RXS experiments in YBa$_{2}$Cu$_{3}$O$_{6+x}$ (extrated from Ref. [@Blanco-Canosa14]).](fig7a "fig:"){width="50mm"} ![\[fig:nernst\] (Color online) upper panel) Temperature-hole doping phase diagram deduced from Nernst effect experiments (from Ref.[@Wang:2006fa]). The gray area corresponds to CDW phase where Nernst coefficient is not zero. lower panel) Temperature-hole doping phase diagram with the superconducting critical temperature (black dots) and the onset temperature of the CDW axial order (red dots) deduced from RXS experiments in YBa$_{2}$Cu$_{3}$O$_{6+x}$ (extrated from Ref. [@Blanco-Canosa14]).](fig7b "fig:"){width="65mm"}
![\[fig:leggett\] (Color online) Leggett mode (highlighted in red), where charge order is induced by $U(1)$ phase fluctuations at the tip of the arcs with the axial wave vector ${\bf{Q}}_{x/y}$.](fig8){width="40mm"}
In this paper, we would like to offer an alternative scenario which to our knowledge has not been proposed yet. It is based on the observation that the CDW dome follows closely the $U(1)$ SC fluctuation dome that can be detected through many probes, like resistivity study [@RullierAlbenque:2000fl; @Alloul:2010ko], Nernst effect [@Wang:2002ke; @Wang:2006fa; @Li:2011cs], Josephson tunneling. The $U(1)$ standard SC fluctuations are usually very weak in SC states because Coulomb interactions push them above the plasma frequency[@Anderson:1963vi]. In the case of the optimally doped cuprates, due to low dimensionality, phase fluctuations are observed in a window of roughly 15K above $T_{c}$ [@Bergeal:2008gf]. For more underdoped compounds, the approach of the Mott insulating states produces a drastic increase of phase fluctuations due to the phase-particle number Heisenberg uncertainty relations [@Emery95]. We would like to exploit this idea here and suggest that the axial CDW order is an effect of the $U(1)$ SC fluctuations. We use a recent remark of Ref. [@Liu:2015uu] where it is noted that after the formation of the PG, the Fermi surface has split into four sections-or arcs, which are now independent from each other. In these conditions, one can get a Leggett mode between the tips of the arcs, which in turn can induce a charge order with the correct wave vectors, see Fig. \[fig:nernst\]. The gap equation for the axial CDW mediated by the Leggett mode writes $$\begin{aligned}
\chi_{\mathbf{k}\sigma,\mathbf{k}+\mathbf{Q_{0}}\sigma} & =T\sum_{\omega_{n}\mathbf{q}}\pi_{\mathbf{Q_{0}},\mathbf{q}}[G_{-\mathbf{k}+\mathbf{q},-\mathbf{k}-\mathbf{Q_{0}}+\mathbf{q}}]_{12},\label{eq:15}\end{aligned}$$ with $\mathbf{Q_{0}}=\mathbf{Q_{x/y}}$ being the axial wave vector, $G_{12}$ given by Eqn.(\[eq:chieq2\]) with the replacement $\xi_{\bf{k}} \rightarrow \xi_{\bf{k}} + \Delta^{PG}_{\bf{k}}$ to take the gapping of the FS into account, whereas $\pi_{\mathbf{Q_{0}},\mathbf{q}}$ is the $U(1)$ correlation of the phase fluctuations at the tip of the arcs, as represented in Fig. \[fig:leggett\] $$\begin{aligned}
\pi_{\mathbf{Q_{0}},\mathbf{q}} & =\left\langle {\cal T}\Delta_{\mathbf{k},\mathbf{-k+q}}\Delta_{\mathbf{k+Q_{0},-k-Q_{0+q}}}^{\dagger}\right\rangle _{U(1)} \nonumber \\
& = \frac{\pi_0}{J_0' \omega_n^2 + J_1' ({\bf{v}} \cdot {\bf{q}})^2 + m_0} ,\end{aligned}$$ with $\mathbf{k}$ being the wave vector at the tip of the one Fermi arc and $\mathbf{k}+\mathbf{Q_{0}}$ being the wave vector at the tip of the adjacent Fermi arc. We use a generic form of the propagator $\pi_{\mathbf{Q_{0}},\mathbf{q}}$, where $\pi_0, J_0', J_1'$ and $m_0$ are mon-universal parameters the dependence on ${\bf{Q}}_0$ is neglected. We have performed a numerical study of Eqn.(\[eq:15\]) which confirms that $U(1)$ SC fluctuations mediated by a Leggett mode produce axial CDW with the desired wave vector. This proposal has the merit to consistently link both the formation of the PG and the observed axial CDW to SC fluctuations, the former being described by the $SU(2)$ non linear $\sigma$-model while the latter are the standard $U(1)$ phase fluctuations.
Discussion {#sectionDiscussion}
==========
In the phase diagram of high temperature cuprates a few key players can be identified [@Norman03; @Lee06]. There is at half-filling the Mott insulating transition with typical energy of 1eV associated to it. Antiferromagnetism is ubiquitous in the whole phase diagram, with an ordered phase of typically $T_{Neel}\approx 700\,K$ at half-filling, very close to the Mott transition, and strong, but short range AF fluctuations in the underdoped regime. In the proposal of this paper, the mysterious PG phase of high temperature cuprates is attributed to a new kind of excitonic state, the RPE, which can be understood as a new type of “liquid” of excitons, with a superposition of degenerate wave vectors. This state is a consequence of integrating out the SC fluctuation, protected by an emergent $SU(2)$ symmetry between the SC and charge channel. In the discussion of this proposal, the first thing to recall is that although antiferromagnetism is not directly responsible for the PG, it is nevertheless the underlying force driving the emergence of precursor orders. In the early version of this theory, the EHS model has been studied as a reference model where the $SU(2)$ symmetry is verified [@Metlitski10b; @Efetov13]. In this model the eight hot spots are singled out of the Fermi surface, and long range AFM fluctuations stabilize the composite $SU(2)$- order parameter, composed by a diagonal quadrupolar density wave and SC. In more generic versions of this theory, the model is extended to “hot regions” of the Brillouin zone - the anti-nodal regions, where AF acts predominantly and the $SU(2)$ symmetry is most strongly verified [@Kloss15]. Antiferromagnetism did not disappear from the phase diagram, but rather has a very special relation to the PG by defining the width of the “hot regions”, thus limiting the domain of action of the RPE state, and also being the driving force both behind SC pairing and the $SU(2)$ symmetry.
The concept of emergent symmetry though, is more robust and general than even the idea of Quantum Criticality and it is under such a generic paradigm that we want to cast out the underdoped region of cuprate superconductivity. The main idea is that charge orders are the natural partner and competitors of SC pairing in the underdoped region of the cuprates, and typical pseudo “spin flops” between the two orders are to be expected, and we believe already observed under magnetic field [@LeBoeuf13].
The experimental consequences of a phase diagram controlled by an $SU(2)$ emergent pseudospin symmetry are numerous, and it is very likely that our proposal for the RPE state may be confirmed or in-firmed within the next few years.
### Spectroscopic signatures
One can first ask about the spectroscopic signature of such an excitonic state. What can be seen in STM or X-rays. Our claim here is that we can reproduce the very recent findings on Bi-2212 [@Hamidian15; @Davis16], that the pure $d$-wave component of the axial CDW extends up to the PG temperature, see Fig. \[fig:d-temp\]. In the RPE state, indeed, the excitons form not only around many degenerate wave vectors, but with a finite width around each wave vector.
a)![\[fig:d-temp\] (Color online) Upper panel a) Tunneling conductance measurements from Ref. [@Hamidian15a] of underdoped cuprates. Two characteristic energies, a lower one for Bogoliubov quasiparticles and a higher one corresponding to the pseudogap are observed. Lower panel b) Energy dependence of the $s$- and $d$- wave form factors, indicating that the higher gap-energy scale corresponds to the $d$-wave form. From Ref. [@Hamidian15a]](fig9a "fig:"){width="50mm"} b)![\[fig:d-temp\] (Color online) Upper panel a) Tunneling conductance measurements from Ref. [@Hamidian15a] of underdoped cuprates. Two characteristic energies, a lower one for Bogoliubov quasiparticles and a higher one corresponding to the pseudogap are observed. Lower panel b) Energy dependence of the $s$- and $d$- wave form factors, indicating that the higher gap-energy scale corresponds to the $d$-wave form. From Ref. [@Hamidian15a]](fig9b "fig:"){width="50mm"}
The real space picture is that the particle-hole pairing is non local in space, and modulated by many wave vectors. When the induced charge on the oxygen is evaluated and Fourier transformed, one finds that it is 90% $d$-wave (100 % for the diagonal wave vector and a bit less for the others), and at the same time, the axial wave vectors are more favored compared to the diagonal due to its nesting properties in the anti-nodal region [@Montiel16]. The consequence is that the charge on the Oxygen shows a preponderant spectrum with axial wave vectors $\mathbf{Q}_{x}$ and $\mathbf{Q}_{y}$. At this stage our conclusion is that the RPE state is already observed by STM and X-rays, which have captured its preponderant contribution on the axial wave vectors.
### Proximity effect
A second remark is that emergent symmetries rotating the SC phase to another type of order predict proximity effects when the PG phase is sandwiched between two optimally doped superconductors. The intensity of the induced current in the junction persists for thickness of the gap material much greater than the superconducting correlation length. This ”Giant” Proximity Effect (GPE) is not explicable by the standard theory of the proximity effect between two SC junction, but can be understood in the situation where the SC state is ”quasi-degenerate” to another phase of matter and Cooper pair can thus be easily injected from the SC state to the other state. The situation is thus very promising for emergent symmetries, and has been extensively studied in the case of the $SO(5)$-symmetry [@Demler04; @denHertog:1999dk], where specific predictions for the current as a function of the phase difference across the junction can be made as well for the $SU(2)$ symmetry, see Fig. \[fig:junction\]. Note that a giant proximity effect has already been observed in various compounds but has not been observed for the specific setup of the $SO(5)$-group. One straightforward application of our theory is to check whether $SU(2)$ fluctuations, where the rotation is between the SC state and the CDW state can account for the experimental data [@Tarutani91; @Yuasa91; @Kasai92; @Meltzow97; @Decca00; @Bozovic04; @Morenzoni11].
![\[fig:junction\] (Color online) Proposition of a SC-PG-SC junction to study the giant proximity effect within the SU(2) theory. At a given temperature $T$ that is homogeneous over the junction, the two outer SC layers are superconducing and at optimal doping such that $T<T_{c}$. The inner PG layer is an underdoped SC in the pseudogap phase ($T^{*}>T>T_{c}$). From the giant proximity effect we expect the PG phase to become superconducting by lowering the thickness of the inner layer. ](fig10){width="30mm"}
### Magnetic field phase diagram
The phase diagram found as a function of magnetic field and temperature, derived with a variety of experiments [@LeBoeuf13; @Doiron-Leyraud07; @Sebastian12; @Wu13a; @Wu:2015bt] is typical for a super “spin-flop” between two states related by a symmetry (see Fig.\[fig:tb\_theo\]). Note that three dimensional CDW has been recently observed by X-ray scattering above $B=17$T [@Chang16].
![\[fig:tb\_theo\] (Color online) $B$-$T$ phase diagram obtained from the spin-fermion model considering order parameter fluctuations around the mean-field value with a nonlinear $\sigma$ model from Ref. [@Meier13].](fig11){width="65mm"}
The CDW and SC orders have the same order of magnitude in this diagram, and the transition between the two is very sudden, like in a generic spin-flop XY model [@Meier13; @Einenkel14]. Moreover, an $SU(2)$ partner of the axial CDW has recently been reported, i.e. the PDW with the same wave vector [@Hamidian16]. Although it is not a direct proof of the underlying symmetry, it seems to rule out other scenarios for the PG state where the PDW is primary while the CDW orders are secondary, and hence occur at twice the same wave vector as the PDW.
### Collective modes
Emergent symmetries also have signatures in terms of collective modes. In a recent work we argued that the $A_{1g}$-mode observed in Raman scattering very close in energy to the neutron mode is such a signature of the SC-CDW $SU(2)$ symmetry [@Montiel15a], see Fig. \[fig:raman\]. The collective mode used in this work was associated with the $\eta$- operator of Eqn.(\[eq:1\]) with axial wave vector, thus associated to the triplet representation Eqn.(\[eq:5-1-1\]). The presence of the two orders in conjunctions was needed in order for the Raman scattering vertex not to vanish. The model could account for the absence of observation of this order in the $B_{1g}$ and $B_{2g}$channels.
![\[fig:raman\] (Color online) Raman scattering response from a collective mode. a) shows the calculated Neutron susceptibilities at the momentum ${\bf{Q}}=(\pi,\pi)$ as a function of the frequency. In b) (and c) ), the experimental (solid line) and calculated (dashed line) Raman response in the A$_{1g}$ ($B_{1g}$) Raman channels. The Raman resonance arises at the same frequency than the Neutron resonance at ${\bf{Q}}$ (Fig (a) and (b)) below the superconducting coherent peak energy observable in the $B_{1g}$ symmetry (Fig (b) and (c)). This collective mode resonance appears in the $A_{1g}$ symmetry since the Raman response is screened by long range Coulomb interaction in this symmetry. From Ref. [@Montiel15a].](fig12){width="60mm"}
Inelastic neutron scattering has reported since the very early days the presence of a collective mode in the underdoped regime, centered around the AFM wave vector $Q=(\pi,\pi)$, and at a finite energy around $E=41$meV for the compound YBCO [@Rossat; @Hinkov04]. Many theories, based on an RPA treatment of a magnetic spinon mode below the SC gap have been produced in order to explain this very characteristic feature of the cuprates [@Demler95; @Demler04; @Tchernyshyov01]. The $SO(5)$ theory was originally devoted to the study of this mode [@Demler95]. The RPA theories, reproduce successfully the position of incommensurate signal around $Q=(\pi,\pi)$, having the typical “hour-glass” shape in the energy-momentum space. The present theories have difficulties to account for the fact that this signal remains inside the PG phase, changing form from the “hour-glass” to a “Y” shape, namely acquiring some extra low energy spectral weight at $Q=(\pi,\pi)$. The proposed RPE state is an excitonic state with excitations around a bunch of $2\mathbf{k}_{F}$- wave vectors in the anti-nodal region. Thus it behaves a little bit as a “charge superconductor”, that in the simplest models, will gap out the electronic degrees of freedom precisely as a superconductor would do. We believe the RPE state can also account for the extra spectral weight at $Q=(\pi,\pi)$, which will be presented in a future work [@Montiel16].
### ARPES
We turn now to angle-resolved photoemission spectroscopy (ARPES), which have been very influential in our understanding of the PG phase of these materials, especially with the seminal observation of Fermi arcs in this phase . A recent ARPES experiment on Bi-2201 has been very important in our understanding of the formation of the PG [@He11; @Shen:2005ir]. Dispersion cuts close to the Zone Edge show that the PG opens at a typical momentum larger than the momentum relating the two Fermi points $2\mathbf{k}_{F}$. Moreover when the dispersion cuts get closer to the nodes, the PG closes from below rather than from above. It has been argued that this set of peculiar features can only be explained by a PDW state (a finite momentum SC state), since only the particle-hole reversal specific to the pairing state can account for the closing of the gap from below [@Lee14; @Wang15a].
![\[fig:gapdisp\] (Color online) Electronic dispersion measured by ARPES in the superconducting (blue and green) and normal state (red). Cuts taken at constant $k_x$: close to the zone edge (O)($k_x=\pi$), close to the nodal region (T, $k_x=0.6 \pi$) and in an intermediate case (R, $k_x=0.8\pi$). The closing of the gap can be observed from the antinodal to the nodal zone. From Ref. [@He11].](fig13){width="80mm"}
We argue that the RPE state provides another explanation for this fascinating set of data. Besides the multiplicity of the wave vectors, the key ingredient is the non locality of the excitons. In particular, in the reciprocal space, they form within a finite window in the anti-nodal region, which can account for the natural closing in energy of the gap, both from above and below (ARPES does not see the positive energies), so that we have only to account for the negative part of the spectrum [@Montiel16].
### Loop currents
The observation of a $\mathbf{Q}=0$ signal in neutron scattering at a temperature line following $T^{*}$ [@Fauque06] is one of the mysteries of the PG phase, which has been interpreted in terms of the formation of intra-unit-cell $\Theta_{II}$-loop currents [@Gull:2013hh]. Although it is commonly understood that a $\mathbf{Q}=0$ phase transition does not open a gap in the electronic spectrum, and thus the $\Theta_{II}$-loop-current phase alone cannot be responsible for the origin of the PG, any proposal for the PG phase has to account for the signal observed in the neutron scattering experiment.
![\[fig:2dcdw\] (Color online) Coexistence between $\Theta_{II}$-loop current order parameter and bidirectional $d$-wave CDW as a function of the interaction strength $\lambda$. From Ref. [@Carvalho15b].](fig14){width="50mm"}
We have produced two studies within the EHS model regarding the possibility of coexistence of charge orders and loop currents [@Carvalho15; @Carvalho15b]. In Ref. [@Carvalho15], we have shown, within a saddle-point approximation, that the $\Theta_{II}$-loop-current order cannot coexist with a d-wave CDW with diagonal wave vectors. As a result, we have offered this scenario as the possible reason, which explains why a d-wave charge order was never observed along the diagonal direction in the cuprates. In a subsequent work [@Carvalho15b], we have demonstrated that a similar behavior is displayed by the d-wave CDW along axial directions described by uni-directional wave vectors (i.e. of the stripe-type), since the $\Theta_{II}$-loop-current order is also detrimental to the latter order. By contrast, we have shown that bi-directional (i.e. checkerboard) d-wave CDW and PDW along axial momenta, which are in turn related by the emergent $SU(2)$ pseudospin symmetry pointed out previously, are compatible with the $\Theta_{II}$-loop-current order, since all these orders can coexist with one another in the phase diagram (see, e.g., Fig. \[fig:2dcdw\]). These theoretical predictions agree, most spectacularly, with recent STM results [[@Hamidian16]]{} and also with x-ray experiments [@Blanco-Canosa14].
### Pump probe experiment
A recent pump probe experiment also gives some evidence of the presence of strong SC fluctuations at an intermediate energy scale [@Kaiser14] in underdoped cuprates. In the first series of pump probe experiments [@Averitt01; @Pashkin10], the cuprate was excited up to $1.5$ eV and relaxation at the pico-second scale - observed in the optical THz regime, destroyed the Cooper pairs and showed two typical energy scales, one related to the PG regime and one associated with the formation of the coherence SC phase. Those two scales are typically the ones observed, for example, in the $dI/dV$ response of STM microscope.
![\[fig:neqgap\] (Color online) Schematic phase diagram for YBCO proposed in Ref. [@Kaiser14]. Under out-of-equilibrium conditions realized by optical pump-probes, a high mobility phase in the blue shaded area can be realized that extends much above the critical temperature of equilibrium SC. From Ref. [@Kaiser14].](fig15){width="50mm"}
But in a recent experiment, the excitation was much weaker, in the mid Infra red regime [@Kaiser14] which enabled to scan the properties of the PG phase without destroying the Cooper pairs. What was found resembles to a pico-second photograph of SC fluctuations with the superfluid density $\rho_{s}\sim\omega\Delta\sigma_{2}$ shooting up in the PG phase, up to temperatures of 300 K , see Fig. \[fig:neqgap\]. This pico-second “photograph” of the superfluid density was shown to follow the PG temperature as a function of doping.
### General phase diagram
A general look at the phase diagram of the cuprates singles out many enduring mysteries, and one of the most enduring one is the linear-in-$T$ resistivity around optimal doping. This phase was identified in a seminal work as a Marginal Fermi Liquid (MFL) [@Varma89; @Varma97], and it is still a challenge for theories to account for this phenomenon. Recent in-depth experiments show a more complex behavior of the resistivity with temperature, with a part linear in $T$ and a residual part going like $T^{2}$ when the strange metal regime is approached from the right hand side of the phase diagram [@Hussey:2008tw; @Hussey:2011kp]. Two schools of ideas have been advanced to explain this very unusual phenomenon. In the first school of ideas, it is believed that the proximity to the Mott transition creates a very strongly correlated electronic medium where the electron mean free path is so weak that we are above the Ioffe-Regel limit for the MFL regime [@Georges96]. The second school of ideas advances that the resistivity slope as a function of temperature is very steep, so that the second MFL regime extends far beyond the Ioffe-Regel limit at low enough temperatures. Within this second viewpoint, the challenge is to suggest a QCP beyond the dome which could produce a very resistive scattering behavior for the conduction electrons. It is precisely what the quantum critical version of the RPE state does.
![\[fig:strangemet\] (Color online) Schematic phase diagram of cuprate superconductors with the RPE state, as proposed in [@Kloss15a]. The quasi one-dimensional scattering in the vicinity of QCP$_{RPE}$ produces the resistivity and the electronic self-energy anomaly observed in the strange metal phase. From Ref. [@Kloss15a].](fig16){width="50mm"}
Electronic scattering through quantum critical excitonic modes shows a quasi-one dimensional behavior, each electron scattering preferentially through its most favorable $2\mathbf{k}_{F}$ wave vector [@Kloss15a], and produces a resistivity that behaves as $\rho\sim T/\log T$ within a Boltzmann semiclassical calculation and the electronic self-energy that reads $\Sigma\left(i\epsilon_{n}\right)\sim i\epsilon_{n}/\log\left|\epsilon_{n}\right|$ in the “Strange Metal” (SM) regime (see Fig. \[fig:strangemet\]). On the same line of thought, maybe one of the most difficult feature of the PG to account for in any theory is that it is insensitive to a large amount of $Zn$-doping or irradiation by electrons [@Alloul09; @RullierAlbenque:2000fl; @RullierAlbenque:2011ji], which locate on $Cu$ sites and produce strong disorder which exclude the doped $Cu$-site in the unitary limit [@Pepin98]. The $T^{*}$-line is not affected and also the slope of the resistivity in the SM regime does not change [@RullierAlbenque:2000fl]. It is difficult for any state of matter to have the sufficient robustness to show no sensitivity to such a strong perturbation. One way the RPE state could resist is through the non-locality of the excitons (i.e. the particles-hole pair), which can typically being created at site $\mathbf{r}$ and removed at site $\mathbf{r'}$ with the typical correlation $\left\langle c_{\mathbf{r}}^{\dagger}c_{\mathbf{r'}}\right\rangle $ [@Montiel16].
Conclusion
==========
To conclude, within the two large views of the cuprates where, on one hand, the physics of the PG is solely determined by strong correlations and the proximity to the Mott transition, and the other view where the qualitative features of the physics of the PG are governed by some hidden symmetry, the present work makes a clear discrimination in favor of the latter. It is claimed here that the physics of the PG and the SM phase are controlled by an emergent $SU(2)$ symmetry. Many properties of the underdoped cuprates can be captured within the pseudo-spin theory, the non-linear $\sigma$-model associated to this symmetry and the pseudo spin-flop physics between the SC and charge channel. We also claim that $SU(2)$ superconducting fluctuations proliferate at intermediate energy scales in the physics of these compounds, and are are the key ingredient to understand the PG phase. At lower energy, they lead to the formation of the RPE state, which we believe has a lot of promising features to be the solution for the PG. At even lower temperatures, the $U(1)$ phase fluctuations enter the game and produce coherent axial CDW mediated by a Leggett-mode. The $SU(2)$ symmetry is present in the background of the whole underdoped region, and it is expected that pseudo-spin partners of the various orders (such as the PDW partner of the CDW order) have recently been observed experimentally [@Hamidian16]. Lastly, what is the influence of the Mott transition on the phase diagram of the cuprates ? We believe it will qualitatively affect the physics up to roughly 6% of doping. Below 6% of doping, techniques adequate to describe the very strongly coupled regime will capture the physics [@LeTacon06; @Gull:2013hh; @Sordi2012] . Beyond 6% doping, the physics is qualitatively protected by the $SU(2)$ symmetry. A very revealing experiment is the variation of the nodal velocity $v_{\Delta}$ as a function of doping extracted from ARPES data (Ref. [@Vishik12]). A tri-sected dome is observed with three distinct regimes, 1) below 6% doping, 2) between 6% and 19% of doping and 3) above 19 % of doping. Within the $SU(2)$ theory, as with all theories controlled by an emergent symmetry, the critical value of 6% of doping is precisely the point where the Mott physics becomes dominant. Within the strongly correlated viewpoint, the typical doping of 6% is difficult to interpret.
![\[fig:tstararpes\] (Color online) ARPES experiments on BSCO performed in Ref. [@Vishik12]. The doping dependence of the gap velocity $v_{\Delta}$ reveals three distinct regime: two regions at low and high doping where $v_{\Delta}$ drops and a third regime in-between, where $v_{\Delta}$ reaches a plateau. From Ref. [@Vishik12].](fig17){width="50mm"}
We are grateful to H. Alloul, Y. Sidis, P. Bourges and A. Ferraz for stimulating and helpful discussions. This work was supported by LabEx PALM (ANR-10-LABX-0039- PALM), ANR project UNESCOS ANR-14-CE05-0007, as well as the grant Ph743-12 of the COFECUB. CP and TK also acknowledge hospitality at the IIP (Natal, Brazil) where parts of this work were done.
[^1]: Note that a different representation was used in the slave-boson approaches [@Kotliar88a; @Lee06], where the rotation was performed from the $d$-wave SC state towards $\pi$-flux phases.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The weakly coupled quasi-one-dimensional spin ladder compound (CH$_3$)$_2$CHNH$_3$CuCl$_3$ is studied by neutron scattering in magnetic fields exceeding the critical field of Bose-Einstein condensation of magnons. Commensurate long-range order and the associated Goldstone mode are detected and found to be similar to those in reference spin-dimer materials. However, for the upper two massive magnon branches the observed behavior is totally different, culminating in a drastic collapse of excitation bandwidth beyond the transition point.'
author:
- 'V. O. Garlea'
- 'A. Zheludev'
- 'T. Masuda'
- 'H. Manaka'
- 'L.-P. Regnault'
- 'E. Ressouche'
- 'B. Grenier'
- 'J.-H. Chung'
- 'Y. Qiu'
- 'K. Habicht'
- 'K. Kiefer'
- 'M. Boehm'
title: 'Excitations from a Bose-Einstein condensate of magnons in coupled spin ladders.'
---
Bose-Einstein condensation (BEC), such as the superfluid transition in liquid $^4$He [@London1938], is the emergence of a collective quantum ground state in a system of interacting Bosons. The condensate is characterized by a macroscopic order parameter that spontaneously breaks a continuous $U(1)$ symmetry. For BEC to occur at $T>0$, the Bosons should be able to freely propagate in 3 dimensions (3D) [@LLIX]. In one dimension BEC is forbidden even at zero temperature. In a striking example of dimensional crossover, even weak 3D coupling can enable BEC in 1D systems, where the normal state itself results from the unique 1D topology. A realization of this peculiar quasi-1D case was proposed only recently [@Giamarchi1999], and involves the condensation of magnetic quasiparticles in weakly coupled antiferromagnetic (AF) spin ladders.
Magnetic BEC can occur in a variety of spin systems [@Batyev1984]. In gapped quantum magnets, for example, an external magnetic field drives the energy of low-lying magnons to zero by virtue of Zeeman effect, prompting them to condense at some critical field $H_c$ [@Affleck91]. This transition is fully equivalent to conventional BEC. The rotational $O(2)\equiv
U(1)$ symmetry is spontaneously broken by the emerging AF long range order. At $T=0$ the density of magnons is zero for $H<H_c$, and vanishingly small just above the transition. The phenomenon is therefore described in the limit of negligible quasiparticle interactions. To date, such transitions were mainly studied in materials composed of coupled structural spin clusters [@Shiramura1997; @Kodama2002; @Ruegg2003; @Jaime2004; @Sebastian2005; @Zapf2006]. The condensing quasiparticles are then local triplet excitation that propagate due to inter-cluster interactions [@Nohadani2004]. This is in contrast to the original model of Ref. [@Giamarchi1999] that deals with coupled extended, translationally invariant objects. Their disordered “spin liquid” normal state is a direct consequence of 1D topology [@Haldane; @Kennedy92]. Even for weak coupling the magnons are fully mobile in 1D, rather than localized. Is the physics of the field-induced BEC in quantum AF spin ladders any different from that in local-cluster spin systems?
![Time-of-flight neutron spectra measured in [IPA-CuCl$_3$]{} in the 1D Haldane-gap spin liquid phase (a) and the 3D ordered BEC phase (b), at $H=0$ and $H=11.5$ T, respectively. Solid white arrows indicate the linearly dispersive gapless Goldstone mode. \[Fig1\]](Fig1.eps){width="8.7cm"}
In the present work we address this issue experimentally, through a neutron scattering investigation of a prototypical spin ladder material (CH$_3$)$_2$CHNH$_3$CuCl$_3$ ([IPA-CuCl$_3$]{}). This compound almost exactly realizes the original theoretical model of Ref. [@Giamarchi1999]. Its spin ladders are built of magnetic $S=1/2$ Cu$^{2+}$ ions and run parallel to the $a$ axis of the triclinic P$\overline{1}$ crystal structure. Conveniently, each one can be viewed as a “composite” Haldane spin chain [@Masuda2006]: pairs of $S=1/2$ spins on each rung are strongly [*ferromagnetically*]{} (FM) correlated and act as effective $S=1$ objects [@Masuda2006; @Manaka2000]. Coupling along the legs of the ladders is antiferromagnetic (AF), and translates into AF interactions between effective spins in the “composite” $S=1$ chains. Such chains are gapped [@Haldane] and are spin liquids with only short-range correlation. For [IPA-CuCl$_3$]{}the energy gap is $\Delta=1.2$ meV [@Manaka97; @Masuda2006].
![(Color online) Background-subtracted neutron spectra measured in [IPA-CuCl$_3$]{} at the 1D AF zone-center $h=0.5$ in various applied magnetic fields (symbols). The field values correspond to the 1D Haldane-gap spin liquid phase (a,b) and the 3D ordered magnon BEC phase (c). Line shapes are entirely due to experimental resolution (solid lines). \[Fig2\] ](Fig2.eps){width="8.5cm"}
![(Color online) (a) Measured gap energies in [IPA-CuCl$_3$]{} as a function of applied magnetic field. (b) Measured field dependencies of the $(0.5,0,0)$ magnetic Bragg reflection at two different temperatures (symbols). The $y$-offset corresponds to the actual background for $T=50$ mK and is arbitrary for the $T=500$ mK data. Lines are power-law fits over a range of 1 T. Inset: critical exponent $\beta$ as a function of the field window used in the least squares fit at $T=50$ mK. \[Fig3\] ](Fig3.eps){width="8.5cm"}
Excitations from this quantum-disordered ground state are revealed in inelastic neutron scattering (INS) experiments that directly probe the pair spin correlation function $S(\mathbf{q},\omega)$. Fig. \[Fig1\]a shows a time-of-flight (TOF) spectrum collected on a 3 g deuterated [IPA-CuCl$_3$]{} single crystal sample at $T=100$ mK in zero magnetic field using the Disc Chopper Spectrometer (DCS) at NCNR and 6.68 meV fixed incident energy neutrons. The magnon, with a steep parabolic dispersion along the $a$ axis and gap at the 1D AF zone-center $h= 0.5$, is clearly visible. $\Delta$ is small compared to the chain-axis magnon bandwidth, but is considerably larger than transverse bandwidths along the $c$ ($0.4$ meV) and $b$ axes ($<0.1$ meV) [@Masuda2006]. Since our main purpose will be to understand the special role that the AF spin ladder structure plays in the BEC phase of [IPA-CuCl$_3$]{}, we shall be comparing our results to those found in literature for TlCuCl$_3$ [@Ruegg2003], a prototypical AF spin-dimer compound. There $\Delta$ is small compared to magnon dispersion bandwidths in all 3 directions [@Cavadini2001; @Oosawa2002]. To date, TlCuCl$_3$ is the only material for which the spectrum of spin excitations in the BEC phase has been measured experimentally.
Figure \[Fig2\] illustrates the effect of magnetic field on [IPA-CuCl$_3$]{}. It shows spectra collected at the 1D AF zone-center $h=0.5$ in several fields using the SPINS 3-axis spectrometer at NCNR, with 3.7 meV fixed-incident energy neutrons, focusing Pyrolitic graphite analyzer and a BeO filter after the sample. As the field is turned on, the single peak at $H=0$ (Fig. \[Fig2\]a) becomes divided into three equidistant components (Fig. \[Fig2\]b). The peak widths are resolution-limited. The measured field dependencies of the gaps are plotted in Fig. \[Fig3\]a, which also includes points obtained using the cold neutron 3-axis spectrometer FLEX at HMI. The gap in the lower mode extrapolates to zero at $H_c=9.6$ T, where a BEC of magnons was previously detected in bulk measurements [@Manaka98].
As the gap softens, commensurate long-range AF order sets in and gives rise to new magnetic Bragg reflections of type $(h+1/2,k,l)$, $h$, $k$ and $l$-integer. The magnetic structure was determined at $H=12$ T in a neutron diffraction experiment at the D23 lifting counter diffractometer at ILL, on a 3$\times$2$\times$9 mm$^3$ single crystal sample, using $\lambda=1.276$ Å neutrons. A good fit to 48 independent magnetic reflections measured at $T=50$ mK was obtained using a collinear model with spins perpendicular to the field, aligned parallel to each other on the rungs, and antiparallel along the legs of the ladders. The refined value of the ordered moment is 0.49(1) $\mu_\mathrm{B}$.
The field dependence of the $(0.5,-1,0)$ peak intensity measured at $T=50$ mK is plotted in Fig. \[Fig3\]b, lower curve. To estimate the order parameter critical exponent $\beta$ we performed power-law fits to the data in a progressively shrinking field window (Fig. \[Fig3\]b, insert). The extrapolated value is $\beta>0.45$, in agreement with expectations. Indeed, at $T\rightarrow 0$, due to vanishing magnon density, one should recover the mean field (MF) result $\beta=0.5$ [@Giamarchi1999; @Matsumoto2002]. Any discrepancies between the observed and MF behavior become more pronounced at elevated $T$, when magnon density increases, and their interactions become relevant. At $T=500$ mK, for example, in [IPA-CuCl$_3$]{} we get $\beta=0.25(3)$ (Fig. \[Fig3\]b, upper curve). BEC critical indexes have also been observed under appropriate conditions in the dimer compound BaCuSi$_2$O$_6$ [@Sebastian2005]. However, recent work showed that the BEC universality of the transition in TlCuCl$_3$ is compromised by deviations from the Heisenberg model. Anisotropy[@Sirker2005] and magnetoelastic coupling[@Johannsen2005] modify the critical indexes and account for a small gap in the ordered phase. To date, in [IPA-CuCl$_3$]{} we found no evidence of lattice distortions at $H_c$ or deviations from BEC behavior.
![(Color online) Dispersion relation of the three excitation branches measured in [IPA-CuCl$_3$]{} at $H=11.5$ T$>H_\mathrm{c}=9.7$ T (solid symbols). Open circles: magnon dispersion at $H=0$ [@Masuda2006] Open squares: Dispersion of the middle branch at $H=9$ T. Lines are guides for the eye. \[Fig4\] ](Fig4.eps){width="8.5cm"}
A key result of this work is a direct measurement of excitations of the magnetic Bose-Einstein condensate in the high-field phase. Data collected above the critical field are shown in Fig. \[Fig2\]c (SPINS) and in Fig. \[Fig1\]b (DCS). Three distinct excitation branches, two gapped and one gapless, are clearly visible, though the overall inelastic intensity is reduced compared to lower fields. For each mode the dispersion relations at $H=11.5$ T were obtained by fitting Gaussian profiles to constant-$h$ cuts through the data in Fig. \[Fig1\]b. The results are plotted in Fig. \[Fig4\] in solid spheres. For comparison, we also plot the dispersion relation of the triplet at $H=0$ (open circles) [@Masuda2006]. Open squares show the dispersion of the middle magnon branch measured just below the transition at $H=9$ T, on the IN14 3-axis spectrometer at ILL under similar conditions.
The gapless mode observed at $H>H_c$ at low energies (arrows in Fig. \[Fig1\]b) is fully analogous to the phonon in conventional BEC, being the Goldstone mode associated with the spontaneous breaking of $O(2)$ symmetry. It is a [*collective*]{} excitation of the magnon condensate and has a linear dispersion relation [@Matsumoto2002]. It takes the place of the massive quadratically dispersive [*single-magnon*]{} excitation below $H_c$ (Fig. \[Fig1\]a). At $H=11.5$ T the fitted velocity of the Goldstone mode $v_\mathrm{G}=1.74(3)$ meV is reduced compared to the spin wave velocity [^1] in zero field $v=2.9(2)$ meV [@Masuda2006]. This behavior is qualitatively similar to that in TlCuCl$_3$ [@Ruegg2003; @Matsumoto2002], if one neglects the tiny anisotropy gap in the latter system [@Sirker2005], which is too small to be detected with INS anyway. The similarity also extends to the field dependence of the gap energies in those magnon branches that do not soften at the transition point. In both compounds the corresponding slope increases abruptly at $H_c$ (Fig. \[Fig3\]a). A bond-operator theoretical treatment of the dimer model [@Matsumoto2002] attributes all these effects to an admixture of the higher-energy triplet modes to the condensate. This interpretation can be qualitatively extended to our case of coupled spin ladders.
While the long-wavelength spectral features in [IPA-CuCl$_3$]{} are very similar to those in simple spin-dimer systems, the [*short-wavelength*]{} spin dynamics at the zone boundary is strikingly different. We find that the two massive excitations in [IPA-CuCl$_3$]{}undergo a qualitative change upon the BEC transition. As seen in Fig. \[Fig1\]b and Fig. \[Fig4\], at $H=11.5$ T, only 20% above $H_\mathrm{c}$, their bandwidths are suppressed by over a factor of two. The collapse occurs abruptly at the critical point: for the middle mode there is virtually no change of dispersion between $H=0$ and $H=9$ T (Fig. \[Fig4\]). Nothing of the sort happens in the spin-dimer compound TlCuCl$_3$, where the bandwidth of the two upper excitation branches evolves continuously with field, and is decreased by only 20% at $H=12$ T, which is more than twice $H_c$ [@Matsumoto2002].
The observed phenomenon can hardly be explained by a simple Zeeman shift of quasiparticle energies. Indeed, the latter is negligible, as small as $\sim 0.2$ meV between $H_c$ and 11.5 T. Instead, we suggest that the abrupt spectrum restructuring is is related to the translational invariance of the ground state wave function for an AF spin ladder or chain below $H_c$. At $H>H_c$ the emergence of long-range AF order breaks an [*additional*]{} discrete symmetry operation, namely a translation by the structural period of the ladder. There is no analogue of this in conventional BEC. Depending on inter-cluster interactions that define the ordering vector, neither does this necessarily happen in spin cluster materials. In particular, in TlCuCl$_3$ the induced magnetic structure retains the periodicity of the underlying crystal lattice. However, in a uniform AF spin ladder the extra symmetry breaking is unavoidable, regardless of inter-ladder coupling.
For [IPA-CuCl$_3$]{} the spontaneous doubling of the period implies that at $H>H_c$ the wave vectors $h=0$, $h=0.5$ and $h=1$ all become equivalent magnetic zone-centers. At the same time, $h=0.25$ and $h=0.75$ emerge as the new boundaries of the Brillouin zone. The result is a formation of anticrossing gaps for all magnons at these wave vectors [@Ziman], where each branch interacts with its own replica from an adjacent zone. This translates into a reduction of the zone-boundary energy for the visible (lower) segments of the two gapped magnons in [IPA-CuCl$_3$]{}. The additional violated symmetry operation is a microscopic one, and therefore plays no role in the long-wavelength physics probed at $h=0.5$.
To summarize, any [*long-wavelength*]{} characteristics of the field-induced magnetic BEC transition and the magnon condensate, such as critical indexes, emergence of the Goldstone mode and behavior of gap energies, appear to be universal. They are not affected by the 1D topological nature of the normal state in spin chains and ladders, and are very similar to those in local-cluster spin systems. In contrast, the [*short-wavelength*]{} properties can be significantly different in these two classes of materials. In coupled AF spin chains or ladders, unlike in many couple dimer systems, and unlike in conventional BEC, the transition breaks an additional discrete symmetry. The result is a radical modification of the excitation spectrum.
We thank A. Chernyshev (U. of California, Irvine) for his theoretical insight and to I. Zaliznyak (Brookhaven National Laboratory) for stressing the significance of the Brilloin zone folding. Research at ORNL was funded by the United States Department of Energy, Office of Basic Energy Sciences- Materials Science, under Contract No. DE-AC05-00OR22725 with UT-Battelle, LLC. T. M. was partially supported by the US - Japan Cooperative Research Program on Neutron Scattering between the US DOE and Japanese MEXT. The work at NIST is supported by the National Science Foundation under Agreement Nos. DMR-9986442, -0086210, and -0454672.
[22]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, ****, ().
, ** (, ), chap. .
, ****, ().
, ****, ().
, ****, ().
, , , , , , , ****, ().
, , , , , , , , , ****, ().
, , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , , , ****, ().
, , , , , , , , , ****, ().
J. Sirker, A. Weiße, and O. P. Sushkov, J. Phys. Soc. Jpn. [**74**]{} Suppl., 129 (2005)
N. Johannsen, A. Vasiliev, A. Oosawa, H. Tanaka, and T. Lorenz, Physical Review Letters [**95**]{}, 017205 (2005).
, ****, ().
, , , , , , ****, ().
, ****, ().
, , , ****, ().
, , , , , , , ****, ().
, , , , , , ****, ().
, , , , , ****, ().
, , , , ****, ().
, ** (, ), chap.
[^1]: For an excitation with gap $\Delta$ we define the velocity as $v={d \sqrt{(\hbar
\omega)^2-\Delta^2}}/{d (2 \pi h)}$, to be measured in energy units.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A non-universal interaction, which involves only the third family leptons induces lepton flavor violating couplings and contributes to the anomalous magnetic moment of muon. In this paper, we study the effects of non-universal interaction on muon (g-2) and rare decay $\tau \to \mu \gamma$ by using an effective lagrangian technique, and a phenomenological $Z^\prime$ model where $Z^\prime$ couples only to the third family lepton. We find that the deviation from the theory can be explained and the induced $\tau \to \mu \gamma$ rate could be very close to the current experimental limit. In the $Z^\prime$ model, $M_{Z^\prime}$ has to be lighter than 2.6 TeV.'
address: |
Institute of High Energy Physics, Chinese Academy of Sciences,\
P.O. Box 918, Beijing, 100039, China
author:
- 'T. Huang, Z.-H. Lin, L.-Y. Shan and X. Zhang\'
title: |
Muon Anomalous Magnetic Moment and\
Lepton Flavor Violation
---
epsf
Recently the Brookhaven AGS experiment 821 group has announced a new measurement of the anomalous magnetic moment with three times higher accuracy than it was previously known [@E821]. The experimental data now is in conflict with the theoretical prediction of the Standard Model (SM) with an excess of 2.6 $\sigma$. This may be an indication of new physics if it is more than a statistical fluctuation. Many authors have considered various possibilities of interpreting the discrepancy in models beyond the Standard Model, such as, supersymmetric version of the standard model, muon with substructure in composite model and anomalous gauge boson couplings [@new].
In this paper we consider a class of models where non-universal interactions exist and study the effects of new physics on muon (g-2) and lepton rare decay. We first take a model independent approach (effective lagrangian) to new physics, then consider a phenomenological $Z^\prime$ model. We find that the non-universal interaction can account for the deviation of muon (g-2) from the SM prediction. Furthermore, our models predict the rare decay rate $\tau \to \mu \gamma$ which can be tested in the planned $\tau$-charm factories and in $Z^\prime$ model $M_{Z^\prime}$ has to be lighter than 2.6 TeV which results in many interesting phenomenon [@nonuni; @lan].
The tau lepton is the heaviest one in the lepton sector and especially the top quark is heavier than all other fermions, with mass close to the electroweak symmetry breaking scale, the new physics, if exists, is expected to become manifest in the effective interaction of the third family fermions [@peccei]. There are also theoretical and phenomenological motivations that there may exist interactions with generation non-universal couplings [@nonuni; @lan]. Furthermore, the possible anomalies in the $Z$- pole $b \bar{b}$ asymmetries may suggest a non-universal $Z^\prime$ [@erler; @chan].
In terms of an effective lagrangian, the new physics is described by higher dimensional operators. At dimension-six, there are two operators [@tau; @esc] which contribute directly to the anomalous magnetic moment of leptons, $$\begin{aligned}
\label{ope}
{\cal O}_{\tau B}&=&{\bar L}\sigma^{\mu\nu} \tau_R \Phi B_{\mu\nu}, \nonumber\\
{\cal O}_{\tau W}&=&{\bar L} \sigma^{\mu\nu} {\vec \sigma} \tau_R \Phi {\vec
W}_{\mu\nu},\end{aligned}$$ where $L=( \nu_{\tau}, ~ \tau_L )$, $\Phi$ is the Higgs scalar, $B_{\mu\nu}$ and $W_{\mu\nu}$ are field strengths of $U_Y(1)$ and $SU_L(2)$, $\vec \sigma$ are Pauli matrices.
In the presence of operators ${\cal O}_{\tau B}$ and ${\cal O}_{\tau W}$, the effective Lagrangian ${\cal L}_{eff}$ can be written as, $$\begin{aligned}
\label{eq5}
{\cal L}_{eff} = {\cal L}_0 + \frac{1}{\Lambda^2} ( C_{\tau B} {\cal O}_{\tau B}
+ C_{\tau W} {\cal O}_{\tau W} + h.c. ),\end{aligned}$$ where ${\cal L}_0$ is the standard model lagrangian and $C_{\tau B}$, $C_{\tau W}$ are constants which represent the coupling strengths of ${\cal O}_{\tau B}$ and ${\cal O}_{\tau W}$. $C_{\tau B}$, $C_{\tau W}$ are expected to be $O(\frac{1}{(4\pi)^2})$ in theories with weakly interaction [@wudka], however could be large if the fundamental theory is strongly interaction such as the models of composite tau lepton.
After the electroweak symmetry is broken and the mass matrices of the fermions and the gauge bosons are diagonalized, the magnetic moment-type couplings of the leptons to gauge boson $Z$ and the photon $\gamma$ are $$\begin{aligned}
\label{lfvlag}
{\cal L}^{Z, \gamma}_{eff}&=&
eg^{Z, \gamma}
{{ \pmatrix{{\overline e}\cr {\overline \mu} \cr{\overline \tau} \cr}}}^T
\left \{
(-\frac{1}{ 2 m_\tau})( i k_\nu \sigma^{\mu\nu}) S^{Z, \gamma}
\right . \nonumber\\
&&\times U_l
\left . \pmatrix{ 0 & & \cr
& 0 & \cr
& & 1 \cr }
U_l^\dagger\right \}
\pmatrix{e \cr \mu \cr \tau \cr} V_{\mu}, \nonumber \\
&&V_{\mu}=Z_{\mu}, A_{\mu},\end{aligned}$$ where $g^Z = 1/(4s_Wc_W), g^\gamma=1$, and $$\begin{aligned}
\label{szsg}
S^Z&=&-g^Z \frac{2\sqrt 2}{e}\frac{m_\tau v}{\Lambda^2}\left [
C_{\tau W }c_W-C_{\tau B}s_W\right ],\nonumber\\
\label{eq15}
S^{\gamma}&=&\frac{2\sqrt 2}{e}\frac{m_\tau v}{\Lambda^2}\left [
C_{\tau W }s_W-C_{\tau B }c_W \right ].\end{aligned}$$
The matrix $U_l$ in Eq. (\[lfvlag\]) is the unitary matrix which diagonalizes the mass matrix of the charged lepton. In the SM, which corresponds to ${\cal L}_{eff}$ in the limit of $\Lambda
\to \infty$, the matrix $U_l$ is not measurable because of the zero neutrino masses. Furthermore, the universality of the gauge interaction guarantees the absence of the flavor changing neutral current and the lepton flavor violation in the lepton sector.
The lagrangian ${\cal L}^{Z, \gamma}_{eff}$ in Eq. (\[lfvlag\]) gives rise to the anomalous magnetic moment of muon (g-2) and the rare decays $\tau \to \mu \gamma$ etc, whose relative size, however, depends on the rotation matrix $U_l$. In Ref. [@tau], we have shown that $\mu \to e \gamma$ puts a very stringent constraint on the product of the tau-electron and muon-electron mixing angles. So in the following discussion we assume that the tau mixes only with the muon, but not with the electron and for simplicity we have assumed that rotation matrices of left-handed leptons and the right-handed leptons are equal [^1], \[matrix\] U\_l=, where $s=\sin\theta$ and $c=\cos\theta$ with $\theta$ being mixing angle. The decay width of $\tau \to \mu \gamma$ is given by ()= m\_ (s c S\^)\^2, and the new contribution to the tau and muon anomalous magnetic moments are given by a\_= c\^2 S\^ ; a\_= s\^2 S\^ . Within the parameter space of $C_{\tau B}, ~C_{\tau W}, \Lambda $ and $\sin\theta$, we have quantitatively studied the new physics effects on the $\tau \to \mu \gamma $, $Z \to \tau\mu $ , $\delta a_{\tau} $ and $\delta a_\mu$. We find that the predicted $Br(\tau \to \mu \gamma)$ is correlated to $\delta a_\mu$, which is shown in Fig. 1. The figure also shows that $Br(Z \to \tau \mu)$ could be quite close to the experimental limit. On the other hand, given the current limits on $Br(\tau \to \mu \gamma)$ and $\delta a_\mu$, the predicted $Br(Z \to \tau \mu)$ and $\delta a_\tau$ are well below the experimental limits. Numerically to fit $\delta a_{\mu}$ to be within 90% C.L. of the new experimental data, we obtain that $s^2$ has to be bigger than 0.46, which is quite reasonable in some models [@xing]. And $Br(\tau \to \mu \gamma)$ could be close to the current experimental limit. For example, taking $s^2\sim 0.5$, $C_{\tau B}, C_{\tau W} \sim 1/{(4 \pi)}^2 $ [@wudka] and $\Lambda \sim 17.5$ TeV, the new physics contribution to muon (g-2) is, a\_= 221 \^[-11]{}, which is within 90% C.L. of the experimental data a\^[exp]{}\_-a\^[SM]{}\_= 426 165 \^[-11]{}, and $ Br( \tau \to \mu \gamma ) = 1 \times {10}^{-6} $ which is close to the current experimental limit $ Br^{exp}( \tau \to \mu \gamma ) = 1.1 \times {10}^{-6} $ [@pdg]. From Fig.1, one can see that increasing the mixing angle $\theta$ will decrease the value of predicted $Br(\tau \to \mu \gamma)$, however for a large range of the mixing angle it remains to be close to the current experimental limit.
2\. truein
Now we consider a phenomenological model where a heavy neutral vector boson $Z^\prime_\mu$ is introduced and coupled to only the tau lepton, \^[int]{} = - g Z\^\_ \^, \[zprime\] where g is the coupling constant. In Eq. (\[zprime\]) we have considered only the vectorial coupling for simplicity. The couplings of $Z^\prime$ to other fermions are not given since they are irrelevant for the purpose of this paper.
After diagonalizing the lepton mass matrix by $U_l$ in Eq. (\[matrix\]), Eq. (\[zprime\]) becomes \^[int]{}&=& - g Z\^\_. \[zpleps\] At one-loop level, the anomalous magnetic moment of the muon is generated as shown in Fig. 2(a) and Fig. 2(b). However compared with Fig. 2(a), contribution to muon (g-2) from Fig. 2(b) is less suppressed by the mixing angles in the vertices and is enhanced by the tau lepton mass in the internal propagator, which gives rise to a\_&=& I( , ),\
I &=& \^1\_0\
&=& + [O]{} ( , ). Taking $ g^2 \le 4\pi $ to make the perturbative calculation reliable and $s^2 \sim 0.5$ to maximize $s^2c^2$, we find that to fit muon (g-2) to be within 90% C.L. of the experimental data, $M_{ Z^\prime} $ has to be lighter than 2.6 TeV. With these input parameters of $g, s$ and $M_{Z^\prime}$, the predicted rate of $ \tau \to \mu \gamma $ is $9.3\times 10^{-7}$. The result holds unchanged for weakly interaction theory if the ratio of $g^2/M^2_{Z^\prime}$ remains the same as of $4\pi/(2.6
TeV)^2$. In this type of models, there are in general two sources which contribute to $Z \to \tau \mu$. One is the radiative vertex correction by the $Z^\prime$ vector boson with the result given by Br(Z|)= Br(Zl |[l]{}) ( ( ))\^2. Taking $ g^2 \le 4\pi $, $s^2 \sim 0.5$ and $M_{ Z^\prime} =$ 2.6 TeV, we find that the branching ratio of $Z\to \tau \bar{\mu}$ is about $1.6\times 10^{-9}$, which is well below the experiment limit $1.2\times 10^{-5}$ [@pdg]. If a $Z - Z^\prime$ mixing $\theta_{Z^\prime}$ is explicitly introduced, we will have [@lan] \[2\] Br(Z|)= Br(Zl |[l]{}) , where $\theta_{Z^\prime}$ is the $Z$-$Z^{\prime}$ mixing angle, $g_Z$, $g_{(L,R)}$ are the $Z$ coupling constants in the SM. Given the experimental limit on $\theta_{Z^\prime}$, $Br(Z \to
\tau \mu)$ is quite safe with the experimental constraint [@pdg]. For example, taking $ g^2 \le 4\pi $, $s^2 \sim 0.5$ and $\sin \theta_{Z^\prime} \sim
10^{-3}$, Eq. (\[2\]) shows that the branching ratio of $Z\to \tau \bar{\mu}$ is about $3\times 10^{-6}$. A full study of the $ Z^\prime $ physics on the electroweak observables depends on the detail of the model parameters. This goes beyond the scope of this paper.
1.3 truein
In summary, we have proposed in this paper a possible solution to the discrepancy of the muon $( g-2 )$ between the standard model prediction and the new experimental data, by introducing the new physics with the third family. Compared to other approaches to $\delta a_\mu$ [@new], ours is the only one which relates the new physics responsible for $\delta a_\mu$ to another exciting possibility, [*i.e.*]{} lepton flavor violation. Furthermore, in model with $Z^\prime$, our results show that $M_{Z^\prime} $ has to be lighter than 2.6 TeV.
[**Acknowledgments**]{}: This work was supported in part by National Natural Science Foundation of China and by the Ministry of Science and Technology of China under Grant No. NKBRSF G19990754.
[99]{}
H.N. Brown et al., hep-ex/0102017.
A. Czarnecki and W.J. Marciano, hep-ph/0102122; K. Lane, hep-ph/0102131; L. Everett, G.L. Kane, S. Rigolin and L.-T. Wang,hep-ph/0102145; J.L. Feng and K.T. Matchev, hep-ph/0102146; E.A. Baltz and P. Gondolo, hep-ph/0102147; U. Chattopadhyay and P. Nath, hep-ph/0102157; U. Mahanta, hep-ph/0102176; D. Chakraverty, D. Choudhury and A. Datta, hep-ph/0102180.
For example, see, C.T. Hill, hep-ph/9702320, hep-ph/9802216; B. Holdom, Phys. Lett. B [**339**]{}, 114 (1994); H.-J He, T. Tait and C.P. Yuan, Phys. Rev. D [**62**]{}, 011702 (2000); S. Chaudhuri, S.-W. Chung, G. Hockney and J. Lykken, Nucl. Phys. B [**456**]{}, 89 (1995).
P. Langacker and M. Plumacher, Phys. Rev. D [**62**]{}, 013006 (2000).
R.D. Peccei and X. Zhang, Nucl. Phys. B[**337**]{}, 269 (1990); R.D. Peccei, S. Peris and X. Zhang, Nucl. Phys. B[**349**]{}, 305 (1991); B.-L. Young and X. Zhang, Phys. Rev. D [**51**]{}, 6584 (1995).
J. Erler and P. Langacker, Phys. Rev. Lett. [**84**]{}, 212 (2000).
M. Chanowitz, hep-ph/0104024, April (2001).
T. Huang, Z.-H. Lin and X. Zhang, Phys. Lett. B[**450**]{}, 257 (1999); hep-ph/0009353, Proceedings of the Symposium on Frontiers of Physics at Millennium, ed. by Y.L. Wu and J.P. Hsu.
R. Escribano and E. Masso, Phys. Lett. B[**395**]{}, 369 (1997); *ibid. Eur. Phys. J. C[**4**]{}, 139 (1998). C. Arzt, M.B. Einhorn and J. Wudka, Nucl. Phys. B[**433**]{}, 41 (1995).*
For example, see, H. Fritzsch and Z. Xing, Phys. Lett. B[**372**]{} (1996).
C.Caso et al., Particle Data Group, Eur. Phys. J. [**C 15**]{} 1(2000).
[^1]: We have also considered the case with different left-handed and the right-handed rotation matrices and find our conclusions unchanged.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
An experiment to search for hypothetical particle dark matter using cryogenic thermal detector, or bolometer is ongoing. The bolometer consists of eight pieces of 21g LiF absorbers and sensitive NTD germanium thermistors attached to them and is installed in the Nokogiriyama underground cell which is a shallow depth site ($\sim
15$m w.e.). We report on the results from the first running for about ten days using this arrayed bolometer system together with appropriate shieldings and muon veto counters. From the obtained energy spectra the exclusion limits for the cross section of the elastic neutralino-proton scattering are derived under commonly accepted astrophysical assumptions. The sensitivity for the light neutralino with a mass below 5GeV is improved by this work.
address: |
$^{\rm a}$RIKEN, The Institute of Physical and Chemical Research, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan\
$^{\rm b}$Department of Physics, School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan\
$^{\rm c}$RESCEU, Research Center for the Early Universe, School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan\
$^{\rm d}$International Center for Elementary Particle Physics, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan\
$^{\rm e}$Cryogenic Center, University of Tokyo, 2-11-16 Yayoi, Bunkyo-ku, Tokyo 113-0033, Japan
author:
- |
$^{\rm a}$[^1], M. Minowa $^{\rm bc}$, K. Miuchi $^{\rm b}$, Y. Inoue $^{\rm d}$,\
T. Watanabe $^{\rm b}$, M. Yoshida $^{\rm d}$, Y. Ito $^{\rm b}$[^2], Y. Ootuka $^{\rm e}$[^3]
title: First Results from Dark Matter Search Experiment in the Nokogiriyama Underground Cell
---
0.5cm
Introduction
============
There are a number of observational evidences to believe that a large fraction of the matter in the Universe exists in the form of non-baryonic particle dark matter. Supersymmetric neutralino is one of the most plausible candidates for such exotic particle dark matter. Various experimental efforts are being made aiming at detection of low energy nuclear recoils caused by the elastic scatterings of the neutralinos off nuclei[@dark; @matter; @search].
Conventional detectors like semiconductor detectors or scintillators generally have a quenching factor less than unity. Here the quenching factor is defined as the ratio of the energy detection efficiency for a nuclear recoil to that for an electron. On the other hand, because the bolometer is sensitive to the whole energy deposited in the absorber the quenching factor of the bolometer should be unity in principle. Actually a quenching factor close to unity has been measured by Milan group[@Alessandrello].
We have been developing bolometers with lithium fluoride absorbers[@Tokyo]. Fluorine is considered to have a large cross section for elastic scattering of the axially-coupled neutralino off the nucleus compared with other nuclei[@Ellis]. Recently we have successfully constructed the bolometer array with a total mass of 168g and installed it in the Nokogiriyama underground cell with a depth of 15m w. e.
In this paper we report on the first results from the experiment performed in the Nokogiriyama underground cell using the bolometer array.
Experimental Set-up and Measurement
===================================
The bolometer array used in this work contains eight 21g LiF bolometers. The schematic drawing of the bolometer array is shown in Fig.\[fig:multi\]. The neutron transmutation doped (NTD) germanium thermistors with the similar temperature dependence of the resistance[@NTD] are attached to the crystals. The thermistor senses a small temperature rise of the absorber crystal caused by the neutralino-nucleus scattering. Each crystal is placed on four copper posts and thermally insulated by the Kapton sheets. Moderate thermal anchoring of the crystal to the copper holder with a temperature of 10mK is realized by a oxygen free copper (OFC) ribbon. The lithium fluoride crystals are checked by a low-background Ge spectrometer prior to the construction of the bolometer. The concentration of radioactive contaminations is less than 0.2ppb for U, 1ppb for Th, and 2ppm for K. The bolometer array is mounted on a mixing chamber of a dilution refrigerator which is mostly made of low-radioactivity materials radio-assayed in advance by low-background Ge spectrometer.
Each thermistor is biased through a 100M$\Omega$ load resistor. The voltage change across the thermistor is fed into the eight channel source follower circuit placed at the 4K stage which include a low noise junction field effect transistor (J-FET), Hitachi 2SK163. Since the J-FET does not work at this low temperature, it is connected to a printed circuit board with thin stainless steel tubes and manganin wires to be thermally isolated from the circuit board with a temperature of 4K and the temperature of the FET is maintained above 100K by the heat produced by itself.
The signal from the source follower circuit is in turn amplified by an eight channel voltage amplifier placed just above the refrigerator. The output of the voltage amplifier is fed into a double pole low-pass filter with a cut-off frequency of 226Hz and in turn into the 16-bit waveform digitizer to record the pulse shape of the signal for off-line analysis.
The passive radiation shielding consists of 10cm-thick oxygen free high conductivity copper layer, 15cm-thick lead layer, 1g cm$^{-2}$-thick boric acid layer and 20cm-thick polyethylene layer. The latter two layers act as a neutron shield. In order to avoid muon-induced background we employ a veto system which consists of 2cm-thick plastic scintillators.
The constructed detector system is installed in the Nokogiriyama underground cell which is located about 100km south from Tokyo and relatively easy to access. The depth of an overburden of sand is inferred to be about 15m w.e. In this work six bolometers of the bolometer array are used and energy spectrum are measured for about ten days. Two bolometers have some problems in the cooling procedure.
Since the detector is enclosed in a cryogenic vacuum can during the measurements it is impossible to place the gamma-ray source close to the detector for energy calibration. The energy calibration during the measurements is, therefore, performed by 662keV gamma-rays from a $^{137}$Cs source and 1333keV and 1173keV gamma-rays from a $^{60}$Co source placed outside a helium dewar of the dilution refrigerator and inside the radiation shieldings. Furthermore, the sharp peak at 4.78MeV due to the neutron capture reaction of $^6$Li observed in the background spectrum is also used for pthe energy calibration. Fig.\[fig:calibration\] shows one of the obtained energy calibration plots. Linearities of the six bolometers up to 5MeV are recognized. It must be noted that linearity down to 60keV gamma-ray is confirmed prior to this measurement using gamma-ray from $^{241}$Am source set inside the cryostat.
Energy Spectra and Dark Matter Limits
=====================================
Fig.\[fig:spectrum\] shows the energy spectra obtained by the six bolometers during ten days. The bump in the low energy region is considered to be due to microphonics caused by a helium liquefier which recondenses evaporated helium gas from the dewar. While the similar spectra are obtained for four bolometers (D3, D5, D6, and D8), the spectra for the other two bolometers (D1 and D4) are affected by microphonics below 30 to 40keV because of their low detector gains.
Comparing the measured energy spectrum with the expected recoil spectrum, the exclusion limits for the cross section for elastic neutralino scattering off the nucleus can be extracted. The calculation is performed in the same manner used in Ref.[@smith]. The theoretical recoil spectrum is calculated assuming a Maxwellian dark matter velocity distribution with rms velocity of 230km/s, and then folded with the measured energy resolution and the nuclear form factor. We also assume the local halo density of the neutralino to be 0.3GeV/cm$^3$. The spin factors calculated assuming an odd group model as a nuclear shell model are 0.75 for $^{19}$F and 0.417 for $^{7}$Li[@Ellis]. Since the detector responses for the six bolometers are not the same, the upper limit of the cross section is evaluated independently from the spectrum of each detector. For a given neutralino mass the lowest value of the cross section is taken as a combined limit from the results of the six detectors.
The calculated exclusion limits in case of the spin-dependent interaction are given in Fig.\[fig:limit\]. For comparison the exclusion limits derived from the data in the other experiments at deep underground sites[@EDELWEISS; @BPRS; @DAMA; @smith; @OSAKA] and the scatter plots predicted in the minimal supersymmetric theories are also shown. Although the other experiments except for Osaka experiment are performed at deep underground laboratories, our experiment gives comparable limits for the light neutralino. This owes to the large cross section for the spin-dependent interaction of $^{19}$F and the low energy threshold of the bolometer. The sensitivity for neutralinos with a mass below 5GeV is improved by this work.
Prospects
=========
Compton scattered gamma-rays from the aperture of the shielding and gamma-rays produced through the interaction of cosmic ray muon within the shielding materials are considered major background sources. In order to reduce the muon-correlated background, the veto efficiency must be improved. The present incompleteness of the veto is due to the penetrations for vacuum tubes and the tube of the helium liquefier. Increasing of the coverage of the plastic scintillator will improve the veto efficiency up to 98%. Against the Compton scattered gamma-ray background, internal lead shielding with a thickness of 20mm surrounding the lithium fluoride bolometer array will be installed. The shielding is made of over 200 year old low-activity lead with a concentration of $^{210}$Pb of less than 0.05pCi/g. The Compton scattered gamma-rays can be reduced by two orders of magnitude by this internal shielding. Since lead fluorescence X-rays are produced mainly by the muon interaction, their contribution can be ignored if muons are sufficiently vetoed. If these improvements are realized, the sensitivity of this experiment will be improved by more than an order of magnitude even at this shallow depth.
The detector system will be installed in a underground facility with a sufficient depth where cosmic muon induced background is expected to be negligible. The long-term measurements in a deep underground site will bring the sensitivity to the spin-dependent interaction below the level predicted by the supersymmetric theory.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Prof. Komura for providing us with the low-radioactivity old lead. This research is supported by the Grant-in-Aid for COE Research by the Japanese Ministry of Education, Science, Sports and Culture. W.O. are grateful to Special Postdoctoral Researchers Program for support of this research.
[9]{} E. W. Kolb and M. S. Turner, [*The Early Universe*]{} (Addison-Wesley, Redwood City, 1989);\
P. F. Smith and J. D. Lewin, [ *Phys. Rep. *]{} [**187**]{}(1990)203;\
R. Bernabei, [*Riv. Nuovo Cimento*]{} 18(1995), No.5. A. Alessandrello et al., [*Phys. Lett. *]{}B [**408**]{}(1997)465. M. Minowa et al., [*Nucl. Instrum. Methods*]{} A [**327**]{}(1993)612;\
W. Ootani et al., [*Astrop. Phys. *]{} [**9**]{}(1998)325;\
W. Ootani, doctoral thesis, University of Tokyo, 1998. J. Ellis and R. A. Flores, [*Phys. Lett. *]{} B [**263**]{}(1991)259;\
J. Ellis and R. A. Flores, [*Phys. Lett. *]{} B [**300**]{}(1993)175. W. Ootani et al., [*Nucl. Instrum. Methods*]{} A [**372**]{}(1996)534. A. de Bellefon et al., [*Astrop. Phys. *]{} [**6**]{}(1996)35. C. Bacci et al., [*Astrop. Phys. *]{} [**2**]{}(1994)117. R. Bernabei et al., [*Astrop. Phys. *]{} [**7**]{}(1997)73;\
R. Bernabei et al., [*Phys. Lett. *]{} B [**389**]{}(1996)73. P. F. Smith et al., [*Phys. Lett. *]{} B [**379**]{}(1996)299. R. Hazama, doctoral thesis, Osaka University, 1998.
[^1]: e-mail: ootani@postman.riken.go.jp
[^2]: Present address: KEK, High Energy Accelerator Research Organization, 3-2-1 Midori-cho, Tanashi-shi, Tokyo 188-8501, Japan
[^3]: Present address: Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-0006, Japan
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We prove that for any $\epsilon>0$ the growth rate $P_n$ of generalized diagonals or periodic orbits of a typical (in the Lebesgue measure sense) triangle billiard satisfies: $P_n<Ce^{n^{\sqrt{3}-1+\epsilon}}$. This provides an explicit sub-exponential estimate on the triangle billiard complexity and answers a long-standing open question for typical triangles. This also makes a progress to the problem 3 in the Katok’s list of “Five most resistant problems in dynamics”. The proof uses essentially new geometric ideas and does not rely on the rational approximations.'
author:
- |
Dmitri Scheglov\
University of Oklahoma
title: Growth of periodic orbits and generalized diagonals for typical triangle billiards
---
Introduction
============
There are several closely related definitions for a complexity of polygonal billiards. Examples include the growth of periodic orbits, growth of generalized diagonals and orbit complexity. More specifically one can also define directional complexity and position complexity ( see, say \[1\], \[2\], \[3\], \[4\], \[12\], \[13\] ). Each definition measures the growth of orbits, satisfying some special property. It is an easy remark that the growth of periodic orbits cannot exceed the growth of generalized diagonals and in the paper \[1\] there is a precise relation between growth of generalized diagonals and the growth of all orbits. This implies that any upper estimate on the growth of generalized diagonals would automatically produce an upper estimate on other complexities. In our paper we estimate the growth rate of generalized diagonals. A generalized diagonal is a billiard orbit which connects two vertices. The complexity function $P_n$ is a total number of generalized diagonals of length no greater than $n$. Here by length of the diagonal we mean a discrete length or the number of reflections, but it is well known that the actual geometric length is uniformly proportional to the discrete one. For a polygon with $k$ sides $P_n\leq k^n$ by trivial combinatorial reasons. In 1987 Katok \[8\] proved the following sub-exponential estimate:
For any polygon: $\lim\frac{\ln(P_n)}{n}=0$.
In 1988-90 Masur \[10\], \[11\] proved more precise estimates for any *rational-angled* polygon:
For any polygon with angles in $\pi\mathbb{Q}$ there are constants $C_1, C_2>0$ such that: $C_1\cdot n^2<P_n<C_2\cdot n^2$.
He used an observation that a billiard in the polygon with rational angles is isomorphic to the geodesic flow on the compact flat surface with a finite number of conical singularities. On such a surface originating from a billiard there is a natural complex structure and moreover a natural choice of holomorphic quadratic differentials which allows to use Teichmuller theory. However for irrational polygons this method can not be applied as the resulting surface is not compact. A well-known open problem is to find an explicit sub-exponential estimate for $P_n$ which is considered to be very difficult by many experts. A. Katok in his “Five most resistant problems in dynamics”\[7\] makes even stronger conjecture:
For any polygon $M$ and any $\epsilon>0$: $P_n<C(M, \epsilon)n^{2+\epsilon}$.
In this strong form the conjecture is quite far from being proven because of the lack of our understanding of the structure of irrational polygonal billiards. However, as we would like to quote A. Katok \[5\] here: “ *Any* effective sub-exponential estimate ( such as $e^{-T^{3/4}}$, say) for arbitrary polygons would be a major advance.” Finding an explicit sub-exponential estimate was a primary motivation for our research. And we would like to thank A. Katok here who mentioned the importance of this problem several times during about 6 years to us, which definitely gave an extra-motivation. One of the reasons why it is so difficult to analyse $P_n$ is that it is a purely discrete counting of different orbits and so it does not take into account any orbit structure such as density of orbits in a particular angular region or distribution of orbits with respect to the natural invariant measure. This distinguishes the complexity growth from other dynamical characteristis. For example ergodicity of some irrational polygons was proven by applying approximating arguments and moreover the result of Vorobets \[14\] explicitely describes some well-approximated ergodic polygons.
For completeness of the exposition we would like to briefly discuss the key ideas of the original paper by Katok \[8\]. He considers a topological subshift on $k$ symbols, naturally associated to the billiard in $k$ - gon and proves that any ergodic invariant measure is supported on the subset, generated by the images of actual billiard orbits. Then he proves that metric entropy of any such measure is equal to zero and then by variational principle it implies that the topological entropy is also zero. As the symbolic cylinder growth in this setting can be reformulated in terms of $P_n$, the fact that topological entropy is zero completes the proof. Even though the proof is elegant it has several non-explicit steps which make it hard to extract more precise information about $P_n$ than sub-exponential growth. First of all it uses ergodic invariant measures and a variational principle and it is not clear how to make this abstract argument constructive. And the second point is that the topological entropy can only distinguish exponential growth and does not ’feel’ any sub-exponential effects, where hypothetically some kind of *slow entropy* is required to extract non-trivial information. Our approach is more geometric and combinatorial and not ergodic-theoretic. The aim of the paper is to prove the following theorem:
For a typical triangle and any $\epsilon>0$ there is a constant $C>0$ such that: $P_n<Ce^{n^{\sqrt{3}-1+\epsilon}}$.
**Acknowledgements.** We would like to thank Dr. John Albert and Dr. Christian Remling for useful discussions and Dr. Andrei Gogolev for useful discussions and constant encouragement along the progress in the work. We would also like to thank Dr. Anatole Katok and Dr. Federico Rodriguez-Hertz for reading the paper, encouragement and a serious help when the gap was found in the first version of the paper. We would also like to express our special acknowledgements to Dr. Serge Troubetzkoy who very carefully read the paper and made several very important remarks which helped to improve its structure and the quality of exposition, and to Dr. Eugene Gutkin for his several important remarks, constant interest to the paper and several useful conversations about billiard dynamics. Without support of these mathematicians this paper would not appear.
Interval partitions
===================
We consider a given triangle, a fixed vertex and the corresponding anglular segment located at the vertex, which we naturally associate with an interval $I$ using the angular distance on it. In this setting points on the interval correspond to rays emanating from the vertex. Now let us create a decreasing sequence of finite indexed partitions $\xi_n $ of $I$ on subintervals as follows. $\xi_0=I$ a trivial partition with one element. Cutting points of partition $\xi_n$ are those corresponding to the generalized diagonals of length no greater than $n$. The following two properties immediately follow from this construction: $1)$ Inside each interval of the partition $\xi_n$ there is at most one point of the partition $\xi_{n+1}$. $2)$ The sequence $\xi_n$ converges to the partition on points. In the other words the union of all cutting points is dense in $I$. By construction the number $P_n$ of generalized diagonals is exactly the number of cutting points of $\xi_n$ and each cutting point has an index, namely the length of the corresponding generalized diagonal.
Points in a good position
-------------------------
Consider a sequence of partitions $\xi_n$ and 3 indexed points $x_p$, $x_q$, $x_r$ as cutting points of corresponding partitions, such that $p<q<r$. **Definition.** The points $x_p$, $x_q$, $x_r$, $p<q<r$ are in a *good position* if: $1)$ In the interval, bounded by $x_p$ and $x_q$ there are no points with index $<r+1$ except $x_r$. $2)$ The point $x_r$ lies between $x_p$ and $x_q$ The picture below shows points in a good position.
=1.00mm
(91.67,14.00) (6.00,10.67)[(1,0)[85.67]{}]{} (28.00,10.67) (49.33,10.67) (73.33,10.67) (28.00,14.00)[(0,0)\[cc\][p]{}]{} (49.33,14.00)[(0,0)\[cc\][r]{}]{} (73.33,14.00)[(0,0)\[cc\][q]{}]{}
=1.00mm
(91.67,14.00) (6.00,10.67)[(1,0)[85.67]{}]{} (28.00,10.67) (49.33,10.67) (73.33,10.67) (28.00,14.00)[(0,0)\[cc\][q]{}]{} (49.33,14.00)[(0,0)\[cc\][r]{}]{} (73.33,14.00)[(0,0)\[cc\][p]{}]{}
Pic 1. Points in good position. Below we present an example of a configuration of points $x_p$, $x_q$, $x_r$, $p<q<r$ which are NOT in good position.
=1.00mm
(91.67,14.00) (6.00,10.67)[(1,0)[85.67]{}]{} (28.00,10.67) (49.33,10.67) (73.33,10.67) (28.00,14.00)[(0,0)\[cc\][r]{}]{} (49.33,14.00)[(0,0)\[cc\][p]{}]{} (73.33,14.00)[(0,0)\[cc\][q]{}]{}
Pic 2. Points NOT in good position.
Consider an interval $J$ belonging to the partition $\xi_n$ and a finite sequence of partitions $\xi_{n+1},\ldots, \xi_{n+c}$. Let $S$ be the set of all cutting points of $\xi_{n+c}$ inside $J$. Assume that the cardinality $|S|\geq 4+2c$. Then there exist points $x_p, x_q, x_r\in S$ in a good position.
Assume that a partition $\xi_{n+m}$ has at least 3 points : $x_{i_1}< x_{i_2}<\ldots< x_{i_p}$ inside $J$. Then the partition $\xi_{n+m+1}$ may only have at most two more points inside $J$: $x_{n+m+1}, y_{n+m+1}$ : $x_{n+m+1}<x_{i_1}$ and $x_{i_p}<y_{n+m+1}$ without producing a triple in good position. So the set $S$ without a triple in good position has at most $3+2c$ points.
Existence of close points in a good position
---------------------------------------------
Consider a finite sequence of partitions $\xi_{n}, \xi_{n+1},\ldots,\xi_{n+c}$. Assume that $P_{n+c}\geq (4+2c)P_n$. We also assume that $c\geq 4$ and $e^c<P_n$. Then there exist 3 points in good position with indices from the range $[n+1,n+c]$ and $e^c/P_n$ - close to each other.
The number of cutting points of $\xi_{n+c}$ inside each interval of the partition $\xi_n$ is bounded by $2^c$. Let $x$ be the number of intervals of the partition $\xi_n$ which have at least $4+2c$ points of $\xi_{n+c}$ inside and $y$ be the number of intervals with less than $4+2c$ points of $\xi_{n+c}$ inside. We then have two obvious relations: 1) $x+y=P_n+1$ 2) $2^cx+(3+2c)y\geq (4+2c)P_n$ from which $x\geq\frac{P_n-3-2c}{2^c-3-2c}>P_n/e^c$ follows. By Lemma 2.1 each interval of $\xi_n$ containing at least $4+2c$ points also contains 3 points in a good position with indices in the range $[n+1,\ldots, n+c]$. As the corresponding intervals do not intersect and their total number is at least $x$ and they all are contained in the interval $[0, 1]$, the estimate on $x$ completes the proof.
Index interval estimates
-------------------------
In this section we estimate the constant $c$ which gives a range for indices of points in a good position. We introduce constants $\mu, \epsilon, \gamma>0$ to be chosen later. Along the way we specify the assumptions to be satisfied for these constants (we use **bold** font for that) and in the end we summarize all the assumptions and choose the particular values for $\mu, \epsilon, \gamma$. We now introduce the following functions: $\phi(n)=n^{\mu}$, $c(n)=n^{\epsilon}$, $k(n)=n^{\gamma}$.
Assume a sequence of partitions $\xi_n$ satisfies inequality $P_n\geq e^{\phi(n)}$ for any $n$ large enough. Then for an appropriate choice of constants $\mu, \gamma, \epsilon$ and any $n$ large enough there exists an integer $N$: $n<N<n+k(n)c(n) $ such that $P_{N+c(n)}/P_{N}\geq 4+2c(n)$.
Divide interval $[n+1,n+k(n)c(n)]$ into $k(n)$ intervals $I_1=[n+1, n+c(n)]$, $I_2=[n+c(n)+1, n+2c(n)]$,$\ldots$, $I_{k(n)}=[n+(k(n)-1)c(n)+1, n+k(n)c(n)]$.
We prove by contradiction that for one of the intervals $I_l$: $P_{N+c(n)}/P_N\geq 4+2c$, where $N=n+lc(n)$. Assume that for all the intervals $I_l$: $P_{n+(l+1)c(n)}/P_{n+lc(n)}<4+2c(n)$. Then $P_{n+k(n)c(n)}/P_{n}<(4+2c(n))^{k(n)}$. On the other hand $P_{n+k(n)c(n)}/P_{n}>e^{\phi(n+k(n)c(n))}/e^n$ which implies: $\phi(n+k(n)c(n))-n<k(n)\ln(4+2c(n))$. We assume that $\gamma\leq 1$ and $\gamma+\epsilon>1$ so $k(n)c(n)$ has degree higher than 1. Then comparing the highest degrees on both sides implies $(\gamma+\epsilon)\mu\leq 1$
which means that we get a contradiction under assumption $(\gamma+\epsilon)\mu> 1$.
Combinatorial geometry of orbits
================================
Unfolding of a billiard trajectory
-----------------------------------
We would like to remind of a useful unfolding construction associated to any polygonal billiard.\[9\] We fix a polygon on the plane and consider a time moment when a particular billiard orbit hits a polygon side. Then instead of reflecting the orbit we continue it as a straight line and reflect the polygon along the line. As we continue this process indefinitely the sequence of polygons obtained this way is called unfolding of the polygon along the orbit. The picture below illustrates the unfolding of a triangle along an orbit. =1.00mm
(94.67,40.33) (9.00,3.67)[(-1,4)[6.67]{}]{} (2.33,29.67)[(2,3)[6.33]{}]{} (8.66,39.34)[(2,-3)[6.67]{}]{} (15.33,29.34)[(-1,-4)[6.00]{}]{} (9.00,3.67)[(1,1)[17.33]{}]{} (26.33,21.00)[(0,1)[13.00]{}]{} (26.33,34.00)[(-5,-2)[11.33]{}]{} (4.00,23.34)[(1,0)[90.67]{}]{} (26.33,34.00)[(6,-1)[11.00]{}]{} (37.33,32.34)[(1,-3)[8.00]{}]{} (45.33,8.34)[(-3,2)[19.00]{}]{} (9.00,4.00)[(0,1)[35.00]{}]{} (9.00,4.33)[(3,5)[18.00]{}]{} (45.33,8.33)[(-3,4)[19.00]{}]{} (37.33,32.33)[(1,1)[8.00]{}]{} (45.33,40.33)[(0,-1)[31.67]{}]{}
Pic 3. Triangle unfolding. For a given triangle the shape obtained from a triangle by reflection about one side is called a *kite*. It is clear that for any triangle unfolding there is an associated kite unfolding. We will use both unfoldings having in mind the natural correspondence between them. The next picture shows the corresponding kite unfolding. =1.00mm
(94.67,42.67) (9.00,3.67)[(-1,4)[6.67]{}]{} (2.33,29.67)[(2,3)[6.33]{}]{} (8.66,39.34)[(2,-3)[6.67]{}]{} (15.33,29.34)[(-1,-4)[6.00]{}]{} (9.00,3.67)[(1,1)[17.33]{}]{} (26.33,21.00)[(0,1)[13.00]{}]{} (26.33,34.00)[(-5,-2)[11.33]{}]{} (4.00,23.34)[(1,0)[90.67]{}]{} (26.33,34.00)[(6,-1)[11.00]{}]{} (37.33,32.34)[(1,-3)[8.00]{}]{} (45.33,8.34)[(1,5)[5.33]{}]{} (45.33,8.34)[(-3,2)[19.00]{}]{} (50.66,35.00)[(-1,1)[7.67]{}]{} (43.33,42.67)[(-3,-5)[6.33]{}]{}
Pic 4. Kite unfolding As we see from the picture above, any kite unfolding along the orbit consists of consecutive rotations of the kite along one of the two kite vertices, corresponding to the angles $\alpha$ and $\beta$ of the original triangle with angles correspondingly $ 2\alpha$ and $2\beta$. We now assume that a kite is located in the standard Euclidean $xy$ coordinate plane and introduce several notations. The *$\alpha$-vertex* and *$\beta$- vertex* are kite vertices corresponding to the angles $2\alpha$ and $2\beta$. Two other vertices are called *side vertices*. The *kite diagonal* is a vector going from the $\alpha$-vertex to the $\beta$-vertex. The *kite angle* is a *counterclockwise* angle between $x$-axis and the kite diagonal. On pic.5 $A$ and $B$ are $\alpha$ and $\beta$ vertices correspondingly, vector $\overrightarrow{AB}$ is a kite diagonal, $C$ and $D$ are side vertices. Pic.6. shows a kite in standard position on the $xy$ plane. Note that any unfolding of $K$ is uniquely characterized by the sequence of angles $\pm 2\alpha$ or $\pm 2\beta$ depending on the kite vertex we rotate about and the direction of rotation. Such a sequence of angles is called the *combinatorics* of a kite unfolding. =1.00mm
(112.33,67.34) (2.00,19.00)[(1,0)[110.33]{}]{} (9.34,4.67)[(0,1)[62.67]{}]{} (62.34,29.34)[(1,1)[34.33]{}]{} (96.67,63.67)[(-1,0)[18.33]{}]{} (78.34,63.67)[(-1,-2)[17.67]{}]{} (96.67,63.67)[(-1,-1)[36.00]{}]{} (60.67,27.67)[(1,1)[36.00]{}]{} (60.67,28.00)[(5,2)[36.00]{}]{} (96.67,42.33)[(0,1)[21.33]{}]{} (61.00,28.33) (96.67,63.67) (78.00,63.67) (96.67,42.33) (61.00,28.33) (56.67,28.00)[(0,0)\[cc\][$A$]{}]{} (100.33,63.67)[(0,0)\[cc\][$B$]{}]{} (73.67,63.67)[(0,0)\[cc\][$C$]{}]{} (100.33,42.33)[(0,0)\[cc\][$D$]{}]{} (110.33,15.33)[(0,0)\[cc\][$x$]{}]{} (6.00,66.00)[(0,0)\[cc\][$y$]{}]{} (68.33,38.67)[(0,0)\[cc\][$\alpha$]{}]{} (71.33,34.67)[(0,0)\[cc\][$\alpha$]{}]{} (89.00,60.67)[(0,0)\[cc\][$\beta$]{}]{} (94.00,56.67)[(0,0)\[cc\][$\beta$]{}]{}
Pic 5. A kite on the $xy$ coordinate plane. =0.5mm
(129.33,114.67) (9.00,52.00)[(1,-1)[29.67]{}]{} (9.33,51.67)[(1,1)[29.33]{}]{} (38.66,81.00)[(1,-2)[14.67]{}]{} (53.33,51.67)[(-1,-2)[14.67]{}]{} (9.33,52.00)[(1,0)[117.67]{}]{} (9.33,12.67)[(0,1)[99.33]{}]{} (15.00,54.00)[(0,0)\[cc\][$\alpha$]{}]{} (15.00,49.67)[(0,0)\[cc\][$\alpha$]{}]{} (49.00,54.33)[(0,0)\[cc\][$\beta$]{}]{} (49.00,49.00)[(0,0)\[cc\][$\beta$]{}]{} (129.33,47.33)[(0,0)\[cc\][$x$]{}]{} (3.33,114.67)[(0,0)\[cc\][$y$]{}]{}
Pic 6. A kite in the standard position on the $xy$ plane.
Assume a kite $K$ with a diagonal length 1 is in standard position and a kite $K'$ is obtained from $K$ by means of a particular combinatorics of length $n$. Let $x^{\alpha}_n$, $y^{\alpha}_n$ and $x^{\beta}_n$, $y^{\beta}_n$ be the coordinates of $\alpha$ and $\beta$ vertices of $K'$ and let $x_n$, $y_n$ be the coordinates of either of the two side vertices of $K'$. Then: $1)$ $x^{\alpha}_n$, $y^{\alpha}_n$, $x^{\beta}_n$, $y^{\beta}_n$ are represented by trigonometric polynomials of angles $\alpha$, $\beta$ with integer coefficients, depending only on the combinatorics and of degree at most $2n-2$. $2)$ $x_n=P_{2n}(\alpha, \beta)+ \frac{\sin (\beta)}{\sin(\alpha+\beta)}\cdot \cos(m\alpha+l\beta)$ $y_n=Q_{2n}(\alpha, \beta)+ \frac{\sin (\beta)}{\sin(\alpha+\beta)}\cdot \sin(m\alpha+l\beta)$, where $P_{2n}(\alpha,\beta)$, $Q_{2n}(\alpha, \beta)$ are trigonometric polynomials with integer coefficients of degree at most $2n-2$ and $|m|+|l|\leq 2n-1$.
The proof of the first statement goes by an easy induction on $n$. For n=1 the statement is trivial. If $\phi_n$ is the kite angle on the $n$-th step and on the $n+1$-th step we rotate, say, about $\alpha$-vertex, then $\phi_{n+1}=\phi_n\pm 2\alpha$ and $x^{\alpha}_{n+1}=x^{\alpha}_n$, $y^{\alpha}_{n+1}=y^{\alpha}_n$, $x^{\beta}_{n+1}=x^{\alpha}_n+\cos(\phi_{n+1})$, $y^{\beta}_{n+1}=y^{\alpha}_{n}+\sin(\phi_{n+1})$. The case when we rotate about $\beta$-vertex is entirely analogous. This completes the induction step. The second statement easily follows from the first one by noticing that the length of the side, adjacent to the $\alpha$-vertex is $\frac{\sin (\beta)}{\sin(\alpha+\beta)}$ and so if $\phi_n$ is the kite angle of $K'$ then $x_n=x^{\alpha}_n+\frac{\sin (\beta)}{\sin(\alpha+\beta)}\cdot \cos(\phi_n\pm\alpha)$ and $y_n=y^{\alpha}_n+\frac{\sin (\beta)}{\sin(\alpha+\beta)}\cdot \sin(\phi_n\pm\alpha)$.
Complexity estimate
===================
In this chapter we assume that the triangle has a fixed side of length 1 and adjacent angles are acute and for some arbitrarily small parameter $\delta>0$ satisfy: $\alpha>\delta$, $\beta>\delta$, $\alpha+\beta<\pi-\delta$. This condition guarantees that there are constants $D_{\delta}, N_{\delta}>0$ such that for any billiard orbit with $n$ reflections : $L(n)/D_{\delta}<n<L(n)D_{\delta}$, for $n>N_{\delta}$, where $L(n)$ is a geometric length of the orbit. As $\delta$ can be chosen arbitrarily small the conclusion of the theorem would hold for a full space of triangles.
For a full measure set of triangles and any $\epsilon>0$: $\lim\inf P_n\cdot e^{-n^{\sqrt{3}-1+\epsilon}}<\infty$.
We consider a triangle and a vertex and fix $n$ large enough. Assume that $P_n\geq e^{\phi(n)}$. Then by Lemma 2.3. we find $N$: $n<N<n+k(n)c(n)$ such that $P_{N+c(n)}/P_{N}\geq 4+2c(n)$. To use Lemma 2.2. we need to make sure that $e^{c(n)}<P_{N}$ which is true if $c(n)<n^{\mu}$ which in turn is satisfied if $\epsilon<\mu$. Now by Lemma 2.2. there are points $x_p, x_q, x_r$ in a good position, where $p<q<r$; $p, q, r\in [N, N+c(n)]$ and with pairwise distances bounded by $e^{c(n)}/P_n$. The good position of points guarantees that there exists a direction $z$ which has the same unfolding combinatorics at times $p, q, r$ as corresponding directions $x_p$, $x_q$ and $x_r$ and such that the directions $x_p$ and $x_r$ lie on the different side from the direction $z$ then the direction $x_q$. It is achieved by taking any direction $z$ lying between $x_q$ and $x_r$. The picture below illustrates this observation. =0.50mm
(288.67,79.34) (18.00,64.00)[(-1,-1)[16.67]{}]{} (1.34,47.34)[(2,-5)[16.67]{}]{} (18.00,5.67)[(0,1)[58.33]{}]{} (1.34,46.67)[(1,0)[287.33]{}]{} (107.34,72.67)[(1,-1)[39.33]{}]{} (146.67,33.34)[(0,1)[29.33]{}]{} (146.67,62.67)[(-4,1)[39.33]{}]{} (200.00,40.67)[(-1,1)[17.33]{}]{} (182.67,58.00)[(-1,-1)[34.67]{}]{} (148.00,23.34)[(3,1)[52.00]{}]{} (241.34,68.00)[(0,-1)[59.33]{}]{} (1.34,46.67)[(1,0)[254.67]{}]{} (241.34,8.67)[(1,2)[23.33]{}]{} (264.67,55.34)[(-1,1)[24.00]{}]{} (241.34,58.67)[(0,1)[20.00]{}]{} (241.34,48.67)[(0,1)[20.67]{}]{} (146.67,63.34) (200.00,40.67) (264.00,55.34) (150.00,65.34)[(0,0)\[cc\][$p$]{}]{} (205.34,38.00)[(0,0)\[cc\][$q$]{}]{} (270.00,58.00)[(0,0)\[cc\][$r$]{}]{}
Pic 7. For points $x_p, x_q, x_r$ in a good position, there is a direction with the same unfolding combinatorics at times $p, q, r$. As directions $x_p, x_q, x_r$ are generalized diagonals, they hit triangle vertices at times $p, q, r$. For simplicity we will denote corresponding vertices as $P, Q, R$. Since $r<n+k(n)c(n)$, and the angular distances are bounded by $e^{c(n)}/P_n$ we obtain that the distances from points $P, Q, R$ to the $z$-trajectory are bounded by $d= D_{\delta}(n+k(n)c(n))e^{c(n)}/P_n<e^{-an^{\mu}}$ for some constant $a>0$. We now look more carefully at the piece of $z$-trajectory between points $P$ and $R$ of length $c(n)$. We complete each triangle to a kite so that a chosen triangle side of length 1 corresponds to the kite diagonal. Now the triangle unfolding of the $z$-trajectory from $P$ to $R$ corresponds to the kite unfolding. Rotate the picture so that the $P$-kite is in standard position. =0.50mm
(288.67,90.67) (7.34,23.00)[(5,-3)[25.33]{}]{} (32.67,7.67)[(3,5)[9.00]{}]{} (41.67,22.67)[(-3,5)[9.33]{}]{} (32.34,38.00)[(-5,-3)[25.00]{}]{} (42.00,22.67)[(1,1)[14.67]{}]{} (56.67,37.33)[(-2,5)[13.33]{}]{} (43.34,70.00)[(-1,-3)[10.67]{}]{} (56.67,37.33)[(5,-3)[22.67]{}]{} (79.34,24.00)[(-3,-2)[22.67]{}]{} (56.67,8.67)[(-1,1)[14.67]{}]{} (136.00,41.33)[(4,3)[26.00]{}]{} (162.00,60.67)[(1,-1)[14.00]{}]{} (176.00,46.67)[(-1,-2)[9.33]{}]{} (166.67,28.67)[(-5,2)[30.00]{}]{} (136.67,40.67)[(1,6)[6.00]{}]{} (142.67,76.67)[(4,1)[19.33]{}]{} (162.00,81.33)[(0,-1)[20.67]{}]{} (162.00,60.67)[(1,2)[14.67]{}]{} (176.67,90.00)[(2,-5)[11.33]{}]{} (188.00,62.00)[(-4,-5)[12.00]{}]{} (136.67,40.67) (8.67,22.67) (6.00,26.67)[(0,0)\[cc\][$P$]{}]{} (134.00,34.67)[(0,0)\[cc\][$Q$]{}]{} (4.00,17.33)[(4,1)[284.67]{}]{} (256.00,85.33)[(-5,-1)[28.67]{}]{} (227.33,79.33)[(-1,-4)[4.00]{}]{} (223.33,62.00)[(4,-1)[13.33]{}]{} (236.67,58.67)[(2,3)[18.00]{}]{} (258.67,90.67)[(0,0)\[cc\][$R$]{}]{} (254.67,85.33)
Pic.8 A $z$-trajectory from $P$ to $R$ starting in standard position. By Lemma 3.1 the $x$ and $y$ coordinates of the points $P, Q, R$ can be represented as $\frac {L(\alpha,\beta)}{\sin(\alpha+\beta)}$, where $L(\alpha, \beta)$ is a trigonometric polynomial with integer coefficients of degree at most $2c(n)$ and so the area of the triangle $PQR$ can be represented as $ \frac{A(\alpha, \beta)}{\sin^2(\alpha+\beta)}$, where $A(\alpha, \beta)$ is a trigonometric polynomial with integer coefficients of degree at most $4c(n)=4n^{\epsilon}$. The estimates on $d$ imply: $|A(\alpha, \beta)|\leq D_{\delta}\sin^2(\alpha+\beta)c(n)d<e^{-bn^{\mu}}$, for some $b>0$. **Note** that $A(\alpha,\beta)\neq 0$ because $p<q<r$ and the points $P$ and $R$ lie on a ***different*** side of $z$ direction than the point $Q$. This is an extremely important observation and it is this particular point of the proof which motivates our definition of a good position. Let $\mathcal{F}_n$ be a set of trigonometric polynomials corresponding to unfolding combinatorics of length $c(n)=4n^{\epsilon}$. Any polynomial is uniquely determined by the combinatorics and a choice of vertices. It implies that the cardinality $|\mathcal{F}_n|< e^{tn^{\epsilon}}$, for some $t>0$. We are now going to estimate the measure of the following set of triangle angles: ${\mathcal{B}}_n=\lbrace (\alpha, \beta)| \exists A\in{\mathcal{F}}_n :|A(\alpha,\beta)|<e^{-bn^{\mu}}\rbrace $. The key point we use here is that a non-trivial trigonometric polynomial with integer coefficients can not be small on the set of large measure. To make this point more precise we refer to the very useful theorem by Kaloshin and Rodnianski \[6\] which can be formulated as follows:
There exist universal constants $R, c>0$ such that any non-zero trigonometric polynomial with integer coefficients $P$ in variables $\alpha, \beta, \gamma\in [0, 2\pi]$ of degree at most $m$ satisfies : $Leb\lbrace (\alpha, \beta, \gamma): |P(\alpha, \beta, \gamma)|<e^{-Rm^{2}}\rbrace<e^{-cm}$.
Any trigonometric polinomial $P(\alpha, \beta)$ in 2 variables can be considered as a polynomial $P(\alpha, \beta, \gamma)$ of three variables of the same degree, where the variable $\gamma$ is not present. Moreover any level set for $P$ in variables $\alpha, \beta, \gamma$ is obtained from the level set for $P$ in variables $\alpha, \beta$ by multiplying on segment $[0, 2\pi]$ in variable $\gamma$. Then an easy use of the Fubini theorem implies the following corollary:
There exist universal positive constants $R, c$ such that any non-zero trigonometric polynomial with integer coefficients $P$ in variables $\alpha, \beta\in [0, 2\pi]$ of degree at most $m$ satisfies the following inequality: $Leb\lbrace (\alpha, \beta): |P(\alpha, \beta)|<e^{-Rm^{2}}\rbrace<e^{-cm}$.
We now pick $A\in{\mathcal{F}}_n$ and take $m=Fn^{\epsilon}$, where $F>4$ is a large enough constant to be chosen later. By corollary 4.1: $Leb\lbrace (\alpha, \beta): |A(\alpha, \beta)|<e^{-RF^{2}n^{2\epsilon}}\rbrace<e^{-cFn^{\epsilon}}$ We need now is to show that for large enough $n$: $e^{-bn^{\mu}}<e^{-RF^2n^{2\epsilon}}$ which by comparing the highest degrees is true if $2\epsilon<\mu$ . As $A\in{\mathcal{F}}_n$ it implies $Leb({\mathcal{B}}_n)< e^{-cFn^{\epsilon}}|{\mathcal{F}}_n|<e^{(t-cF)n^{\epsilon}}$ and so if we choose $F>t/c$ then $\sum Leb({\mathcal{B}}_n)<\infty$. From the argument above it follows that under assumption $P_n>e^{\phi(n)}$ for large enough $n$ the pair $(\alpha, \beta)\in {\mathcal{B}}_n$ and a standard Borel-Cantelly argument completes the proof for the appropriate choice of $\mu, \epsilon, \gamma$.
Choice of constants
--------------------
Here we summarize all the assumptions on constants $\mu, \epsilon, \gamma>0$ which we met in the proof and choose a minimal $\mu$ satisfying them. $\begin{cases}\gamma\leq 1\\ \gamma+\epsilon>1\\ (\gamma+\epsilon)\mu>1 \\ \epsilon<\mu \\ 2\epsilon<\mu \end{cases}$ It is clear that we may take $\gamma =1$ and then the problem reduces to minimizing $\mu$ satisfying: $\begin{cases} (1+\epsilon)\mu>1 \\ 2\epsilon<\mu \end{cases}$ Taking the extreme case we get: $(1+\mu/2)\mu=1$, which in turn implies that all the conditions above can be satisfied for any $\mu>\sqrt{ 3}-1$.
Complexity estimate.
---------------------
In this section we use theorem 4.1 to get a global complexity estimate. Let us fix arbitrary $\mu>\sqrt{3}-1$. By the theorem 4.1 for any triangle $\Delta$ from a full measure set $X$ of triangles and any vertex there exists a monotone sequence of times $n_i$ characterized by the property: $P_{n_i}<e^{n_i^{\mu}}$. Our aim now is to estimate the gap $n_{i+1}-n_i$.
For any triangle $\Delta\in X$ and any $\epsilon>0$ under assumptions above for all $i$ large enough: $n_{i+1}-n_i<n_i^{1+\epsilon}$.
In the proof we repeat the previous arguments with mild changes. First we introduce the following notations: $\phi(n)=n^{\mu}, k(n)=n^{\mu}, c(n)=n^{1-\mu+\epsilon}$.
Assume that for a fixed triangle and for some $i$ large enough $n_{i+1}-n_{i}>n_i^{1+\epsilon}$. Then there exists an integer $N$: $n_i<N<n_{i+1} $ such that $P_{N+c(n_i)}/P_{N}\geq 4+2c(n_i)$.
From the Lemma assumptions it follows that $n_i+k(n_i)c(n_i)<n_{i+1}$. Divide the interval $[n_i+1, n_i+k(n_i)c(n_i)]$ into $k(n_i)$ intervals of length $c(n_i)$: $I_1=[n_i+1, n_i+c(n_i)]$, $I_2=[n_i+c(n_i)+1, n_i+2c(n_i)]$,$\ldots$, $I_{k(n_i)}=[n_i+(k(n_i)-1)c(n_i)+1, n_i+k(n_i)c(n_i)]$. We prove by contradiction that for one of the intervals $I_l$: $P_{N+c(n_i)}/P_N\geq 4+2c(n_i)$, where $N=n_i+lc(n_i)$. Assume that for all the intervals $I_l$: $P_{n_i+(l+1)c(n_i)}/P_{n_i+lc(n_i)}<4+2c(n_i)$. Then $P_{n_i+k(n_i)c(n_i)}/P_{n_i}<(4+2c(n_i))^{k(n_i)}$. On the other hand $P_{n_i+k(n_i)c(n_i)}/P_{n_i}>e^{\phi(n_i+k(n_i)c(n_i))}/e^{\phi(n_i)}$ which implies: $\phi(n_i+k(n_i)c(n_i))-\phi(n_i)<k(n_i)\ln(4+2c(n_i))$ **Remark.** Notice the difference of this inequality from the similar one in the proof of Lemma 2.3. Comparing the highest degrees of both sides implies $(1+\epsilon)\mu\leq \mu$ and so we get a contradiction.
In order to use Lemma 2.2. we need to make sure that $e^{c(n_i)}<P_{N}$. Since $N>n_i$ it is enough to establish that $c(n_i)<P_{n_i+1}$. Since $P_{n_i+1}>\phi(n_i+1)> (n_i+1)^{\mu}$ and $c(n_i)=n_i^{1-\mu+\epsilon}$ comparing the highest degrees gives: $1-\mu+\epsilon<\mu$ which is true for any $\epsilon$ small enough as $\mu>\sqrt{3}-1$. We apply Lemma 2.2 to the sequence of partitions ${\xi}_N, {\xi}_{N+1},\ldots, {\xi}_{N+c(n_i)}$ and find that there are three points $x_p, x_q, x_r$ in a good position with indexes $p, q, r$ in the range $[N+1,\ldots,N+c(n_i)]$ and with pairwise distances bounded by $d=e^{c(n_i)}/P_{n_i+1}<e^{n_i^{1-\mu+\epsilon}}/e^{N^{\mu}}\leq e^{n_i^{1-\mu+\epsilon}}/e^{{(n_i+1)}^{\mu}}$. As in the proof of theorem 4.1. we consider vertices $P, Q, R$ corresponding to the unfolding along generalized diagonals $x_p, x_q, x_r$ and entirely repeating the argument of theorem 4.1. we get that the area of the triangle $PQR$ can be represented as $\frac{A(\alpha,\beta)}{{\sin}^2(\alpha+\beta)}$, where $A(\alpha, \beta)$ is a trigonometric polynomial with integer coefficients of degree at most $4c(n_i)$. Moreover, again, by repeating the argument of Theorem 4.1. we have the estimate: $0\neq |A(\alpha, \beta)|<D_{\delta}{\sin}^2(\alpha+\beta)c(n_i)d<e^{-bn_i^{\mu}}$ for some $b>0$. Let $\mathcal{G}_n$ be a set of trigonometric polynomials corresponding to unfolding combinatorics of length $c(n)=4n^{1-\mu+\epsilon}$. Any polynomial is determined by the combinatorics and a choice of vertices. It implies $|\mathcal{G}_n|< e^{tn^{1-\mu+\epsilon}}$, for some $t>0$. We are now going to estimate the measure of the following set ${\mathcal{C}}_n$ of triangle angles: ${\mathcal{C}}_n=\lbrace (\alpha, \beta)| \exists A\in{\mathcal{G}}_n :|A(\alpha,\beta)|<e^{-bn^{\mu}}\rbrace $. We now pick $A\in{\mathcal{G}}_n$ and take $m=Fn^{1-\mu+\epsilon}$, where $F>4$ is a large enough constant to be chosen later. By corollary 4.1: $Leb\lbrace (\alpha, \beta): |A(\alpha, \beta)|<e^{-RF^{2}n^{2(1-\mu+\epsilon)}}\rbrace<e^{-cFn^{(1-\mu+\epsilon)}}$. We need now to show that for large enough $n$: $e^{-bn^{\mu}}<e^{-RF^2n^{2(1-\mu+\epsilon)}}$ . Comparing the degrees we get: $2(1-\mu+\epsilon)<\mu$ which is true for any $\mu>\sqrt{3}-1$ and $\epsilon<0.01$. The argument above implies: $Leb({\mathcal{C}}_n)<e^{-cFn^{(1-\mu+\epsilon)}}|{\mathcal{G}}_n|<e^{(t-cF)n^{(1-\mu+\epsilon)}}$ If we choose $F>t/c$ then $\sum Leb({\mathcal{C}}_n)<\infty$. Let $Y$ be a subset of $X$ such that for any triangle $\Delta\in Y$ there is an infinite subsequence of times $n_i$ such that $n_{i+1}-n_{i}>n_i^{1+\epsilon}$. Then $\Delta\in{\mathcal{C}}_{n_i}$ for infinitely many $n_i$ and so by Borel-Cantelly argument $Leb(Y)=0$.
We fiinally have all the tools to prove the main theorem.
For any $\epsilon>0$ and a typical triangle: $P_n<Ce^{n^{\sqrt{3}-1+\epsilon}}$ for some $C>0$.
Consider a triangle $\Delta\in X\setminus Y$. It is enough to prove an estimate in case of one vertex. Pick any $\epsilon>0$ small enough and then pick positive $\delta\ll\epsilon$ small enough. Consider a sequence $n_i$ corresponding to $\mu=\sqrt{3}-1+\delta$. By Theorem 4.3 for all $i$ large enough: $n_{i+1}<n_{i}+n_{i}^{1+\delta}$. As we are able to slightly perturb $\delta$ if needed we may assume: $n_{i+1}<n_i^{1+\delta}$. We then pick $n$ large enough and find $i$ such that $n_i\leq n<n_{i+1}$. By monotonicity $P_{n_i}\leq P_n\leq P_{n_{i+1}}$, so
$P_n\leq e^{n_{i+1}^{\sqrt{3}-1+\delta}}\leq e^{n_i^{(1+\delta)(\sqrt{3}-1+\delta)}}\leq e^{n^{(1+\delta)(\sqrt{3}-1+\delta)}}\leq e^{n^{\sqrt{3}-1+\epsilon}}$.
**Remark.** It is a well-known observation (see, say \[8\]) that the number of different combinatorial types of periodic orbits of length not greater than $n$ for a polygonal billiard is bounded above by $P_n$ and so as a corollary of Theorem 4.4 we have:
For any $\epsilon>0$ and a typical triangle: $Per_n<Ce^{n^{\sqrt{3}-1+\epsilon}}$ for some $C>0$, where $Per_n$ is a number of different combinatorial types of periodic orbits of length not greater than $n$.
[99]{}
J. Cassaigne, P. Hubert, S. Troubetzkoy, Complexity and growth for polygonal billiards, Ann.Inst. Fourier (Grenoble), 52(3):835-847, 2002 E. Gutkin, M. Rams, Growth rates for geometric complexities and counting functions in polygonal billiards, Ergodic Theory Dynam. Systems, 2009, vol. 29, pp. 1163–1183. E. Gutkin, S. Tabachnikov, Complexity of piecewise convex transformations in two dimensions, with applications to polygonal billiards on surfaces of constant curvature. (English summary) Mosc. Math. J. 6 (2006), no. 4, 701-772. E. Gutkin, S. Troubetzkoy, ‘Directional flows and strong recurrence for polygonal billiards, Proceedings of the International Congress of Dynamical Systems, Montevideo, Uruguay. Dynamics, Ergodic Theory and Geometry, Editor: B. Hasselblatt, MSRI Publications, 2007
V. Kaloshin, I. Rodnianski, Diophantine properties of elements of SO(3). GAFA. 11, 953–970 (2001) A. Katok, Five most resistant problems in dynamics, www.math.psu.edu/katok a/pub/5problems-expanded.pdf A. Katok, The growth rate for the number of singular and periodic orbits for a polygonal billiard. Comm. Math. Phys. 111 (1987) 151–160 A. Katok, A. Zemlyakov, Topological transitivity of billiards in polygons, Mat. Zametki, 18:2 (1975), 291–300 H. Masur, The growth rate of trajectories of a quadratic differential, Ergod. Th. Dyn. Sys. 10 (1990), 151-176. H. Masur, Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential In Holomorphic functions and moduli, vol. 1, D. Drasin ed., Springer-Verlag 1988.
D. Scheglov, Lower bounds on directional complexity for irrational triangle billiards, preprint. S. Troubetzkoy, Complexity lower bounds for polygonal billiards, Chaos 8 (1998) 242-244 Ya. Vorobets, Ergodicity of billiards in polygons, Mat. Sb. 188 (1997) 65-112.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We show that if a sequence of piecewise affine linear processes converges in the strong sense with a positive rate to a stochastic process which is strongly Hölder continuous in time, then this sequence converges in the strong sense even with respect to much stronger Hölder norms and the convergence rate is essentially reduced by the Hölder exponent. Our first application hereof establishes pathwise convergence rates for spectral Galerkin approximations of stochastic partial differential equations. Our second application derives strong convergence rates of multilevel Monte Carlo approximations of expectations of Banach space valued stochastic processes.'
author:
- |
Sonja Cox$^1$, Martin Hutzenthaler$^2$, Arnulf Jentzen$^3$,\
Jan van Neerven$^4$, and Timo Welti$^3$\
\
\
\
\
\
\
\
bibliography:
- 'bibfile.bib'
title: |
Convergence in Hölder norms with applications\
to Monte Carlo methods in infinite dimensions
---
Introduction {#sec:introduction}
============
In this article we study convergence rates for general stochastic processes in Hölder norms. In particular, in the main results of this work (see Corollary \[cor:hoelder1\] and Corollary \[cor:hoelder2\] in Subsection \[subsec:holder2\] below) we reveal estimates for uniform Hölder errors of general stochastic processes. In this introductory section we now sketch these results and thereafter outline several applications of the general estimates, which can be found in subsequent sections of this article (see Corollary \[cor:hoelder3\] in Subsection \[subsec:holder2\], Corollary \[cor:convLocLip\] in Subsection \[subsec:convLocLip\], and Corollary \[c:mlmc.conv\] in Subsection \[sec:mlmc\] below). To illustrate the key results of this work, we consider the following framework throughout this section. Let $
T \in (0,\infty)
$ be a real number, let $
\left( \Omega, \mathscr{F},
{{\mathbb P}}\right)
$ be a probability space, let $
\left(
E, \left\| \cdot \right\|_E
\right)
$ be an $ {\mathbb{R}}$-Banach space, and for every function $ f \colon [ 0, T ] \to E $ and every natural number $ N \in {\mathbb{N}}= \{ 1, 2, 3, \ldots \} $ let $ [ f ]_N \colon [0, T] \to E $ be the function which satisfies for all $ n \in \{ 0, 1, \dots, N - 1 \} $, $ t \in \bigl[ \frac{nT}{N}, \frac{(n+1) T}{N} \bigr] $ that $$[ f ]_N ( t ) = \bigl( n + 1 - \tfrac{tN}{T} \bigr) \cdot f \bigl( \tfrac{nT}{N} \bigr) + \bigl( \tfrac{tN}{T} - n \bigr) \cdot f \bigl( \tfrac{(n+1)T}{N} \bigr)$$ (the piecewise affine linear interpolation of $ f |_{ \{ 0, \nicefrac{ T }{ N } , \nicefrac{ 2 T }{ N } , \ldots, \nicefrac{ ( N - 1 ) T }{ N } , T \} } $, cf. below).
\[thm:intorduction\] Assume the above setting. Then for all $ p \in (1,\infty) $, $ {\varepsilon}\in ( \nicefrac{ 1 }{ p }, 1 ] $, $ \alpha \in [ 0, {\varepsilon}- \nicefrac{ 1 }{ p } ) $ there exists $ C \in [ 0, \infty ) $ such that it holds for all $ \beta \in [ {\varepsilon}, 1 ] $, $ N \in {\mathbb{N}}$ and all $ ( \mathscr{F}, \left\| \cdot \right\|_E ) $-strongly measurable stochastic processes $ X, Y \colon [0,T] \times \Omega \rightarrow E $ with continuous sample paths that $$\label{eq:thmintro}
\begin{split}
& \| X - [ Y ]_N \|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) }
) }
\\ &
\leq
C N^{\varepsilon}\biggl(
\sup_{ n \in \{ 0, 1, \ldots, N \}}
\bigl\| X_{ \frac{nT}{N} } - Y_{ \frac{nT}{N} } \bigr\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
+ N^{ -\beta }
\| X \|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
} \biggr).
\end{split}$$
The Hölder and $ \mathscr{L}^p $-norms in are to be understood in the usual sense (see Subsection \[notation\] below for details). Theorem \[thm:intorduction\] is a direct consequence of the more general result in Corollary \[cor:hoelder2\] in Subsection \[subsec:holder2\] below, which establishes an estimate similar to also for the case of non-equidistant time grids. Moreover, Corollary \[cor:hoelder1\] in Subsection \[subsec:holder2\] provides an estimate similar to but with $ \| X - Y \|_{ \mathscr{L}^p( {{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) } ) } $ instead of $ \| X - [ Y ]_N \|_{ \mathscr{L}^p( {{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) } ) } $ on the left hand side and with an appropriate Hölder norm of $ Y $ occurring on the right hand side. Theorem \[thm:intorduction\] has a number of applications in the numerical approximation of stochastic processes, as the next corollary, Corollary \[cor:hoelder4\], clarifies. Corollary \[cor:hoelder4\] follows immediately from Theorem \[thm:intorduction\].
\[cor:hoelder4\] Assume the above setting, let $ \beta \in ( 0, 1 ] $, let $ X \colon [0,T] \times \Omega \rightarrow E $ and $ Y^N \colon [0,T] \times \Omega \rightarrow E $, $ N \in {\mathbb{N}}$, be $ ( \mathscr{F}, \left\| \cdot \right\|_E ) $-strongly measurable stochastic processes with continuous sample paths which satisfy for all $ p \in ( 1, \infty ) $ that $ \forall \, N \in {\mathbb{N}}\colon Y^N = [ Y^N ]_N $ and $$\label{eq:assumptions}
\| X \|_{
{\mathscr{C}}^{ \beta }( [0,T] , \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
+
\sup_{ N \in {\mathbb{N}}}
\biggl[ N^\beta \sup_{ n \in \{ 0, 1, \ldots, N \}}
\bigl\| X_{ \frac{nT}{N} } - Y_{ \frac{nT}{N} }^N \bigr\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
\biggr] < \infty.$$ Then it holds for all $ p, {\varepsilon}\in ( 0, \infty ) $ that $$\sup_{ N \in {\mathbb{N}}}
\Biggl[
N^{ \beta - {\varepsilon}}
\biggl( {{\mathbb E}}\biggl[
\sup_{ t \in [ 0, T ] }
\| X_t - Y_t^N \|_E^p
\biggr] \biggr)^{ \nicefrac{1}{p} }
\Biggr] < \infty.$$
It is assumed in that a sequence of affine linearly interpolated $ ( \mathscr{F}, \left\| \cdot \right\|_E ) $-strongly measurable stochastic processes $ ( Y^N )_{ N \in {\mathbb{N}}} $ converges for every $ p \in ( 1, \infty ) $ in $ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) $ to a $ ( \mathscr{F}, \left\| \cdot \right\|_E ) $-strongly measurable stochastic process $ X $ with a positive rate uniformly on all grid points and that this process $ X $ admits corresponding temporal Hölder regularity. Corollary \[cor:hoelder4\] then shows that these assumptions are sufficient to obtain convergence for every $ p \in ( 1, \infty ) $ in the uniform $ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ C( [ 0, T ], \left\| \cdot \right\|_E ) } )$-norm with essentially the same rate. Corollary \[cor:hoelder3\] in Subsection \[subsec:holder2\] below implies this result as a special case and includes the case of non-equidistant time grids. Moreover, Corollary \[cor:hoelder3\] proves an analogous conclusion for convergence in uniform Hölder norms, where the obtained convergence rate is reduced by the considered Hölder exponent. Corollary \[cor:eulermethod\] below demonstrates how this principle can be applied to Euler-Maryuama approximations for stochastic differential equations (SDEs) with globally Lipschitz coefficients. Arguments related to Corollary \[cor:hoelder3\] can be found in Lemma A1 in Bally, Millet & Sanz-Solé [@BallyMilletSanzSole1995] and in the second display on page 325 in [@CoxVanNeerven2013].
Corollary \[cor:hoelder4\] is particularly useful for the study of stochastic partial differential equations (SPDEs). In general, a solution of an SPDE fails to be a semimartingale. As a consequence, Doob’s maximal inequality cannot be applied to obtain estimates with respect to the $ \mathscr{L}^2( {{\mathbb P}}; \left\| \cdot \right\|_{ C( [ 0, T ], \left\| \cdot \right\|_E ) } )$-norm. However, convergence rates with respect to the $ C( [ 0, T ], \left\| \cdot \right\|_{ \mathscr{L}^2( {{\mathbb P}}; \left\| \cdot \right\|_E ) } ) $-norm are often feasible and Corollary \[cor:hoelder4\] can then be applied to obtain convergence rates with respect to the $ \mathscr{L}^2( {{\mathbb P}}; \left\| \cdot \right\|_{ C( [ 0, T ], \left\| \cdot \right\|_E ) } )$-norm. Estimates with respect to the $ \mathscr{L}^2( {{\mathbb P}}; \left\| \cdot \right\|_{ C( [ 0, T ], \left\| \cdot \right\|_E ) } )$-norm are useful for using standard localisation arguments in order to extend results for SPDEs with globally Lipschitz continuous nonlinearities to results for SPDEs with nonlinearities that are only Lipschitz continuous on bounded sets. We demonstrate this in Corollary \[cor:convLocLip\] in Subsection \[subsec:convLocLip\] below in the case of pathwise convergence rates for Galerkin approximations. To be more specific, Corollary \[cor:convLocLip\] proves essentially sharp pathwise convergence rates for spatial Galerkin and noise approximations for a large class of SPDEs with non-globally Lipschitz continuous nonlinearities. For example, Corollary \[cor:convLocLip\] applies to stochastic Burgers’, stochastic Ginzburg-Landau, stochastic Kuramoto-Sivashinsky, and Cahn-Hilliard-Cook equations.
Another prominent application of Corollary \[cor:hoelder4\] are multilevel Monte Carlo methods in Banach spaces. For a random variable $ X \in \mathscr{L}^2( {{\mathbb P}}; \left\| \cdot \right\|_E ) $ convergence in $ \mathscr{L}^2( {{\mathbb P}}; \left\| \cdot \right\|_E ) $ of Monte Carlo approximations of the expectation $ {{\mathbb E}}[X] \in E $ has only been established if $ E $ has so-called (Rademacher) type $ p $ for some $ p \in ( 1, 2 ] $ and in this case the convergence rate is given by $ 1 - \nicefrac{1}{p} $ (see, e.g., Heinrich [@h01] or Corollary \[cor:monteCarloBanach\] in Subsection \[sec:montecarlo\] below). However, the space $ C( [ 0, T ], E ) $ fails to have type $ p $ for any $ p \in ( 1, 2 ] $. If $ X $ has more sample path regularity, this problem can nevertheless be bypassed. More precisely, if it holds for some $ \alpha \in ( 0, 1 ] $, $ p \in ( \nicefrac{1}{\alpha}, \infty) $ that $ X \in \mathscr{L}^2( {{\mathbb P}}; \left\| \cdot \right\|_{ W^{ \alpha, p }( [ 0, T ], E ) } ) $, then Monte Carlo approximations of $ {{\mathbb E}}[X] \in W^{ \alpha, p }( [ 0, T ], E ) $ have been shown to converge in $ \mathscr{L}^2( {{\mathbb P}}; \left\| \cdot \right\|_{ W^{ \alpha, p }( [ 0, T ], E ) } ) $ with rate $ 1 - \nicefrac{1}{ \min\{ 2, p \} } $ and, by the Sobolev embedding theorem, also converge in $ \mathscr{L}^2( {{\mathbb P}}; \left\| \cdot \right\|_{ C( [ 0, T ], \left\| \cdot \right\|_E ) } )$ with the same rate. Here for any real numbers $ \alpha \in ( 0, 1 ] $, $ p \in [ 1, \infty ) $ we denote by $ W^{ \alpha, p } ( [ 0, T ], E ) $ the Sobolev space with regularity parameter $ \alpha $ and integrability parameter $ p $ of functions from $ [ 0, T ] $ to $ E $. Informally speaking, in order to gain control over the variances appearing in multilevel Monte Carlo approximations it is therefore sufficient for the approximations to converge with respect to the $ \mathscr{L}^2( {{\mathbb P}}; \left\| \cdot \right\|_{ {\mathscr{C}}^\alpha( [ 0, T ], \left\| \cdot \right\|_E ) } )$-norm for some $ \alpha \in ( 0, 1] $. For more details, we refer the reader to Section \[sec:cubature\] and, in particular, to Corollary \[c:mlmc.conv\], which formalizes this approach for the case of multilevel Monte Carlo approximations of expectations of Banach space valued stochastic processes.
Finally, we mention a few results in the literature which employ some findings from this article. In particular, Corollary \[cor:hoelder23\] in this article is applied in the proof of Corollary 6.3 in Jentzen & Pušnik [@JentzenPusnik2015] to prove uniform convergence in probability for spatial spectral Galerkin approximations of stochastic evolution equations (SEEs) with semi-globally Lipschitz continuous coefficients (see Proposition 6.4 in Jentzen & Pušnik [@JentzenPusnik2015]). Moreover, Corollary \[cor:convGlobLip\] in this article is employed in Subsection 5.2 and Subsection 5.3 in [@CoxHutzenthalerJentzen2013] for transferring initial value regularity results for finite-dimensional stochastic differential equations to the case of infinite-dimensional SPDEs using the example of the stochastic Burgers’ equation and the Cahn-Hilliard-Cook equation. Furthermore, Corollary \[cor:hoelder3\] in this article is used in the proof of Corollary 5.10 in Hutzenthaler, Jentzen & Salimova [@HutzenthalerJentzenSalimova2016arXiv] to establish essentially sharp uniform strong convergence rates for spatial spectral Galerkin approximations of linear stochastic heat equations.
Notation
--------
The following notation is used throughout this article. For two sets $ A $ and $ B $ we denote by $ \mathbb{M}( A , B ) $ the set of all mappings from $ A $ to $ B $. For measurable spaces $ ( \Omega_1, {\mathscr{F}}_1 ) $ and $ ( \Omega_2, {\mathscr{F}}_2 ) $ we denote by $ \mathscr{M}( {\mathscr{F}}_1, {\mathscr{F}}_2 ) $ the set of all $ {\mathscr{F}}_1 / {\mathscr{F}}_2 $-measurable mappings from $ \Omega_1 $ to $ \Omega_2 $. For topological spaces $ ( E, \mathscr{E} ) $ and $ ( F, \mathscr{F} ) $ we denote by $ \mathscr{B}( E ) $ the Borel $\sigma$-algebra on $ ( E, \mathscr{E} ) $ and we denote by $ C( E, F ) $ the set of all continuous functions from $ E $ to $ F $. We denote by $ \left| \cdot \right| \colon {\mathbb{R}}\to [ 0, \infty ) $ the absolute value function on $ {\mathbb{R}}$. We denote by $ \Gamma \colon ( 0 , \infty ) \to ( 0 , \infty ) $ the Gamma function, that is, we denote by $ \Gamma \colon ( 0 , \infty ) \to ( 0 , \infty ) $ the function which satisfies for all $ x \in ( 0 , \infty ) $ that $ \Gamma ( x ) = \int_0^{ \infty } t^{ ( x - 1 ) } \, e^{ - t } {\,\mathrm{d}}t $. We denote by $ \mathscr{E}_r \colon [ 0 , \infty ) \to [ 0 , \infty )$, $ r \in ( 0 , \infty ) $, the mappings which satisfy for all $ r \in ( 0 , \infty ) $, $ x \in [ 0 , \infty ) $ that $$\mathscr{E}_{r} [x] = \biggl( \sum_{ n = 0 }^{ \infty } \frac{ x^{2n} ( \Gamma (r) )^n }{ \Gamma ( nr + 1 ) } \biggr)^{ \nicefrac{1}{2} }
=
\biggl(
1
+ \frac{ x^2 \, \Gamma (r) }{ \Gamma ( r + 1 ) }
+ \frac{ x^4 ( \Gamma (r) )^2 }{ \Gamma ( 2r + 1 ) }
+ \ \ldots \
\biggr)^{ \nicefrac{1}{2} }$$ (cf. Chapter 7 in Henry [@Henry1981]). For a field $ \mathbb{K} \in \{ {\mathbb{R}}, {\mathbb{C}}\} $, a $ \mathbb{K} $-vector space $ V $, and a mapping $ \left\| \cdot \right\| \colon V \to [0,\infty] $ which satisfies for all $ v, w \in \left\{ u \in V \colon \| u \| < \infty \right\} $, $ \lambda \in \mathbb{K} \setminus \{ 0 \} $ that $( \| v \| = 0 \Leftrightarrow v = 0 )$, $ \left\| \lambda v \right\| = \sqrt{ [ \mathrm{Re}( \lambda ) ]^2 + [ \mathrm{Im}( \lambda ) ]^2 } \left\| v \right\| $, and $ \left\| v + w \right\| \leq \left\| v \right\| + \left\| w \right\| $ we call $ \left\| \cdot \right\| $ an extended norm on $ V $ and we call $ ( V , \left\| \cdot \right\| ) $ an extendedly normed vector space. For a metric space $ ( M, d ) $, an extendedly normed vector space $ ( E, \left\| \cdot \right\| ) $, a real number $ r \in [0,1] $, and a set $ A \subseteq (0,\infty) $ we denote by $
\left|
\cdot
\right|_{
{\mathscr{C}}^{ r, A }( M, \left\| \cdot \right\| )
},
\left| \cdot \right|_{
{\mathscr{C}}^{ r }( M, \left\| \cdot \right\| )
},
\left\| \cdot \right\|_{
C( M, \left\| \cdot \right\| )
}, $ $
\left\| \cdot \right\|_{
{\mathscr{C}}^{ r }( M, \left\| \cdot \right\| )
} \colon
\mathbb{M}( M, E ) \to [0, \infty] $ the mappings which satisfy for all $ f \in \mathbb{M}( M, E ) $ that $$\begin{aligned}
\left|
f
\right|_{
{\mathscr{C}}^{ r, A }( M, \left\| \cdot \right\| )
}
&
=
\sup\!\left(
\left\{
\tfrac{
\left\| f(e_1) - f(e_2) \right\|
}{
\left| d(e_1, e_2) \right|^{ r }
}
\colon
e_1, e_2 \in M,
d(e_1,e_2) \in A
\right\} \cup \left\{ 0 \right\}
\right)
\in [0,\infty],
\\
\left| f \right|_{
{\mathscr{C}}^{ r }( M, \left\| \cdot \right\| )
}
&
=
\left| f \right|_{
{\mathscr{C}}^{ r, (0,\infty) }( M, \left\| \cdot \right\| )
}
\in [0,\infty],
\\
\left\| f \right\|_{
C( M, \left\| \cdot \right\| )
}
&
=
\sup\!\left(
\left\{
\left\| f(e) \right\|
\colon
e\in M
\right\}
\cup
\{ 0 \}
\right)
\in [0,\infty] ,
\\
\left\| f \right\|_{
{\mathscr{C}}^r( M, \left\| \cdot \right\| )
}
&
=
\left\| f \right\|_{ C( M, \left\| \cdot \right\| ) }
+
\left| f \right|_{ {\mathscr{C}}^r( M, \left\| \cdot \right\| ) }
\in [0,\infty]\end{aligned}$$ and we denote by $
{\mathscr{C}}^r( M, \left\| \cdot \right\| )
$ the set given by $${\mathscr{C}}^r( M, \left\| \cdot \right\| )
=
\{
f \in
C(M,E)
\colon
\| f \|_{ {\mathscr{C}}^r( M, \left\| \cdot \right\| ) } < \infty
\}.$$ Note that for every $ r \in [0,1] $, every metric space $ ( M, d ) $, and every extendedly normed vector space $ ( E, \left\| \cdot \right\| ) $ it holds that $
\bigl(
\mathbb{M}( M, E ),
\left\| \cdot \right\|_{
{\mathscr{C}}^r( M, \left\| \cdot \right\| )
}
\bigr)
$, $
\bigl(
C( M, E ),
\left\| \cdot \right\|_{
{\mathscr{C}}^r( M, \left\| \cdot \right\| )
} \bigr|_{ C( M, E ) }
\bigr)
$, and $
\bigl(
{\mathscr{C}}^r( M, \left\| \cdot \right\| ),
\left\| \cdot \right\|_{
{\mathscr{C}}^r( M, \left\| \cdot \right\| )
} \bigr|_{ {\mathscr{C}}^r( M, \left\| \cdot \right\| ) }
\bigr)
$ are extendedly normed vector spaces. For Hilbert spaces $ ( H_i, \langle \cdot, \cdot \rangle_{H_i}, \left\| \cdot \right\|_{H_i} ) $, $ i \in \{ 1, 2 \} $, we denote by $ ( \mathrm{HS}( H_1, H_2 ), \langle \cdot, \cdot \rangle_{\mathrm{HS}( H_1, H_2 )}, \left\| \cdot \right\|_{\mathrm{HS}( H_1, H_2 )} ) $ the Hilbert space of Hilbert-Schmidt operators from $ H_1 $ to $ H_2 $. For a measure space $ ( \Omega, \mathscr{F}, \mu ) $, a measurable space $ ( S, \mathscr{S} ) $, a set $ R \subseteq S $, and a function $ f \colon \Omega \to R $ we denote by $ [ f ]_{ \mu, \mathscr{S} } $ the set given by $$[ f ]_{ \mu, \mathscr{S} }
= \bigl\{ g \in \mathscr{M}( \mathscr{F}, \mathscr{S} ) \colon ( \exists A \in \mathscr{F} \colon \mu(A) = 0
\text{ and } \{ \omega \in \Omega \colon f(\omega) \neq g(\omega) \} \subseteq A ) \bigr\}.$$ For a measure space $ ( \Omega, \mathscr{F}, \mu ) $, an extendedly normed vector space $ ( V , \left\| \cdot \right\| ) $, and real numbers $ p \in [ 0, \infty ) $, $ q \in ( 0, \infty ) $ we denote by $ \mathscr{L}^0( \mu; \left\| \cdot \right\| ) $ the set given by $$\mathscr{L}^0( \mu; \left\| \cdot \right\| )
= \bigl\{ f \in \mathbb{M}( \Omega, V ) \colon
f \text{ is } ( \mathscr{F}, \left\| \cdot \right\| ) \text{-strongly measurable} \bigr\},$$ we denote by $ \left\| \cdot \right\|_{ \mathscr{L}^q( \mu; \left\| \cdot \right\| ) } \colon
\mathscr{L}^0( \mu; \left\| \cdot \right\| ) \to [0,\infty] $ the mapping which satisfies for all $ f \in \mathscr{L}^0( \mu; \left\| \cdot \right\| ) $ that $$\left\| f \right\|_{ \mathscr{L}^q( \mu; \left\| \cdot \right\| ) }
= \left[ \int_{ \Omega }
\left\| f( \omega ) \right\|^q \mu( {\mathrm{d}}\omega )
\right]^{ \nicefrac{ 1 }{ q } }
\in [0,\infty],$$ we denote by $ \mathscr{L}^q( \mu; \left\| \cdot \right\| ) $ the set given by $$\mathscr{L}^q( \mu; \left\| \cdot \right\| )
= \bigl\{ f \in \mathscr{L}^0( \mu; \left\| \cdot \right\| ) \colon
\left\| f \right\|_{ \mathscr{L}^q( \mu; \left\| \cdot \right\| ) }
< \infty \bigr\},$$ we denote by $ L^p ( \mu; \left\| \cdot \right\| ) $ the set given by $$L^p ( \mu; \left\| \cdot \right\| )
= \bigl\{ \{ g \in \mathscr{L}^0( \mu; \left\| \cdot \right\| ) \colon
\mu( f \neq g ) = 0 \} \subseteq \mathscr{L}^0( \mu; \left\| \cdot \right\| ) \colon
f \in \mathscr{L}^p( \mu; \left\| \cdot \right\| ) \bigr\},$$ and we denote by $ \left\| \cdot \right\|_{ L^q( \mu; \left\| \cdot \right\| ) } \colon
L^0( \mu; \left\| \cdot \right\| ) \to [0,\infty] $ the function which satisfies for all $ f \in \mathscr{L}^0( \mu; \left\| \cdot \right\| ) $ that $$\bigl\| \{ g \in \mathscr{L}^0( \mu; \left\| \cdot \right\| ) \colon
\mu( f \neq g ) = 0 \} \bigr\|_{ L^q( \mu; \left\| \cdot \right\| ) }
= \left\| f \right\|_{ \mathscr{L}^q( \mu; \left\| \cdot \right\| ) } \in [0,\infty].$$ For a real number $ T \in (0,\infty) $, a measurable space $ ( S, \mathscr{S} ) $, a normed vector space $ ( V, \left\| \cdot \right\|_V ) $, and a mapping $ X \colon [ 0, T ] \times S \to V $ which satisfies for all $ t \in [ 0, T ] $ that $ X_t \colon S \to V $ is an $ ( \mathscr{S}, \left\| \cdot \right\|_V ) $-strongly measurable mapping we call $ X $ an $ ( \mathscr{S}, \left\| \cdot \right\|_V ) $-strongly measurable stochastic process. For a real number $ T \in (0,\infty) $ we denote by $ {\mathscr{P}}_T $ the set given by $${\mathscr{P}}_T
= \bigl\{ \theta \subseteq [0,T] \colon
\{ 0, T \} \subseteq \theta
\text{ and } \#( \theta ) < \infty \bigr\}.$$ We denote by $ d_{ \max }, d_{ \min } \colon
\cup_{ T \in (0,\infty) } {\mathscr{P}}_T \to {\mathbb{R}}$ the functions which satisfy for all $ \theta = \{ \theta_0 , \theta_1, \ldots, \theta_{ \#( \theta ) - 1 } \} \in \cup_{ T \in (0,\infty) } {\mathscr{P}}_T $ with $ \theta_0 < \theta_1 < \ldots < \theta_{ \#( \theta ) - 1 } $ that $$d_{\max}( \theta )
= \max_{ j \in \{ 1, 2, \ldots, \#( \theta ) - 1 \} }
| \theta_j - \theta_{ j - 1 } |
\qquad \text{and} \qquad
d_{\min}( \theta )
= \min_{ j \in \{1, 2, \ldots, \#( \theta ) - 1 \} }
| \theta_j - \theta_{ j - 1 } |.$$ For a normed vector space $ ( E , \left\| \cdot \right\|_E ) $, an element $ \theta = \{ \theta_0 , \theta_1, \ldots, \theta_{ \#( \theta ) - 1 } \} \in \cup_{ T \in (0,\infty) } {\mathscr{P}}_T $ with $ \theta_0 < \theta_1 < \ldots < \theta_{ \#( \theta ) - 1 } $, and a function $ f \colon [ 0, \theta_{ \#( \theta ) - 1 } ] \to E $ we denote by $ [ f ]_{ \theta } \colon $$ [ 0, \theta_{ \#( \theta ) - 1 } ] \to E $ the piecewise affine linear interpolation of $ f |_{ \{ \theta_0 , \theta_1, \ldots, \theta_{ \#( \theta ) - 1 } \} } $, that is, we denote by $ [ f ]_{ \theta } \colon
[ 0, \theta_{ \#( \theta ) - 1 } ] \to E $ the function which satisfies for all $ j \in \{ 1, 2, \ldots, \theta_{ \#( \theta ) - 1 } \} $, $ s \in [ \theta_{ j - 1 } , \theta_j ] $ that $$\label{eq:lin_intpolation}
[ f ]_{ \theta }( s )
= \frac{ ( \theta_j - s ) f( \theta_{ j - 1 } ) } { ( \theta_j - \theta_{ j - 1 } ) }
+ \frac{ ( s - \theta_{ j - 1 } ) f( \theta_j ) } { ( \theta_j - \theta_{j-1} ) } .$$
Convergence in Hölder norms for Banach space valued stochastic processes {#sec:holder1}
========================================================================
Error bounds for the Hölder norm {#subsec:holder1}
--------------------------------
\[lem:hoelderconv\] Consider the notation in Subsection \[notation\], let $ ( E, \left\| \cdot \right\|_E ) $ be a normed vector space, let $ (M,d) $ be a metric space, let $ f \colon M \to E $ be a function, and let $ c \in (0,\infty) $, $ \alpha, \beta, \gamma \in [0,1] $ satisfy $ \alpha \leq \beta \leq \gamma $. Then $$\label{eq:toshow0}
\left|
f
\right|_{
{\mathscr{C}}^{ \beta }(
M, \left\| \cdot \right\|_E
)
}
\leq
\max\!\left\{
c^{ \alpha - \beta }
\left|
f
\right|_{
{\mathscr{C}}^{\alpha, (c,\infty)
}( M, \left\| \cdot \right\|_E )
}
,
c^{ \gamma - \beta }
\left| f \right|_{
{\mathscr{C}}^{ \gamma, (0,c]
}( M, \left\| \cdot \right\|_E )
}
\right\}$$ and $$\left|
f
\right|_{
{\mathscr{C}}^{ \beta }(
M, \left\| \cdot \right\|_E
)
}
\leq
\max\!\left\{
c^{ \alpha - \beta }
\left|
f
\right|_{
{\mathscr{C}}^{\alpha, [c,\infty)
}( M, \left\| \cdot \right\|_E )
}
,
c^{ \gamma - \beta }
\left| f \right|_{
{\mathscr{C}}^{ \gamma, (0,c)
}( M, \left\| \cdot \right\|_E )
}
\right\}
.
\label{eq:toshow1}$$
First of all, note that it holds for all $ e_1, e_2 \in M $ with $ d(e_1,e_2) \in (c,\infty) $ that $$\label{eq:use1}
\begin{aligned}
&
\frac{
\left\|
f( e_1 )
-
f( e_2 )
\right\|_{
E
}
}{
\left| d( e_1, e_2 ) \right|^{ \beta }
}
\leq
\left| d( e_1, e_2 ) \right|^{
\alpha - \beta
}
\left|
f
\right|_{
{\mathscr{C}}^{\alpha, (c,\infty)
}( M, \left\| \cdot \right\|_E )
}
\leq
c^{ \alpha - \beta }
\left|
f
\right|_{
{\mathscr{C}}^{\alpha, (c,\infty)
}( M, \left\| \cdot \right\|_E )
}.
\end{aligned}$$ In addition, observe that it holds for all $ e_1, e_2 \in M $ with $
d( e_1, e_2 ) \in (0, c]
$ that $$\label{eq:use2}
\begin{aligned}
\frac{
\|
f( e_1 )
-
f( e_2 )
\|_{
E
}
}{
\left| d( e_1, e_2 ) \right|^{ \beta }
}
& \leq
\left| d( e_1, e_2 ) \right|^{
\gamma - \beta
}
\left| f \right|_{
{\mathscr{C}}^{ \gamma, (0,c] }( M, \left\| \cdot \right\|_E
)
}
\leq
c^{ \gamma - \beta }
\left| f \right|_{
{\mathscr{C}}^{ \gamma, (0,c] }( M, \left\| \cdot \right\|_E
)
}.
\end{aligned}$$ Combining and shows . The proof of is analogous. This finishes the proof of Lemma \[lem:hoelderconv\].
\[lem:estNaffine\] Consider the notation in Subsection \[notation\], let $ T \in (0,\infty) $, $ \theta \in {\mathscr{P}}_T $, $ \alpha \in [0,1] $, let $ ( E, \left\| \cdot \right\|_E ) $ be a normed vector space, and let $
f
\colon
[0,T]
\rightarrow E
$ be a function. Then $$\label{eq:appPiecewise}
\left\|
f - [ f ]_{ \theta }
\right\|_{
C([0,T],\left\| \cdot \right\|_E)
}
\leq
\big|
\tfrac{d_{\max}(\theta)}{2 }
\big|^{\alpha}
\left|
f
\right|_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}
.$$
Throughout this proof let $ N \in {\mathbb{N}}$ be a natural number, let $ s \in [0,T] $, $ \theta_0, \theta_1 , \dots, \theta_N
\in [0,T] $ be real numbers, and let $ j \in \{ 1, 2, \ldots, N \} $ be a natural number such that $ 0 = \theta_0 < \theta_1 < \ldots < \theta_N = T $, $
\theta = \{
\theta_0 , \theta_1, \dots, \theta_N
\}
$, and $ s \in [ \theta_{ j - 1 } , \theta_j ] $ and let $
g \colon [0,1] \rightarrow {\mathbb{R}}$ be the function which satisfies for all $ u \in [0,1] $ that $
g( u ) = (1 - u) \, u^\alpha + u \, ( 1 - u )^\alpha
$. Observe that the concavity of the function $ [ 0, \infty ) \ni x \mapsto x^\alpha \in {\mathbb{R}}$ shows for all $ u \in [ 0, 1 ] $ that $$\begin{split}
2^\alpha g(u)
& =
( 1 - u ) \, ( 2 u )^\alpha
+ u \, ( 2 ( 1 - u ) )^\alpha
\leq
( (1-u) \, 2 u + u \, 2 ( 1 - u ) )^\alpha
\\ &
= ( 4 u ( 1 - u ) )^\alpha
= ( 1 - ( 2 u - 1 )^2 )^\alpha
\leq 1.
\end{split}$$ Note that this proves that $$\label{eq:f_minus_ftheta}
\begin{aligned}
&
\| f(s) - [ f ]_{ \theta }( s ) \|_E
\leq
\tfrac{ ( \theta_j - s ) }{ ( \theta_j - \theta_{ j - 1 } ) }
\| f(s) - f( \theta_{j-1} ) \|_E
+
\tfrac{ ( s - \theta_{ j - 1 } ) }{ ( \theta_j - \theta_{ j - 1 } ) }
\| f(s) - f( \theta_j ) \|_E
\\ &
\leq
\tfrac{ ( \theta_j - s ) }{ ( \theta_j - \theta_{ j - 1 } ) }
\big(
s - \theta_{j-1}
\big)^{ \alpha }
| f |_{
{\mathscr{C}}^{ \alpha }([0,T],\left\| \cdot \right\|_E)
}
+
\tfrac{ ( s - \theta_{ j - 1 } ) }{ ( \theta_j - \theta_{j-1} ) }
\big(
\theta_j - s
\big)^\alpha
| f |_{
{\mathscr{C}}^\alpha( [0,T], \left\| \cdot \right\|_E )
}
\\ &
=
\big(
\tfrac{
( \theta_j - s )
}{
( \theta_j - \theta_{j-1} )
}
\big(
\tfrac{
( s - \theta_{j-1} )
}{
( \theta_j - \theta_{j-1} )
}
\big)^{ \alpha }
+
\tfrac{ ( s - \theta_{j-1} ) }{ ( \theta_j - \theta_{j-1} ) }
\big(
\tfrac{
( \theta_j - s )
}{
( \theta_j - \theta_{j-1} )
}
\big)^{ \alpha }
\big)
\big(
\theta_j - \theta_{j-1}
\big)^{ \alpha }
| f |_{ {\mathscr{C}}^{ \alpha }([0,T],\left\| \cdot \right\|_E) }
\\ &
=
g\big(
\tfrac{ s - \theta_{ j - 1 } }{ \theta_j - \theta_{ j - 1 } }
\big)
\,
\big(
\theta_j - \theta_{ j - 1 }
\big)^\alpha
\left| f \right|_{
{\mathscr{C}}^\alpha( [0,T] , \left\| \cdot \right\|_E )
}
\leq
\big(
\tfrac{ \theta_j - \theta_{ j - 1 } }{ 2 }
\big)^\alpha
\left| f \right|_{
{\mathscr{C}}^\alpha( [0,T], \left\| \cdot \right\|_E )
}
.
\end{aligned}$$ The proof of Lemma \[lem:estNaffine\] is thus completed.
The next result, Corollary \[cor:HoelderEstFunc\], provides estimates for the Hölder norm differences of two functions by using the difference of the two functions on suitable grid points. Corollary \[cor:HoelderEstFunc\] is a consequence of Lemma \[lem:hoelderconv\] and of Lemma \[lem:estNaffine\].
\[cor:HoelderEstFunc\] Consider the notation in Subsection \[notation\], let $ T \in (0,\infty) $, $ \theta \in {\mathscr{P}}_T $, $ \beta \in [0,1] $, $ \alpha \in [0,\beta] $, let $ ( E, \left\| \cdot \right\|_E ) $ be a normed vector space, and let $ f, g \colon [0,T] \rightarrow E $ be functions. Then $$\begin{split}
&
\label{eq:HoederEstFunc}
\left| f - g \right|_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}
\\ &
\leq
\tfrac{
2
}{
|
d_{ \max }( \theta )
|^{ \alpha }
}
\bigg[
\sup_{
t \in \theta
}
\big\|
f(
t
)
-
g(
t
)
\big\|_E
+
\tfrac{
|
d_{ \max }( \theta )
|^{ \beta }
}{ 2^{ \beta } }
\,
\big(
| f |_{ {\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_E) }
+
| g |_{ {\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_E) }
\big)
\bigg]
\end{split}$$ and $$\begin{aligned}
\label{eq:HoederEstFunc2}
&
\| f - g \|_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}
\\ & \nonumber
\leq
\Big[
\tfrac{ 2 }{
|
d_{ \max }( \theta )
|^{ \alpha }
}
+ 1
\Big]
\bigg[
\sup_{
t \in \theta
}
\big\|
f(
t
)
-
g(
t
)
\big\|_E
+
\tfrac{
|
d_{ \max }( \theta )
|^{ \beta }
}{
2^{ \beta }
}
\,
\big(
| f |_{
{\mathscr{C}}^{\beta}([0,T],\left\| \cdot \right\|_E)
}
+
| g |_{
{\mathscr{C}}^{\beta}([0,T],\left\| \cdot \right\|_E)
}
\big)
\bigg]
.\end{aligned}$$
Lemma \[lem:hoelderconv\] and the triangle inequality ensure that $$\begin{aligned}
\label{eq:HoelderEsth1}
&
|
f - g
|_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}
\\ & {\refstepcounter{equation}\tag{\theequation}}\leq
\max\!\left\{
|
d_{ \max }( \theta )
|^{ - \alpha }
\,
| f - g |_{
{\mathscr{C}}^{ 0, ( d_{ \max }( \theta ), \infty ) }( [0,T], \left\| \cdot \right\|_E )
}
,
|
d_{ \max }( \theta )
|^{ \beta - \alpha }
\,
| f - g |_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_E )
}
\right\}
\\ &
\leq
\max\!\left\{
2
\,
|
d_{ \max }( \theta )
|^{ - \alpha }
\,
\| f - g \|_{ C( [0,T], \left\| \cdot \right\|_E ) }
,
|
d_{ \max }( \theta )
|^{ \beta - \alpha }
\left(
| f |_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_E )
}
+
| g |_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_E )
}
\right)
\right\}
.\end{aligned}$$ In addition, observe that Lemma \[lem:estNaffine\] and the triangle inequality assure that $$\begin{aligned}
\label{eq:HoelderEsth2}
& \nonumber
\big\|
f - g
\big\|_{ C( [0,T], \left\| \cdot \right\|_E ) }
\leq
\big\|
f - [ f ]_{ \theta }
\big\|_{ C( [0,T], \left\| \cdot \right\|_E ) }
+
\big\|
[ f ]_{ \theta } - [ g ]_{ \theta }
\big\|_{ C( [0,T], \left\| \cdot \right\|_E ) }
+
\big\|
[ g ]_{ \theta } - g
\big\|_{ C( [0,T], \left\| \cdot \right\|_E ) }
\\ &
\leq
\sup_{
t \in \theta
}
\|
f( t )
-
g( t )
\|_E
+
\big|
\tfrac{
d_{ \max }(\theta)
}{ 2 }
\big|^{ \beta }
\left(
| f |_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_E )
}
+
| g |_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_E )
}
\right).\end{aligned}$$ Inserting into yields inequality . Moreover, adding inequality and results in inequality . This finishes the proof of Corollary \[cor:HoelderEstFunc\].
\[lem:estcagrid\] Consider the notation in Subsection \[notation\], let $( E, \left\| \cdot \right\|_E ) $ be a normed vector space, let $ T, c \in (0,\infty) $, $ \alpha \in [0,1] $, $ \theta \in \mathscr{P}_T $, $ N \in {\mathbb{N}}$, $
\theta_0, \dots, \theta_N \in [0,T]
$ satisfy $
0 = \theta_0 < \dots < \theta_N = T
$ and $ \theta = \{ \theta_0, \dots, \theta_N \} $, and let $
f
\colon
[0,T]
\rightarrow E
$ be a function. Then $$\label{eq:grid_linear_interpolation}
\left|
[ f ]_{ \theta }
\right|_{
{\mathscr{C}}^{ \alpha, (0,c]
}( [0,T], \left\| \cdot \right\|_E )
}
\leq
\tfrac{
c^{ 1 - \alpha }
}{
d_{ \min }( \theta )
}
\big[
\sup\nolimits_{ j \in \{ 1, 2, \ldots, N \} }
\| f( \theta_j ) - f( \theta_{ j - 1 } ) \|_E
\big]
.$$
Observe that it holds for all $ s, t \in [0,T] $ with $ t - s \in (0, c] $ that $$\begin{aligned}
&
\frac{
\| [ f ]_{ \theta }( t ) - [ f ]_{ \theta }( s ) \|_E
}{
| t-s |^{\alpha}
}
=
\frac{
\|
\int_{ (s,t) \setminus \theta } ( [ f ]_{ \theta } )'( u ) {\,\mathrm{d}}u
\|_E
}{
| t-s |^{ \alpha }
}
\\ & \leq
\frac{
| t - s|
\,
\big[
\sup_{
u \in
(s, t) \setminus
\theta
}
\|
( [ f ]_{ \theta } )'(u)
\|_E
\big]
}{
| t - s |^{ \alpha }
}
\\ &
\leq
| t - s |^{ 1 - \alpha }
\left[
\sup_{
j \in \{
1, 2, \ldots, N
\}
}
\frac{
\| f( \theta_j ) - f( \theta_{ j - 1 } ) \|_E
}{
| \theta_j - \theta_{ j - 1 } |
}
\right]
\\ &
\leq
\frac{
c^{ 1 - \alpha }
}{
d_{ \min }( \theta )
}
\left[
\sup_{
j \in \{ 1, 2, \ldots, N \}
}
\| f( \theta_j ) - f( \theta_{ j - 1 } ) \|_E
\right]
.
\end{aligned}$$ This completes the proof of Lemma \[lem:estcagrid\].
\[lem:est\_hoelder\_semi\_norm\] Consider the notation in Subsection \[notation\], let $ ( E, \left\| \cdot \right\|_E ) $ be a normed vector space, let $ T \in (0,\infty) $, $ \alpha \in [0,1] $, $ \theta \in {\mathscr{P}}_T $, and let $
f \colon [0,T] \rightarrow E
$ be a function. Then $
\left| [ f ]_{ \theta } \right|_{
{\mathscr{C}}^{ \alpha }( [0,T] , \left\| \cdot \right\|_E )
}
\leq
\left| f \right|_{
{\mathscr{C}}^{ \alpha }( [0,T] , \left\| \cdot \right\|_E )
}
$.
Throughout this proof let $ N \in {\mathbb{N}}$, $ \theta_0, \theta_1, \dots, \theta_N \in [0,T] $ be the real numbers which satisfy $
0 = \theta_0 < \theta_1 < \dots < \theta_N = T
$ and $
\theta = \{ \theta_0, \theta_1, \dots, \theta_N \}
$ and let $ n \colon [0,T] \to {\mathbb{N}}$ and $
\rho \colon [0,T] \to [0,1]
$ be the functions which satisfy for all $ t \in [0,T] $ that $$n( t ) = \min\!\big\{
k \in \{ 1, 2, \dots, N \}
\colon
t \in [ \theta_{ k - 1 } , \theta_k ]
\big\}
\qquad
\text{and}
\qquad
\rho( t ) =
\frac{
t - \theta_{ n(t) - 1 }
}{
\theta_{ n(t) } - \theta_{ n(t) - 1 }
}
.$$ Note that it holds for all $ t \in [0,T] $ that $$\label{eq:f_theta_rep}
[ f ]_{ \theta }( t )
=
\left( 1 - \rho(t) \right)
\cdot
f( \theta_{ n(t) - 1 } )
+
\rho(t)
\cdot f( \theta_{ n(t) } )
=
f( \theta_{ n(t) - 1 } )
+
\rho( t ) \cdot
\big(
f( \theta_{ n(t) } )
-
f( \theta_{ n(t) - 1 } )
\big)
.$$ Hence, we obtain for all $ t_1, t_2 \in [0,T] $ with $ t_1 < t_2 $ and $ n( t_1 ) = n( t_2 ) $ that $$\begin{split}
\label{eq:compare_f_affine_same_grid}
&
\left\|
[ f ]_{ \theta }( t_1 )
-
[ f ]_{ \theta }( t_2 )
\right\|_E
=
\bigl\|
\left[
\left( 1 - \rho( t_1 ) \right)
\cdot
f( \theta_{ n(t_1) - 1 } )
+
\rho( t_1 )
\cdot
f( \theta_{ n(t_1) } )
\right]
\\ & \hphantom{= \bigl\|}
-
\left[
\left( 1 - \rho( t_2 ) \right)
\cdot
f( \theta_{ n(t_1) - 1 } )
+
\rho( t_2 )
\cdot
f( \theta_{ n(t_1) } )
\right]
\bigr\|_E
\\ & =
\left\|
\left( \rho( t_2 ) - \rho( t_1 ) \right)
\cdot
f( \theta_{ n(t_1) - 1 } )
+
\left( \rho( t_1 ) - \rho( t_2 ) \right)
\cdot
f( \theta_{ n(t_1) } )
\right\|_E
\\ & =
\left|
\rho( t_1 ) - \rho( t_2 )
\right|
\cdot
\left\|
f( \theta_{ n(t_1) - 1 } )
-
f( \theta_{ n(t_1) } )
\right\|_E
\\ &
\leq
\left|
\rho( t_1 ) - \rho( t_2 )
\right|
\left| f \right|_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}
\left|
\theta_{ n(t_1) - 1 }
-
\theta_{ n(t_1) }
\right|^{ \alpha }
\\ & =
\left|
\rho( t_1 ) - \rho( t_2 )
\right|^{ 1 - \alpha }
\left| f \right|_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}
\left|
\left(
\rho( t_1 ) - \rho( t_2 )
\right)
\cdot
\left(
\theta_{ n(t_1) } - \theta_{ n(t_1) - 1 }
\right)
\right|^{ \alpha }
\\ & \leq
\left| f \right|_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}
\left|
\left(
\rho( t_1 ) - \rho( t_2 )
\right)
\cdot
\left(
\theta_{ n(t_1) } - \theta_{ n(t_1) - 1 }
\right)
\right|^{ \alpha }
\\ & =
\left| f \right|_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}
\left|
t_1 - \theta_{ n(t_1) - 1 } - \left( t_2 - \theta_{ n(t_1) - 1 } \right)
\right|^{ \alpha }
\\ & =
\left| f \right|_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}
\left|
t_1 - t_2
\right|^{ \alpha }
.
\end{split}$$ Moreover, ensures for all $ t_1, t_2 \in [0,T] $ with $ n( t_1 ) < n( t_2 ) $ that $$\begin{aligned}
& \nonumber
\left\|
[ f ]_{ \theta }( t_1 )
-
[ f ]_{ \theta }( t_2 )
\right\|_E
=
\bigl\|
\left[
\left( 1 - \rho( t_1 ) \right)
\cdot
f( \theta_{ n(t_1) - 1 } )
+
\rho( t_1 )
\cdot
f( \theta_{ n(t_1) } )
\right]
\\ \nonumber & \hphantom{= \bigl\|}
-
\left[
\left( 1 - \rho( t_2 ) \right)
\cdot
f( \theta_{ n(t_2) - 1 } )
+
\rho( t_2 )
\cdot
f( \theta_{ n(t_2) } )
\right]
\bigr\|_E
\\ \nonumber & \leq
\left( 1 - \rho( t_1 ) \right)
\left( 1 - \rho( t_2 ) \right)
\left\|
f( \theta_{ n(t_1) - 1 } )
-
f( \theta_{ n(t_2) - 1 } )
\right\|_E
+
\rho( t_1 )
\,
\rho( t_2 )
\left\|
f( \theta_{ n(t_1) } )
-
f( \theta_{ n(t_2) } )
\right\|_E
\\ \nonumber &
+
\left( 1 - \rho( t_1 ) \right)
\rho( t_2 )
\left\|
f( \theta_{ n(t_1) - 1 } )
-
f( \theta_{ n(t_2) } )
\right\|_E
+
\rho( t_1 )
\left( 1 - \rho( t_2 ) \right)
\left\|
f( \theta_{ n(t_1) } )
-
f( \theta_{ n(t_2) - 1 } )
\right\|_E
\\ \nonumber & \leq
\left| f \right|_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}
\big\{
\left( 1 - \rho( t_1 ) \right)
\left( 1 - \rho( t_2 ) \right)
|
\theta_{ n(t_1) - 1 }
-
\theta_{ n(t_2) - 1 }
|^{ \alpha }
+
\rho( t_1 )
\,
\rho( t_2 )
\,
|
\theta_{ n(t_1) }
-
\theta_{ n(t_2) }
|^{ \alpha }
\\ & \quad
+
\left( 1 - \rho( t_1 ) \right)
\rho( t_2 )
\,
|
\theta_{ n(t_1) - 1 }
-
\theta_{ n(t_2) }
|^{ \alpha }
+
\rho( t_1 )
\left( 1 - \rho( t_2 ) \right)
|
\theta_{ n(t_1) }
-
\theta_{ n(t_2) - 1 }
|^{ \alpha }
\big\}
.\end{aligned}$$ The concavity of the function $
( - \infty, 0 ] \ni x \mapsto | x |^{ \alpha } \in {\mathbb{R}}$ hence proves for all $ t_1, t_2 \in [0,T] $ with $ n( t_1 ) < n( t_2 ) $ that $$\begin{aligned}
& \nonumber
\left\|
[ f ]_{ \theta }( t_1 )
-
[ f ]_{ \theta }( t_2 )
\right\|_E
\\ \nonumber & \leq
\left| f \right|_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}
\big|
\left( 1 - \rho( t_1 ) \right)
\left( 1 - \rho( t_2 ) \right)
\left(
\theta_{ n(t_1) - 1 }
-
\theta_{ n(t_2) - 1 }
\right)
+
\rho( t_1 )
\,
\rho( t_2 )
\left(
\theta_{ n(t_1) }
-
\theta_{ n(t_2) }
\right)
\\ \nonumber & \quad
+
\left( 1 - \rho( t_1 ) \right)
\rho( t_2 )
\left(
\theta_{ n(t_1) - 1 }
-
\theta_{ n(t_2) }
\right)
+
\rho( t_1 )
\left( 1 - \rho( t_2 ) \right)
\left(
\theta_{ n(t_1) }
-
\theta_{ n(t_2) - 1 }
\right)
\big|^{ \alpha }
\\ \nonumber & =
\left| f \right|_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}
\big|
\left( 1 - \rho( t_1 ) \right)
\theta_{ n(t_1) - 1 }
+
\rho( t_1 )
\,
\theta_{ n(t_1) }
-
\left( 1 - \rho( t_2 ) \right)
\theta_{ n(t_2) - 1 }
-
\rho( t_2 )
\,
\theta_{ n(t_2) }
\big|^{ \alpha }
\\ & =
\left| f \right|_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}
\\ \nonumber & \quad \cdot
\left|
\left\{
\theta_{ n(t_1) - 1 }
+
\rho( t_1 )
\left[
\theta_{ n(t_1) }
-
\theta_{ n(t_1) - 1 }
\right]
\right\}
-
\left\{
\theta_{ n(t_2) - 1 }
+
\rho( t_2 )
\left[
\theta_{ n(t_2) }
-
\theta_{ n(t_2) - 1 }
\right]
\right\}
\right|^{ \alpha }
\\ \nonumber & =
\left| f \right|_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}
\left|
t_1
-
t_2
\right|^{ \alpha }
.\end{aligned}$$ Combining this and completes the proof of Lemma \[lem:est\_hoelder\_semi\_norm\].
\[lem:hoelderconv2\] Consider the notation in Subsection \[notation\], let $ ( E, \left\| \cdot \right\|_E ) $ be a normed vector space, let $ T \in (0,\infty) $, $ \alpha \in [0,1] $, $ \beta \in [\alpha,1] $, $ \theta \in {\mathscr{P}}_T $, and let $
f, g \colon [0,T] \rightarrow E
$ be functions. Then $$\label{eq:hoelderconv1}
\left|
f - [ g ]_{ \theta }
\right|_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}
\leq
\tfrac{
2
\,
\left|
d_{ \max }( \theta )
\right|^{ 1 - \alpha }
}{
d_{ \min }( \theta )
}
\sup_{ t \in \theta }
\left\|
f( t )
-
g( t )
\right\|_E
+
2
\left|
d_{ \max }( \theta )
\right|^{ \beta - \alpha }
\left|
f
\right|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_E )
}$$ and $$\begin{split}
\label{eq:hoelderconv2}
&
\left\|
f - [ g ]_{ \theta }
\right\|_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}
\\ &
\leq
\left(
\tfrac{
2
\,
\left|
d_{ \max }( \theta )
\right|^{ 1 - \alpha }
}{
d_{ \min }( \theta )
}
+ 1
\right)
\sup_{ t \in \theta }
\left\|
f( t )
-
g( t )
\right\|_E
+
\left(
\tfrac{
2
}{
\left|
d_{\max}(\theta)
\right|^{ \alpha }
}
+
\tfrac{ 1 }{
2^{ \beta }
}
\right)
\left|
d_{\max}(\theta)
\right|^{ \beta }
\left|
f
\right|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_E )
}
.
\end{split}$$
Throughout this proof let $ N \in {\mathbb{N}}$, $ \theta_0, \theta_1, \dots, \theta_N \in [0,T] $ be the real numbers which satisfy $
0 = \theta_0 < \theta_1 < \dots < \theta_N = T
$ and $
\theta = \{ \theta_0, \theta_1, \dots, \theta_N \}
$. Note that Lemma \[lem:hoelderconv\] implies that $$\label{eq:split2}
\begin{split}
\left|
f - [ g ]_{ \theta }
\right|_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}
& \leq
\max\Bigl\{
\left|
d_{ \max }( \theta )
\right|^{ - \alpha }
\left|
f - [ g ]_{ \theta }
\right|_{
{\mathscr{C}}^{
0, ( d_{ \max }( \theta ) , \infty )
}( [0,T], \left\| \cdot \right\|_E )
}
,
\\ & \hphantom{\leq\max\Big\{}
\left|
d_{ \max }( \theta )
\right|^{ \beta - \alpha }
\left|
f - [ g ]_{ \theta }
\right|_{
{\mathscr{C}}^{
\beta, ( 0, d_{ \max }( \theta ) ]
}(
[0,T], \left\| \cdot \right\|_E
)
}
\Bigr\}
.
\end{split}$$ Next note that Lemma \[lem:estNaffine\] ensures that $$\label{eq:splitparttwo}
\begin{aligned}
&
\left|
f - [ g ]_{ \theta }
\right|_{
{\mathscr{C}}^{ 0, ( d_{\max}(\theta),\infty )
}( [0,T], \left\| \cdot \right\|_E )
}
\leq
2
\left\|
f - [ g ]_{ \theta }
\right\|_{
C( [0,T], \left\| \cdot \right\|_E )
}
\\ & \leq
2
\left\|
f - [ f ]_{ \theta }
\right\|_{
C( [0,T], \left\| \cdot \right\|_E )
}
+
2
\left\|
[ f ]_{ \theta }
-
[ g ]_{ \theta }
\right\|_{
C( [0,T], \left\| \cdot \right\|_E )
}
\\ & \leq
2 \,
\big|
\tfrac{
d_{ \max }( \theta )
}{ 2 }
\big|^{ \beta }
\left|
f
\right|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_E )
}
+
2
\sup_{
t \in \theta
}
\left\|
f(
t
)
-
g(
t
)
\right\|_E
\\ & \leq
2 \,
\big|
d_{ \max }( \theta )
\big|^{ \beta }
\left|
f
\right|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_E )
}
+
2
\cdot
\frac{
d_{ \max }( \theta )
}{
d_{ \min }( \theta )
}
\cdot
\sup_{
t \in \theta
}
\left\|
f(
t
)
-
g(
t
)
\right\|_E
.
\end{aligned}$$ Moreover, observe that Lemma \[lem:estcagrid\] and Lemma \[lem:est\_hoelder\_semi\_norm\] imply that $$\begin{aligned}
& \nonumber
\left|
f - [ g ]_{ \theta }
\right|_{
{\mathscr{C}}^{ \beta, ( 0,d_{\max}(\theta) ]
}( [0,T], \left\| \cdot \right\|_E )
}
\leq
\left|
f - [ f ]_{ \theta }
\right|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_E
)
}
+
\left|
[ f - g ]_{ \theta }
\right|_{
{\mathscr{C}}^{
\beta ,
( 0, d_{\max}(\theta) ]
}( [0,T], \left\| \cdot \right\|_E )
}
\\ \nonumber & \leq
\left|
f
\right|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_E
)
}
+
\left|
[ f ]_{ \theta }
\right|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_E
)
}
\\ \label{eq:splitpartone} & \quad
+
\tfrac{
\left| d_{\max}(\theta) \right|^{1-\beta}
}{ d_{\min}(\theta) }
\Big[
\sup_{
j \in \{ 1, 2, \ldots, N \}
}
\big\|
\big[
f( \theta_j ) - g( \theta_j )
\big]
-
\big[
f( \theta_{ j - 1 } ) - g( \theta_{ j - 1 } )
\big]
\big\|_E
\Big]
\\ \nonumber & \leq
2
\left|
f
\right|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_E
)
}
+
\frac{ 2 }{
\left|
d_{ \max }( \theta )
\right|^{ \beta }
}
\cdot
\frac{
d_{ \max }( \theta )
}{
d_{ \min }( \theta )
}
\cdot
\sup_{
t \in \theta
}
\|
f( t ) - g( t )
\|_E
.\end{aligned}$$ Substituting and into proves . It thus remains to prove estimate . For this note that Lemma \[lem:estNaffine\] yields that $$\label{eq:hoelderconvuse2}
\begin{aligned}
\left\|
f
-
[ g ]_{ \theta }
\right\|_{
C( [0,T], \left\| \cdot \right\|_E )
}
& \leq
\left\|
f
-
[ f ]_{ \theta }
\right\|_{
C( [0,T], \left\| \cdot \right\|_E )
}
+
\left\|
[ f ]_{ \theta }
-
[ g ]_{ \theta }
\right\|_{
C( [0,T], \left\| \cdot \right\|_E )
}
\\ & \leq
\left|
\tfrac{ d_{ \max }( \theta ) }{ 2 }
\right|^{ \beta }
\left|
f
\right|_{
{\mathscr{C}}^{ \beta }(
[0,T], \left\| \cdot \right\|_E
)
}
+
\sup_{
t \in \theta
}
{\Vert}f( t ) - g( t ) {\Vert}_E .
\end{aligned}$$ Combining and shows . The proof of Lemma \[lem:hoelderconv2\] is thus completed.
Upper error bounds for stochastic processes with Hölder continuous sample paths {#subsec:holder2}
-------------------------------------------------------------------------------
We now turn to the result announced in the introduction which provides convergence of stochastic processes in Hölder norms given convergence on the grid-points. For this we first recall the Kolmogorov-Chentsov continuity theorem, cf., e.g., Theorem I.2.1 in Revuz & Yor [@RevuzYor1994].
\[thm:Kolmogorov\] Consider the notation in Subsection \[notation\]. There exists a function $
\Xi =
( \Xi_{ T, p, \alpha, \beta } )_{
T, p, \alpha, \beta \in {\mathbb{R}}}
\colon {\mathbb{R}}^4 \to {\mathbb{R}}$ such that for every $ T \in [0,\infty) $, $ p \in (1,\infty) $, $ \beta \in ( \nicefrac{ 1 }{ p } , 1 ] $, every Banach space $ ( E, \left\| \cdot \right\|_E ) $, every probability space $
( \Omega, \mathscr{F}, {{\mathbb P}})
$, and every $ X \in {\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } ) $ there exists an $ ( \mathscr{F}, \left\| \cdot \right\|_E ) $-strongly measurable stochastic process $
Y \colon [0,T] \times \Omega \to E
$ such that it holds for every $ \alpha \in [ 0, \beta - \nicefrac{ 1 }{ p } ) $ that $$\begin{split}
& \| Y \|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) } )
}
\leq
\Xi_{ T, p, \alpha, \beta }
\,
\| X \|_{
{\mathscr{C}}^{ \beta }( [0,T] , \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
< \infty
\quad
\text{and} \\
& \, \forall \, t \in [0,T] \colon
{{\mathbb P}}( X_t = Y_t ) = 1
.
\end{split}$$
The next result, Corollary \[cor:hoelder1\], follows directly from Corollary \[cor:HoelderEstFunc\] and the Kolmogorov-Chentsov continuity theorem (see Theorem \[thm:Kolmogorov\] above).
\[cor:hoelder1\] Consider the notation in Subsection \[notation\], let $ T \in (0,\infty) $, $ \theta \in {\mathscr{P}}_{T} $, let $
\left( \Omega, \mathscr{F},
{{\mathbb P}}\right)
$ be a probability space, and let $
\left(
E, \left\| \cdot \right\|_E
\right)
$ be a Banach space. Then
(i) it holds for all $ p \in [1,\infty) $, $ \beta \in [0,1] $, $ \gamma \in [ \beta , 1 ] $ and all $ ( \mathscr{F}, \left\| \cdot \right\|_E ) $-strongly measurable stochastic processes $
X, Y \colon [0,T] \times
\Omega \rightarrow E
$ that $$\begin{aligned}
\nonumber
& \left\|
X - Y
\right\|_{
{\mathscr{C}}^{ \beta }(
[0,T],
\left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
)
}
\leq
\left(
2
\left|
d_{\max}( \theta )
\right|^{ - \beta }
+
1
\right)
\Big[
\sup\nolimits_{ t \in \theta }
\|
X_t
-
Y_t
\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
\\ & \quad
+
\left|
d_{\max}( \theta )
\right|^{ \gamma }
\big(
\left|
X
\right|_{
{\mathscr{C}}^{ \gamma }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
+
\left|
Y
\right|_{
{\mathscr{C}}^{ \gamma }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
\big)
\Big]\end{aligned}$$
(ii) \[item:II\_cor\_hoelder2\] and it holds for all $ p \in (1,\infty) $, $ \beta \in ( \nicefrac{ 1 }{ p } , 1 ] $, $ \alpha \in [ 0, \beta - \nicefrac{ 1 }{ p } ) $, $ \gamma \in [ \beta, 1 ] $ and all $ ( \mathscr{F}, \left\| \cdot \right\|_E ) $-strongly measurable stochastic processes $
X, Y \colon [0,T] \times \Omega
\rightarrow E
$ with continuous sample paths that $$\begin{split}
&
\left\|
X - Y
\right\|_{
\mathscr{L}^p({{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) }
)
}
\leq
\Xi_{ T, p, \alpha, \beta }
\left\|
X - Y
\right\|_{
{\mathscr{C}}^{ \beta }(
[0,T],
\left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
)
} \\
& \leq
\Xi_{ T, p, \alpha, \beta }
\left(
2
\left|
d_{ \max }( \theta )
\right|^{ - \beta }
+
1
\right)
\Big[
\sup\nolimits_{ t \in \theta }
\|
X_t
-
Y_t
\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
\\ & \quad
+
\left|
d_{\max}( \theta )
\right|^{ \gamma }
\big(
\left|
X
\right|_{
{\mathscr{C}}^{ \gamma }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
+
\left|
Y
\right|_{
{\mathscr{C}}^{ \gamma }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
\big)
\Big]
.
\end{split}$$
The next result, Corollary \[cor:hoelder2\], follows directly from Lemma \[lem:hoelderconv2\] and the Kolmogorov-Chentsov continuity theorem (see Theorem \[thm:Kolmogorov\] above).
\[cor:hoelder2\] Consider the notation in Subsection \[notation\], let $ T \in (0,\infty) $, $
\theta \in \mathscr{P}_T
$, let $
\left( \Omega, \mathscr{F},
{{\mathbb P}}\right)
$ be a probability space, and let $
\left(
E, \left\| \cdot \right\|_E
\right)
$ be a Banach space. Then
(i) it holds for all $ p \in [1,\infty) $, $ \beta \in [0,1] $, $ \gamma \in [ \beta , 1 ] $ and all $ ( \mathscr{F}, \left\| \cdot \right\|_E ) $-strongly measurable stochastic processes $
X, Y \colon [0,T] \times
\Omega \rightarrow E
$ that $$\begin{aligned}
& \left\|
X - [ Y ]_{ \theta }
\right\|_{
{\mathscr{C}}^{ \beta }(
[0,T],
\left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
)
}
\leq
\Big[
\tfrac{
2
\,
\left|
d_{ \max }( \theta )
\right|^{ 1 - \beta }
}{
d_{ \min }( \theta )
}
+ 1
\Big]
\sup_{ t \in \theta }
\|
X_t
-
Y_t
\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
\\ & \quad
+
\left[
2 \, | d_{ \max }( \theta ) |^{ - \beta }
+ 2^{ - \gamma }
\right]
\left|
d_{ \max }( \theta )
\right|^{ \gamma }
\left|
X
\right|_{
{\mathscr{C}}^{ \gamma }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
\end{aligned}$$
(ii) \[item:2\_cor\_hoelder2\] and it holds for all $ p \in (1,\infty) $, $ \beta \in ( \nicefrac{ 1 }{ p }, 1 ] $, $ \alpha \in [ 0, \beta - \nicefrac{ 1 }{ p } ) $, $ \gamma \in [ \beta, 1 ] $ and all $ ( \mathscr{F}, \left\| \cdot \right\|_E ) $-strongly measurable stochastic processes $
X, Y \colon [0,T] \times
\Omega \rightarrow E
$ with continuous sample paths that $$\begin{aligned}
& \left\|
X - [ Y ]_{ \theta }
\right\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) }
)
}
\leq
\Xi_{ T, p, \alpha, \beta }
\Big(
\Big[
\tfrac{
2
\,
\left|
d_{ \max }( \theta )
\right|^{ 1 - \beta }
}{
d_{ \min }( \theta )
}
+ 1
\Big]
\sup_{ t \in \theta }
\|
X_t
-
Y_t
\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
\nonumber \\ & \quad
+
\left[
2 \, | d_{ \max }( \theta ) |^{ - \beta }
+ 2^{ - \gamma }
\right]
\left|
d_{ \max }( \theta )
\right|^{ \gamma }
\left|
X
\right|_{
{\mathscr{C}}^{ \gamma }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
\Big)
. \label{eq:cor_hoelder2}\end{aligned}$$
In in Corollary \[cor:hoelder2\] we assume beside other assumptions that $ \alpha $ is strictly smaller than $ \gamma $. In general, this assumption can not be omitted. To give an example, let $ ( \Omega, {\mathscr{F}}, {{\mathbb P}}) $ be a probability space and let $
W \colon [0,1] \times \Omega \to {\mathbb{R}}$ be a one-dimensional standard Brownian motion. Then it clearly holds for all $ p \in (0,\infty) $ that $
\| W \|_{ {\mathscr{C}}^{ \nicefrac{1}{2} }( [0,1], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; | \cdot | ) } ) } < \infty
$. However, the law of the iterated logarithm (see, e.g., Corollary 3.1 in Arcones [@Arcones1995]) ensures that the sample paths of the Brownian motion are $ {{\mathbb P}}$-a.s.not $ \nicefrac{ 1 }{ 2 } $-Hölder continuous, so that it holds for all $ p \in (0,\infty) $ that $
\|
W - [ W ]_{ \theta }
\|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ {\mathscr{C}}^{ \nicefrac{1}{2} }( [0,1], | \cdot | ) } )
}
= \infty
$. The following corollary is related to Lemma A1 in Bally, Millet & Sanz-Solé [@BallyMilletSanzSole1995].
\[$ \mathscr{L}^p $–convergence in Hölder norms for a fixed $ p \in [ 1,\infty) $\] \[cor:hoelder23\] Consider the notation in Subsection \[notation\], let $
T \in (0,\infty)
$, $ p \in [1,\infty) $, $
\beta \in [ 0 , 1 ]
$, let $
\left( \Omega, \mathscr{F},
{{\mathbb P}}\right)
$ be a probability space, let $
\left(
E, \left\| \cdot \right\|_E
\right)
$ be a Banach space, and let $
Y^N
\colon [0,T] \times
\Omega \rightarrow E
$, $ N \in {\mathbb{N}}_0 $, be $ ( \mathscr{F}, \left\| \cdot \right\|_E ) $-strongly measurable stochastic processes with continuous sample paths which satisfy $
\limsup_{ N \to \infty }
|
Y^N
|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
< \infty
$ and $
\forall \, t \in [0,T] \colon
\limsup_{ N \to \infty }
\|
Y^0_t -
Y_t^N
\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
= 0
$. Then
(i) \[item:i\_a\_priori\] it holds that $
| Y^0 |_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
\leq
\limsup_{ N \to \infty }
|
Y^N
|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
<
\infty
$,
(ii) \[item:ii\_convergence\] it holds for all $
\alpha \in [0,1]
\cap
( - \infty , \beta )
$ that $
\limsup_{ N \to \infty }
\|
Y^0 - Y^N
\|_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
= 0
$,
(iii) \[item:iii\_convergence\] and it holds for all $
\alpha \in
[0,1]
\cap
( - \infty , \beta - \nicefrac{ 1 }{ p } )
$ that\
$
\limsup_{ N \to \infty }
\|
Y^0 - Y^N
\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) }
)
}
= 0
$.
Throughout this proof let $ \theta^n \in \mathscr{P}_T $, $ n \in {\mathbb{N}}$, be the sequence which satisfies for all $ n \in {\mathbb{N}}$ that $
\theta^n
=
\{ 0, \frac{ T }{ n } , \frac{ 2 T }{ n } , \ldots,
\frac{ ( n - 1 ) T }{ n } , T \}
\in \mathscr{P}_T
$. Observe that the assumption that $
\forall \, t \in [0,T] \colon
\limsup_{ N \to \infty }
\|
Y^0_t -
Y_t^N
\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
= 0
$ and the assumption that $
\limsup_{ N \to \infty }
|
Y^N
|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
< \infty
$ ensure that $$\label{eq:XinCbeta}
\begin{split}
&
| Y^0 |_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
=
\sup_{
\substack{
s, t \in [0,T],
\\
s \neq t
}
}
\left[
\tfrac{
\| Y^0_t - Y^0_s \|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
}{
| t - s |^{ \beta }
}
\right]
\\ & =
\sup_{
\substack{
s, t \in [0,T],
\\
s \neq t
}
}
\left[
\tfrac{
\limsup_{ N \to \infty }
\|
( Y_t^N - Y_s^N )
+
( Y^0_t - Y^N_t )
+
( Y^N_s - Y^0_s )
\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
}{
| t - s |^{ \beta }
}
\right]
\\ &
\leq
\sup_{
\substack{
s, t \in [0,T],
\\
s \neq t
}
}
\limsup_{ N \to \infty }
\left[
\tfrac{
\|
Y_t^N - Y_s^N
\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
}{
| t - s |^{ \beta }
}
\right]
\leq
\limsup_{ N \to \infty }
|
Y^N
|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
< \infty
.
\end{split}$$ This establishes Item . In the next step we prove Item . We apply Item (i) in Corollary \[cor:hoelder1\] to obtain for all $
\alpha \in [0, \beta ]
$, $ n, N \in {\mathbb{N}}$ that $$\begin{aligned}
& \nonumber \|
Y^0 - Y^N
\|_{
{\mathscr{C}}^{ \alpha }(
[0,T],
\left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
)
}
\leq
\left(
2
\left|
d_{\max}( \theta^n )
\right|^{ - \alpha }
+
1
\right)
\Big[
\sup\nolimits_{ t \in \theta^n }
\|
Y^0_t
-
Y^N_t
\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
\\ & \nonumber \quad
+
\left|
d_{\max}( \theta^n )
\right|^{ \beta }
\big(
|
Y^0
|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
+
|
Y^N
|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
\big)
\Big]
\\ &
\leq
\left(
\tfrac{ 2 \, T^{ - \alpha } }{ n^{ - \alpha } }
+
1
\right)
\sup\nolimits_{ t \in \theta^n }
\|
Y^0_t
-
Y^N_t
\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
\\ & \nonumber \quad
+
\left(
\tfrac{ 2 \, T^{ \beta - \alpha } }{ n^{ \beta - \alpha } }
+
\tfrac{ T^{ \beta } }{ n^{ \beta } }
\right)
\big(
|
Y^0
|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
+
|
Y^N
|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
\big)
.\end{aligned}$$ Item and the assumption that $
\forall \, t \in [0,T] \colon
\limsup_{ N \to \infty }
\|
Y^0_t -
Y_t^N
\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
= 0
$ hence imply for all $
\alpha \in [0, \beta ]
$, $ n \in {\mathbb{N}}$ that $$\begin{aligned}
& \nonumber \limsup_{ N \to \infty }
\|
Y^0 - Y^N
\|_{
{\mathscr{C}}^{ \alpha }(
[0,T],
\left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
)
}
\leq
\Big[
\tfrac{ 2 \, T^{ - \alpha } }{ n^{ - \alpha } }
+
1
\Big]
\Big[
\limsup_{ N \to \infty }
\sup_{ t \in \theta^n }
\|
Y^0_t
-
Y^N_t
\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
\Big]
\\ & \quad
+
\Big[
\tfrac{ 4 \, T^{ \beta - \alpha } }{ n^{ \beta - \alpha } }
+
\tfrac{ 2 \, T^{ \beta } }{ n^{ \beta } }
\Big]
\limsup_{ N \to \infty }
|
Y^N
|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
\\ \nonumber & =
\Big[
\tfrac{ 4 \, T^{ \beta - \alpha } }{ n^{ \beta - \alpha } }
+
\tfrac{ 2 \, T^{ \beta } }{ n^{ \beta } }
\Big]
\limsup_{ N \to \infty }
|
Y^N
|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
< \infty
.\end{aligned}$$ Hence, we obtain for all $
\alpha \in [0,\infty) \cap ( - \infty , \beta )
$ that $$\begin{aligned}
& \nonumber
\limsup_{ N \to \infty }
\|
Y^0 - Y^N
\|_{
{\mathscr{C}}^{ \alpha }(
[0,T],
\left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
)
}
=
\limsup_{ n \to \infty }
\limsup_{ N \to \infty }
\|
Y^0 - Y^N
\|_{
{\mathscr{C}}^{ \alpha }(
[0,T],
\left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
)
}
\\ & \leq
\Big[
\limsup_{ n \to \infty }
\tfrac{ 4 \, T^{ \beta - \alpha } }{ n^{ \beta - \alpha } }
+
\limsup_{ n \to \infty }
\tfrac{ 2 \, T^{ \beta } }{ n^{ \beta } }
\Big]
\limsup_{ N \to \infty }
|
Y^N
|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
= 0
.\end{aligned}$$ This shows Item . It thus remains to establish Item to complete the proof of Corollary \[cor:hoelder23\]. For this we apply the first inequality in Item (ii) in Corollary \[cor:hoelder1\] to obtain for all $ r \in ( \nicefrac{ 1 }{ p } , \infty ) \cap ( - \infty , \beta ] $, $ \alpha \in [ 0, r - \nicefrac{ 1 }{ p } ) $, $ N \in {\mathbb{N}}$ that $$\begin{aligned}
\|
Y^0 - Y^N
\|_{
\mathscr{L}^p({{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) }
)
}
\leq
\Xi_{ T, p, \alpha, r }
\,
\|
Y^0 - Y^N
\|_{
{\mathscr{C}}^r( [0,T] , \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
.\end{aligned}$$ This and Item imply for all $ r \in ( \nicefrac{ 1 }{ p } , \infty ) \cap ( - \infty , \beta ) $, $ \alpha \in [ 0, r - \nicefrac{ 1 }{ p } ) $ that $$\begin{aligned}
\limsup_{ N \to \infty }
\|
Y^0 - Y^N
\|_{
\mathscr{L}^p({{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) }
)
}
\leq
\Xi_{ T, p, \alpha, r }
\,
\limsup_{ N \to \infty }
\|
Y^0 - Y^N
\|_{
{\mathscr{C}}^r( [0,T] , \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
= 0
.\end{aligned}$$ This establishes Item . The proof of Corollary \[cor:hoelder23\] is thus completed.
The next result, Corollary \[cor:hoelder3\] below, is a consequence from Corollary \[cor:hoelder1\] and Corollary \[cor:hoelder2\].
\[cor:hoelder3\] Consider the notation in Subsection \[notation\], let $
T \in (0,\infty)
$, $
p \in (1,\infty)
$, $
\beta \in ( \nicefrac{ 1 }{ p } , 1 ]
$, $ ( \theta^N )_{ N \in {\mathbb{N}}} \subseteq {\mathscr{P}}_T $ satisfy $
\limsup_{ N \to \infty }
d_{ \max }( \theta^N ) = 0
$, let $
\left( \Omega, \mathscr{F},
{{\mathbb P}}\right)
$ be a probability space, let $
\left(
E, \left\| \cdot \right\|_E
\right)
$ be a Banach space, let $
Y^N
\colon [0,T] \times
\Omega \rightarrow E
$, $ N \in {\mathbb{N}}_0 $, be $ ( \mathscr{F}, \left\| \cdot \right\|_E ) $-strongly measurable stochastic processes with continuous sample paths which satisfy $ Y_0^0 \in \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) $ and $$\label{eq:ass_convergencegrid1}
| Y^0 |_{
{\mathscr{C}}^{ \beta }( [0,T] , \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
+
\sup_{ N \in {\mathbb{N}}}
\left[
\left|
d_{ \max }( \theta^N )
\right|^{ - \beta }
\sup\nolimits_{
t \in \theta^N
}
\|
Y_t^0 -
Y_t^N
\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
\right]
< \infty ,$$ and assume $
\big(
\big[
\sup_{ N \in {\mathbb{N}}}
|
Y^N
|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
< \infty
\big]
$ or $
\big[
\sup_{ N \in {\mathbb{N}}}
\nicefrac{ d_{ \max }( \theta^N ) }{ d_{ \min }( \theta^N ) }
< \infty
$ and $
\forall \, N \in {\mathbb{N}}\colon
Y^N
=
[ Y^N ]_{
\theta^N
}
\big]
\big)
$. Then it holds for all $ \alpha \in [ 0, \beta - \nicefrac{ 1 }{ p } ) $, $ {\varepsilon}\in (0,\infty) $ that $$\label{eq:convHoelder1}
\sup_{ N \in {\mathbb{N}}}
\left[
\|
Y^N
\|_{
\mathscr{L}^p({{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) }
)
}
+
\left|
d_{ \max }( \theta^N )
\right|^{ - ( \beta - \alpha - \nicefrac{1}{p} - {\varepsilon}) }
\|
Y^0 - Y^N
\|_{
\mathscr{L}^p({{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) }
)
}
\right]
< \infty .$$
Throughout this proof let $ c_0 \in [0,\infty) $, $ c_1, c_2 \in [0,\infty] $ be the extended real numbers given by $$\begin{split}
c_0
& =
| Y^0 |_{
{\mathscr{C}}^{ \beta }( [0,T] , \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
+
\sup_{ N \in {\mathbb{N}}}
\left[
\left|
d_{ \max }( \theta^N )
\right|^{ - \beta }
\sup\nolimits_{
t \in \theta^N
}
\|
Y_t^0 -
Y_t^N
\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
\right]
,
\\
c_1
& =
\sup_{ N \in {\mathbb{N}}}
\left[
\frac{ d_{ \max }( \theta^N ) }{ d_{ \min }( \theta^N ) }
\right]
,
\qquad
\text{and}
\qquad
c_2
=
\sup_{ N \in {\mathbb{N}}}
|
Y^N
|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
\,
.
\end{split}$$ Next we observe that Item in Corollary \[cor:hoelder1\] ensures for all $ r \in ( \nicefrac{ 1 }{ p } , \beta ] $, $ \alpha \in [ 0, r - \nicefrac{ 1 }{ p } ) $, $ N \in {\mathbb{N}}$ that $$\begin{aligned}
\nonumber &
\left\|
Y^0 - Y^N
\right\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) }
)
}
\leq
\Xi_{ T, p, \alpha, r }
\left(
2
\left|
d_{ \max }( \theta^N )
\right|^{ - r }
+
1
\right)
\bigg[
\sup_{ t \in \theta^N }
\|
Y^0_t
-
Y^N_t
\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
\\ \nonumber & \quad
+
\left|
d_{ \max }( \theta^N )
\right|^{ \beta }
\big(
|
Y^0
|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
+
|
Y^N
|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
\big)
\bigg]
\\ & \leq
\Xi_{ T, p, \alpha, r }
\left(
2
\left|
d_{ \max }( \theta^N )
\right|^{ ( \beta - r ) }
+
\left|
d_{ \max }( \theta^N )
\right|^{ \beta }
\right)
\Big[
c_0
+
|
Y^N
|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
\Big]
\\ \nonumber & \leq
\Xi_{ T, p, \alpha, r }
\left(
2
\left|
d_{ \max }( \theta^N )
\right|^{ ( \beta - r ) }
+
\left|
d_{ \max }( \theta^N )
\right|^{ \beta }
\right)
\left[
c_0
+
c_2
\right]
\\ \nonumber & =
\Xi_{ T, p, \alpha, r }
\left(
2
+
\left|
d_{ \max }( \theta^N )
\right|^{ r }
\right)
\left|
d_{ \max }( \theta^N )
\right|^{ ( \beta - r ) }
\left[
c_0
+
c_2
\right]
.\end{aligned}$$ This implies for all $ r \in ( \nicefrac{ 1 }{ p } , \beta ] $, $ \alpha \in [ 0, r - \nicefrac{ 1 }{ p } ) $ that $$\begin{split}
\sup_{ N \in {\mathbb{N}}}
\left[
\left|
d_{ \max }( \theta^N )
\right|^{ - ( \beta - r ) }
\left\|
Y^0 - Y^N
\right\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) }
)
}
\right]
\leq
\Xi_{ T, p, \alpha, r }
\left(
2
+
T^r
\right)
\left[
c_0
+
c_2
\right]
.
\end{split}$$ Hence, we obtain for all $ \alpha \in [ 0, \beta - \nicefrac{ 1 }{ p } ) $, $ r \in ( \alpha + \nicefrac{ 1 }{ p } , \beta ] $ that $$\begin{split}
&
\sup_{ N \in {\mathbb{N}}}
\left[
\left|
d_{ \max }( \theta^N )
\right|^{ - ( \beta - \alpha - \nicefrac{1}{p} - [ r - \alpha - \nicefrac{1}{p} ] ) }
\left\|
Y^0 - Y^N
\right\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) }
)
}
\right]
\\ & \leq
\Xi_{ T, p, \alpha, \alpha + \nicefrac{1}{p} + [ r - \alpha - \nicefrac{1}{p} ] }
\left(
3
+
T
\right)
\left(
c_0
+
c_2
\right)
.
\end{split}$$ This shows for all $ \alpha \in [ 0, \beta - \nicefrac{ 1 }{ p } ) $, $ \varepsilon \in ( 0 , \beta - \alpha - \nicefrac{ 1 }{ p } ] $ that $$\label{eq:RATE_first_limit}
\begin{split}
& \sup_{ N \in {\mathbb{N}}}
\left[
\left|
d_{ \max }( \theta^N )
\right|^{ - ( \beta - \alpha - \nicefrac{1}{p} - \varepsilon ) }
\left\|
Y^0 - Y^N
\right\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) }
)
}
\right]
\\ & \leq
\Xi_{ T, p, \alpha, \alpha + \nicefrac{1}{p} + \varepsilon }
\left(
3
+
T
\right)
\left(
c_0
+
c_2
\right)
.
\end{split}$$ In the next step we note that Item in Corollary \[cor:hoelder2\] proves for all $ r \in ( \nicefrac{ 1 }{ p }, \beta ] $, $ \alpha \in [ 0, r - \nicefrac{ 1 }{ p } ) $, $ N \in {\mathbb{N}}$ that $$\begin{aligned}
& \nonumber
\left\|
Y^0 - [ Y^N ]_{ \theta^N }
\right\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) }
)
}
\leq
\Xi_{ T, p, \alpha, r }
\Big(
\Big[
\tfrac{
2
\,
|
d_{ \max }( \theta^N )
|^{ 1 - r }
}{
d_{ \min }( \theta^N )
}
+ 1
\Big]
\sup_{ t \in \theta^N }
\|
Y^0_t
-
Y^N_t
\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
\\ & \quad
+
\left[
2 \, | d_{ \max }( \theta^N ) |^{ - r }
+ 2^{ - \beta }
\right]
\left|
d_{ \max }( \theta^N )
\right|^{ \beta }
\left|
Y^0
\right|_{
{\mathscr{C}}^{ \beta }( [0,T], \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
\Big)
.\end{aligned}$$ This implies for all $ r \in ( \nicefrac{ 1 }{ p }, \beta ] $, $ \alpha \in [ 0, r - \nicefrac{ 1 }{ p } ) $, $ N \in {\mathbb{N}}$ that $$\begin{split}
&
\left\|
Y^0 - [ Y^N ]_{ \theta^N }
\right\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) }
)
}
\\ & \leq
c_0
\left|
d_{ \max }( \theta^N )
\right|^{ \beta }
\Xi_{ T, p, \alpha, r }
\Big(
\tfrac{
2
\,
|
d_{ \max }( \theta^N )
|^{ 1 - r }
}{
d_{ \min }( \theta^N )
}
+ 1
+
2 \, | d_{ \max }( \theta^N ) |^{ - r }
+ 2^{ - \beta }
\Big)
\\ & \leq
2 \, c_0
\left|
d_{ \max }( \theta^N )
\right|^{ \beta }
\Xi_{ T, p, \alpha, r }
\left(
\left[ c_1 + 1 \right]
\left|
d_{ \max }( \theta^N )
\right|^{ - r }
+ 1
\right)
.
\end{split}$$ Hence, we obtain for all $ r \in ( \nicefrac{ 1 }{ p }, \beta ] $, $ \alpha \in [ 0, r - \nicefrac{ 1 }{ p } ) $ that $$\begin{split}
&
\sup_{ N \in {\mathbb{N}}}
\left[
\left|
d_{ \max }( \theta^N )
\right|^{ - ( \beta - r ) }
\left\|
Y^0 - [ Y^N ]_{ \theta^N }
\right\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) }
)
}
\right]
\\ & \leq
2 \, c_0 \,
\Xi_{ T, p, \alpha, r }
\left(
c_1 + 1
+ T^r
\right)
\leq
2 \, c_0 \,
\Xi_{ T, p, \alpha, r }
\left(
2 + T + c_1
\right)
.
\end{split}$$ This shows for all $ \alpha \in [ 0, \beta - \nicefrac{ 1 }{ p } ) $, $ r \in ( \alpha + \nicefrac{ 1 }{ p }, \beta ] $ that $$\begin{split}
&
\sup_{ N \in {\mathbb{N}}}
\left[
\left|
d_{ \max }( \theta^N )
\right|^{ - ( \beta - \alpha - \nicefrac{1}{p} - [ r - \alpha - \nicefrac{1}{p} ] ) }
\left\|
Y^0 - [ Y^N ]_{ \theta^N }
\right\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) }
)
}
\right]
\\ & \leq
2 \, c_0 \,
\Xi_{ T, p, \alpha, \alpha + \nicefrac{1}{p} + [ r - \alpha - \nicefrac{1}{p} ] }
\left(
2 + T + c_1
\right)
.
\end{split}$$ This establishes for all $ \alpha \in [ 0, \beta - \nicefrac{ 1 }{ p } ) $, $ \varepsilon \in ( 0 , \beta - \alpha - \nicefrac{ 1 }{ p } ] $ that $$\label{eq:RATE_second_limit}
\begin{split}
&
\sup_{ N \in {\mathbb{N}}}
\left[
\left|
d_{ \max }( \theta^N )
\right|^{ - ( \beta - \alpha - \nicefrac{1}{p} - \varepsilon ) }
\left\|
Y^0 - [ Y^N ]_{ \theta^N }
\right\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) }
)
}
\right]
\\ & \leq
2 \, c_0 \,
\Xi_{ T, p, \alpha, \alpha + \nicefrac{1}{p} + \varepsilon }
\left(
2 + T + c_1
\right)
.
\end{split}$$ Combining and assures for all $ \alpha \in [ 0, \beta - \nicefrac{ 1 }{ p } ) $, $ {\varepsilon}\in (0,\infty) $ that $$\label{eq:RATE_both_limits}
\sup_{ N \in {\mathbb{N}}}
\left[
\left|
d_{ \max }( \theta^N )
\right|^{ - ( \beta - \alpha - \nicefrac{1}{p} - {\varepsilon}) }
\|
Y^0 - Y^N
\|_{
\mathscr{L}^p({{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) }
)
}
\right]
< \infty .$$ In addition, note that the assumption that $ Y_0^0 \in \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) $, the assumption that $
| Y^0 |_{
{\mathscr{C}}^{ \beta }( [0,T] , \left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) } )
}
< \infty
$, the assumption that $ Y^0 $ has continuous sample paths, and Theorem \[thm:Kolmogorov\] ensure for all $ \alpha \in [ 0, \beta - \nicefrac{ 1 }{ p } ) $ that $
\|
Y^0
\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) }
)
}
< \infty
$. This and complete the proof of Corollary \[cor:hoelder3\].
The next result, Corollary \[cor:eulermethod\] below, illustrates Corollary \[cor:hoelder3\] through a simple example. For this note that standard results for the Euler-Maruyama method show for every $ p \in [2,\infty) $, $ \beta \in [0,\nicefrac{ 1 }{ 2 } ] $ that assumption in Corollary \[cor:hoelder3\] with uniform time steps is satisfied (cf., e.g., Section 10.6 in Kloeden & Platen [@kp92]). The convergence rate established in Corollary \[cor:eulermethod\] (see below) is essentially sharp; see Proposition \[prop:brownian\] below. Corollary \[cor:eulermethod\] is related to Theorem 1.2 in [@CoxVanNeerven2010] and Theorem 1.1 in [@CoxVanNeerven2013].
\[cor:eulermethod\] Consider the notation in Subsection \[notation\], let $ T \in (0,\infty) $, $ d, m \in {\mathbb{N}}$, let $
\left( \Omega, \mathscr{F},
{{\mathbb P}}\right)
$ be a probability space with a normal filtration $
\left( \mathscr{F}_t \right)_{
t \in [0,T]
}
$, let $
W \colon [0,T] \times
\Omega \rightarrow \mathbb{R}^m
$ be an $m$-dimensional standard $ ( \mathscr{F}_t )_{ t \in [0,T] } $-Brownian motion, let $
\mu \colon \mathbb{R}^d
\rightarrow \mathbb{R}^d
$ and $
\sigma \colon \mathbb{R}^d
\rightarrow
\mathbb{R}^{ d \times m }
$ be globally Lipschitz continuous functions, let $
X \colon [0,T] \times
\Omega \rightarrow \mathbb{R}^d
$ be an $
( \mathscr{F}_t )_{ t \in [0,T] }
$/$ \mathscr{B}( \mathbb{R}^d ) $-adapted stochastic process with continuous sample paths which satisfies $
\forall \, p \in [1,\infty)
\colon
{{\mathbb E}}\!\left[
\left\| X_0
\right\|^p_{ \mathbb{R}^d }
\right]
< \infty
$ and which satisfies for all $ t \in [0,T] $ that $$[ X_t ]_{ {{\mathbb P}}, \mathscr{B}( {\mathbb{R}}^d ) } = \biggl[ X_0+\int_0^t\mu( X_s ) {{\,\mathrm{d}}s}\biggr]_{ {{\mathbb P}}, \mathscr{B}( {\mathbb{R}}^d ) }
+\int_0^t \sigma( X_s ) {{\,\mathrm{d}}W_s},$$ and let $
Y^N \colon [0,T] \times
\Omega \rightarrow \mathbb{R}^d
$, $ N \in {\mathbb{N}}$, be mappings which satisfy for all $ N \in {\mathbb{N}}$, $ n \in \{ 0, 1, \dots, N-1 \}
$, $
t \in
[
\frac{ n T }{ N },
\frac{ (n+1) T }{ N }
]
$ that $ Y^N_0 = X_0 $ and $$Y_t^N
=
Y_{ \frac{nT}{N} }^N
+
\left(
t - \tfrac{ n T }{ N }
\right) \cdot
\mu\big(
Y_{
\frac{nT}{N}
}^N
\big)
+
\left(
\tfrac{ t N }{ T } - n
\right) \cdot
\sigma\big(
Y_{
\frac{nT}{N}
}^N
\big)
\big(
W_{
\frac{(n + 1) T}{N}
}
-
W_{
\frac{nT}{N}
}
\big).$$ Then it holds for all $ \alpha \in [ 0, \nicefrac{ 1 }{ 2 } ) $, $ {\varepsilon}\in (0,\infty) $, $ p \in [1,\infty) $ that $$\begin{split}
\label{eq:convSup5}
\sup_{ N \in {\mathbb{N}}}
\big(
N^{
\nicefrac{ 1 }{ 2 } - \alpha - {\varepsilon}}
\,
\|
X - Y^N
\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_{\mathbb{R}^d} ) }
)
}
\big)
< \infty
.
\end{split}$$
Lower error bounds for stochastic processes with Hölder continuous sample paths {#sec:holder3}
-------------------------------------------------------------------------------
In this subsection we comment on the optimality of the convergence rate provided by Corollary \[cor:hoelder3\] and Corollary \[cor:eulermethod\], respectively. In particular, in the setting of Corollary \[cor:eulermethod\], Theorem 3 in Müller-Gronbach [@m02] shows in the case $ \alpha = 0 $ that there exists a class of SDEs for which the factors $ N^{ \nicefrac{1}{2} - {\varepsilon}} $, $ N \in {\mathbb{N}}$, on the left hand side of the estimate can at best – up to a constant – be replaced by the factors $
\frac{ N^{ \nicefrac{1}{2} } }{ \log( N ) }
$, $ N \in {\mathbb{N}}$. In Proposition \[prop:brownian\] below we show for every $ \alpha \in [ 0, \nicefrac{ 1 }{ 2 } ) $ in the simple example $ \mu = 0 $ and $ \sigma = \big( \mathbb{R} \ni x \mapsto 1 \in \mathbb{R} \big) $ in Corollary \[cor:eulermethod\] that the factors $ N^{ \nicefrac{1}{2} - \alpha - {\varepsilon}} $, $ N \in {\mathbb{N}}$, on the left hand side of the estimate can at best – up to a constant – be replaced by the factors $
N^{ \nicefrac{1}{2} - \alpha }
$, $ N \in {\mathbb{N}}$.
\[prop:brownian\] Consider the notation in Subsection \[notation\], let $ T \in (0,\infty) $, let $
\left( \Omega, \mathscr{F},
\mathbb{P} \right)
$ be a probability space, let $
W \colon [0,T] \times
\Omega \rightarrow \mathbb{R}
$ be a one-dimensional standard Brownian motion, and let $
W^N \colon [0,T] \times
\Omega \rightarrow \mathbb{R}
$, $ N \in \mathbb{N} $, be mappings which satisfy for all $ N \in \mathbb{N} $, $ n \in \{ 0, 1, \dots, N-1 \} $, $
t \in
\big[
\frac{ n T }{ N },
\frac{ (n+1) T }{ N }
\big]
$ that $$\label{eq:def_WN_t}
W^N_t
=
\left(
n + 1 - \tfrac{ t N }{ T }
\right) \cdot
W_{ \frac{ n T }{ N } }
+
\left(
\tfrac{ t N }{ T } - n
\right) \cdot
W_{ \frac{ (n+1) T }{ N } }
.$$ Then it holds for all $ \alpha \in [0, \nicefrac{ 1 }{ 2 } ] $, $ p \in [1,\infty) $, $ N \in {\mathbb{N}}$ that $$\label{eq:lower_bound_I}
\|
W - W^N
\|_{
C(
[0,T],
\left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left| \cdot \right| ) }
)
}
=
\tfrac{
\left\| W_T \right\|_{
\mathscr{L}^p( {{\mathbb P}}; \left| \cdot \right| )
}
}{
2 \sqrt{ N }
} ,$$ $$\label{eq:lower_bound_II}
\tfrac{
|
W - W^N
|_{
{\mathscr{C}}^{ \alpha }(
[0,T],
\left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left| \cdot \right| ) }
)
}
}{
N^{
\left( \alpha - \nicefrac{ 1 }{ 2 } \right)
}
\,
T^{ - \alpha }
\,
\left\| W_T \right\|_{
\mathscr{L}^p( {{\mathbb P}}; \left| \cdot \right| )
}
}
=
\tfrac{
\left( \nicefrac{1}{2} - \alpha \right)^{
\left( \nicefrac{ 1 }{ 2 } - \alpha \right)
}
}{
2^{ \alpha }
\,
\left(
1 - \alpha
\right)^{ \left( 1 - \alpha \right) }
}
\in
\left[
\tfrac{ 1 }{ \sqrt{2} }, 1
\right] ,$$ $$\label{eq:lower_bound_III}
\tfrac{
\|
W - W^N
\|_{
{\mathscr{C}}^{ \alpha }(
[0,T],
\left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left| \cdot \right| ) }
)
}
}{
N^{
\left( \alpha - \nicefrac{ 1 }{ 2 } \right)
}
\,
T^{ - \alpha }
\,
\left\| W_T \right\|_{
\mathscr{L}^p( {{\mathbb P}}; \left| \cdot \right| )
}
}
=
\tfrac{
T^{ \alpha }
}{
2 N^{ \alpha }
}
+
\tfrac{
\left( \nicefrac{ 1 }{ 2 } - \alpha \right)^{
\left( \nicefrac{ 1 }{ 2 } - \alpha \right)
}
}{
2^{ \alpha }
\,
\left(
1 - \alpha
\right)^{ \left( 1 - \alpha \right) }
}
\in
\left[
\tfrac{ 1 }{ \sqrt{2} }
,
\tfrac{
2 + T^{ \alpha }
}{2}
\right]
,$$ $$\label{eq:lower_bound_IIII}
\tfrac{
\|
W - W^N
\|_{
\mathscr{L}^p( {{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }(
[0,T],
\left| \cdot \right|
) }
)
}
}{
N^{
\left( \alpha - \nicefrac{ 1 }{ 2 } \right)
}
\,
T^{ - \alpha }
\,
\left\| W_T \right\|_{
\mathscr{L}^p( {{\mathbb P}}; \left| \cdot \right| )
}
}
\geq
\tfrac{
\left( \nicefrac{1}{2} - \alpha \right)^{
\left( \nicefrac{ 1 }{ 2 } - \alpha \right)
}
}{
2^{ \alpha }
\,
\left(
1 - \alpha
\right)^{ \left( 1 - \alpha \right) }
}
\geq
\tfrac{ 1 }{ \sqrt{2} }
.$$
Convergence in Hölder norms for Galerkin approximations {#sec:Galerkin}
=======================================================
Setting {#ssec:setting}
-------
Consider the notation in Subsection \[notation\], let $
(
H,
\langle \cdot, \cdot \rangle_H,
\left\| \cdot \right\|_H
)
$ and $
(
U ,
\langle \cdot, \cdot \rangle_U ,
\left\| \cdot \right\|_U
)
$ be separable $ \mathbb{R} $-Hilbert spaces, let $ \mathbb{H} \subseteq H $ be a non-empty orthonormal basis of $ H $, let $ T, \iota \in (0,\infty) $, let $ ( \Omega, {\mathscr{F}}, {{\mathbb P}}) $ be a probability space with a normal filtration $ ( {\mathscr{F}}_t )_{ t \in [0,T] } $, let $
( W_t )_{ t \in [0,T] }
$ be an $ \operatorname{Id}_U $-cylindrical $ ( \Omega, {\mathscr{F}}, {{\mathbb P}}, ( {\mathscr{F}}_t )_{ t \in [0,T] } ) $-Wiener process, let $ \lambda \colon \mathbb{H} \to \mathbb{R} $ be a function with $
\sup_{ h \in \mathbb{H} }
\lambda_h < 0
$, let $
A \colon D(A) \subseteq H \rightarrow H
$ be the linear operator which satisfies $$D(A)
=
\biggl\{
v \in H
\colon
\sum_{ h \in \mathbb{H} }
\left|
\lambda_h
\langle h, v \rangle_H
\right|^2
< \infty
\biggr\}$$ and which satisfies for all $ v \in D(A) $ that $$Av
=
\sum_{ h \in \mathbb{H} }
\lambda_h \langle h, v \rangle_H h,$$ let $
( H_r , \left< \cdot , \cdot \right>_{ H_r }, \left\| \cdot \right\|_{ H_r } )
$, $ r \in {\mathbb{R}}$, be a family of interpolation spaces associated to $ - A $, let $ \gamma \in {\mathbb{R}}$, $ \alpha \in [0,1) $, $ \beta \in [ 0, \nicefrac{ 1 }{ 2 } ) $, $ \chi \in [ \beta, \nicefrac{ 1 }{ 2 } ) $, $ F \in C( H_{ \gamma }, H_{ \gamma - \alpha } ) $, $ B \in C( H_{ \gamma }, \mathrm{HS}( U, H_{ \gamma - \beta } ) ) $ satisfy for all bounded sets $ E \subseteq H_{ \gamma } $ that $$| F|_E |_{
{\mathscr{C}}^1( E, \left\| \cdot \right\|_{ H_{ \gamma - \alpha } } )
}
+
| B|_E |_{
{\mathscr{C}}^1( E, \left\| \cdot \right\|_{ \mathrm{HS}( U, H_{ \gamma - \beta } ) } )
}
< \infty,$$ let $ \mathbb{H}_N \subseteq \mathbb{H} $, $ N \in {\mathbb{N}}_0 $, be sets which satisfy $
\mathbb{H}_0 = \mathbb{H}
$ and $
\sup_{ N \in \mathbb{N} }
N^{ \iota }
\sup\!\big(
\{
\nicefrac{ 1 }{ | \lambda_h | }
\colon
h \in \mathbb{H} \backslash \mathbb{H}_N
\} \cup \{ 0 \}
\big)
< \infty
$, let $ P_N \in L( H_{ \min\{ 0, \gamma - 1 \} } ) $, $ N \in {\mathbb{N}}_0 $, and $ \mathscr{P}_N \in L( U ) $, $ N \in {\mathbb{N}}_0 $, be linear operators which satisfy for all $ N \in {\mathbb{N}}_0 $, $ v \in H $ that $$P_N( v )
= \sum_{ h \in \mathbb{H}_N }
\left< h, v \right>_H h,$$ and let $
X^N \colon [0,T] \times \Omega \rightarrow H_{ \gamma }
$, $ N \in {\mathbb{N}}_0 $, be $
( \mathscr{F}_t )_{ t \in [0,T] }
$/$ \mathscr{B}( H_{ \gamma } ) $-adapted stochastic processes with continuous sample paths which satisfy for all $ N \in {\mathbb{N}}_0 $, $ t \in [0,T] $ that $$\label{eq:solution}
\bigl[ X_t^N \bigr]_{ {{\mathbb P}}, \mathscr{B}( H_\gamma ) }
=
\biggl[ e^{ t A } P_N X_0^0
+
\int_0^t
e^{ ( t - s ) A }
P_N F( X^N_s )
{{\,\mathrm{d}}s}\biggr]_{ {{\mathbb P}}, \mathscr{B}( H_\gamma ) }
+
\int_0^t
e^{ ( t - s ) A }
P_N B( X_s^N ) \mathscr{P}_N
{{\,\mathrm{d}}W_s}.$$
Strong convergence in Hölder norms for Galerkin approximations of SEEs with globally Lipschitz continuous nonlinearities
------------------------------------------------------------------------------------------------------------------------
The next lemma, Lemma \[lem:a\_priori\_XN\] below, follows directly from, e.g.,
\[lem:a\_priori\_XN\] Assume the setting in Subsection \[ssec:setting\], let $ p \in [2,\infty) $, $
\eta \in [ \max\{ \alpha, 2 \beta \}, 1 )
$, $ N \in {\mathbb{N}}_0 $, and assume that $${{\mathbb E}}\big[
\| X_0^0 \|_{ H_{ \gamma } }^p
\big]
+
| F |_{
{\mathscr{C}}^1( H_{ \gamma }, \left\| \cdot \right\|_{ H_{ \gamma - \alpha } } )
}
+
| B |_{
{\mathscr{C}}^1( H_{ \gamma }, \left\| \cdot \right\|_{ \mathrm{HS}( U, H_{ \gamma - \beta } ) } )
}
< \infty.$$ Then $$\begin{aligned}
&
\sup_{ t \in [0,T] }
\left\|
\max\{ 1, \| X_t^N \|_{ H_{ \gamma } } \}
\right\|_{
\mathscr{L}^p( {{\mathbb P}}; \left| \cdot \right| )
}
\leq
\sqrt{ 2 }
\left\|
\max\{ 1, \| X^0_0 \|_{ H_{ \gamma } } \}
\right\|_{
\mathscr{L}^p( {{\mathbb P}}; \left| \cdot \right| )
}
\\ \nonumber &
\cdot
\mathscr{E}_{ ( 1 - \eta ) }\biggl[
\tfrac{
T^{ 1 - \eta }
\sqrt{ 2 }
}{
\sqrt{ 1 - \eta }
}
\biggl(
\sup_{ v \in H_{ \gamma } }
\tfrac{
\| F( v ) \|_{ H_{ \gamma - \eta } }
}{
\max\{ 1, \| v \|_{ H_{ \gamma } } \}
}
\biggr)
+
\sqrt{
T^{ 1 - \eta } p ( p - 1 )
}
\biggl(
\sup_{ v \in H_{ \gamma } }
\tfrac{
\| B( v ) \mathscr{P}_N \|_{ \mathrm{HS}( U, H_{ \gamma - \nicefrac{\eta}{2} } ) }
}{
\max\{ 1, \| v \|_{ H_{ \gamma } } \}
}
\biggr)
\biggr]
.\end{aligned}$$
\[lem:spectral\_noise\] Assume the setting in Subsection \[ssec:setting\], let $ p \in [2,\infty) $, $
\eta \in [ \max\{ \alpha, 2 \beta \}, 1 )
$, $ N \in {\mathbb{N}}_0 $, and assume that $${{\mathbb E}}\big[
\| X_0^0 \|_{ H_{ \gamma } }^p
\big]
+
| F |_{
{\mathscr{C}}^1( H_{ \gamma }, \left\| \cdot \right\|_{ H_{ \gamma - \alpha } } )
}
+
| B |_{
{\mathscr{C}}^1( H_{ \gamma }, \left\| \cdot \right\|_{ \mathrm{HS}( U, H_{ \gamma - \beta } ) } )
}
< \infty.$$ Then $$\begin{split}
&
\sup_{ t \in [0,T] }
\left\|
X^0_t - X^N_t
\right\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ H_{ \gamma } }
)
}
\leq
\Biggl[
\sqrt{ 2 }
\sup_{ t \in [0,T] }
\left\| ( P_0 - P_N ) X_t^0 \right\|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma } } )
}
\\ &
+
\tfrac{
T^{ \nicefrac{1}{2} - \chi }
\sqrt{ p \, ( p - 1 ) }
}{
\sqrt{ 1 - 2 \chi }
}
\biggl(
1 +
\sup_{ t \in [0,T] }
\| X^N_t \|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma } } )
}
\biggr)
\Biggl(
\sup_{ v \in H_{ \gamma } }
\frac{
\| B( v ) ( \mathscr{P}_0 - \mathscr{P}_N ) \|_{
\mathrm{HS}( U, H_{ \gamma - \chi } )
}
}{
1 + \| v \|_{ H_{ \gamma } }
}
\Biggr)
\Biggr]
\\ &
\cdot
\mathscr{E}_{ (1 - \eta) } \biggl[
\tfrac{
T^{ 1 - \eta }
\sqrt{ 2 }
\,
\left| F \right|_{
{\mathscr{C}}^1(
H_{ \gamma }, \left\| \cdot \right\|_{ H_{ \gamma - \eta } }
)
}
}{ \sqrt{ 1 - \eta } }
+
\sqrt{
T^{ 1 - \eta }
p ( p - 1 )
}
\left| B \right|_{
{\mathscr{C}}^1(
H_{ \gamma }, \left\| \cdot \right\|_{ \mathrm{HS}( U, H_{ \gamma - \nicefrac{\eta}{2} } ) }
)
}
\| \mathscr{P}_0 \|_{ L( U ) }
\biggr]
< \infty
.
\end{split}$$
First of all, observe that Lemma \[lem:a\_priori\_XN\] ensures that $$\label{eq:finiteness_XN}
\sup_{ t \in [0,T] }
\max \bigl\{
\| X^0_t \|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma } } ) },
\| X^N_t \|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma } } ) }
\bigr\}
< \infty
.$$ We can hence apply to obtain that $$\begin{split}
&
\sup_{ t \in [0,T] }
\left\|
X^0_t - X^N_t
\right\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ H_{ \gamma } }
)
}
\\ & \leq
\mathscr{E}_{ (1 - \eta) } \biggl[
\tfrac{
T^{ 1 - \eta }
\sqrt{ 2 }
\,
\left| P_N F( \cdot ) \right|_{
{\mathscr{C}}^1(
H_{ \gamma }, \left\| \cdot \right\|_{ H_{ \gamma - \eta } }
)
}
}{ \sqrt{ 1 - \eta } }
+
\sqrt{
T^{ 1 - \eta }
p ( p - 1 )
}
\left| P_N B( \cdot ) \mathscr{P}_0 \right|_{
{\mathscr{C}}^1(
H_{ \gamma }, \left\| \cdot \right\|_{ \mathrm{HS}( U, H_{ \gamma - \nicefrac{\eta}{2} } ) }
)
}
\biggr]
\\ & \quad
\cdot
\sqrt{2}
\,
\sup_{ t \in [0,T] }
\bigg\|
\biggl[
X_t^0
-
\int_0^t
e^{ ( t - s ) A } P_N F( X^0_s )
{{\,\mathrm{d}}s}\biggr]_{ {{\mathbb P}}, \mathscr{B}( H_\gamma ) }
-
\int_0^t
e^{ ( t - s ) A } P_N B( X^0_s ) \mathscr{P}_0
{{\,\mathrm{d}}W_s}\\ & \quad
+
\biggl[
\int_0^t
e^{ ( t - s ) A } P_N F( X^N_s )
{{\,\mathrm{d}}s}- X^N_t
\biggr]_{ {{\mathbb P}}, \mathscr{B}( H_\gamma ) }
+
\int_0^t
e^{ ( t - s ) A } P_N B( X^N_s ) \mathscr{P}_0
{{\,\mathrm{d}}W_s}\bigg\|_{
L^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ H_{ \gamma } }
)
}
.
\end{split}$$ This shows that $$\begin{split}
&
\sup_{ t \in [0,T] }
\left\|
X^0_t - X^N_t
\right\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ H_{ \gamma } }
)
}
\\ & \leq
\mathscr{E}_{ (1 - \eta) } \biggl[
\tfrac{
T^{ 1 - \eta }
\sqrt{ 2 }
\,
\left| F \right|_{
{\mathscr{C}}^1(
H_{ \gamma }, \left\| \cdot \right\|_{ H_{ \gamma - \eta } }
)
}
}{ \sqrt{ 1 - \eta } }
+
\sqrt{
T^{ 1 - \eta }
p ( p - 1 )
}
\left| B \right|_{
{\mathscr{C}}^1(
H_{ \gamma }, \left\| \cdot \right\|_{ \mathrm{HS}( U, H_{ \gamma - \nicefrac{\eta}{2} } ) }
)
}
\| \mathscr{P}_0 \|_{ L( U ) }
\biggr]
\\ & \quad
\cdot
\sqrt{2}
\,
\sup_{ t \in [0,T] }
\left\|
\bigl[ ( P_0 - P_N ) X^0_t \bigr]_{ {{\mathbb P}}, \mathscr{B}( H_\gamma ) }
+
\int_0^t
e^{ ( t - s ) A } P_N B( X^N_s ) ( \mathscr{P}_0 - \mathscr{P}_N )
{{\,\mathrm{d}}W_s}\right\|_{
L^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ H_{ \gamma } }
)
}
.
\end{split}$$ The Burkholder-Davis-Gundy type inequality in Lemma 7.7 in Da Prato & Zabczyk [@dz92] hence implies that $$\begin{aligned}
& \nonumber
\sup_{ t \in [0,T] }
\left\|
X^0_t - X^N_t
\right\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{
H_{ \gamma }
}
)
}
\\ \nonumber & \leq
\mathscr{E}_{ (1 - \eta) } \biggl[
\tfrac{
T^{ 1 - \eta }
\sqrt{ 2 }
\,
\left| F \right|_{
{\mathscr{C}}^1(
H_{ \gamma }, \left\| \cdot \right\|_{ H_{ \gamma - \eta } }
)
}
}{ \sqrt{ 1 - \eta } }
+
\sqrt{
T^{ 1 - \eta }
p ( p - 1 )
}
\left| B \right|_{
{\mathscr{C}}^1(
H_{ \gamma }, \left\| \cdot \right\|_{ \mathrm{HS}( U, H_{ \gamma - \nicefrac{\eta}{2} } ) }
)
}
\| \mathscr{P}_0 \|_{ L( U ) }
\biggr]
\\ & \quad
\cdot
\sqrt{2}
\,
\Biggl[
\sup_{ t \in [0,T] }
\left\| ( P_0 - P_N ) X_t^0 \right\|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma } } )
}
\\ \nonumber & \quad
+
\sup_{ s \in [0,T] }
\left\|
B( X^N_s ) [ \mathscr{P}_0 - \mathscr{P}_N ]
\right\|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ \mathrm{HS}( U, H_{ \gamma - \chi } ) } )
}
\sqrt{
\tfrac{ p \, ( p - 1 ) }{ 2 }
\sup_{ t \in [0,T] }
\int_0^t
\left( t - s \right)^{ - 2 \chi }
{{\,\mathrm{d}}s}}
\Biggr]
.\end{aligned}$$ This and complete the proof of Proposition \[lem:spectral\_noise\].
\[cor:convGlobLip\_0\] Assume the setting in Subsection \[ssec:setting\], let $
\vartheta \in [ 0, \min\{ 1 - \alpha, \nicefrac{ 1 }{ 2 } - \beta \} )
$, $ p \in [2,\infty) $, and assume that $ X^0_0( \Omega ) \subseteq H_{ \gamma + \vartheta } $ and $$\begin{aligned}
\label{eq:ass_FB0}
{{\mathbb E}}\big[
\| X_0^0 \|_{ H_{ \gamma + \vartheta } }^p
\big]
+
| F |_{
{\mathscr{C}}^1( H_{ \gamma }, \left\| \cdot \right\|_{ H_{ \gamma - \alpha } } )
}
+
| B |_{
{\mathscr{C}}^1( H_{ \gamma }, \left\| \cdot \right\|_{ \mathrm{HS}( U, H_{ \gamma - \beta } ) } )
} & < \infty,
\\
\label{eq:ass_B1.general_0}
\sup_{ N \in {\mathbb{N}}}
\sup_{ v \in H_{ \gamma } }
\left[
\frac{
N^{ \iota \vartheta }
\,
\|
B( v ) ( \operatorname{Id}_U - \mathscr{P}_N )
\|_{ \mathrm{HS}( U, H_{ \gamma - \chi } ) }
}{
1 + \| v \|_{ H_{ \gamma } }
}
\right]
& < \infty
.\end{aligned}$$ Then it holds that $$\sup_{ N \in {\mathbb{N}}_0 }
\sup_{ t \in [0,T] }
\bigl(
\| F( X^N_t ) \|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ H_{ \gamma - \alpha } }
)
}
+
\| B( X^N_t ) \mathscr{P}_N \|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ \mathrm{HS}( U, H_{ \gamma - \chi } ) }
)
} \bigr) < \infty$$ and $$\sup_{ N \in {\mathbb{N}}_0 }
\sup_{ t \in [0,T] }
\bigl( N^{ \iota \vartheta }
\,
\|
X^0_t - X^N_t
\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ H_{ \gamma } }
)
}
\bigr)
<
\infty.$$
Combining the assumptions that $ X^0_0( \Omega ) \subseteq H_{ \gamma + \vartheta } $ and $
{{\mathbb E}}\big[
\| X^0_0 \|_{ H_{ \gamma + \vartheta } }^p
\big] $ $
< \infty
$ with, e.g., ensures that $
\forall \, t \in [0,T] \colon
{{\mathbb P}}\big(
X^0_t \in H_{ \gamma + \vartheta }
\big) = 1
$ and $
\sup_{ t \in [0,T] }
{{\mathbb E}}\big[ \| \mathbbm{1}_{ \{ X^0_t \in H_{ \gamma + \vartheta } \} } X_t^0 \|_{ H_{ \gamma + \vartheta } }^p \big] < \infty
$. This and the assumption that $
\sup_{ N \in \mathbb{N} }
N^{ \iota }
\sup\!\big(
\{
\nicefrac{ 1 }{ | \lambda_h | }
\colon
h \in \mathbb{H} \backslash \mathbb{H}_N
\} \cup \{ 0 \}
\big)
< \infty
$ imply that $$\begin{aligned}
& \nonumber
\sup_{ N \in {\mathbb{N}}}
\sup_{ t \in [0,T] }
\left[
N^{ \iota \vartheta }
\left\|
( P_0 - P_N ) X^0_t
\right\|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma } } )
}
\right]
\\ \nonumber & \leq
\sup_{ N \in {\mathbb{N}}}
\sup_{ t \in [0,T] }
\left[
N^{ \iota \vartheta }
\left\|
( - A )^{ - \vartheta }
( P_0 - P_N )
\right\|_{
L( H_{ \gamma } )
}
\big\|
\mathbbm{1}_{ \{ X^0_t \in H_{ \gamma + \vartheta } \} } X^0_t
\big\|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma + \vartheta } } )
}
\right]
\\ \label{eq:N_iota_rate} & \leq
\left[
\sup_{ N \in {\mathbb{N}}}
N^{ \iota \vartheta }
\left\|
( - A )^{ - 1 }
( \operatorname{Id}_{ H_{ \gamma } } - P_N )
\right\|_{
L( H_{ \gamma } )
}^{ \vartheta }
\right]
\left[
\sup_{ t \in [0,T] }
\big\|
\mathbbm{1}_{ \{ X^0_t \in H_{ \gamma + \vartheta } \} } X^0_t
\big\|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma + \vartheta } } )
}
\right]
\\ \nonumber & =
\left[
\sup_{ N \in {\mathbb{N}}}
N^{ \iota \vartheta }
\left[
\sup\!\big(
\{
\nicefrac{ 1 }{ | \lambda_h | }
\colon
h \in \mathbb{H} \backslash \mathbb{H}_N
\} \cup \{ 0 \}
\big)
\right]^{ \vartheta }
\right]
\left[
\sup_{ t \in [0,T] }
\big\|
\mathbbm{1}_{ \{ X^0_t \in H_{ \gamma + \vartheta } \} } X^0_t
\big\|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma + \vartheta } } )
}
\right]
\\ \nonumber & < \infty .\end{aligned}$$ In addition, observe that , , and Lemma \[lem:a\_priori\_XN\] ensure that $$\label{eq:X_bound}
\sup_{ N \in {\mathbb{N}}_0 }
\sup_{ t \in [0,T] }
\| X_t^N \|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma } } )
}
< \infty
.$$ The triangle inequality and again hence prove that $$\label{eq:X_B_bound}
\begin{split}
&
\sup_{ N \in {\mathbb{N}}_0 }
\sup_{ t \in [0,T] }
\| B( X^N_t ) \mathscr{P}_N \|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ \mathrm{HS}( U, H_{ \gamma - \chi } ) }
)
}
\\ &
\leq
\biggl(
1 +
\sup_{ N \in {\mathbb{N}}_0 }
\sup_{ t \in [0,T] }
\left\| X^N_t \right\|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma } } )
}
\biggr)
\biggl(
\sup_{ N \in {\mathbb{N}}_0 }
\sup_{ v \in H_{ \gamma } }
\tfrac{
\|
B( v ) \mathscr{P}_N
\|_{ \mathrm{HS}( U, H_{ \gamma - \chi } )
}
}{
1 + \| v \|_{ H_{ \gamma } }
}
\biggr)
\\ & \leq
\biggl(
1 +
\sup_{ N \in {\mathbb{N}}_0 }
\sup_{ t \in [0,T] }
\left\| X^N_t \right\|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma } } )
}
\biggr)
\\ & \quad \cdot
\biggl(
\sup_{ v \in H_{ \gamma } }
\tfrac{
\|
B( v )
\|_{\mathrm{HS}( U, H_{ \gamma - \chi } )}
}{
1 + \| v \|_{ H_{ \gamma } }
}
+
\sup_{ N \in {\mathbb{N}}_0 }
\sup_{ v \in H_{ \gamma } }
\tfrac{
\|
B( v ) ( \operatorname{Id}_U - \mathscr{P}_N )
\|_{\mathrm{HS}( U, H_{ \gamma - \chi } )}
}{
1 + \| v \|_{ H_{ \gamma } }
}
\biggr)
< \infty
.
\end{split}$$ In the next step we combine , , and with Lemma \[lem:spectral\_noise\] to obtain that $$\label{eq:X_rate_zero}
\sup\nolimits_{ N \in {\mathbb{N}}_0 }
\sup\nolimits_{ t \in [0,T] }
\big(
N^{ \iota \vartheta }
\,
\|
X^0_t - X^N_t
\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ H_{ \gamma } }
)
}
\big)
<
\infty
.$$ Furthermore, observe that assures that $
\sup_{ N \in {\mathbb{N}}_0 }
\sup_{ t \in [0,T] }
\| F( X^N_t ) \|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma - \alpha } } )
}
< \infty
$. This, , and complete the proof of Corollary \[cor:convGlobLip\_0\].
The next result, Corollary \[cor:convGlobLip\], proves strong convergence rates in Hölder norms for spatial spectral Galerkin approximations of SEEs with globally Lipschitz continuous nonlinearities. Note in the setting of Corollary \[cor:convGlobLip\] that, e.g., Lemma 6.2 in Conus, Jentzen & Kurniawan [@ConusJentzenKurniawan2014arXiv] shows in the case $ \iota = 2 $, $ \delta = 0 $ that the convergence rate established in is essentially sharp.
\[cor:convGlobLip\] Assume the setting in Subsection \[ssec:setting\], let $
\vartheta \in ( 0, \min\{ 1 - \alpha, \nicefrac{ 1 }{ 2 } - \beta \} )
$, $ p \in ( \nicefrac{ 1 }{ \vartheta }, \infty ) $, and assume that $ X^0_0( \Omega ) \subseteq H_{ \gamma + \vartheta } $, $
{{\mathbb E}}\big[
\| X_0^0 \|_{ H_{ \gamma + \vartheta } }^p
\big] < \infty $, $
| F |_{
{\mathscr{C}}^1( H_{ \gamma }, \left\| \cdot \right\|_{ H_{ \gamma - \alpha } } )
} < \infty $, $
| B |_{
{\mathscr{C}}^1( H_{ \gamma }, \left\| \cdot \right\|_{ \mathrm{HS}( U, H_{ \gamma - \beta } ) } )
} < \infty $, and $$\label{eq:ass_B1.general_0b}
\sup_{ N \in {\mathbb{N}}}
\sup_{ v \in H_{ \gamma } }
\left[
\frac{
\|
B( v ) \mathscr{P}_N
\|_{ \mathrm{HS}( U, H_{ \gamma - \beta } ) }
+
N^{ \iota \vartheta }
\,
\|
B( v ) ( \operatorname{Id}_U - \mathscr{P}_N )
\|_{ \mathrm{HS}( U, H_{ \gamma - \chi } ) }
}{
1 + \| v \|_{ H_{ \gamma } }
}
\right]
< \infty
.$$ Then it holds for all $ \delta \in [ 0, \vartheta - \nicefrac{ 1 }{ p } ) $, $ \varepsilon \in (0,\infty) $ that $$\label{eq:convGlobLip}
\sup\nolimits_{ N \in {\mathbb{N}}}
\big(
\|
X^N
\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{
{\mathscr{C}}^{ \delta }( [0,T], \left\| \cdot \right\|_{ H_{ \gamma } } )
}
)
}
+
N^{ \iota \, ( \vartheta - \delta - \nicefrac{1}{p} - {\varepsilon}) }
\,
\|
X^0 - X^N
\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{
{\mathscr{C}}^{ \delta }( [0,T], \left\| \cdot \right\|_{ H_{ \gamma } } )
}
)
}
\big)
< \infty
.$$
Throughout this proof let $ \eta \in {\mathbb{R}}$ be the real number given by $
\eta = \max\{ \alpha, 2 \beta \}
$ and let $ \theta^N \in \mathscr{P}_T $, $ N \in {\mathbb{N}}$, be a sequence of sets such that $$\sup_{ N \in {\mathbb{N}}}
\biggl[
\frac{d_{ \max }( \theta^N )}{ N^{ - \iota } }
+ \frac{ N^{ - \iota } }{d_{ \min }( \theta^N )}
\biggr]
< \infty.$$ In particular, this ensures that $
\limsup_{ N \to \infty }
d_{ \max }( \theta^N ) = 0
$. In addition, Corollary \[cor:convGlobLip\_0\] proves that $$\label{eq:proof_Lip_to_apply_hoelder}
\begin{split}
&
\sup_{ N \in {\mathbb{N}}}
\left[
\left| d_{ \max }( \theta^N ) \right|^{ - \vartheta }
\sup_{ t \in \theta^N }
\|
X^0_t - X^N_t
\|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma } } )
}
\right]
\\ &
\leq
\biggl[
\sup_{ N \in {\mathbb{N}}}
\tfrac{
\left|
d_{ \max }( \theta^N )
\right|^{ - \vartheta }
}{
N^{ \iota \vartheta }
}
\biggr]
\biggl(
\sup_{ N \in {\mathbb{N}}}
\sup_{ t \in \theta^N }
N^{ \iota \vartheta }
\,
\|
X^0_t - X^N_t
\|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma } } )
}
\biggr)
\\ &
\leq
\biggl[
\sup_{ N \in {\mathbb{N}}}
\tfrac{
N^{ - \iota }
}{
d_{ \min }( \theta^N )
}
\biggr]^{ \vartheta }
\biggl(
\sup_{ N \in {\mathbb{N}}}
\sup_{ t \in [0,T] }
N^{ \iota \vartheta }
\,
\|
X^0_t - X^N_t
\|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma } } )
}
\biggr)
< \infty
.
\end{split}$$ Next note that, e.g., shows for all $ N \in {\mathbb{N}}_0 $, $
\varepsilon \in
(
0, \min\{ 1 - \eta , \nicefrac{ 1 }{ 2 } - \beta \}
)
$ that $$\begin{aligned}
&
\sup_{
\substack{
t_1, t_2 \in [0,T]
,
\\
t_1 \neq t_2
}
}
\Biggl(
\frac{
\left|
\min\{ t_1, t_2 \}
\right|^{ \max\{ \gamma + \varepsilon - ( \gamma + \vartheta ) , 0 \} }
\| X^N_{ t_1 } - X^N_{ t_2 } \|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma } } )
}
}{
\left| t_1 - t_2 \right|^{ \varepsilon }
}
\Biggr)
\\ \nonumber &
\leq
\|
X_0^N
\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ H_{ \min\{ \gamma + \vartheta, \gamma + \varepsilon \} } }
)
}
+
\biggl[
\sup_{ s \in [0,T] }
\left\|
P_N F( X_s^N )
\right\|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma - \eta } } )
}
\biggr]
\frac{
2
\,
T^{
( 1 + \gamma - \eta - \min\{ \gamma + \vartheta, \gamma + \varepsilon \} )
}
}{
\left( 1 - \eta - \varepsilon \right)
}
\\ \nonumber & \quad
+
\biggl[
\sup_{ s \in [0,T] }
\left\|
P_N B( X_s^N ) \mathscr{P}_N
\right\|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ \mathrm{HS}( U, H_{ \gamma - \beta } ) } )
}
\biggr]
\frac{
\sqrt{ p \, ( p - 1 ) }
\,
T^{ ( \nicefrac{1}{2} + \gamma - \beta - \min\{ \gamma + \vartheta , \gamma + \varepsilon \} ) }
}{
\left(
1 - 2 \beta - 2 \varepsilon
\right)^{ \nicefrac{1}{2} }
}
< \infty
.\end{aligned}$$ This and the fact that $
\min\{ 1 - \eta , \nicefrac{ 1 }{ 2 } - \beta \}
=
\min\{ 1 - \max\{ \alpha, 2 \beta \} , \nicefrac{ 1 }{ 2 } - \beta \}
=
\min\{ 1 - \alpha, \nicefrac{ 1 }{ 2 } - \beta \}
> \vartheta > 0
$ imply that $$\begin{split}
&
\sup_{ N \in {\mathbb{N}}_0 }
\sup_{
\substack{
t_1, t_2 \in [0,T]
,
\\
t_1 \neq t_2
}
}
\Biggl(
\frac{
\| X^N_{ t_1 } - X^N_{ t_2 } \|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma } } )
}
}{
\left| t_1 - t_2 \right|^{ \vartheta }
}
\Biggr)
\\ &
\leq
\sup_{ N \in {\mathbb{N}}_0 }
\|
X_0^N
\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ H_{ \gamma + \vartheta } }
)
}
+
\biggl[
\sup_{ N \in {\mathbb{N}}_0 }
\sup_{ s \in [0,T] }
\|
F( X_s^N )
\|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma - \eta } } )
}
\biggr]
\frac{
2
\,
T^{
( 1 - \eta - \vartheta )
}
}{
\left( 1 - \eta - \vartheta \right)
}
\\ & \quad
+
\biggl[
\sup_{ N \in {\mathbb{N}}_0 }
\sup_{ s \in [0,T] }
\|
B( X_s^N ) \mathscr{P}_N
\|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ \mathrm{HS}( U, H_{ \gamma - \beta } ) } )
}
\biggr]
\frac{
\sqrt{ p \, ( p - 1 ) }
\,
T^{
(
\nicefrac{1}{2} - \beta - \vartheta
)
}
}{
\left(
1 - 2 \beta - 2 \vartheta
\right)^{ \nicefrac{1}{2} }
}
.
\end{split}$$ Corollary \[cor:convGlobLip\_0\] and estimate hence prove that $$\begin{split}
&
\sup_{ N \in {\mathbb{N}}_0 }
|
X^N
|_{
{\mathscr{C}}^{ \vartheta }( [0,T] ,
\left\| \cdot \right\|_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma } } ) } )
}
\\ &
\leq
\|
X_0^0
\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{ H_{ \gamma + \vartheta } }
)
}
+
\biggl[
\sup_{ N \in {\mathbb{N}}_0 }
\sup_{ s \in [0,T] }
\|
F( X_s^N )
\|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ H_{ \gamma - \eta } } )
}
\biggr]
\frac{
2
\,
T^{
( 1 - \eta - \vartheta )
}
}{
\left( 1 - \eta - \vartheta \right)
}
\\ & \quad
+
\biggl[
\sup_{ N \in {\mathbb{N}}_0 }
\sup_{ s \in [0,T] }
\|
B( X_s^N ) \mathscr{P}_N
\|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ \mathrm{HS}( U, H_{ \gamma - \beta } ) } )
}
\biggr]
\frac{
\sqrt{ p \, ( p - 1 ) }
\,
T^{
(
\nicefrac{1}{2} - \beta - \vartheta
)
}
}{
\left(
1 - 2 \beta - 2 \vartheta
\right)^{ \nicefrac{1}{2} }
}
< \infty
.
\end{split}$$ This, , and the fact that $
\vartheta \in ( \nicefrac{ 1 }{ p } , 1 ]
$ allow us to apply Corollary \[cor:hoelder3\] to obtain for all $ \delta \in [ 0, \vartheta - \nicefrac{ 1 }{ p } ) $, $ \varepsilon \in (0,\infty) $ that $$\begin{split}
\sup_{ N \in {\mathbb{N}}} \,
\Big[
& \|
X^N
\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{
{\mathscr{C}}^{ \delta }( [0,T],\left\| \cdot \right\|_{ H_{ \gamma } } )
}
)
}
\\ &
+
\left|
d_{ \max }( \theta^N )
\right|^{ - ( \vartheta - \delta - \nicefrac{1}{p} - {\varepsilon}) }
\|
X^0 - X^N
\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{
{\mathscr{C}}^{ \delta }( [0,T], \left\| \cdot \right\|_{ H_{ \gamma } } )
}\texttt{}
)
}
\Big]
< \infty .
\end{split}$$ Combining this with the fact that $
\sup_{ N \in {\mathbb{N}}}
\bigl[
\frac{d_{ \max }( \theta^N )}{ N^{ - \iota } }
\bigr] < \infty
$ completes the proof of Corollary \[cor:convGlobLip\].
Almost sure convergence in Hölder norms for Galerkin approximations of SEEs with semi-globally Lipschitz continuous nonlinearities {#subsec:convLocLip}
----------------------------------------------------------------------------------------------------------------------------------
The proof of the following corollary employs a standard localisation argument; see, e.g., [@g98; @p01].
\[cor:convLocLip\] Assume the setting in Subsection \[ssec:setting\], let $
\vartheta \in
( 0, \min\{ 1 - \alpha , \nicefrac{ 1 }{ 2 } - \beta \} )
$, assume that $
{{\mathbb P}}\big( X_0^0 \in H_{ \gamma + \vartheta }
\big) = 1
$, and assume for all bounded sets $ E \subseteq H_{ \gamma } $ that $$\label{eq:convLocLip_assumption}
\sup_{ N \in {\mathbb{N}}}
\sup_{ v \in E }
\left[
\frac{
\|
B( v ) \mathscr{P}_N
\|_{ \mathrm{HS}( U, H_{ \gamma - \beta } ) }
+
N^{ \iota \vartheta }
\,
\|
B( v ) ( \operatorname{Id}_U - \mathscr{P}_N )
\|_{ \mathrm{HS}( U, H_{ \gamma - \chi } ) }
}{
1 + \| v \|_{ H_{ \gamma } }
}
\right]
< \infty
.$$ Then it holds for all $ \delta \in [0,\vartheta) $, $ {\varepsilon}\in (0,\infty) $ that $${{\mathbb P}}\biggl(
\sup_{ N \in {\mathbb{N}}}
\big[
N^{ \iota ( \vartheta - \delta - {\varepsilon}) }
\,
\|
X^0 - X^N
\|_{
{\mathscr{C}}^{ \delta }( [0,T], \left\| \cdot \right\|_{ H_{ \gamma } } )
}
\big]
<
\infty
\biggr) = 1.$$
Throughout this proof we assume w.l.o.g. that $ X^0_0( \Omega ) \subseteq H_{ \gamma + \vartheta } $ and throughout this proof let $
\delta \in [ 0, \vartheta )
$, let $
\phi_{ r, M } \colon H_r \to H_r
$, $ r \in {\mathbb{R}}$, $ M \in (0,\infty) $, be the mappings which satisfy for all $ r \in {\mathbb{R}}$, $ M \in (0,\infty) $, $ v \in H_r $ that $$\phi_{ r, M }( v )
=
v \cdot
\min\biggl\{
1, \frac{M + 1}{1 + \| v \|_{ H_r }}
\biggr\},$$ let $ \xi_M \colon \Omega \to H_{ \gamma } $, $ M \in {\mathbb{N}}$, be the mappings which satisfy for all $ M \in {\mathbb{N}}$ that $
\xi_M =
\phi_{ \gamma + \vartheta, M }( X^0_0 )
$, let $ F_M \colon H_{ \gamma } \to H_{ \gamma - \alpha } $, $ M \in {\mathbb{N}}$, and $ B_M \colon H_{ \gamma } \to \mathrm{HS}( U, H_{ \gamma - \beta } ) $, $ M \in {\mathbb{N}}$, be the mappings which satisfy for all $ M \in {\mathbb{N}}$ that $
F_M = F \circ \phi_{ \gamma, M }
$ and $
B_M = B \circ \phi_{ \gamma, M }
$, and let $
S_M \subseteq H_{ \gamma }
$, $ M \in {\mathbb{N}}$, be the sets which satisfy for all $ M \in {\mathbb{N}}$ that $
S_M = \{
v \in H_{ \gamma } \colon
\| v \|_{ H_{ \gamma } } \leq M + 1
\}
$. Observe that it holds for all $ v, w \in H_{ \gamma } $, $ M \in {\mathbb{N}}$ that $$\begin{split}
&
\left\|
\phi_{ \gamma, M }( v )
-
\phi_{ \gamma, M }( w )
\right\|_{ H_{ \gamma } }
\\ & =
\left\|
\frac{
v
\,
( 1 + \| w \|_{ H_{ \gamma } } )
\min\{ 1 + \| v \|_{ H_{ \gamma } } , M + 1 \}
-
w
\,
( 1 + \| v \|_{ H_{ \gamma } } )
\min\{ 1 + \| w \|_{ H_{ \gamma } } , M + 1 \}
}{
( 1 + \| v \|_{ H_{ \gamma } } )
\,
( 1 + \| w \|_{ H_{ \gamma } } )
}
\right\|_{ H_{ \gamma } }
\\ & \leq
\left\| v - w \right\|_{ H_{ \gamma } }
\\ &
+
\left\|
\frac{
w
\,
\big[
( 1 + \| w \|_{ H_{ \gamma } } )
\min\{ 1 + \| v \|_{ H_{ \gamma } } , M + 1 \}
-
( 1 + \| v \|_{ H_{ \gamma } } )
\min\{ 1 + \| w \|_{ H_{ \gamma } } , M + 1 \}
\big]
}{
( 1 + \| v \|_{ H_{ \gamma } } )
\,
( 1 + \| w \|_{ H_{ \gamma } } )
}
\right\|_{ H_{ \gamma } }
\\ & \leq
\left\| v - w \right\|_{ H_{ \gamma } }
\\ & \quad
+
\frac{
\left|
( 1 + \| w \|_{ H_{ \gamma } } )
\min\{ 1 + \| v \|_{ H_{ \gamma } } , M + 1 \}
-
( 1 + \| v \|_{ H_{ \gamma } } )
\min\{ 1 + \| w \|_{ H_{ \gamma } } , M + 1 \}
\right|
}{
( 1 + \| v \|_{ H_{ \gamma } } )
}
.
\end{split}$$ This ensures for all $ v, w \in H_{ \gamma } $, $ M \in {\mathbb{N}}$ that $$\begin{split}
&
\left\|
\phi_{ \gamma, M }( v )
-
\phi_{ \gamma, M }( w )
\right\|_{ H_{ \gamma } }
\\ & \leq
\left\| v - w \right\|_{ H_{ \gamma } }
+
\frac{
\big|
\| w \|_{ H_{ \gamma } }
-
\| v \|_{ H_{ \gamma } }
\big|
\min\{ 1 + \| v \|_{ H_{ \gamma } } , M + 1 \}
}{
( 1 + \| v \|_{ H_{ \gamma } } )
}
\\ & \quad
+
\frac{
( 1 + \| v \|_{ H_{ \gamma } } )
\left|
\min\{ 1 + \| v \|_{ H_{ \gamma } } , M + 1 \}
-
\min\{ 1 + \| w \|_{ H_{ \gamma } } , M + 1 \}
\right|
}{
( 1 + \| v \|_{ H_{ \gamma } } )
}
\\ & \leq
\left\| v - w \right\|_{ H_{ \gamma } }
+
\left|
\| w \|_{ H_{ \gamma } }
-
\| v \|_{ H_{ \gamma } }
\right|
\\ & \quad
+
\left|
\min\{ 1 + \| v \|_{ H_{ \gamma } } , M + 1 \}
-
\min\{ 1 + \| w \|_{ H_{ \gamma } } , M + 1 \}
\right|
\\ & \leq
3
\left\| v - w \right\|_{ H_{ \gamma } }
.
\end{split}$$ Hence, we obtain for all $ M \in {\mathbb{N}}$ that $
|
\phi_{ \gamma, M }
|_{
{\mathscr{C}}^1( H_{ \gamma }, \left\| \cdot \right\|_{ H_{ \gamma } } )
}
\leq 3
$. This, the fact that $
\forall \, M \in {\mathbb{N}}\colon
|
F|_{ S_M }
|_{
{\mathscr{C}}^1( S_M, \left\| \cdot \right\|_{ H_{ \gamma - \alpha } } )
}
+
|
B|_{ S_M }
|_{
{\mathscr{C}}^1( S_M, \left\| \cdot \right\|_{ \mathrm{HS}( U, H_{ \gamma - \beta } ) } )
}
+
|
\phi_{ \gamma, M }
|_{
{\mathscr{C}}^1( H_{ \gamma }, \left\| \cdot \right\|_{ H_{ \gamma } } )
}
< \infty
$, and the fact that $
\forall \, M \in {\mathbb{N}}\colon
\phi_{ \gamma, M }( H_{ \gamma } )
\subseteq
S_M
$ ensure that it holds for all $ M \in {\mathbb{N}}$, $ p \in [1,\infty) $ that $$|
F_M
|_{
{\mathscr{C}}^1( H_{ \gamma }, \left\| \cdot \right\|_{ H_{ \gamma - \alpha } } )
}
+
|
B_M
|_{
{\mathscr{C}}^1( H_{ \gamma }, \left\| \cdot \right\|_{ \mathrm{HS}( U, H_{ \gamma - \beta } ) } )
}
+
{{\mathbb E}}\big[
\| \xi_M \|^p_{ H_{ \gamma + \vartheta } }
\big]
< \infty.$$ E.g., hence proves that there exist $
( \mathscr{F}_t )_{ t \in [0,T] }
$/$ \mathscr{B}( H_{ \gamma } ) $-adapted stochastic processes $
\mathscr{X}^{ N, M } \colon [0,T] \times \Omega \rightarrow H_{ \gamma }
$, $ N \in {\mathbb{N}}_0 $, $ M \in {\mathbb{N}}$, with continuous sample paths such that it holds for all $ N \in {\mathbb{N}}_0 $, $ M \in {\mathbb{N}}$, $ t \in [0,T] $ that $$\label{eq:GlobLipGalerkin_solution2}
\begin{split}
\bigl[ \mathscr{X}_t^{ N, M } \bigr]_{ {{\mathbb P}}, \mathscr{B}( H_\gamma ) }
& =
\biggl[ e^{ t A } P_N \xi_M
+
\int_0^t
e^{ ( t - s ) A }
P_N F_M( \mathscr{X}^{ N, M }_s )
{{\,\mathrm{d}}s}\biggr]_{ {{\mathbb P}}, \mathscr{B}( H_\gamma ) }
\\ & \quad
+
\int_0^t
e^{ ( t - s ) A }
P_N B_M( \mathscr{X}_s^{ N, M } ) \mathscr{P}_N
{{\,\mathrm{d}}W_s}\end{split}$$ (cf., e.g., Theorem 7.1 in van Neerven, Veraar & Weis [@VanNeervenVeraarWeis2008]). We now introduce a bit more notation. Let $
\tau_{ N, M } \colon \Omega \rightarrow [0,T]
$, $ M \in {\mathbb{N}}$, $ N \in {\mathbb{N}}_0 $, be the mappings which satisfy for all $ M \in {\mathbb{N}}$, $ N \in {\mathbb{N}}_0 $ that $$\label{eq:def_tau_NM}
\tau_{ N, M }
=
\min\!\left\{
T
\,
\mathbbm{1}_{
\{
\| X_0^0 \|_{ H_{ \gamma + \vartheta } } \leq M
\}
}
,
\inf\!\left(
\big\{
t \in [0,T]
\colon
\| \mathscr{X}_t^{ N, M } \|_{ H_{ \gamma } }
\geq M
\big\}
\cup
\{ T \}
\right)
\right\}
,$$ let $ \varUpsilon \in {\mathscr{F}}$ be the set given by $$\begin{aligned}
\label{eq:def_varOmega}
&
\varUpsilon
=
\\ &
\left[
\cap_{ N \in {\mathbb{N}}_0 }
\cup_{ M \in {\mathbb{N}}}
\cap_{ m \in \{ M, M + 1, \dots \} }
\{ \tau_{ N, m } = T \}
\right]
\cap
\left[
\cap_{
M \in {\mathbb{N}},
N \in {\mathbb{N}}_0
}
\left\{
\| \mathscr{X}^{ N, M } \|_{
{\mathscr{C}}^{ \delta }( [0,T], \left\| \cdot \right\|_{ H_{ \gamma } } )
}
< \infty
\right\}
\right]
\\ &
\cap
\left[
\cap_{
M \in {\mathbb{N}}, N \in {\mathbb{N}}_0
}
\left(
\left\{
\| X_0^0 \|_{ H_{ \gamma + \vartheta } }
>
M
\right\}
\cup
\big\{
\forall \, t \in [0,T] \colon
\mathscr{X}^{ N, M }_{ \min\{ t, \tau_{ N, M } \} }
=
X^N_{ \min\{ t, \tau_{ N, M } \} }
\big\}
\right)
\right]
\\ & {\refstepcounter{equation}\tag{\theequation}}\cap
\left[
\cap_{ M, n \in {\mathbb{N}}}
\left\{
\sup\nolimits_{ N \in {\mathbb{N}}}
\big(
N^{ \iota ( \vartheta - \delta - \nicefrac{ 1 }{ n } ) }
\,
\|
\mathscr{X}^{ 0, M }
-
\mathscr{X}^{ N, M }
\|_{
{\mathscr{C}}^{ \delta }( [0,T], \left\| \cdot \right\|_{ H_{ \gamma } } )
}
\big)
< \infty
\right\}
\right]
,\end{aligned}$$ let $
\mathscr{M} \colon
\varUpsilon
\to {\mathbb{N}}$ be the mapping which satisfies for all $ \omega \in \varUpsilon $ that $$\label{eq:def_calM}
\mathscr{M}( \omega )
=
\min\!\big\{
M \in {\mathbb{N}}\cap
\big(
\| X^0_0( \omega ) \|_{ H_{ \gamma + \vartheta } }
,
\infty
\big)
\colon
\forall \, m \in \{ M, M + 1, \dots \} \colon
\tau_{ 0, m }( \omega ) = T
\big\}
,$$ and let $
\mathscr{N} \colon
\varUpsilon
\to {\mathbb{N}}$ be the mapping which satisfies for all $$\omega \in
\varUpsilon
\subseteq
\left\{
\omega \in \Omega
\colon
\left[
\forall \, M \in {\mathbb{N}}\colon
\limsup\nolimits_{ N \to \infty }
\|
\mathscr{X}^{ 0, M }( \omega )
-
\mathscr{X}^{ N, M }( \omega )
\|_{
{\mathscr{C}}^{ \delta }( [0,T], \left\| \cdot \right\|_{ H_{ \gamma } } )
}
= 0
\right]
\right\}$$ that $$\label{eq:def_calN}
\mathscr{N}( \omega )
=
\min\!\left\{
N \in {\mathbb{N}}\colon
\sup\nolimits_{ n \in \{ N, N + 1, \dots \} }
\|
\mathscr{X}^{ 0, 2 \mathscr{M}( \omega ) }( \omega )
-
\mathscr{X}^{ n, 2 \mathscr{M}( \omega ) }( \omega )
\|_{
C( [0,T] , \left\| \cdot \right\|_{ H_{ \gamma } } )
}
< 1
\right\}
.$$ Observe that ensures for all $ \omega \in \varUpsilon $, $ N \in {\mathbb{N}}_0 $, $ M \in {\mathbb{N}}$, $ t \in [ 0, \tau_{ N, M }( \omega) ] $ with $
M \geq \| X^0_0( \omega ) \|_{ H_{ \gamma + \vartheta } }
$ that $$\label{eq:X_the_same_0}
\mathscr{X}^{ N, M }_t( \omega )
=
X^N_t( \omega )
.$$ This, the fact that $
\forall \, \omega \in \varUpsilon ,
N \in {\mathbb{N}}_0
\colon
\exists \, M \in {\mathbb{N}}\colon
\forall \, m \in \{ M, M + 1, \dots \} \colon
\tau_{ N, m }( \omega ) = T
$, and the fact that $
\forall \, \omega \in \varUpsilon,
N \in {\mathbb{N}}_0, m \in {\mathbb{N}}\colon
\|
\mathscr{X}^{ N, m }( \omega )
\|_{
{\mathscr{C}}^{ \delta }( [0,T] , \left\| \cdot \right\|_{ H_{ \gamma } } )
}
< \infty
$ prove that it holds for all $ \omega \in \varUpsilon $, $ N \in {\mathbb{N}}_0 $ that $$\label{eq:Hoelder_reg_X}
\| X^N( \omega ) \|_{
{\mathscr{C}}^{ \delta }( [0,T] , \left\| \cdot \right\|_{ H_{ \gamma } } )
}
< \infty
.$$ Next note that ensures for all $ \omega \in \varUpsilon $, $ M \in \{ \mathscr{M}( \omega ), \mathscr{M}( \omega ) + 1, \dots \} $ that $$\label{eq:property_calM}
\tau_{ 0, M }( \omega ) = T
\qquad
\text{and}
\qquad
M \geq
\mathscr{M}( \omega )
>
\| X^0_0( \omega ) \|_{ H_{ \gamma + \vartheta } }
.$$ This and show for all $ \omega \in \varUpsilon $, $ M \in \{ \mathscr{M}( \omega ) , \mathscr{M}( \omega ) + 1, \dots \} $, $ t \in [ 0, T ] $ that $$\label{eq:X_the_same_1}
\mathscr{X}^{ 0, M }_t( \omega )
=
X^0_t( \omega )
=
\mathscr{X}^{ 0, \mathscr{M}( \omega ) }_t( \omega )
.$$ This, , and prove for all $ \omega \in \varUpsilon $ that $$\sup\nolimits_{ t \in [0,T] }
\| \mathscr{X}^{ 0, 2 \mathscr{M}( \omega ) }_t \|_{
H_{ \gamma }
}
=
\sup\nolimits_{ t \in [0,T] }
\| \mathscr{X}^{ 0, \mathscr{M}( \omega ) }_t \|_{
H_{ \gamma }
}
\leq \mathscr{M}( \omega ).$$ The triangle inequality and hence assure for all $ \omega \in \varUpsilon $, $
N \in
\{
\mathscr{N}( \omega ),
\mathscr{N}( \omega ) + 1 ,
\ldots
\}
$ that $$\begin{aligned}
&
\sup\nolimits_{ t \in [0,T] }
\|
\mathscr{X}^{ N, 2 \mathscr{M}( \omega ) }_t( \omega )
\|_{
H_{ \gamma }
}
\\ &
\leq
\sup\nolimits_{ t \in [0,T] }
\|
\mathscr{X}^{ 0 , 2 \mathscr{M}( \omega ) }_t( \omega )
\|_{
H_{ \gamma }
}
+
\sup\nolimits_{ t \in [0,T] }
\|
\mathscr{X}^{ 0 , 2 \mathscr{M}( \omega ) }_t( \omega )
-
\mathscr{X}^{ N , 2 \mathscr{M}( \omega ) }_t( \omega )
\|_{
H_{ \gamma }
}
\\ & <
\sup\nolimits_{ t \in [0,T] }
\|
\mathscr{X}^{ 0 , 2 \mathscr{M}( \omega ) }_t( \omega )
\|_{
H_{ \gamma }
}
+
1
\leq
\mathscr{M}( \omega )
+
1
\leq
2 \mathscr{M}( \omega )
.
\end{aligned}$$ This and the fact that $
\forall \, \omega \in \varUpsilon \colon
\| X^0_0( \omega ) \|_{ H_{ \gamma + \vartheta } }
<
\mathscr{M}( \omega )
\leq
2 \mathscr{M}( \omega )
$ prove for all $ \omega \in \varUpsilon $, $
N \in
\{
\mathscr{N}( \omega ), \mathscr{N}( \omega ) + 1 , \ldots
\}
$ that $
\tau_{ N, 2 \mathscr{M}( \omega ) } = T
$. Again the fact that $
\forall \, \omega \in \varUpsilon \colon
\| X^0_0( \omega ) \|_{ H_{ \gamma + \vartheta } }
<
\mathscr{M}( \omega )
\leq
2 \mathscr{M}( \omega )
$ and hence show for all $
\omega \in \varUpsilon
$, $
N \in
\{
\mathscr{N}( \omega ), \mathscr{N}( \omega ) + 1 , \dots
\}
$, $ t \in [0,T] $ that $
\mathscr{X}^{ N, 2 \mathscr{M}( \omega ) }_t( \omega )
=
X^N_t( \omega )
$. This and prove for all $ \omega \in \varUpsilon $, $ \varepsilon \in ( 0 , \infty ) $ that $$\begin{split}
&
\sup\nolimits_{ N \in {\mathbb{N}}}
N^{ \iota ( \vartheta - \delta - {\varepsilon}) }
\,
\|
X^0( \omega ) - X^N( \omega )
\|_{
{\mathscr{C}}^{ \delta }( [0,T] , \left\| \cdot \right\|_{ H_{ \gamma } } )
}
\\ & \leq
\sup\nolimits_{
N \in \{ 1, 2, \dots, \mathscr{N}( \omega ) \}
}
N^{ \iota \vartheta }
\,
\|
X^0( \omega ) - X^N( \omega )
\|_{
{\mathscr{C}}^{ \delta }( [0,T] , \left\| \cdot \right\|_{ H_{ \gamma } } )
}
\\ & \quad
+
\sup\nolimits_{
N \in \{ \mathscr{N}( \omega ) , \mathscr{N}( \omega ) + 1 , \dots \}
}
N^{ \iota ( \vartheta - \delta - {\varepsilon}) }
\,
\|
X^0( \omega ) - X^N( \omega )
\|_{
{\mathscr{C}}^{ \delta }( [0,T] , \left\| \cdot \right\|_{ H_{ \gamma } } )
}
\\ &
\leq
\left[
\mathscr{N}( \omega )
\right]^{ \iota \vartheta }
\left[
\|
X^0( \omega )
\|_{
{\mathscr{C}}^{ \delta }( [0,T], \left\| \cdot \right\|_{ H_{ \gamma } } )
}
+
\sup\nolimits_{
N \in \{ 1, 2, \ldots, \mathscr{N}( \omega ) \}
}
\|
X^N( \omega )
\|_{
{\mathscr{C}}^{ \delta }( [0,T], \left\| \cdot \right\|_{ H_{ \gamma } } )
}
\right]
\\ & \quad
+
\sup\nolimits_{
N \in \{ \mathscr{N}( \omega ) , \mathscr{N}( \omega ) + 1 , \dots \}
}
N^{ \iota ( \vartheta - \delta - {\varepsilon}) }
\,
\|
\mathscr{X}^{ 0, 2 \mathscr{M}( \omega ) }( \omega )
-
\mathscr{X}^{ N, 2 \mathscr{M}( \omega ) }( \omega )
\|_{
{\mathscr{C}}^{ \delta }( [0,T], \left\| \cdot \right\|_{ H_{ \gamma } } )
}
.
\end{split}$$ Combining this with and ensures for all $ \omega \in \varUpsilon $, $ \varepsilon \in ( 0 , \infty ) $ that $$\begin{aligned}
&
\sup\nolimits_{ N \in {\mathbb{N}}}
N^{ \iota ( \vartheta - \delta - {\varepsilon}) }
\,
\|
X^0( \omega ) - X^N( \omega )
\|_{
{\mathscr{C}}^{ \delta }( [0,T] , \left\| \cdot \right\|_{ H_{ \gamma } } )
}
\\ & \leq {\refstepcounter{equation}\tag{\theequation}}\left[
\mathscr{N}( \omega )
\right]^{ \iota \vartheta }
\textstyle
\sum_{ N = 0 }^{ \mathscr{N}( \omega ) }
\displaystyle
\|
X^N( \omega )
\|_{
{\mathscr{C}}^{ \delta }( [0,T], \left\| \cdot \right\|_{ H_{ \gamma } } )
}
\\ & \quad
+
\sup\nolimits_{
N \in \{ \mathscr{N}( \omega ) , \mathscr{N}( \omega ) + 1 , \dots \}
}
N^{ \iota ( \vartheta - \delta - {\varepsilon}) }
\,
\|
\mathscr{X}^{ 0, 2 \mathscr{M}( \omega ) }( \omega )
-
\mathscr{X}^{ N, 2 \mathscr{M}( \omega ) }( \omega )
\|_{
{\mathscr{C}}^{ \delta }( [0,T], \left\| \cdot \right\|_{ H_{ \gamma } } )
}
< \infty
.\end{aligned}$$ It thus remains to prove that $ {{\mathbb P}}\big( \varUpsilon ) = 1 $ to complete the proof of Corollary \[cor:convLocLip\]. For this observe that the assumption shows for all $ M \in {\mathbb{N}}$ that $$\begin{split}
&
\sup_{ N \in {\mathbb{N}}}
\sup_{ v \in H_{ \gamma } }
\left[
\frac{
\|
B_M( v ) \mathscr{P}_N
\|_{ \mathrm{HS}( U, H_{ \gamma - \beta } ) }
+
N^{ \iota \vartheta }
\,
\|
B_M( v ) ( \operatorname{Id}_U - \mathscr{P}_N )
\|_{ \mathrm{HS}( U, H_{ \gamma - \chi } ) }
}{
1 + \| v \|_{ H_{ \gamma } }
}
\right]
\\
&
\leq
\sup_{ N \in {\mathbb{N}}}
\sup_{ v \in S_M }
\left[
\frac{
\|
B( v ) \mathscr{P}_N
\|_{ \mathrm{HS}( U, H_{ \gamma - \beta } ) }
+
N^{ \iota \vartheta }
\,
\|
B( v ) ( \operatorname{Id}_U - \mathscr{P}_N )
\|_{ \mathrm{HS}( U, H_{ \gamma - \chi } ) }
}{
1 + \| v \|_{ H_{ \gamma } }
}
\right]
< \infty
.
\end{split}$$ Corollary \[cor:convGlobLip\] hence proves for all $ p \in ( \nicefrac{ 1 }{ \vartheta } , \infty) $, $ r \in [ 0, \vartheta - \nicefrac{ 1 }{ p } ) $, $ \varepsilon \in (0,\infty) $, $ M \in {\mathbb{N}}$ that $$\label{eq:apply_CorLip}
\sup_{ N \in {\mathbb{N}}}
\big(
\|
\mathscr{X}^{ N, M }
\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{
{\mathscr{C}}^{ r }( [0,T], \left\| \cdot \right\|_{ H_{ \gamma } } )
}
)
}
+
N^{ \iota \, ( \vartheta - r - {\varepsilon}) }
\,
\|
\mathscr{X}^{ 0, M } - \mathscr{X}^{ N, M }
\|_{
\mathscr{L}^p(
{{\mathbb P}};
\left\| \cdot \right\|_{
{\mathscr{C}}^{ r }( [0,T], \left\| \cdot \right\|_{ H_{ \gamma } } )
}
)
}
\big)
< \infty
.$$ A standard Borel-Cantelli-type argument (see, e.g., Lemma 2.1 in Kloeden & Neuenkirch [@kn07]) hence ensures for all $ {\varepsilon}\in (0, \infty ) $, $ M \in {\mathbb{N}}$ that $${{\mathbb P}}\Bigl(
\sup\nolimits_{ N \in {\mathbb{N}}}
\big(
N^{ \iota ( \vartheta - \delta - {\varepsilon}) }
\|
\mathscr{X}^{ 0, M } - \mathscr{X}^{ N, M }
\|_{
{\mathscr{C}}^{ \delta }( [0,T], \left\| \cdot \right\|_{ H_{ \gamma } } )
}
\big)
< \infty
\Bigr) = 1.$$ Hence, we obtain that $$\label{eq:ASconvergence_GalApprox}
{{\mathbb P}}\!\left(
\forall \, M, n \in {\mathbb{N}}\colon
\sup\nolimits_{ N \in {\mathbb{N}}}
\big[
N^{ \iota ( \vartheta - \delta - \nicefrac{ 1 }{ n } ) }
\,
\|
\mathscr{X}^{ 0, M } - \mathscr{X}^{ N, M }
\|_{
{\mathscr{C}}^{ \delta }( [0,T], \left\| \cdot \right\|_{ H_{ \gamma } } )
}
\big]
< \infty
\right) = 1
.$$ In addition, proves for all $ N \in {\mathbb{N}}_0 $, $ M \in {\mathbb{N}}$ that $
{{\mathbb P}}\big(
\mathscr{X}^{ N, M } \in {\mathscr{C}}^{ \delta }( [0,T], \left\| \cdot \right\|_{ H_{ \gamma } } )
\big) = 1
$. This, in turn, ensures that $$\label{eq:local_Lip_in_Hoelder}
{{\mathbb P}}\!\left(
\forall \, M \in {\mathbb{N}}, N \in {\mathbb{N}}_0 \colon
\mathscr{X}^{ N, M } \in {\mathscr{C}}^{ \delta }( [0,T], \left\| \cdot \right\|_{ H_{ \gamma } } )
\right) = 1 .\iftoggle{arXiv:v2}{\pagebreak[4]}{}$$ Next observe that it holds for all $ t \in [0,T] $, $ M \in {\mathbb{N}}$, $ N \in {\mathbb{N}}_0 $ that $$\begin{aligned}
\nonumber
&
\bigl[
\mathscr{X}^{ N, M }_t
-
e^{ t A }
P_N
\mathscr{X}_0^{ 0, M }
\bigr]_{ {{\mathbb P}}, \mathscr{B}( H_\gamma ) }
\mathbbm{1}_{
\{ t \leq \tau_{ N, M } \}
}
\\ & \nonumber
=
\biggl( \biggl[
\int_0^t
e^{ ( t - s ) A }
P_N
F_M( \mathscr{X}^{ N, M }_s )
{{\,\mathrm{d}}s}\biggr]_{ {{\mathbb P}}, \mathscr{B}( H_\gamma ) }
+
\int_0^t
e^{ ( t - s ) A }
P_N
B_M( \mathscr{X}^{ N, M }_s )
\mathscr{P}_N
{{\,\mathrm{d}}W_s}\biggr)
\mathbbm{1}_{
\{ t \leq \tau_{ N, M } \}
}
\\ & \nonumber
=
\biggl( \biggl[
\int_0^t
\mathbbm{1}_{
\{ s < \tau_{ N, M } \}
}
\,
e^{ ( t - s ) A }
P_N
F_M( \mathscr{X}^{ N, M }_s )
{{\,\mathrm{d}}s}\biggr]_{ {{\mathbb P}}, \mathscr{B}( H_\gamma ) }
\\ & \qquad
+
\int_0^t
\mathbbm{1}_{
\{ s < \tau_{ N, M } \}
}
e^{ ( t - s ) A }
P_N
B_M( \mathscr{X}^{ N, M }_s )
\mathscr{P}_N
{{\,\mathrm{d}}W_s}\biggr)
\mathbbm{1}_{
\{ t \leq \tau_{ N, M } \}
}
\\ \nonumber &
= \biggl( \biggl[
\int_0^t
\mathbbm{1}_{
\{ s < \tau_{ N, M } \}
}
\,
e^{ ( t - s ) A }
P_N
F( \mathscr{X}^{ N, M }_s )
{{\,\mathrm{d}}s}\biggr]_{ {{\mathbb P}}, \mathscr{B}( H_\gamma ) }
\\ & \qquad \nonumber
+
\int_0^t
\mathbbm{1}_{
\{ s < \tau_{ N, M } \}
}
\,
e^{ ( t - s ) A }
P_N
B( \mathscr{X}^{ N, M }_s )
\mathscr{P}_N
{{\,\mathrm{d}}W_s}\biggr)
\mathbbm{1}_{
\{ t \leq \tau_{ N, M } \}
}.\end{aligned}$$ E.g., hence shows for all $ N \in {\mathbb{N}}_0 $, $ M \in {\mathbb{N}}$ that $${{\mathbb P}}\Bigl(
\forall \, t \in [0,T] \colon
\mathbbm{1}_{
\{ \mathscr{X}^{ N, M }_0 = X^N_0 \}
}
\mathscr{X}^{ N, M }_{ \min\{ t, \tau_{ N, M } \} }
=
\mathbbm{1}_{
\{ \mathscr{X}^{ N, M }_0 = X^N_0 \}
}
X^N_{ \min\{ t, \tau_{ N, M } \} }
\Bigr)
= 1$$ (cf., e.g., Lemma 8.2 in van Neerven, Veraar & Weis [@VanNeervenVeraarWeis2008]). This implies for all $ N \in {\mathbb{N}}_0 $, $ M \in {\mathbb{N}}$ that $${{\mathbb P}}\big(
\{
\| X_0^0 \|_{ H_{ \gamma + \vartheta } }
> M
\}
\cup
\big\{
\forall \, t \in [0,T] \colon
\mathscr{X}^{ N, M }_{ \min\{ t, \tau_{ N, M } \} }
=
X^N_{ \min\{ t, \tau_{ N, M } \} }
\big\}
\big)
= 1.$$ Hence, we obtain that $$\label{eq:zero_set_1}
{{\mathbb P}}\Big(
\cap_{ M \in {\mathbb{N}}, N \in {\mathbb{N}}_0 }
\Big[
\{
\| X_0^0 \|_{ H_{ \gamma + \vartheta } }
> M
\}
\cup
\big\{
\forall \, t \in [0,T] \colon
\mathscr{X}^{ N, M }_{ \min\{ t, \tau_{ N, M } \} }
=
X^N_{ \min\{ t, \tau_{ N, M } \} }
\big\}
\Big]
\Big)
= 1
.$$ In the next step we combine this with to obtain for all $ M \in {\mathbb{N}}$, $ N \in {\mathbb{N}}_0 $ that $$\label{eq:prop_tau_NM}
{{\mathbb P}}\Bigl(
\tau_{ N, M }
=
\min\!\left\{
T
\,
\mathbbm{1}_{
\{
\| X_0^0 \|_{ H_{ \gamma + \vartheta } } \leq M
\}
}
,
\inf\!\left(
\big\{
t \in [0,T]
\colon
\| X_t^N \|_{ H_{ \gamma } }
\geq M
\big\}
\cup
\{ T \}
\right)
\right\}
\Bigr) = 1.$$ This shows for all $ N \in {\mathbb{N}}_0 $, $ M_1, M_2 \in {\mathbb{N}}$ with $ M_1 \leq M_2 $ that $
{{\mathbb P}}\big( \tau_{ N, M_1 } \leq \tau_{ N, M_2 } \big) = 1
$. This, , and the fact that $
\forall \, \omega \in \Omega,
N \in {\mathbb{N}}_0
\colon
\sup_{ t \in [0,T] }
\| X_t^N( \omega ) \|_{ H_{ \gamma } }
< \infty
$ imply that it holds for all $ N \in {\mathbb{N}}_0 $ that $
{{\mathbb P}}\big(
\cup_{ M \in {\mathbb{N}}}
\cap_{ m \in \{ M, M + 1, \dots \} }
\{ \tau_{ N, m } = T \}
\big)
= 1
$. This, in turn, proves that $$\label{eq:zero_set_2}
{{\mathbb P}}\big(
\cap_{
N \in {\mathbb{N}}_0
}
\cup_{
M \in {\mathbb{N}}}
\cap_{ m \in \{ M, M + 1, \dots \} }
\{ \tau_{ N, m } = T \}
\big)
= 1
.$$ Combining , , , and proves that $ {{\mathbb P}}\big( \varUpsilon ) = 1 $. The proof of Corollary \[cor:convLocLip\] is thus completed.
Cubature methods in Banach spaces {#sec:cubature}
=================================
We first discuss in Subsection \[sec:preliminaries\] a number of preliminary definitions related to the Monte Carlo method in Banach spaces. In Subsection \[sec:montecarlo\] we present an elementary error estimate for the Monte Carlo method in Corollary \[cor:BanachMC\]. In Subsection \[sec:mlmc\] we then illustrate how expectations of Banach space valued functions of stochastic processes can be approximated.
Preliminaries {#sec:preliminaries}
-------------
As mentioned in the introduction, the rate of convergence of Monte Carlo approximations in a Banach space depends on the so-called *type* of the Banach space; cf., e.g., Section 9.2 in Ledoux & Talagrand [@lt91]. In order to define the type of a Banach space, we first reconsider a few concepts from the literature.
Let $ ( \Omega, {\mathscr{F}}, {{\mathbb P}}) $ be a probability space, let $ J $ be a set, and let $
r_j \colon \Omega \to \{ -1, 1 \}
$, $ j \in J $, be a family of independent random variables with $
\forall \, j \in J \colon
{{\mathbb P}}\big( r_j = 1 \big) = {{\mathbb P}}\big( r_j = - 1 \big)
$. Then we say that $ ( r_j )_{ j \in J } $ is a $ {{\mathbb P}}$-Rademacher family.
Let $ p \in (0,\infty) $ and let $ ( E, \left\| \cdot \right\|_E ) $ be an $ {\mathbb{R}}$-Banach space. Then we denote by $
\mathscr{T}_p( E ) \in [0,\infty]
$ the extended real number given by $$\begin{aligned}
&
\mathscr{T}_p( E )
=
\sup\!\left(
\left\{
r \in [0,\infty) \colon
\begin{array}{c}
\exists \, \text{probability space } ( \Omega, \mathscr{F}, {{\mathbb P}}) \colon
\\
\exists \, \text{$ {{\mathbb P}}$-Rademacher family } ( r_j )_{ j \in {\mathbb{N}}} \colon
\\
\exists \, k \in {\mathbb{N}}\colon
\exists \, x_1, x_2, \dots, x_k \in E \backslash \{ 0 \} \colon
\\
r =
\frac{
\left(
{{\mathbb E}}\left[
\|
\sum_{ j = 1 }^k
r_j x_j
\|^p_E
\right]
\right)^{ 1 / p }
}{
\left(
\sum_{ j = 1 }^k
\left\| x_j \right\|_E^p
\right)^{ 1 / p }
}
\end{array}
\right\} \cup \{ 0 \}
\right)\end{aligned}$$ and we call $ \mathscr{T}_p( E ) $ the type $ p $-constant of $ E $.
Let $ p \in (0,\infty) $ and let $ ( E, \left\| \cdot \right\|_E ) $ be an $ {\mathbb{R}}$-Banach space which satisfies $ \mathscr{T}_p( E ) < \infty $. Then we say that $ E $ has type $ p $.
Note that it holds for all $ p \in (0,\infty) $, all $ {\mathbb{R}}$-Banach spaces $ ( E, \left\| \cdot \right\|_E ) $ with type $ p $, all probability spaces $ ( \Omega, \mathscr{F}, {{\mathbb P}}) $, all $ {{\mathbb P}}$-Rademacher families $ ( r_j )_{ j \in {\mathbb{N}}} $, and all $ k \in {\mathbb{N}}$, $ x_1, x_2,\linebreak \dots, x_k \in E $ that $$\biggl\|
\sum\limits_{ j = 1 }^k
r_j x_j
\biggr\|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E )
}
\leq
\mathscr{T}_p( E )
\biggl(
\sum_{ j = 1 }^k
\left\| x_j \right\|^p_E
\biggr)^{ \! \nicefrac{1}{p} }
.$$ In addition, observe that it holds for all $ {\mathbb{R}}$-Banach spaces $ ( E, \left\| \cdot \right\|_E ) $, all probability spaces $ ( \Omega, \mathscr{F}, {{\mathbb P}}) $, all $ {{\mathbb P}}$-Rademacher families $ ( r_j )_{ j \in {\mathbb{N}}} $, and all $ p \in [2,\infty) $, $ k \in {\mathbb{N}}$, $ x \in E \backslash \{ 0 \} $ that $$\begin{split}
&
\mathscr{T}_p( E )
\geq
\frac{
\Bigl\|
\sum_{ j = 1 }^k
r_j x
\Bigr\|_{
\mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E )
}
}{
\left[
\sum_{ j = 1 }^k
\|
x
\|^p_E
\right]^{ \nicefrac{1}{p} }
}
\geq
\frac{
\left\| x \right\|_E
\Bigl\|
\sum_{ j = 1 }^k
r_j
\Bigr\|_{
\mathscr{L}^2( {{\mathbb P}}; \left| \cdot \right| )
}
}{
k^{ \nicefrac{ 1 }{ p } }
\left\| x \right\|_E
}
=
\frac{
k^{ \nicefrac{ 1 }{ 2 } }
\left\| x \right\|_E
}{
k^{ \nicefrac{ 1 }{ p } }
\left\| x \right\|_E
}
=
k^{
( \nicefrac{ 1 }{ 2 } - \nicefrac{ 1 }{ p } )
}
.
\end{split}$$ In particular, it holds for all $ p \in (2,\infty) $ and all $ {\mathbb{R}}$-Banach spaces $ ( E, \left\| \cdot \right\|_E ) $ with $ E \neq \{ 0 \} $ that $
\mathscr{T}_p( E ) = \infty
$. Furthermore, observe that Jensen’s inequality together with the fact that it holds for all normed $ {\mathbb{R}}$-vector spaces $ ( E , \left\| \cdot \right\|_E ) $ and all $ p \in (0,\infty) $, $ q \in [p,\infty) $, $ k \in {\mathbb{N}}$, $ x_1, \dots, x_k \in E $ that $$\textstyle
\left(
\sum_{ j = 1 }^k
\left\| x_j \right\|^q_E
\right)^{ \! \nicefrac{1}{q} }
\leq
\left(
\sum_{ j = 1 }^k
\left\| x_j \right\|^p_E
\right)^{ \! \nicefrac{1}{p} }$$ assures that it holds for all $ {\mathbb{R}}$-Banach spaces $ ( E, \left\| \cdot \right\|_E ) $ and all $ p, q \in (0,\infty) $ with $ p \leq q $ that $ \mathscr{T}_p( E ) \leq \mathscr{T}_q( E ) $. Hence, it holds for every $ {\mathbb{R}}$-Banach space $ ( E, \left\| \cdot \right\|_E ) $ that the function $
(0,\infty) \ni p \mapsto \mathscr{T}_p( E ) \in [0,\infty]
$ is non-decreasing. This and the triangle inequality ensure for all $ p \in (0,1] $ and all $ {\mathbb{R}}$-Banach spaces $ ( E, \left\| \cdot \right\|_E ) $ with $ E \neq \{ 0 \} $ that $ \mathscr{T}_p( E ) = 1 $. In particular, note that it holds for all $ {\mathbb{R}}$-Banach spaces $ ( E , \left\| \cdot \right\|_E ) $ that $ \sup_{ p \in (0,1] } \mathscr{T}_p( E ) \leq 1 < \infty $. Additionally, observe that it holds for all $ p \in (0,2] $ and all $ {\mathbb{R}}$-Hilbert spaces $ ( H, \left< \cdot , \cdot \right>_H , \left\| \cdot \right\|_H ) $ with $ H \neq \{ 0 \} $ that $
\mathscr{T}_p( H ) = 1
$. Furthermore, we note that it holds for every probability space $ ( \Omega , \mathscr{F}, {{\mathbb P}}) $, every $ p, q \in [1,\infty) $, and every $ {\mathbb{R}}$-Banach space $ ( E, \left\| \cdot \right\|_E ) $ with type $ q $ that $
L^p( {{\mathbb P}}; \left\| \cdot \right\|_E )
$ has type $ \min\{ p, q \} $; cf., e.g., Proposition 7.4 in Hytönen et al. [@HytonenNeervenVeraarWeis2016], Section 9.2 in Ledoux & Talagrand [@lt91], or Theorem 6.2.14 in Albiac & Kalton [@ak06]. In particular, it holds for every $ p \in [1,\infty) $ and every probability space $
( \Omega, \mathscr{F}, {{\mathbb P}})
$ that $
L^p( {{\mathbb P}}; \left| \cdot \right| )
$ has type $ \min\{ p, 2 \} $.
Let $ p, q \in (0, \infty) $. Then we denote by $
\mathscr{K}_{ p, q } \in [0,\infty]
$ the extended real number given by $$\begin{aligned}
\mathscr{K}_{ p, q } =
\sup
\left\{
r \in [0,\infty) \colon
\begin{array}{c}
\exists \, \text{$ {\mathbb{R}}$-Banach space } ( E, \left\| \cdot \right\|_E ) \colon
\\
\exists \, \text{probability space } ( \Omega, \mathscr{F}, {{\mathbb P}}) \colon \! \! \!
\\
\exists \, \text{$ {{\mathbb P}}$-Rademacher family } ( r_j )_{ j \in {\mathbb{N}}} \colon
\exists \, k \in {\mathbb{N}}\colon
\\
\exists \, x_1, x_2, \dots, x_k \in E \backslash \{ 0 \} \colon
r = \frac{
\left(
{{\mathbb E}}\left[
\|
\sum_{ j = 1 }^k
r_j x_j
\|^p_E
\right]
\right)^{ 1 / p }
}{
\left(
{{\mathbb E}}\left[
\|
\sum_{ j = 1 }^k
r_j x_j
\|^q_E
\right]
\right)^{ 1 / q }
}
\end{array}
\right\}\end{aligned}$$ and we call $ \mathscr{K}_{ p, q } $ the $ (p,q) $-Kahane-Khintchine constant.
The celebrated [*Kahane-Khintchine inequality*]{} asserts that it holds for all $ p , q \in (0,\infty) $ that $ \mathscr{K}_{ p, q } < \infty $; see, e.g., Theorem 6.2.5 in Albiac & Kalton [@ak06]. Observe that Jensen’s inequality ensures for all $ p, q \in (0,\infty) $ with $ p \leq q $ that $
\mathscr{K}_{ p, q } = 1
$. The nontrivial assertion of the Kahane-Khintchine inequality is the fact that it holds for all $ p, q \in (0,\infty) $ with $ p > q $ that $
\mathscr{K}_{ p, q } < \infty
$. In our analysis below we also use the following two abbreviations.
Let $ p, q \in (0,\infty) $ and let $ ( E, \left\| \cdot \right\|_E ) $ be an $ {\mathbb{R}}$-Banach space. Then we denote by $
\varTheta_{ p, q }( E ) \in [0,\infty]
$ the extended real number given by $
\varTheta_{ p, q }( E )
=
2 \mathscr{T}_q( E ) \mathscr{K}_{ p, q }
$.
Let $ ( \Omega, \mathscr{F}, {{\mathbb P}}) $ be a probability space, let $ p \in (0,\infty) $, let $ ( E, \left\| \cdot \right\|_E ) $ be an $ {\mathbb{R}}$-Banach space, and let $ X \in \mathscr{L}^1( {{\mathbb P}}; \left\| \cdot \right\|_E ) $. Then we denote by $
\sigma_{p,E}( X ) \in [0,\infty]
$ the extended real number given by $
\sigma_{p,E}( X ) =
\bigl(
{{\mathbb E}}\bigl[
\| X - {{\mathbb E}}[ X ] \|_E^p
\bigr]
\bigr)^{ \! \nicefrac{1}{p} }
$.
Monte Carlo methods in Banach spaces {#sec:montecarlo}
------------------------------------
In this subsection we collect a few elementary results on sums of random variables with values in Banach spaces. The next result, Lemma \[lem:symm\] below, can be found, e.g., in Section 2.2 of Ledoux & Talagrand [@lt91].
\[lem:symm\] Consider the notation in Subsection \[notation\], let$ ( E, \norm{\cdot}_E ) $ be an $ {\mathbb{R}}$-Banach space, let $ ( \Omega, {\mathscr{F}}, {{\mathbb P}}) $ be a probability space, let $ \xi, \tilde{\xi} \in \mathscr{L}^0( {{\mathbb P}}; \left\| \cdot \right\|_E ) $ be independent mappings which satisfy $ {{\mathbb E}}\bigl[ \norm{ \tilde{\xi} }_E \bigr] < \infty $ and $ {{\mathbb E}}[ \tilde{\xi} ] = 0 $, and let $ \varphi \colon [0,\infty) \to [0,\infty) $ be a convex and non-decreasing function. Then $${{\mathbb E}}\bigl[ \varphi( \norm{ \xi }_E ) \bigr]
\leq {{\mathbb E}}\bigl[ \varphi( \norm{ \xi - \tilde{\xi} }_E ) \bigr].$$
Jensen’s inequality assures that $$\begin{split}
{{\mathbb E}}\bigl[ \varphi( \norm{ \xi }_E ) \bigr]
& = {{\mathbb E}}\bigl[ \varphi( \norm{ \xi - {{\mathbb E}}[ \tilde{\xi} ] }_E ) \bigr]
= \int_\Omega \varphi \biggl( \norm[\bigg]{
\int_\Omega \xi(\omega)
- \tilde{\xi}( \tilde{\omega} ) \,{{\mathbb P}}( {\mathrm{d}}\tilde{\omega} )
}_E \biggr) \, {{\mathbb P}}( {\mathrm{d}}\omega ) \\
& \leq \int_\Omega \varphi \biggl(
\int_\Omega \norm{ \xi(\omega)
- \tilde{\xi}( \tilde{\omega} ) }_E \, {{\mathbb P}}( {\mathrm{d}}\tilde{\omega} )
\biggr) \, {{\mathbb P}}( {\mathrm{d}}\omega ) \\
& \leq \int_\Omega \int_\Omega \varphi( \norm{ \xi(\omega)
- \tilde{\xi}( \tilde{\omega} ) }_E )
\,{{\mathbb P}}( {\mathrm{d}}\tilde{\omega} ) {{\mathbb P}}( {\mathrm{d}}\omega ) \\
& = \int_E \int_E \varphi( \norm{ x - y }_E ) \, \bigl( \tilde{\xi} ({{\mathbb P}}) \bigr)( {\mathrm{d}}y ) \bigl( \xi ({{\mathbb P}}) \bigr)( {\mathrm{d}}x ) \\
& = \int_{E \times E} \varphi( \norm{ x - y }_E ) \, \bigl( ( \xi, \tilde{\xi} ) ({{\mathbb P}}) \bigr)( {\mathrm{d}}x, {\mathrm{d}}y )
= {{\mathbb E}}\bigl[ \varphi( \norm{ \xi - \tilde{\xi} }_E ) \bigr]. \\
\end{split}$$ This completes the proof of Lemma \[lem:symm\].
\[cor:symm\] Let $ ( E, \norm{\cdot}_E ) $ be an $ {\mathbb{R}}$-Banach space, let $ ( \Omega, {\mathscr{F}}, {{\mathbb P}}) $ be a probability space, let $ \xi, \tilde{\xi} \in \mathscr{L}^1( {{\mathbb P}}; \left\| \cdot \right\|_E ) $ be independent and identically distributed mappings which satisfy $ {{\mathbb E}}[ \xi ] = 0 $, and let $ \varphi \colon [0,\infty) \to [0,\infty) $ be a convex and non-decreasing function. Then $${{\mathbb E}}\bigl[ \varphi( \norm{ \xi }_E ) \bigr]
\leq {{\mathbb E}}\bigl[ \varphi( \norm{ \xi - \tilde{\xi} }_E ) \bigr]
\leq {{\mathbb E}}\bigl[ \varphi( 2 \, \norm{ \xi }_E ) \bigr].$$
Lemma \[lem:symm\] shows that $$\begin{split}
{{\mathbb E}}\bigl[ \varphi( \norm{ \xi }_E ) \bigr]
& \leq
{{\mathbb E}}\bigl[ \varphi( \norm{ \xi - \tilde{\xi} }_E ) \bigr]
\leq
{{\mathbb E}}\bigl[ \varphi( \norm{ \xi }_E + \norm{ \tilde{\xi} }_E ) \bigr]
=
{{\mathbb E}}\bigl[ \varphi( \tfrac{1}{2} \, 2 \, \norm{ \xi }_E + \tfrac{1}{2} \, 2 \, \norm{ \tilde{\xi} }_E ) \bigr]
\\ & \leq
{{\mathbb E}}\bigl[ \tfrac{1}{2} \, \varphi( 2 \, \norm{ \xi }_E ) + \tfrac{1}{2} \, \varphi( 2 \, \norm{ \tilde{\xi} }_E ) \bigr]
=
\tfrac{1}{2} \, {{\mathbb E}}\bigl[ \varphi( 2 \, \norm{ \xi }_E ) \bigr]
+ \tfrac{1}{2} \, {{\mathbb E}}\bigl[ \varphi( 2 \, \norm{ \tilde{\xi} }_E ) \bigr]
\\ & =
\tfrac{1}{2} \, {{\mathbb E}}\bigl[ \varphi( 2 \, \norm{ \xi }_E ) \bigr]
+ \tfrac{1}{2} \, {{\mathbb E}}\bigl[ \varphi( 2 \, \norm{ \xi }_E ) \bigr]
=
{{\mathbb E}}\bigl[ \varphi( 2 \, \norm{ \xi }_E ) \bigr].
\end{split}$$ The proof of Corollary \[cor:symm\] is thus completed.
As a straightforward application we obtain the following randomisation result, cf., e.g., Lemma 6.3 in Ledoux & Talagrand [@lt91].
\[lem:rand\] Let $ ( E, \norm{\cdot}_E ) $ be an $ {\mathbb{R}}$-Banach space, let $ ( \Omega, {\mathscr{F}}, {{\mathbb P}}) $ be a probability space, let $ k \in {\mathbb{N}}$, let $ \xi_j \in \mathscr{L}^1( {{\mathbb P}}; \left\| \cdot \right\|_E ) $, $ j \in \{1, \ldots, k \} $, satisfy for all $ j \in \{1, \ldots, k \} $ that $ {{\mathbb E}}[ \xi_j ] = 0 $, and let $ r_j \colon \Omega \to \{ -1, 1 \} $, $ j \in \{1, \ldots, k \} $, be a $ {{\mathbb P}}$-Rademacher family such that $ \xi_1, \xi_2, \ldots, \xi_k, \ r_1, r_2, \ldots, r_k $ are independent. Then it holds for all $p \in [ 1, \infty ) $ that $$\norm[\bigg]{ \sum_{j=1}^k \xi_j }_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
\leq 2 \, \norm[\bigg]{ \sum_{j=1}^k r_j \xi_j }_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }.$$
Throughout this proof let $ \mathbf{\Omega} = \Omega \times \Omega $, let $ \bm{\xi}_j \colon \mathbf{\Omega} \to E $, $ j \in \{1, \ldots, k \} $, $ \bm{\tilde{\xi}}_j \colon \mathbf{\Omega} \to E $, $ j \in \{1, \ldots, k \} $, and $ \mathbf{r}_j \colon \mathbf{\Omega} \to \{ -1, 1 \} $, $ j \in \{1, \ldots, k \} $, be the mappings which satisfy for all $ \bm{\omega} = ( \omega, \tilde{\omega} ) \in \mathbf{\Omega} $, $ j \in \{ 1, \ldots, k \} $ that $ \bm{\xi}_j( \bm{\omega} ) = \xi_j( \omega ) $, $ \bm{\tilde{\xi}}_j( \bm{\omega} ) = \xi_j( \tilde{\omega} ) $, and $ \mathbf{r}_j( \bm{\omega} ) = r_j( \omega ) $, and let $ \mathbf{P} = {{\mathbb P}}\otimes {{\mathbb P}}$. The fact that $ \bm{\xi}_j - \bm{\tilde{\xi}}_j \colon \mathbf{\Omega} \to E $, $ j \in \{1, \ldots, k \} $, and $ \mathbf{r}_j \colon \mathbf{\Omega} \to \{ -1, 1 \} $, $ j \in \{1, \ldots, k \} $ are independent and the symmetry of $ \bm{\xi}_j - \bm{\tilde{\xi}}_j $, $ j \in \{1, \ldots, k \} $, prove for all $p \in [ 1, \infty ) $ that $$\begin{aligned}
& \int_{ \mathbf{\Omega} } \norm[\bigg]{ \sum_{j=1}^{k}
\mathbf{r}_j( \bm{\omega} ) \bigl( \bm{\xi}_j( \bm{\omega} ) - \bm{\tilde{\xi}}_j( \bm{\omega} ) \bigr)
}_E^p \, \mathbf{P}( {\mathrm{d}}\bm{\omega} ) \nonumber \\
& = \int_{ (\{ -1, 1 \} \times E )^k } \norm[\bigg]{ \sum_{j=1}^{k} z_j x_j }_E^p
\bigl( ( \mathbf{r}_1, \bm{\xi}_1 - \bm{\tilde{\xi}}_1, \ldots, \mathbf{r}_k, \bm{\xi}_k - \bm{\tilde{\xi}}_k ) ( \mathbf{P} ) \bigr)
( {\mathrm{d}}z_1, {\mathrm{d}}x_1, \ldots, {\mathrm{d}}z_k, {\mathrm{d}}x_k ) \nonumber \\
& = \int_{\{ -1, 1 \}}\int_E \cdots \int_{\{ -1, 1 \}}\int_E \ \raisebox{-3mm}{\ldots} \label{eq:rand,1} \\
& \quad \norm[\bigg]{ \sum_{j=1}^{k} x_j }_E^p
\bigl( ( \bm{\xi}_1 - \bm{\tilde{\xi}}_1 ) ( \mathbf{P} ) \bigr) ( {\mathrm{d}}x_1 ) \bigl( ( \mathbf{r}_1 ) ( \mathbf{P} ) \bigr) ( {\mathrm{d}}z_1 ) \, \ldots \, \bigl( ( \bm{\xi}_k - \bm{\tilde{\xi}}_k ) ( \mathbf{P} ) \bigr) ( {\mathrm{d}}x_k ) \bigl( ( \mathbf{r}_k ) ( \mathbf{P} ) \bigr) ( {\mathrm{d}}z_k ) \nonumber \\
& = \int_{ E^k } \norm[\bigg]{ \sum_{j=1}^{k} x_j }_E^p
\bigl( ( \bm{\xi}_1 - \bm{\tilde{\xi}}_1, \ldots, \bm{\xi}_k - \bm{\tilde{\xi}}_k ) ( \mathbf{P} ) \bigr) ( x_1, \ldots, x_k ) \nonumber \\
& = \int_{ \mathbf{\Omega} } \norm[\bigg]{ \sum_{j=1}^{k}
\bigl( \bm{\xi}_j( \bm{\omega} ) - \bm{\tilde{\xi}}_j( \bm{\omega} ) \bigr)
}_E^p \, \mathbf{P}( {\mathrm{d}}\bm{\omega} ). \nonumber\end{aligned}$$ Furthermore, the fact that $ \sum_{j=1}^{k} \bm{\xi}_j \colon \mathbf{\Omega} \to E $ and $ \sum_{j=1}^{k} \bm{\tilde{\xi}}_j \colon \mathbf{\Omega} \to E $ are independent and identically distributed, the facts that $ {{\mathbb E}}\bigl[ \norm[\big]{ \sum_{j=1}^{k} \bm{\xi}_j }_E \bigr] < \infty $ and $ {{\mathbb E}}\bigl[ \sum_{j=1}^{k} \bm{\xi}_j \bigr] = 0 $, Lemma \[lem:symm\], and imply that it holds for all $p \in [ 1, \infty ) $ that $$\begin{split}
& \norm[\bigg]{ \sum_{j=1}^k \xi_j }_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
= \norm[\bigg]{ \sum_{j=1}^k \bm{\xi}_j }_{ \mathscr{L}^p( \mathbf{P}; \left\| \cdot \right\|_E ) }
\leq \norm[\bigg]{ \sum_{j=1}^k ( \bm{\xi}_j - \bm{\tilde{\xi}}_j ) }_{ \mathscr{L}^p( \mathbf{P}; \left\| \cdot \right\|_E ) } \\
& = \norm[\bigg]{ \sum_{j=1}^k \mathbf{r}_j ( \bm{\xi}_j - \bm{\tilde{\xi}}_j ) }_{ \mathscr{L}^p( \mathbf{P}; \left\| \cdot \right\|_E ) }
\leq \norm[\bigg]{ \sum_{j=1}^k \mathbf{r}_j \bm{\xi}_j }_{ \mathscr{L}^p( \mathbf{P}; \left\| \cdot \right\|_E ) }
+ \norm[\bigg]{ \sum_{j=1}^k \mathbf{r}_j \bm{\tilde{\xi}}_j }_{ \mathscr{L}^p( \mathbf{P}; \left\| \cdot \right\|_E ) } \\
& = 2 \, \norm[\bigg]{ \sum_{j=1}^k r_j \xi_j }_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }.
\end{split}$$ The proof of Lemma \[lem:rand\] is thus completed.
The next result, Proposition \[prop:type\] below, is the key to estimate the statistical error term in the Banach space valued Monte Carlo method in the next subsection. Proposition \[prop:type\] is similar to, e.g., Proposition 9.11 in Ledoux & Talagrand [@lt91].
\[prop:type\] Let $ k \in \mathbb{N} $, $ q \in [1,2] $, let $ ( E, \norm{\cdot}_E ) $ be an $ {\mathbb{R}}$-Banach space with type $ q $, let $ ( \Omega, {\mathscr{F}}, {{\mathbb P}}) $ be a probability space, and let $ \xi_j \in \mathscr{L}^1( {{\mathbb P}}; \left\| \cdot \right\|_E ) $, $ j \in \{1, \ldots, k \} $, be independent mappings which satisfy for all $ j \in \{1, \ldots, k \} $ that $ {{\mathbb E}}[ \xi_j ] = 0 $. Then it holds for all $ p \in [q,\infty) $ that $$\norm[\bigg]{ \sum_{j=1}^k \xi_j }_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
\leq \varTheta_{ p, q }( E ) \biggl( \sum_{j=1}^k \norm{ \xi_j }_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }^q \biggr)^{\nicefrac{1}{q}}.$$
Throughout this proof let $ ( \tilde{\Omega}, \tilde{{\mathscr{F}}}, \tilde{{{\mathbb P}}} ) $ be a probability space, let $ r_j \colon \tilde{\Omega} \to \{ -1, 1 \} $, $ j \in \{1, \ldots, k \} $, be a $ \tilde{{{\mathbb P}}} $-Rademacher family, and let $ \bm{\xi}_j \colon \Omega \times \tilde{\Omega} \to E $, $ j \in \{1, \ldots, k \} $, and $ \mathbf{r}_j \colon \Omega \times \tilde{\Omega} \to \{ -1, 1 \} $, $ j \in \{1, \ldots, k \} $, be the mappings which satisfy for all $ \bm{\omega} = ( \omega, \tilde{\omega} ) \in \Omega \times \tilde{\Omega} $, $ j \in \{ 1, \ldots, k \} $ that $ \bm{\xi}_j( \bm{\omega} ) = \xi_j( \omega ) $ and $ \mathbf{r}_j( \bm{\omega} ) = r_j( \tilde{\omega} ) $. Lemma \[lem:rand\] and the triangle inequality show for all $ p \in [ q, \infty ) $ that $$\begin{split}
& \norm[\bigg]{ \sum_{j=1}^k \xi_j }_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
= \norm[\bigg]{ \sum_{j=1}^k \bm{\xi}_j }_{ \mathscr{L}^p( {{\mathbb P}}\otimes \tilde{{{\mathbb P}}}; \left\| \cdot \right\|_E ) }
\leq 2 \, \norm[\bigg]{ \sum_{j=1}^k \mathbf{r}_j \bm{\xi}_j }_{ \mathscr{L}^p( {{\mathbb P}}\otimes \tilde{{{\mathbb P}}}; \left\| \cdot \right\|_E ) } \\
& = 2 \, \norm[\bigg]{ \sum_{j=1}^k r_j \xi_j }_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ \mathscr{L}^p( \tilde{{{\mathbb P}}}; \left\| \cdot \right\|_E ) } ) }
\leq 2 \mathscr{K}_{ p, q }
\norm[\bigg]{ \sum_{j=1}^k r_j \xi_j }_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ \mathscr{L}^q( \tilde{{{\mathbb P}}}; \left\| \cdot \right\|_E ) } ) } \\
& \leq 2 \mathscr{K}_{ p, q } \mathscr{T}_q( E )
\norm[\bigg]{ \biggl( \sum_{j=1}^k \norm{ \xi_j }_E^q \biggr)^{\nicefrac{1}{q}} }_{ \mathscr{L}^p( {{\mathbb P}}; \left| \cdot \right| ) }
= 2 \mathscr{K}_{ p, q } \mathscr{T}_q( E )
\norm[\bigg]{ \sum_{j=1}^k \norm{ \xi_j }_E^q }_{ \mathscr{L}^{\nicefrac{p}{q}}( {{\mathbb P}}; \left| \cdot \right| ) }^{ \nicefrac{1}{q} } \\
& \leq 2 \mathscr{K}_{ p, q } \mathscr{T}_q( E )
\biggl( \sum_{j=1}^k \norm{ \xi_j }_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }^q \biggr)^{\nicefrac{1}{q}}
\end{split}$$ (cf., e.g., Proposition 7.4 in Hytönen et al. [@HytonenNeervenVeraarWeis2016]). This finishes the proof of Proposition \[prop:type\].
The result in Corollary \[cor:sums\] below is a direct consequence of Proposition \[prop:type\].
\[cor:sums\] Let $ M \in \mathbb{N} $, $ q \in [1,2] $, let $ ( E, \norm{\cdot}_E ) $ be an $ {\mathbb{R}}$-Banach space with type $ q $, let $ ( \Omega, {\mathscr{F}}, {{\mathbb P}}) $ be a probability space, and let $ \xi_j \in \mathscr{L}^1( {{\mathbb P}}; \left\| \cdot \right\|_E ) $, $ j \in \{1, \ldots, M \} $, be independent. Then it holds for all $ p \in [q,\infty) $ that $$\sigma_{p,E} \biggl( \sum_{j=1}^M \xi_j \biggr)
\leq \varTheta_{ p, q }( E ) \biggl( \sum_{j=1}^M \abs{ \sigma_{p,E}( \xi_j ) }^q \biggr)^{\nicefrac{1}{q}}.$$
\[cor:BanachMC\] \[cor:monteCarloBanach\] Let $ M \in \mathbb{N} $, $ q \in [1,2] $, let $ ( E, \norm{\cdot}_E ) $ be an $ {\mathbb{R}}$-Banach space with type $ q $, let $ ( \Omega, {\mathscr{F}}, {{\mathbb P}}) $ be a probability space, and let $ \xi_j \in \mathscr{L}^1( {{\mathbb P}}; \left\| \cdot \right\|_E ) $, $ j \in \{1, \ldots, M \} $, be independent and identically distributed. Then it holds for all $ p \in [q,\infty) $ that $$\norm[\bigg]{ {{\mathbb E}}[ \xi_1 ] - \frac{1}{M} \sum_{j=1}^{M} \xi_j }_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_E ) }
= \frac{ \sigma_{p,E} \bigl( \sum_{j=1}^{M} \xi_j \bigr) }{ M }
\leq \frac{ \varTheta_{ p, q }( E ) \, \sigma_{p,E} ( \xi_1 ) }{ M^{ 1 - \nicefrac{1}{q} } }.$$
Results on lower and upper error bounds related to Corollary \[cor:BanachMC\] can be found, e.g., in Theorem 1 in Daun & Heinrich [@DaunHeinrich2013] and in Corollary 2 in Heinrich & Hinrichs [@HeinrichHinrichs2014]. Note that Corollary \[cor:BanachMC\] does not imply convergence if the underlying Banach space $ ( E, \norm{\cdot}_E ) $ has only type $1$, in the sense that it holds for all $ q \in ( 1, \infty ) $ that $ \mathscr{T}_q( E ) = \infty $.
Multilevel Monte Carlo methods in Banach spaces {#sec:mlmc}
-----------------------------------------------
In many situations the work required to obtain a certain accuracy of an approximation using the Monte Carlo method can be improved by using a multilevel Monte Carlo method. Heinrich [@h98; @h01] was first to observe this and established multilevel Monte Carlo methods concerning convergence in a Banach (function) space. However, these methods do not apply to SDEs. Then Giles [@g08a] derived the complexity reduction of multilevel Monte Carlo methods for SDEs. The minor contribution of Proposition \[prop:multilevelMC\] below to the literature on multilevel Monte Carlo methods is to combine the approaches of Heinrich [@h98] and of Giles [@g08a] into a single result on multilevel Monte Carlo methods in Banach spaces. The useful observation of Proposition \[prop:multilevelMC\] generalizes the discussion in Section 4 of Heinrich [@h01].
\[Abstract multilevel Monte Carlo methods in Banach spaces\] \[prop:multilevelMC\] Let $ q \in [1,2] $, let $
(
\Omega, \mathscr{F}, {{\mathbb P}})
$ be a probability space, let $
(
V_1, \left\| \cdot \right\|_{V_1}
)
$ be an $ {\mathbb{R}}$-Banach space with type $ q $, let $
(
V_2, \left\| \cdot \right\|_{V_2}
)
$ be an $ {\mathbb{R}}$-Banach space with $ V_1 \subseteq V_2 $ continuously, let $ v \in V_2 $, $ L \in {\mathbb{N}}$, $ M_1, \dots, M_L \in {\mathbb{N}}$, and for every $ \ell \in \{ 1, \dots, L \} $ let $
D_{ \ell, k } \in \mathscr{L}^1( {{\mathbb P}}; \left\| \cdot \right\|_{ V_1 } )
$, $ k \in \{ 1, \ldots, M_\ell \} $, be independent and identically distributed. Then it holds for all $ p \in [q,\infty) $ that $$\begin{split}
& \left\|
v -
\sum_{ \ell = 1 }^{ L }
\frac{1}{ M_\ell }
\sum_{ k = 1 }^{ M_\ell }
D_{ \ell, k }
\right\|_{
\mathscr{L}^p({{\mathbb P}}; \left\| \cdot \right\|_{V_2} )
} \\
& \leq
\left\|
v
-
\sum_{ \ell = 1 }^L
{{\mathbb E}}\big[
D_{ \ell, 1 }
\big]
\right\|_{ V_2 }
+
\left\|
\operatorname{Id}_{V_1}
\right\|_{
\mathscr{L}( V_1 , V_2 )
}
\varTheta_{ p, q }( V_1 )
\sum_{ \ell = 1 }^L
\frac{
\sigma_{ p, V_1 }( D_{ \ell, 1 } )
}{
(
M_\ell
)^{ 1 - \nicefrac{ 1 }{ q } }
}.
\end{split}$$
The triangle inequality and Corollary \[cor:BanachMC\] imply for all $ p \in [q,\infty) $ that $$\begin{split}
&
\left\|
v -
\sum_{ \ell = 1 }^{ L }
\frac{1}{ M_\ell }
\sum_{ k = 1 }^{ M_\ell }
D_{ \ell, k }
\right\|_{
\mathscr{L}^p({{\mathbb P}}; \left\| \cdot \right\|_{V_2} )
}
\\ & \leq
\left\|
v
-
\sum_{ \ell = 1 }^L
{{\mathbb E}}\!\left[
D_{ \ell, 1 }
\right]
\right\|_{ V_2 }
+
\left\|
\sum_{ \ell = 1 }^{ L }
{{\mathbb E}}\!\left[
D_{ \ell, 1 }
\right]
-
\sum_{ \ell = 1 }^{ L }
\frac{1}{M_\ell}
\sum_{ k = 1 }^{ M_\ell }
D_{ \ell, k }
\right\|_{
\mathscr{L}^p({{\mathbb P}}; \left\| \cdot \right\|_{V_2} )
}
\\ & \leq
\left\|
v
-
\sum_{ \ell = 1 }^L
{{\mathbb E}}\!\left[
D_{ \ell, 1 }
\right]
\right\|_{ V_2 }
+
\left\|
\operatorname{Id}_{ V_1 }
\right\|_{
\mathscr{L}( V_1, V_2 )
}
\sum_{ \ell = 1 }^{ L }
\left\|
{{\mathbb E}}\!\left[
D_{ \ell, 1 }
\right]
-
\frac{1}{M_\ell}
\sum_{ k = 1 }^{ M_\ell }
D_{ \ell, k }
\right\|_{
\mathscr{L}^p({{\mathbb P}}; \left\| \cdot \right\|_{V_1} )
}
\\ & \leq
\left\|
v
-
\sum_{ \ell = 1 }^L
{{\mathbb E}}\!\left[
D_{ \ell, 1 }
\right]
\right\|_{V_2}
+
\left\|
\operatorname{Id}_{ V_1 }
\right\|_{
\mathscr{L}( V_1, V_2 )
}
\varTheta_{p,q}( V_1 )
\sum_{ \ell = 1 }^L
\frac{
\sigma_{p,V_1}( D_{ \ell, 1 } )
}{
(M_\ell)^{1-\nicefrac{1}{q}}
}.
\end{split}$$ This completes the proof of Proposition \[prop:multilevelMC\].
\[lem:montecarlo\] Consider the notation in Subsection \[notation\], let $ q \in [ 1, 2 ] $, let $
\left(
\Omega, \mathscr{F}, {{\mathbb P}}\right)
$ be a probability space, let $
\left( V_i, \left\| \cdot \right\|_{V_i}
\right)
$, $ i \in \{ 1, 2, 3 \} $, be $ {\mathbb{R}}$-Banach spaces such that $
\left(
V_1, \left\| \cdot \right\|_{ V_1 }
\right)
$ has type $ q $ and such that $
V_1 \subseteq V_2
$ continuously, let $
f \colon V_3 \rightarrow V_2
$ be a $ \mathscr{B}( V_3 ) / \mathscr{B}( V_2 ) $-measurable mapping, let $
g
\colon V_3 \rightarrow V_1
$ be a $ \mathscr{B}( V_3 ) / \mathscr{B}( V_1) $-measurable mapping, let $ X \in \mathscr{L}^0( {{\mathbb P}}; \left\| \cdot \right\|_{ V_3 } ) $ satisfy $
{{\mathbb E}}\bigl[
\norm{ f( X ) }_{ V_2 }
\bigr]
< \infty
$, for every $ N \in {\mathbb{N}}$ let $
Y^{ N, \ell, k } \in \mathscr{L}^0( {{\mathbb P}}; \left\| \cdot \right\|_{ V_3 } )
$, $ \ell \in {\mathbb{N}}_0 $, $ k \in {\mathbb{N}}$, be independent and identically distributed mappings which satisfy $
{{\mathbb E}}\bigl[
\norm{ g( Y^{N,0,1} ) }_{ V_1 }
\bigr]
< \infty
$, and let $ L \in {\mathbb{N}}_0 $, $ M_0, M_1, \dots, M_{L+1}, N_0, N_1, \dots, N_L \in {\mathbb{N}}$. Then it holds for all $ p \in [q, \infty ) $ that $$\begin{aligned}
\label{eq:multimontecarlo}
&
\left\|
{{\mathbb E}}\!\left[ f(X) \right]
-
\tfrac{
1
}{
M_0
}
\sum_{ k = 1 }^{ M_0 }
g( Y^{ N_0, 0, k } )
-
\sum_{ \ell = 1 }^{ L }
\tfrac{
1
}{ M_\ell }
\sum_{ k = 1 }^{ M_\ell }
\left[
g( Y^{ N_\ell, \ell, k } ) -
g( Y^{ N_{ (\ell - 1) }, \ell, k } )
\right]
\right\|_{
\mathscr{L}^p({{\mathbb P}}; \left\| \cdot \right\|_{V_2} )
}
\nonumber \\
& \leq
\left\|
{{\mathbb E}}\!\left[ f(X) \right]
-
{{\mathbb E}}\!\left[
g( Y^{N_L,0,1} )
\right]
\right\|_{ V_2 }
\\&\quad\nonumber
+
\left\|
\operatorname{Id}_{ V_1 }
\right\|_{
\mathscr{L}( V_1, V_2 )
}
\varTheta_{ p, q }( V_1 )
\biggl(
\tfrac{
\sigma_{p,V_1}\!\left(
g( Y^{ N_0, 0, 1 } )
\right)
}{ (M_0)^{1-\nicefrac{1}{q}} }
+
\sum_{ \ell = 1 }^{ L }
\tfrac{
\sigma_{p,V_1}\!\left(
g( Y^{ N_\ell, 0, 1 } ) -
g( Y^{ N_{ (\ell - 1) }, 0, 1 } )
\right)
}{
(M_\ell)^{1-\nicefrac{1}{q}}
}
\biggr)
\\ & \leq
\nonumber
\left\|
{{\mathbb E}}\!\left[ f(X) \right]
-
{{\mathbb E}}\!\left[ g( Y^{N_L,0,1} )
\right]
\right\|_{ V_2 }
\\&
\quad\nonumber
+
\left\|
\operatorname{Id}_{ V_1 }
\right\|_{
\mathscr{L}( V_1, V_2 )
}
\varTheta_{ p, q }( V_1 )
\biggl(
\tfrac{
2\left\|
g( X )
\right\|_{
\mathscr{L}^p({{\mathbb P}}; \left\| \cdot \right\|_{V_1} )
}
}{ (M_0)^{1-\nicefrac{1}{q}} }
+
\sum_{ \ell = 0 }^{ L }
\tfrac{
4\left\|
g( Y^{ N_\ell, 0, 1 } ) -
g( X )
\right\|_{
\mathscr{L}^p({{\mathbb P}}; \left\| \cdot \right\|_{V_1} )
}
}{
(\min\{ M_\ell, M_{\ell+1} \})^{1-\nicefrac{1}{q}}
}
\biggr).\end{aligned}$$
Proposition \[prop:multilevelMC\] and the identity $${{\mathbb E}}\!\left[ g( Y^{N_L,0,1} )
\right]
=
{{\mathbb E}}\!\left[
g( Y^{N_0,0,1} )
\right]
+
\sum_{ \ell = 1 }^{ L }
{{\mathbb E}}\!\left[
g( Y^{N_\ell,0,1} ) -
g( Y^{N_{(\ell-1)},0,1} )
\right]$$ imply the first inequality in . Next note that the triangle inequality implies for all $ \xi \in \mathscr{L}^1( {{\mathbb P}}; \left\| \cdot \right\|_{ V_1 } ) $, $ p \in [q, \infty ) $ that $ \sigma_{p,V_1}( \xi ) \leq 2 \| \xi \|_{\mathscr{L}^p({{\mathbb P}};\left\| \cdot \right\|_{V_1})} $. This and again the triangle inequality show for all $ p \in [q, \infty ) $ that $$\begin{split}
&
\tfrac{
\sigma_{p,V_1}\!\left(
g( Y^{ N_0, 0, 1 } )
\right)
}{ (M_0)^{1-\nicefrac{1}{q}} }
+
\sum_{ \ell = 1 }^{ L }
\tfrac{
\sigma_{p,V_1}\!\left(
g( Y^{ N_\ell, 0, 1 } ) -
g( Y^{ N_{ (\ell - 1) }, 0, 1 } )
\right)
}{
(M_\ell)^{1-\nicefrac{1}{q}}
}
\\
& \leq
\tfrac{
2 \, \left\|
g( Y^{ N_0, 0, 1 } )
\right\|_{\mathscr{L}^p({{\mathbb P}};\left\| \cdot \right\|_{V_1})}
}{ (M_0)^{1-\nicefrac{1}{q}} }
+
\sum_{ \ell = 1 }^{ L }
\tfrac{
2 \, \left\|
g( Y^{ N_\ell, 0, 1 } ) -
g( Y^{ N_{ (\ell - 1) }, 0, 1 } )
\right\|_{\mathscr{L}^p({{\mathbb P}};\left\| \cdot \right\|_{V_1})}
}{
(M_\ell)^{1-\nicefrac{1}{q}}
}
\\
&\leq
\tfrac{
2 \, \left\|
g( X )
\right\|_{\mathscr{L}^p({{\mathbb P}};\left\| \cdot \right\|_{V_1})}
+
2 \, \left\|
g( Y^{ N_0, 0, 1 } ) - g(X)
\right\|_{\mathscr{L}^p({{\mathbb P}};\left\| \cdot \right\|_{V_1})}
}{ (M_0)^{1-\nicefrac{1}{q}} }
\\ & \quad
+
\sum_{ \ell = 1 }^{ L }
\tfrac{
2 \, \left\|
g( Y^{ N_\ell, 0, 1 } ) -
g( X )
\right\|_{\mathscr{L}^p({{\mathbb P}};\left\| \cdot \right\|_{V_1})}
+
2 \, \left\|
g( Y^{ N_{ (\ell - 1) }, 0, 1 } )
-
g( X )
\right\|_{\mathscr{L}^p({{\mathbb P}};\left\| \cdot \right\|_{V_1})}
}{
(M_\ell)^{1-\nicefrac{1}{q}}
}
\\ &
\leq
\tfrac{
2\left\|
g( X )
\right\|_{\mathscr{L}^p({{\mathbb P}};\left\| \cdot \right\|_{V_1})}
}{ (M_0)^{1-\nicefrac{1}{q}} }
+
\sum_{ \ell = 0 }^{ L }
\tfrac{
4 \left\|
g( Y^{ N_\ell, 0, 1 } ) -
g( X )
\right\|_{\mathscr{L}^p({{\mathbb P}};\left\| \cdot \right\|_{V_1})}
}{
(\min\{M_{\ell},M_{\ell+1}\})^{1-\nicefrac{1}{q}}
}.
\end{split}$$ This implies the second inequality in . The proof of Corollary \[lem:montecarlo\] is thus completed.
\[c:mlmc.conv\] Consider the notation in Subsection \[notation\], let $ T \in (0,\infty) $, $ \beta \in (0,1] $, $
\alpha \in (0,\beta)
$, $
c,r \in [0,\infty)
$, let $ ( \Omega, \mathscr{F}, {{\mathbb P}}) $ be a probability space, let $ ( E, \norm{\cdot}_{E} ) $ be a separable $ {\mathbb{R}}$-Banach space with type $ 2 $, let $
X \colon [0,T] \times \Omega
\rightarrow E
$ be a stochastic process with continuous sample paths which satisfies for all $p\in[1,\infty)$, $\gamma\in[0,\beta)$ that $
X \in {\mathscr{C}}^{\gamma}\bigl([0,T], \left\| \cdot \right\|_{ \mathscr{L}^p({{\mathbb P}}; \left\| \cdot \right\|_E ) } \bigr)
$, for every $ N \in {\mathbb{N}}$ let $ Y^{ N, \ell, k } \colon
[0,T] \times \Omega
\rightarrow E $, $ \ell \in {\mathbb{N}}_0 $, $ k \in {\mathbb{N}}$, be independent and identically distributed stochastic processes which satisfy for all $ k \in {\mathbb{N}}$, $ \ell \in {\mathbb{N}}_0 $, $
n \in \{ 0, 1, \dots, N-1 \}
$, $
t \in
\big[
\frac{ n T }{ N },
\frac{ (n+1) T }{ N }
\big]
$, $ p \in [1,\infty) $, $ \rho \in [0,\beta) $ that $$\begin{aligned}
& Y_{t}^{ N, \ell, k }
=
\bigl(
n + 1 - \tfrac{ t N }{ T }
\bigr) \cdot
Y_{ \frac{ n T }{ N } }^{ N, \ell, k }
+
\bigl(
\tfrac{ t N }{ T } - n
\bigr) \cdot
Y_{ \frac{ (n+1) T }{ N } }^{ N, \ell, k }, \\
& \sup_{ M \in {\mathbb{N}}}
\sup_{ m\in\{0,1,\ldots,M\} }
\Bigl(
M^{ \rho } \,
\norm[\big]{
X_{ \frac{ m T }{ M } } -
Y_{ \frac{ m T }{ M } }^{ M, 0, 1 }
}_{
\mathscr{L}^p({{\mathbb P}}; \left\| \cdot \right\|_E )
}
\Bigr)
< \infty, \label{eq:strong.conv}\end{aligned}$$ and let $
f \colon
C( [0,T], E)
\rightarrow
C( [0,T], E)
$ be a $ \mathscr{B}\bigl( C( [0,T], E) \bigr) / \mathscr{B}\bigl( C( [0,T], E) \bigr) $-measurable function which satisfies for all $ v, w \in {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) $ that $ f(v), f(w) \in {\mathscr{C}}^{\alpha}( [ 0, T ], \left\| \cdot \right\|_E ) $ and $$\label{eq:f.local.lipschitz}
\norm{ f(v) - f(w) }_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}
\leq
c \,
\Bigl(
1
+
\norm{v}_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}^r
+
\norm{w}_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}^r
\Bigr)
\norm{ v - w }_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}.$$ Then it holds that $${{\mathbb E}}\bigl[ \norm{ f(X) }_{{\mathscr{C}}^{\alpha}([0,T],\left\| \cdot \right\|_E)}\bigr]<\infty,$$ it holds for all $ p \in [1,\infty) $, $ \rho \in [0,\beta-\alpha) $ that $$\sup_{ N \in {\mathbb{N}}}
\bigl(
N^{ \rho } \,
\norm{
f(X) - f(Y^{N,0,1})
}_{ \mathscr{L}^p{
( {{\mathbb P}};
\left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) } )
} }
\bigr) < \infty,$$ and it holds for all $p\in[1,\infty)$, $\gamma\in[0,\alpha) $, $\rho \in [0, \beta-\alpha)$ that $$\label{eq:mlmc.conv}
\begin{multlined}[c][0.9\textwidth]
\sup_{L\in{\mathbb{N}}}
\Biggl[
2^{L \cdot \min\{ \rho, \frac{1}{2} \} }
L^{ -\mathbbm{1}_{ \{ \nicefrac{1}{2} \} }( \rho ) }
\biggl\|
{{\mathbb E}}[ f( X ) ]
-
{\textstyle \sum\limits_{ k = 1 }^{ 2^L } }
\tfrac{
f( Y^{ 1 , 0, k } )
}{ 2^L }
\\
-
{\textstyle \sum\limits_{ \ell = 1 }^{L} }
{\textstyle \sum\limits_{ k = 1 }^{ 2^{L-\ell} } }
\tfrac{
f( Y^{ 2^\ell , \ell, k } )
-
f( Y^{ 2^{(\ell-1)} , \ell, k } )
}{
2^{L-\ell}
}
\biggr\|_{ \mathscr{L}^p({{\mathbb P}}; \left\| \cdot \right\|_{ {\mathscr{C}}^{ \gamma }( [0,T], \left\| \cdot \right\|_E ) } ) }
\Biggr]
<\infty.
\end{multlined}$$
Throughout this proof let $ \gamma \in [ 0, \alpha ) $, let $ W^{ \alpha, \nicefrac{2}{(\alpha - \gamma)} }( [ 0, T ], E ) $ be the Sobolev space with regularity parameter $ \alpha \in ( 0, 1 ) $ and integrability parameter $ \nicefrac{2}{(\alpha - \gamma)} \in ( 2, \infty ) $ of functions from $ [ 0, T ] $ to $ E $, let $ ( V_i, \norm{\cdot}_{V_i} ) $, $ i \in \{ 1, 2, 3 \} $, be the $ {\mathbb{R}}$-Banach spaces which satisfy $$\begin{aligned}
( V_1, \norm{\cdot}_{V_1} )
& = \bigl( W^{ \alpha, \nicefrac{2}{(\alpha - \gamma)} }( [ 0, T ], E ),
\norm{\cdot}_{ W^{ \alpha, \nicefrac{2}{(\alpha - \gamma)} }( [ 0, T ], E ) } \bigr), \\
( V_2, \norm{\cdot}_{V_2} ) & = \Bigl( {\mathscr{C}}^\gamma( [ 0, T ], \left\| \cdot \right\|_E ),
\norm{\cdot}_{ {\mathscr{C}}^\gamma( [ 0, T ], \left\| \cdot \right\|_E ) }
\text{\raisebox{-0.1\baselineskip}{$\bigr|_{ {\mathscr{C}}^\gamma( [ 0, T ], \left\| \cdot \right\|_E ) }$}} \Bigr), \\
( V_3, \norm{\cdot}_{V_3} ) & = \Bigl( C( [ 0, T ], E ), \norm{\cdot}_{ C( [ 0, T ], \left\| \cdot \right\|_E ) }
\text{\raisebox{-0.1\baselineskip}{$\bigr|_{ C( [ 0, T ], E ) }$}} \Bigr),\end{aligned}$$ and let $ \tilde{f} \colon V_3 \to V_2 $ and $ g \colon V_3 \to V_1 $ be the functions which satisfy for all $ v \in V_3 $ that $ \tilde{f}(v) = g(v) = \mathbbm{1}_{ {\mathscr{C}}^\alpha( [ 0, T ], \left\| \cdot \right\|_E ) }(v) f(v) $. The Kolmogorov-Chentsov continuity theorem (see Theorem \[thm:Kolmogorov\]) together with the assumptions that $
X \in \cap_{p\in[1,\infty)}\cap_{\eta\in[0,\beta)}
{\mathscr{C}}^{\eta}\bigl([0,T], \left\| \cdot \right\|_{ \mathscr{L}^p({{\mathbb P}}; \left\| \cdot \right\|_E ) } \bigr)
$ and that $ X $ has continuous sample paths implies that $X \in
\cap_{p\in[1,\infty)} \
\mathscr{L}^p({{\mathbb P}}; \left\| \cdot \right\|_{ {\mathscr{C}}^{\alpha}([0,T], \left\| \cdot \right\|_E) } ) $. Next observe that assumption , Hölder’s inequality, and Corollary \[cor:hoelder3\] show for all $ p \in [1,\infty) $, $ \rho \in [0,\beta-\alpha) $ that $$\begin{aligned}
& \sup_{ N \in {\mathbb{N}}}
\Bigl(
N^{ \rho } \,
{{\mathbb E}}\Bigl[
\norm[\big]{
\tilde{f}(X) - g(Y^{N,0,1})
}_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}
\Bigr]
\Bigr) \nonumber \\
& \leq \sup_{ N \in {\mathbb{N}}}
\biggl(
N^{ \rho } \,
\Bigl( {{\mathbb E}}\Bigl[
\norm{
f(X) - f(Y^{N,0,1})
}_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}^p
\Bigr]
\Bigr)^{\nicefrac{1}{p}} \biggr) \nonumber
\\&
\nonumber
\leq
\sup_{ N \in {\mathbb{N}}}
\biggl(
N^{ \rho } \,
\Bigl| {{\mathbb E}}\Bigl[ \Bigl(
c \,
\bigl(
1
+
\norm{
X
}_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}^r
+
\norm{
Y^{N,0,1}
}_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}^r
\bigr)
\\ & \label{eq:Calpha.fX-fY} \qquad\qquad\qquad\quad \ \, \, \cdot
\norm{
X - Y^{N,0,1}
}_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E )
}
\Bigr)^p \, \Bigr] \Bigr|^{ \nicefrac{1}{p} }
\biggr)
\\& \nonumber
\leq
c \,
\biggl(
1+
\norm{
X
}_{\mathscr{L}^{2pr} ({{\mathbb P}}; \left\| \cdot \right\|_{ {\mathscr{C}}^{\alpha}([0,T],\left\| \cdot \right\|_E) } ) }^{r}
+
\sup_{ N \in {\mathbb{N}}} \,
\norm{
Y^{N,0,1}
}_{\mathscr{L}^{2pr} ({{\mathbb P}}; \left\| \cdot \right\|_{ {\mathscr{C}}^{\alpha}([0,T],\left\| \cdot \right\|_E) } ) }^{r}
\biggr)
\\&
\qquad
\cdot
\sup_{ N \in {\mathbb{N}}}
\Bigl(
N^{ \rho } \,
\norm{
X - Y^{N,0,1}
}_{
\mathscr{L}^{2p}( {{\mathbb P}};
\left\| \cdot \right\|_{
{\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) }
)
}
\Bigr)
< \infty.
\nonumber\end{aligned}$$ Again assumption implies for all $ p \in [ 1, \infty ) $ that $$\begin{aligned}
& {{\mathbb E}}\Bigl[ \norm{ f(X) }_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) } \Bigr]
\leq \norm{ f( X ) }_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) } ) } \label{eq:Calpha.fX} \\
& \leq \norm{ f(0) }_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) }
+ c \, \Bigl( \norm{ X }_{ \mathscr{L}^p( {{\mathbb P}}; \left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) } ) }
+ \norm{ X }_{ \mathscr{L}^{(r+1)p}( {{\mathbb P}}; \left\| \cdot \right\|_{ {\mathscr{C}}^{ \alpha }( [0,T], \left\| \cdot \right\|_E ) } ) }^{r+1} \Bigr)
< \infty. \nonumber\end{aligned}$$ Next observe that $ ( V_1, \norm{\cdot}_{V_1} ) $ has type 2 and note that the Sobolev embedding theorem proves that $ V_1 \subseteq V_2 $ continuously. Combining with and the fact that $ {\mathscr{C}}^{\alpha}([ 0, T ], \left\| \cdot \right\|_E ) \subseteq V_1 $ continuously hence implies for all $ N \in {\mathbb{N}}$, $ p \in [1,\infty) $, $ \rho \in [0,\beta-\alpha) $ that $ {{\mathbb E}}\bigl[ \norm{ \tilde{f}(X) }_{V_2} \bigr]
+ {{\mathbb E}}\bigl[ \norm{ g(Y^{N,0,1}) }_{V_1} \bigr] < \infty $, $ \norm{ g( X ) }_{ \mathscr{L}^p({{\mathbb P}}; \left\| \cdot \right\|_{V_1} ) } < \infty $, and $$\label{eq:convergence}
\sup_{ N \in {\mathbb{N}}} \Bigl( N^{ \rho } \,
{{\mathbb E}}\Bigl[ \norm[\big]{ \tilde{f}( X ) - g( Y^{N,0,1} ) }_{V_2} \Bigr] \Bigr)
+ \sup_{ N \in {\mathbb{N}}} \Bigl( N^{ \rho } \,
\norm{ g( X ) - g( Y^{N,0,1} ) }_{ \mathscr{L}^p({{\mathbb P}}; \left\| \cdot \right\|_{V_1} ) } \Bigr)
< \infty.$$ In addition, observe that it holds for all $ L \in {\mathbb{N}}$, $ \rho \in [ 0, \beta-\alpha ) \setminus \{ \tfrac{1}{2} \} $ that $$\label{eq:sum}
\begin{split}
\sum_{\ell=1}^{L} (2^{\ell})^{-\rho} \, 2^{-\frac12(L-\ell)}
& = 2^{-\frac{L}{2}}
\sum_{\ell=1}^{L} 2^{(\frac{1}{2}-\rho)\ell}
= 2^{-\frac{L}{2}}
\tfrac{ 1-2^{(\frac12-\rho)L} }{ 2^{\rho - \frac{1}{2}} - 1 }
= 2^{- L \cdot \min \{ \rho, \frac{1}{2} \} }
\tfrac{ 1-2^{-|\frac12-\rho| L} }{ {\displaystyle |} 1-2^{\rho-\frac{1}{2}} {\displaystyle |} }
\leq \tfrac{ 2^{- L \cdot \min \{ \rho, \frac{1}{2} \} } }{ {\displaystyle |} 1-2^{\rho-\frac{1}{2}} {\displaystyle |} }
\end{split}$$ and $$\label{eq:sum2}
\begin{split}
\sum_{\ell=1}^{L} (2^{\ell})^{-\frac12} \, 2^{-\frac12(L-\ell)}
& =
2^{-\frac{L}{2}} L
.
\end{split}$$ Combining Corollary \[lem:montecarlo\] with , , and implies . This finishes the proof of Corollary \[c:mlmc.conv\].
Corollary \[c:mlmc.conv\] can be applied to many SDEs. Under general conditions on the coefficient functions of the SDEs (see, e.g., Theorem 1.3 and Subsection 3.1 in [@HutzenthalerJentzen2014]), suitable stopped-tamed Euler approximations (cf. (6) in [@HutzenthalerJentzenWang2013] or (10) in [@HutzenthalerJentzenKloeden2012]) converge in the strong sense with convergence rate $\nicefrac{1}{2}$. We note that the classical Euler-Maruyama approximations do not satisfy condition for most SDEs with superlinearly growing coefficients; see Theorem 2.1 in [@hjk11] and Theorem 2.1 in [@HutzenthalerJentzenKloeden2013]. Moreover, under general conditions on the coefficients it holds that the solution process is strongly $\nicefrac{1}{2}$-Hölder continuous in time. In conclusion, provided that a suitable numerical scheme is employed, Corollary \[c:mlmc.conv\] can be applied to many SDEs with $\beta=\nicefrac{1}{2}$.
Acknowledgements {#acknowledgements .unnumbered}
----------------
This project has been partially supported by the research project “Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients” funded by the German Research Foundation and by the ETH Research Grant “Mild stochastic calculus and numerical approximations for nonlinear stochastic evolution equations with Lévy noise”.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Information diffusion in networks can be used to model many real-world phenomena, including rumor spreading on online social networks, epidemics in human beings, and malware on the Internet. Informally speaking, the source localization problem is to identify a node in the network that provides the best explanation of the observed diffusion. Despite significant efforts and successes over last few years, theoretical guarantees of source localization algorithms were established only for tree networks due to the complexity of the problem. This paper presents a new source localization algorithm, called the Short-Fat Tree (SFT) algorithm. Loosely speaking, the algorithm selects the node such that the breadth-first search (BFS) tree from the node has the minimum depth but the maximum number of leaf nodes. Performance guarantees of SFT under the independent cascade (IC) model are established for both tree networks and the Erdos-Renyi (ER) random graph. On tree networks, SFT is the maximum a posterior (MAP) estimator. On the ER random graph, the following fundamental limits have been obtained: $(i)$ when the infection duration $<\frac{2}{3}t_u,$ SFT identifies the source with probability one asymptotically, where $t_u=\left\lceil\frac{\log n}{\log \mu}\right\rceil+2$ and $\mu$ is the average node degree, $(ii)$ when the infection duration $>t_u,$ the probability of identifying the source approaches zero asymptotically under any algorithm; and $(iii)$ when infection duration $<t_u,$ the BFS tree starting from the source is a fat tree. Numerical experiments on tree networks, the ER random graphs and real world networks with different evaluation metrics show that the SFT algorithm outperforms existing algorithms.'
author:
- '\'
bibliography:
- 'ERcitation.bib'
title: 'Source Localization in Networks: Trees and Beyond'
---
Introduction {#sec:introduction}
============
The information source detection problem (or called rumor source detection problem) is to identify the source of information diffusion in networks based on available observations like the states of the nodes and the timestamps at which nodes adopted the information (or called infected). The solution of the problem can be used to answer a wide range of important questions. For example, in epidemiology, the knowledge of the epidemic source has been used to understand the transmission media of the disease [@John_1854]. For a computer virus spreading on the Internet, tracing the source helps locate the virus creator. For the news over the social media, locating the sources helps users verify the credibility of the news.
Because of its wide range of applications, the problem has gained a lot of attention in the last few years since the seminal work by Shah and Zaman [@ShaZam_11]. A number of effective information source detection algorithms have been proposed under different diffusion models. Despite significant efforts and successes, theoretical guarantees have been established only for tree networks due to the complexity of the problem in non-tree networks. In this paper, we first develop a new information source detection algorithm, called the Short-Fat Tree algorithm, and then present a comprehensive performance analysis of the algorithm under the IC model for both tree networks and the ER random graph. To the best of our knowledge, SFT is the first algorithm that has provable performance guarantees on both tree networks and the ER random graph [@ErdRen_59] (non-tree networks).
The fundamental possibility and impossibility results are summarized as follows.
1. For tree networks, we prove that the Jordan infection center with the maximum weighted boundary node degree (WBND) is the MAP estimator of the source under the heterogeneous IC model. Based on the derivation, we propose an algorithm called the Short-Fat Tree (SFT) algorithm which is applicable to both tree and general networks.
2. We analyze the performance of the SFT algorithm on the ER random graph. Under some mild conditions on the average node degree, we establish the following three results:
- Assume the infection duration $<\frac{\log n}{(1+\alpha)\log\mu}$ for some $\alpha>0.5,$ SFT identifies the source with probability 1 (w.p.1) asymptotically (as network size increases to infinity).
- Assume the infection duration $\geq \left\lceil\frac{\log n}{\log \mu}\right\rceil+2,$ the probability of identifying the source approaches zero asymptotically under any information source detection algorithm, i.e., it is impossible to detect the source with non-zero probability.
- Assume the infection duration $<\frac{\log n}{(1+\alpha)\log\mu}$ for some $\alpha>0,$ asymptotically, at least $1-\delta$ fraction of the nodes on the BFS-tree starting from the source are leaf-nodes, where $\delta>3\sqrt{\frac{\log n}{\mu}}$. This result does not provide any guarantee on the probability of correctly localizing the source, but states that the BFS-tree starting from the true source is a “fat” tree, which further justifies the SFT algorithm.
The results are summarized in Figure \[fig:PerformanceSummary\].
![Summary of the main results. This figure summarizes the key results in terms of $t,$ the infection time, and $\mu,$ the average node degree. In the figure, $q$ is the lower bound on the infection probability; “fat tree” means that there are $1-\delta$ fraction of nodes are boundary nodes on the BFS tree rooted at the source; and $t_u=\left\lceil\frac{\log n}{\log \mu}\right\rceil+2,$ which is the lower bound of the observation time (we proved that all algorithms fail when $t>t_u.$)[]{data-label="fig:PerformanceSummary"}](PerformanceSummary){width="0.8\columnwidth"}
We remark that results (i) and (iii) are highly nontrivial because a subgraph of the ER random graph is a tree with high probability only when the diameter is $\frac{\log n}{2\log \mu}$, and (i) and (iii) deal with subgraphs that are not trees. To the best of our knowledge, these are the first theoretical results on information source detection on non-tree networks under probabilistic diffusion models.
3. One drawback of the WBND tie-breaking is that it requires the infection probabilities of all edges in the IC model. We simplify WBND to BND by using the boundary node degree in SFT. As shown in Section \[sec:performance-evaluation\], the performance of BND tie-breaking is very close to WBND tie-breaking. We conducted extensive simulations on trees, ER random graphs and real world networks. SFT outperforms existing algorithms by having a higher detection rate and being closer to the actual source. We further evaluated the scalability of the algorithm by measuring the running time. Our results demonstrate that SFT achieves a better performance with a reasonably short execution time.
Related Work {#sec:relatedWork}
------------
[@ShaZam_11] is one of the first papers that study the information source detection problem, in which a new graph centrality called rumor centrality was proposed and proved to be the maximum likelihood estimator (MLE) on regular trees under the susceptible-infected (SI) model. In addition, the detection probability (the probability that the estimator is the source) for regular trees was proved to be greater than zero and the detection probability for geometric trees approaches one asymptotically as the increase of the spreading time. Later, [@ShaZam_12] quantified the detection probability of the rumor centrality on general random trees.
The rumor centrality has been further studied under different scenarios: 1) [@LuoTayLen_13] extended the rumor centrality to multiple sources and showed that the detection probability goes to one as the number of infected nodes increases for geometric trees when there are at most two sources; 2) [@KarFra_13] proved a similar performance guarantee for the single source case when only a subset of infected nodes are observed; 3) [@DonZhaTan_13] studied the detection probability when the prior knowledge of suspect nodes is available in the single source detection problem for trees; 4) [@WanDonZha_14] analyzed the detection probability of the rumor centrality for tree networks when there are multiple observations of independent diffusion processes from the same source.
[@ZhuYin_14_2] proposed the sample path based approach for the single source detection problem. Define the infection eccentricity of a node to be the maximum distance between the node and the infected nodes. [@ZhuYin_14_2] proved that on tree networks, under the homogeneous susceptible-infected-recovered (SIR) model, the root of the most likely sample path is a node with the minimum infection eccentricity (a Jordan infection center), which is within a constant distance to the actual source with a high probability. The approach has been extended to several directions: 1) [@ZhuYin_14] extended the approach to the case with partial observations and under the heterogeneous SIR model; 2) [@CheZhuYin_14] extended the analysis to multiple sources under the SIR model and proved that the distance between the estimator and its closest actual source is bounded by a constant with a high probability in tree networks; 3) [@LuoTay_13_2; @LuoTay_13] proved that the Jordan infection centers are the optimal sample path estimators under the SI model [@LuoTay_13_2] and the susceptible-infected-susceptible (SIS) model [@LuoTay_13] for tree networks, respectively.
Besides the rumor centrality and the Jordan infection center, several other heuristic algorithms based on a single snapshot of the network have been proposed in the literature: 1) [@LapTerGun_10] studied a similar problem under the independent cascade (IC) model [@GolLibMul_01] to minimize the l1 distance between the expected states and observed states of the nodes. A dynamic programming algorithm was proposed to solve the problem for tree networks and a Steiner tree heuristic was used for general networks; 2) [@PraVreFal_12] proposed an algorithm called NETSLEUTH which ranks the nodes according to an eigen vector based metric under the SI model. The algorithm was designed based on the Minimum Description Length principle; 3) [@LokMezOht_14] proposed a dynamic message passing algorithm based on the mean field approximation of the maximum likelihood estimation (MLE) of the source.
In addition, there exist several other algorithms which tackled the problem under the assumption that a subset of the infection timestamps are known: 1) [@PinThiVet_12] solved the MLE problem with partial timestamps for tree networks and extended the algorithm to general networks using a BFS tree heuristic; 2) [@ZhuCheYin_15] proposed two rank based algorithms using a modified BFS tree heuristic for general graphs; 3) [@AgaLu_13] proposed a simulation based Monte Carlo algorithm which utilizes the states of the sparsely placed observers within a fixed time window; 4) [@ZejGomSin_13] obtained sufficient conditions on the number of timestamps needed to locate the source correctly under the deterministic slotted SI models.
Our paper establishes possibility and impossibility results of SFT beyond tree networks, which differs it from the existing work mentioned above. The rest of the paper is organized as follows. In Section \[sec:model\], we first introduce the IC model and formulate a MAP problem for information source detection and SFT will be presented in Section \[sec:algorithm\]. Section \[sec:mainresult\] summarizes the main theoretical results of the paper including the analysis on both tree networks and the ER random graph. The simulation based performance evaluation will be presented in Section \[sec:performance-evaluation\]. All the proofs are provided in the appendices.
Model and Algorithm {#sec:model-algorithm-mainresult}
===================
Model {#sec:model}
-----
Given an undirected graph $g,$ denote by ${{\cal E}}(g)$ the set of edges in $g$ and denote by ${{\cal V}}(g)$ the set of nodes in $g.$ We consider the IC model [@GolLibMul_01] for information diffusion and assume a time-slotted system. Each node has two possible states: active (or called infected) and inactive (or called susceptible). At time slot $0,$ all nodes are inactive except the source. At the beginning of each time slot, an active node attempts to activate its inactive neighbors. If an attempt is successful, the corresponding node becomes active at next time slot; otherwise, the node remains inactive. The weight of each edge represents the success probability of the attempt, called the *infection probability* of the edge and each attempt is independent of others. Each active node only attempts to activate each of its inactive neighbors once. Denote by $q_{uv}$ the infection probability of edge $(u,v)$ and we assume $q_{uv}=q_{vu}$ throughout the paper since the graph is undirected. We assume that a complete snapshot ${{\cal O}}=\{{{\cal I}},{{\cal H}}\}$ of the network at time $t$ (called the *observation time*) is given, where ${{\cal I}}$ is the set of active nodes and ${{\cal H}}$ is the set of inactive nodes. Based on ${{\cal O}},$ we want to detect the source. We further assume the observation time ${t}$ is unknown. The problem can be formulated as a MAP problem as follows, $$\arg\max_{v\in{{\cal V}}(g)} \Pr(v|{{\cal O}}).$$ where $\Pr(v|{{\cal O}})$ is the probability that $v$ is the source given the snapshot ${{\cal O}}.$ The infected nodes form a connected component under the IC model, called the *infection subgraph* and denoted by $g_i.$ Since the source must be an infected node, the MAP problem can be simplified to $$\arg\max_{v\in{{\cal I}}} \Pr(v|{{\cal O}}),$$ and the search of the information source can be restricted to the infection subgraph. We assume the observation time ${t},$ which itself is a random variable, is independent of the source node.
The Short-Fat Tree Algorithm {#sec:algorithm}
----------------------------
In this section, we first present the SFT algorithm. We will show in Theorem \[thm:treepartialMAP\] that the algorithm outputs the MAP estimator for tree networks, which motivates the algorithm. The performance on the ER random graph is studied in Theorems \[thm:sourceIsJordanER\] and \[thm:approximateRatio\].
We first introduce several necessary definitions. Denote by ${d}^g_{uv}$ the distance from node $u$ to node $v$ in graph $g,$ where the distance is the minimum number of hops between two nodes. Define the *infection eccentricity* of an infected node to be the maximum distance from the node to all infected nodes on the infection subgraph $g_i,$ denote by ${e}(v,{{\cal I}}),$ $${e}(v,{{\cal I}})=\max_{u\in {{\cal I}}}{d}^{g_i}_{uv}.$$ Recall that the *Jordan infection centers* of a graph are the nodes with the minimum infection eccentricity [@ZhuYin_14_2].
Consider a BFS tree $T_v$ rooted at node $v$ on the infection subgraph $g_i.$ Denote by $\hbox{par}_v(u)$ the parent of node $u$ in $T_v.$ Define the set of *boundary nodes* of $T_v$ to be $${{\cal B}}(v,{{\cal I}})=\{w\in {{\cal I}}|{d}^{T_v}_{vw}={e}(v,{{\cal I}})\},$$which are the set of active nodes furthest away from node $v$ in the infection subgraph.
The weighted boundary node degree (WBND) with respect to node $v$ is defined to be $$\begin{aligned}
\sum_{(u,w)\in{{\cal F}'_v}} |\log (1-q_{uw})|,\label{eqn:boundaryDegreeCountNew}\end{aligned}$$ where $$\begin{aligned}
{\cal F}'_v=\{(u,w)|(u,w)\in{{\cal E}}(g),w\neq \hbox{par}_v(u), u\in {{\cal B}}(v,{{\cal I}})\}.\label{eqn:FvDefinition}\end{aligned}$$
The SFT algorithm, presented in Algorithm \[alg:RI\], identifies the source based on the BFS trees on the infection subgraph. The algorithm is called the *Short-Fat Tree* algorithm because (1) it first identifies the *shortest* BFS tree; and (2) the shortest BFS tree that maximizes the WBND is then selected in tie-breaking, which is usually the tree with a large number of leaf-nodes, i.e., a *fat* tree. The pseudo codes of the algorithms are presented in Algorithm \[alg:RI\] and \[alg:WBNDC\], which can be executed in a parallel fashion.
A simple example is presented in Figure \[fig:AlgorithmExample\] to illustrate algorithm. Each node has a unique node ID. The red nodes are infected and the white nodes are healthy. For simplicity, we assume the weights of all edges equal to $|\log(0.5)|$. The vector next to each infected node records the distance from it to all infected nodes. Initially at Iteration 0, each infected node only knows the distance to itself. For example, $[0$ $*$ $*$ $*]$ next to node 1 means that the distance from node 1 to itself is 0 and the distance from node 1 to node 2 is unknown. At Iteration 1, each infected node broadcasts its ID to its neighbors in next iteration. Upon receiving the node ID from node 1, node 2 updates its vector to $[1$ $0$ $*$ $*],$ and broadcasts node 1’s ID to its neighbors. The figure in the middle shows the updated vectors after all node ID exchanges occur at Iteration 1. At Iteration 2, node 1 and 2 do not receive any new node IDs. Therefore, node 1 and node 2 report themselves as the Jordan infection centers which are circled with blue in Figure \[fig:AlgorithmExample\]. The boundary nodes of the BFS tree rooted at node 1 are 2,3,4. The WBND of node 1 is $13|\log(0.5)|$. Similarly, the boundary nodes of the BFS tree rooted at node 2 are 1,3,4 and the WBND is $9|\log(0.5)|.$ Therefore, node 1 has a larger WBND and is chosen to be our estimator of the information source.
{width="75.00000%"}
Set subgraph $g_i$ to be a subgraph of $g$ induced by node set ${{\cal I}}.$
Each node receives its own node ID at time slot $0.$
Set time slot $t=1.$
Set ${\cal S}$ to be the set of nodes who receive $|{\cal I}|$ distinct node IDs.
$v^\dag\in{\cal S}$ with the maximum WBND.
Set ${\cal B}$ to be empty.
Set $x=0;$
$x.$
[**Remark:**]{} Note Equation (\[eqn:boundaryDegreeCountNew\]) requires the infection probabilities of all edges in the network which could be hard to obtain in practice. When the infection probabilities are not available, we can assume each edge has the same infection probability $q$ and WBND becomes, $$\begin{aligned}
\left(\sum_{u\in{{\cal B}}(v,{{\cal I}})}{{\hbox{deg}}}(u)-|{{\cal B}}(v,{{\cal I}})|\right)|\log(1-q)|,\end{aligned}$$ where ${{\hbox{deg}}}(u)$ is the degree of node $u.$
Define the boundary node degree (BND) of node $v$ to be $$\begin{aligned}
\sum_{u\in{{\cal B}}(v,{{\cal I}})}{{\hbox{deg}}}(u)-|{{\cal B}}(v,{{\cal I}})|\label{eqn:boundaryNodeDegreeCountHomo}\end{aligned}$$ which is only related to the degree of the boundary nodes and can be used to replace WBND as the tie-breaking among the Jordan infection center in SFT when the infection probabilities are unknown. As shown in Section \[sec:performance-evaluation\], the performance using BND and WBND are similar. To differentiate the two algorithms, we call the algorithm which uses WBND as wSFT and the one which uses BND as SFT. Next, we analyze the complexity of the algorithm.
\[thm:RunningTime\] The worst case computational complexity of the SFT algorithm is $O(|{{\cal I}}|{{\hbox{deg}}}({{\cal I}}))$ where ${{\hbox{deg}}}({{\cal I}})$ is the total degree of nodes in ${{\cal I}}$ in graph $g.$
The detailed proof can be found in Appendix \[proof:RunningTime\].
Main Results {#sec:mainresult}
============
In this section, we summarize the main results of the paper and present the intuitions of the proofs.
Main Result 1 (The MAP Estimator on Tree Networks)
--------------------------------------------------
On tree networks, the Jordan infection center of the infection subgraph with the maximum WBND is a MAP estimator.
\[thm:treepartialMAP\] Consider a tree network. Assume the following conditions hold.
- The probability distribution of the observation time satisfies $\Pr({t})\geq \Pr({t}+1)$ for all ${t}.$
- The source is uniformly and randomly selected, i.e., $\Pr(u)=\Pr(v).$
Denote by ${\cal J}$ the set of Jordan infection centers of the infection subgraph $g_i$. We have $$\begin{aligned}
\arg\max_{u\in{\cal J}} \sum_{(v,w)\in{{\cal F}'_u}}|\log(1-q_{vw})|\subset \arg\max_u\Pr(u|{{\cal O}}).\label{eqn:boundaryDegreeCount}\end{aligned}$$ where ${\cal F}'_u$ is defined in Equation (\[eqn:FvDefinition\]).
The detailed proof can be found in Appendix \[sec:TreeJordanIsMAP\]. The theorem has been proved in two steps: 1) We show that one of the Jordan infection centers maximizes the posterior probability on tree networks following similar arguments in [@ZhuYin_14_2]. In particular, for two neighboring nodes, we show the one with smaller infection eccentricity has a larger posterior probability of being the source. Since there exists a path from any node to a Jordan infection center on the infection subgraph, along which the infection eccentricity strictly decreases, we conclude that a MAP estimator of the source must be a Jordan infection center; 2) Consider the case where the tree network has more than one (at most two according to [@Har_91]) Jordan infection centers. When the observation time is larger than the infection eccentricity of the Jordan infection center, the probability of having the observed infected subgraph from any Jordan infection center is the same. When the observation time equals the infection eccentricity, we prove that the probability for a Jordan infection center to be the source is an increasing function of WBND of the BFS tree starting from it.
Main Result 2 (Detection with Probability One on the ER Random Graph)
---------------------------------------------------------------------
Denote by $n$ the number of nodes in the ER random graph and $p$ the wiring probability of the ER random graph. Let $\mu=np.$ Recall that $t$ is the observation time. We show that the Jordan infection center is the actual source in the ER random graph with probability one asymptotically when $t<\frac{\log n}{(1+\alpha)\log \mu},$ which implies that SFT can locate the source w.p.1 asymptotically.
\[thm:sourceIsJordanER\] If the following conditions hold, source ${s}$ is the only Jordan infection center on the infection subgraph with probability one asymptotically.
- $\mu> 3\log n.$
- ${t}\leq \frac{\log n}{(1+\alpha)\log\mu},$ for some $\alpha\in(\frac{1}{2},1).$
We present a brief overview of the proof and the details can be found in Appendix \[sec:ERJordanIsSource\]. Note the infection eccentricity of the actual source is no larger than the observation time $t.$ We show in the proof that the infection eccentricity of an infected node other than the source is larger than $t.$ Consider the BFS tree ${T}^\dag$ rooted at the actual source ${s}.$ A node is said to be on level $i$ if its distance to the source is $i.$ Consider another infected node ${s}'.$ Denote by $a({s}')$ the ancestor of ${s}'$ on level $1$ of ${T}^\dag.$ As shown in Figure \[fig:onebyoneexploration\], the yellow area shows the level $t$ infected nodes on subtree $T_u^{-s},$ which is the subtree of $T^\dag$ rooted at node $u,$ and the distance from ${s}'$ to a node in the yellow area is larger than $t$ if any path between the two nodes can only traverse the edges on tree ${T}^\dag$. If ${s}'$ has an infection eccentricity no larger than $t,$ there must exist a path from ${s}'$ to each node in the yellow area with length no larger than $t.$ Such a path must contain edges that are not in ${T}^\dag$ (we call these edges *collision edges*). We show in the proof that the number of nodes that are within $t$ hops from ${s}'$ via collision edges are strictly less than the number of nodes in the yellow area. Therefore, the infection eccentricity of ${s}'$ must be larger than $t$, which implies that ${s}$ is the only Jordan infection center.
Existing theoretical results in the literature on information source detection problems are only for tree networks. As shown in the proof of Theorem \[thm:sourceIsJordanER\], the infection subgraph of the ER random graph is not a tree when $t>\frac{\log n}{2\log\mu}.$ From the best of our knowledge, this result is the first one on non-tree networks.
![A pictorial example of ${\cal Z}^{t}_{t}(u)$ in BFS tree ${T}^\dag$[]{data-label="fig:onebyoneexploration"}](ER_lower_level_tree){width="0.8\columnwidth"}
Main Result 3 (The Fat Tree Result on the ER Random Graph)
-----------------------------------------------------------
\[thm:approximateRatio\] If the following conditions hold,
- $\mu> \frac{9}{\delta^2}\log n$.
- ${t}\leq \frac{\log n}{(1+\alpha)\log\mu},$ for some $\alpha\in(0,1)$.
the leaf-nodes of the BFS tree starting from the actual source consists of at least $1-\delta$ fraction of the BFS tree asymptotically.
The detailed proof can be found in Appendix \[sec:approximateRate\]. Consider the BFS tree from the source ${s}$ in graph $g.$ The boundary nodes are the nodes at level $t$ and all boundary nodes must be infected at time $t.$ If we ignore the presence of collision edges, the number of infected nodes roughly increases by a factor of $q\mu$ at each level where $q=\min_{(u,v)\in {{\cal E}}(g)}q_{uv}.$ Due to this exponential growth nature, the total number of infected nodes is dominated by those infected at the last time slot. We show this property holds with the presence of collision edges. Theorem \[thm:approximateRatio\] suggests that the BFS tree rooted at the actual source is a “fat" tree and the BND of the actual source is large. Hence, in the tie breaking, the SFT algorithm has a good chance to select the actual source, which suggests that BND is a good tie breaking rule for the ER random graph.
Main Result 4 (The Impossibility Result on the ER Random Graph)
---------------------------------------------------------------
We next present the threshold of $t$ after which it is impossible for any algorithm to find the actual source with a non-zero probability asymptotically. The result is based on the analysis of the diameter of an ER random graph in Theorem 4.2 in [@DraMas_10]. For clarity purpose, we rephrase that theorem with our notation in the following lemma.
\[thm:diameterER\] If $24\log n<np<<\sqrt{n},$ we have $$\lim_{n\rightarrow \infty} \Pr(\hbox{Diameter}(g)\leq D+2)=1,$$ where $D=\lceil\frac{\log n}{\log np}\rceil.$
We remark that in [@DraMas_10], the condition is $\log n<<(n-1)p<<\sqrt{n}.$ We explicitly calculated the lower bound according to the proof in [@DraMas_10]. For the sake of completeness, we present the proof in Appendix \[sec:proofImpossibility\].
Based on Lemma \[thm:diameterER\], we obtain the following impossibility result.
\[thm:impossibility\] If $24\log n<q\mu <<\sqrt{n}$ and $q>0$ is a constant, $$\lim_{n\rightarrow \infty}\Pr({{\cal I}}={{\cal V}}(g))=1$$ when the observation time $$\begin{aligned}
t\geq\left\lceil\frac{\log n}{\log \mu+\log q}\right\rceil+2 \triangleq t_u.\label{eqn:tRange}\end{aligned}$$ In other words the entire network is infected. In such a case, asymptotically, the probability of any node being the source is $1/n.$
The process to generate the ER random graph and the process of the information diffusion under the IC model can be viewed as a combined process. In this combined process, an edge exits only when the edge exists in the ER random graph and is live in the IC model. The detailed definition of the live edge could be found in Appendix \[sec:TreeJordanIsMAP\]. Loosely speaking, an edge $(u,v)$ is said to be live if node $v$ is infected by node $u$ under the IC model. When the observation time is larger than or equal to the diameter of the coupled ER random graph, all nodes in the network are infected. In such a case, the probability of a node being the source is $1/n$ as the source was uniformly chosen. Based on Lemma \[thm:diameterER\], the diameter of the combine network is smaller than $\lceil\frac{\log n}{\log q+\log \mu}\rceil+2$ w.p.1 asymptotically.
[**Remark 1:**]{} We compare $t_u$ in Equation (\[eqn:tRange\]) and the upper bound in Theorem \[thm:sourceIsJordanER\]. Since $q$ is a constant, the ratio between $t_u$ and the upper bound becomes $\frac{1}{1+\alpha}$ asymptotically. Since $\alpha$ can be arbitrarily close to $\frac{1}{2},$ the ratio becomes $\frac{2}{3}.$ Therefore, the Jordan infection center is the actual source when the observation time is in the range of $(0,\frac{2}{3}t_u)$ and it is impossible to locate the source when the observation time is $(t_u,\infty).$
[**Remark 2:**]{} We compare $t_u$ and the upper bound in Theorem \[thm:approximateRatio\] and asymptotically the ratio between $t_u$ and the upper bound becomes $\frac{1}{1+\alpha}$ where $\alpha \in (0,1).$ Since $\alpha$ can be arbitrarily close to $0$ and the ratio are close to $1$ which means the BFS tree from the source has large BND before it becomes impossible to locate the source. While the theorem does not provide any guarantee on the detection rate, it justifies the tie-breaking using BND and WBND.
Performance Evaluation {#sec:performance-evaluation}
======================
In this section, we compare the proposed algorithms with existing algorithms on different networks such as tree networks, the ER random graphs and real world networks.
Algorithms
----------
Among all the existing algorithms discussed in Section \[sec:introduction\], we choose the algorithms which require only a single snapshot of the network but not the infection probabilities which could be difficult to obtain in practice. We compared SFT and wSFT with the algorithms summarized as follows.
- [**ECCE:**]{} Select the node with minimum infection eccentricity. Ties are breaking randomly. Recall the definition of the infection eccentricity is the maximum distance from the node to all infected nodes. [@ZhuYin_14_2] showed that the optimal sample path estimator on tree networks is the Jordan infection center of the graph under the SIR model.
- [**RUM:**]{} Select the node with maximum rumor centrality proposed in [@ShaZam_11]. The rumor centrality was proved to be the maximum likelihood estimator on regular trees under the continuous time SI model in which the infection time follows exponential distribution.
- [**NETSLEUTH:**]{} Select the node with maximum value in the eigenvector corresponding to the largest eigenvalue of a submatrix which is constructed from the infected nodes based on the graph Laplacian matrix. The algorithm was proposed in [@PraVreFal_12].
Among the selected algorithms, only wSFT requires the infection probabilities. We included wSFT to evaluate the importance of the knowledge of edge weights to our algorithm. We will see that the performance of SFT is almost identical to wSFT, so the infection probabilities are not important for our detection algorithm.
Evaluation Metrics
------------------
We evaluated the performance of the algorithms with three different metrics.
- Detection rate is the probability that the node identified by the algorithm is the actual source. A desired goal of the information source detection is to have a high detection rate.
- Distance is the number of hops from the source estimator to the actual source. The distance is an often used metric for information source detection.
- $\gamma\%$-accuracy is the probability with which the source is ranked among top $\gamma$ percent. Note that besides providing a source estimator, an information source algorithm can also be used to rank the infected nodes according to their likelihood to be the source. For example, SFT can rank the nodes in an ascendant order according to their infection eccentricity and then breaks ties using BND. Other algorithms can be used to rank nodes as well. $\gamma\%$-accuracy is a less ambitious alternation to the detection rate. When the detection rates of all algorithms are low, it is useful to compare $\gamma\%$-accuracy as a high $\gamma\%$-accuracy guarantee that the actual source is among the top ranked nodes with a high probability.
Binomial Trees
--------------
In this section, we evaluate the algorithms on binomial trees. Denote by $\hbox{Bi}(m,\beta)$ the binomial distribution with $m$ number of trials and each trial succeeds with probability $\beta.$ A binomial tree is a tree where the number of children of each node follows a binomial distribution $\hbox{Bi}(m,\beta).$ In the experiments, we set $m=20$ and $\beta=0.5.$ We adopted the IC model where the infection probability of each edge is assigned with a uniform distribution in $(0.2,0.5).$ The lower bound on the infection probability is set to be $0.2$ to prevent the diffusion process dies out quickly. We evaluated the performance for different infection size $x.$ Under a discrete infection model, it is hard to obtain the diffusion snapshots with exact $x$ infected nodes. Therefore, for each infection size $x,$ we generate the diffusion samples where the number of infected nodes are in range $[0.75x, 1.25x].$ The source was chosen uniformly at random among all nodes in the network. We varied $x$ from $200$ to $2000$ with a step size $200.$ For each infection size, we generate $400$ diffusion samples.
Figure \[fig:binomialDetectionRate\] shows the detection rates for different infection sizes. The detection rates of ECCE, SFT and wSFT do not change for different infection sizes since the structure of the binomial tree is simple. SFT, wSFT and ECCE have the highest detection rate (more than 0.9) while the detection rate of RUM and NETSLEUTH are much lower.
The distance results are shown in Figure \[fig:binomialDist\]. As expected, SFT, wSFT and ECCE outperform RUM, which are all much better than NETSLEUTH.
Figure \[fig:binomialcdf\] shows the $\gamma\%$-accuracy versus the rank percentage $\gamma.$ We picked infection size 1,000. As shown in Figure \[fig:binomialcdf\], all three algorithms based on infection eccentricity (ECCE, SFT, wSFT) have better performance than RUM and NETSLEUTH. Recall that the node identified by wSFT is a MAP estimator of the actual source.
The ER random graph {#subsec:ERsimulation}
-------------------
In this section, we compared the performance of the algorithms on the ER random graph. In the experiments, we generated the ER random graph with $n=5,000$ and wiring probability $p = 0.002.$ We again varied the infection network size from $200$ to $2,000$. The infection probability of each edge is assigned with a uniform distribution in $(0.2,0.5).$ We generated $400$ diffusion samples.
Figure \[fig:erDetectionRate\] shows the detection rate versus the infection size. The detection rate decreases as the infection size increases. SFT and wSFT have higher detection rates compared to other algorithms. Figure \[fig:erDist\] shows the results on distance. As we expected, SFT and wSFT outperform other algorithms when the infection size is less than 1,600 nodes. As the size of the infected nodes increase, SFT and wSFT become close to RUM in term of distance to the source. However, the detection rate of both algorithms are still much higher than that of RUM. Another observation is that SFT and wSFT have identical performance which indicates that the performance of SFT is robust to edge weights.
Figure \[fig:ercdf1000\] shows the $\gamma\%$-accuracy versus the rank percentage $\gamma$ with 1000 infected nodes. SFT and wSFT have similar or better performance compared to all other algorithms.
Although the performance of ECCE and SFT algorithms are similar in tree networks, SFT outperforms ECCE significantly on the ER random graphs. The observation indicates that BND is an effective tie breaking rule and increases the detection accuracy.
The Internet Autonomous System Network
---------------------------------------
The Internet autonomous systems (IAS) network [^1] is the Internet autonomous system from Oregon route-views on March, 31st, 2001 with 10,670 nodes and 22,002 edges. The IAS network is a small world network. We adopted similar settings as in Section \[subsec:ERsimulation\].
The detection rates are shown in Figure \[fig:iasDetectionRate\]. The detection rate of ECCE is low since the IAS graph is a small world network and there are multiple Jordan infection centers due to the small diameter of the network. With the tie breaking rule BND, the detection rate doubles in most cases which demonstrates the effectiveness of BND. While the detection rate of SFT is only $10\%$ when the infection size is 1,000, the distance to the actual source is slightly more than one-hop away as shown in Figure \[fig:iasDist\]. In addition, the $\gamma\%$-accuracy versus $\gamma$ for 1,000 infection size is shown in Figure \[fig:ercdf1000\]. The $10\%$-accuracies of SFT and wSFT are close to $70\%$ which are significantly higher than that of other algorithms.
Running Time vs Performance
---------------------------
In this section, we evaluated the scalability of the algorithms by comparing the running time. The experiments were conducted on an Intel Core i5-3210M CPU with four cores and 8G RAM with a Windows 7 Professional 64 bit system. All algorithms were implemented with python 2.7. The ER random graphs with 5,000 nodes and $p=0.002$ edge generation probability were used in the experiments. The infection probability of each edge is uniformly distributed over $(0.2,0.5).$ We generated 100 diffusion samples for the experiments. Figure \[fig:TimeVsDetection\] show the average running time versus the detection rate. The infection size is chosen to be 1,000. SFT and wSFT took 1.11 seconds and achieves 0.87 detection rate while NETSLEUTH took 0.62 seconds with 0 detection rate and RUM took 14.86 seconds with 0.7 detection rate. The detection rate of SFT is much higher than NETSLEUTH and SFT is 14 times faster than RUM.
Conclusions
===========
In this paper, we derived the MAP estimator of the information source on tree networks under the IC model. Based on that, the SFT algorithm has been proposed. We proved that the SFT algorithm identifies the information source with probability one asymptotically in the ER random graph when the observation time $t\leq \frac{2}{3}t_u,$ which is the first theoretical guarantee on non-tree networks to our best knowledge. We evaluated the performance of SFT on tree networks, the ER random graph and the IAS network.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work was supported in part by the U.S. Army Research Laboratory’s Army Research Office (ARO Grant No. W911NF1310279).
![Detection rate versus running time in the ER random graph[]{data-label="fig:TimeVsDetection"}](TimeVsDetectionRate1000){width="35.00000%"}
Proof of Theorem \[thm:RunningTime\] {#proof:RunningTime}
====================================
Follow the argument in [@ZhuYin_14_2], the computational complexity of the node ID broadcasting phase of Algorithm \[alg:RI\] is $O(|{{\cal V}}(g_i)||{{\cal E}}(g_i)|)$ since the node IDs are passed on the subgraph $g_i.$ The complexity of Algorithm \[alg:WBNDC\] is $O({{\hbox{deg}}}({{\cal I}}))$ since the number of boundary nodes is bounded by $|{\cal I}|.$ In addition, Algorithms \[alg:WBNDC\] are called at most $|{\cal S}|$ times in Algorithm \[alg:RI\] and $|{\cal S}|\leq |{{\cal V}}(g_i)|.$ Therefore, the complexity of Algorithm \[alg:RI\] is $$O(|{{\cal V}}(g_i)||{{\cal E}}(g_i)|+|{{\cal V}}(g_i)|{{\hbox{deg}}}({{\cal I}}))$$ Note ${{\cal V}}(g_i) = {{\cal I}}$ and $|{{\cal E}}(g_i)|\leq {{\hbox{deg}}}({{\cal I}}).$ The complexity becomes $$O(|{{\cal I}}|{{\hbox{deg}}}({{\cal I}})).$$
Proof of Theorem \[thm:treepartialMAP\] {#sec:TreeJordanIsMAP}
=======================================
First, we prove the following lemma for neighboring nodes.
\[cor:MAPneighbors\] [**Neighboring nodes inequality**]{} Consider nodes $u,v$ on tree $\tilde{{T}}$ satisfying the following conditions:
- $(u,v)\in{{\cal E}}(\tilde{{T}}).$
- The observation time follows a distribution such that $\Pr({t})\geq \Pr({t}+1)$ for all ${t}.$
- The source is uniformly chosen among all nodes, i.e., $\Pr(u)=\Pr(v).$
- ${e}(v,{{\cal I}})>{e}(u,{{\cal I}}).$
We have $$\Pr(v|{{\cal O}})\leq \Pr(u|{{\cal O}})$$
Consider nodes $u,v$ on a tree $\tilde{{T}}$ where $(u,v)\in{{\cal E}}(\tilde{{T}})$. Let ${t}_u,{t}_v$ be the observation times associated with $u,v.$ We will show that when ${e}(v,{{\cal I}})>{e}(u,{{\cal I}}),$ $$\begin{aligned}
\Pr({{\cal O}}|v,{t}_v = t+1)\leq \Pr({{\cal O}}|u,{t}_u = t),\label{eqn:neighborInequality}\end{aligned}$$ where $\Pr({{\cal O}}|v,{t}_v = t)$ is the probability of the snapshot ${{\cal O}}$ given $v$ is the source and the observation time is at time slot $t.$ .
We adopt an equivalent view of the IC model called *live edge* model [@KemKleTar_03]. In the IC model, after $u$ becomes infected, it attempts to infect its neighbor $w$ with probability $q_{uw}$ once. Therefore, we can assume that a biased coin with parameter $q_{uw}$ is flipped for edge $(u,w)\in {{\cal E}}(g)$ when $u$ tries to infect $w$ in the IC model. Note that the probability of node $w$ is infected by node $u$ remains the same whether the coin is flipped at the moment when node $u$ attempts to infect $w$ or prior to the infection but is revealed for the attempt. Assume the coins of all edges are flipped at the beginning of the infection process. When one node attempts to infect one of its neighbors, we check the stored coin realization to determine whether the infection succeeds. This process is called live edge model and it is equivalent to the IC model since we only change the time of the coin flippings, so the probability of an infection trace remains the same. In the live edge model, the infection process consists of two steps. First, each edge $(u,w)$ flips a biased coin with probability $q_{uw}$ to be a *live edge* prior to the infection starts. After all coin flippings, the graph formed by live edges is called the *live edge graph*. In the second step, the infection spreads over all live edges deterministically, starting from the source. We now analyze SFT under the live-edge model.
Denote by ${{\cal T}}$ the set of all live edge graphs of $\tilde{{T}}$, i.e., $${{\cal T}}=\{{T}|{{\cal E}}({T})\subset{{\cal E}}(\tilde{{T}}),{{\cal V}}({T})={{\cal V}}(\tilde{{T}})\}$$ Note there are no loops in ${T}\in {{\cal T}}$ since ${T}$ is a subgraph of tree $\tilde{{T}}.$
Denote by ${\cal K}({{\cal O}},v,{t}_v)$ the set of all live edge graphs on which the observation ${{\cal O}}$ is feasible if the source is $v$ and the observation time is ${t}_v.$ All infected nodes must be within $t_v$ hops from the source and all the observed healthy nodes must be more than $t_v$ hops away from the source in a feasible live edge graph. Formally, we have $${\cal K}({{\cal O}},v,{t}_v)=\{{T}\in {{\cal T}}| \forall w \in {{\cal I}}, {d}^{T}_{vw}\leq {t}_v, \forall w \in {{\cal H}}, {d}^{T}_{vw}>{t}_v\}.$$
$\Pr({{\cal O}}| v, {t}_v)$ equals to the probability a live edge graph is in set ${\cal K}({{\cal O}},v,{t}_v)$ due to the equivalence between the IC model and the live-edge model. The probability of a specific live edge graph is the product of edge live/dead probabilities.
Hence we have $$\Pr({{\cal O}}|v,{t}_v)=\sum_{{T}\in {\cal K}({{\cal O}},v,{t}_v)}\Pr({T}),$$ To prove the lemma, we will prove the following claim, $${\cal K}({{\cal O}},v,{t}_v = t+1)\subset{\cal K}({{\cal O}},u,{t}_u = t).$$ To simplify notation, we next assume $t_u=t_v-1=t\geq e(u, {\cal I}),$ and ignore $t$ in the equations. Note that we only consider $t_u\geq e(u, {\cal I})$ because $\Pr({\cal O}|u, t_u) =0$ otherwise.
Consider ${T}\in {\cal K}({{\cal O}},v,{t}_v).$ Denote by ${T}^{-u}_v$ the tree rooted at $v$ without the subtree starting from $u$ where $(u,v)\in{{\cal E}}({T}).$ Since ${e}(v,{{\cal I}})>{e}(u,{{\cal I}}),$ according to Lemma 2 in [@ZhuYin_14_2], there exists $w^\dag\in{{\cal I}}\cap\tilde{{T}}^{-v}_{u}$ which has ${d}^{\tilde{{T}}}_{vw^\dag}={d}^{\tilde{{T}}}_{uw^\dag}+1={e}(v,{{\cal I}}).$ If $(v,u)\not \in {{\cal E}}({T}),$ we have ${d}^{T}_{vw^\dag}=\infty>t_v$ which is a contradiction to ${T}\in {\cal K}({{\cal O}},v,{t}_v).$ Hence, we have $(v,u)\in {{\cal E}}({T}).$
- Consider $\tilde{{T}}^{-v}_u$ part.
- For any $w \in \tilde{{T}}^{-v}_u \cap {\cal I},$ we have $${d}^{{T}}_{vw}\leq {e}(v, {\cal I})\leq {t}_v.$$ Since $(v,u)\in{{\cal E}}(g)$ and ${T}$ have no loops, we have $${d}^{{T}}_{uw}={d}^{{T}}_{vw}-1\leq {t}_v-1={t}_u.$$
- For any $w \in \tilde{{T}}^{-v}_u \cap {{\cal H}},$ we have $${d}^{{T}}_{vw} > e(v, {\cal I})\geq {t}_v.$$ Since $(v,u)\in{{\cal E}}(g)$ and ${T}$ has no loops, we have $${d}^{{T}}_{uw}={d}^{{T}}_{vw}-1> {t}_v-1={t}_u.$$
- Consider $\tilde{{T}}^{-u}_v$ part.
- For any $w \in \tilde{{T}}^{-u}_v \cap {\cal I},$ based on the proof of Lemma 2 in [@ZhuYin_14_2], we have $${d}^{\tilde{{T}}}_{vw}\leq {e}(u,{{\cal I}})-1.$$ There is only one path ${\cal P}_{vw}$ from $v$ to $w$ in tree $\tilde{{T}}.$ If ${\cal P}_{vw}\not\subset {{\cal E}}({T}),$ $v,w$ are disconnected in ${T}$ which contradicts the fact that ${d}^{T}_{vw}\leq {t}_v.$ Hence ${\cal P}_{vw}\subset {{\cal E}}({T}).$ In addition, we have $(v,u)\subset {{\cal E}}({T}).$ Hence $${d}^{{T}}_{uw}={d}^{{T}}_{vw}+1={d}^{\tilde{{T}}}_{vw}+1\leq {e}(u,{{\cal I}})\leq {t}_u.$$
- For any $w \in \tilde{{T}}^{-u}_v \cap {\cal H},$ there is only one path ${\cal P}_{vw}$ from $v$ to $w$ in tree $\tilde{{T}}.$ If ${\cal P}_{vw}\not\subset {{\cal E}}({T}),$ we have $${d}^{T}_{uw}=\infty>t_u.$$ If ${\cal P}_{vw}\subset {{\cal E}}({T}),$ $${d}^{{T}}_{uw}={d}^{\tilde{{T}}}_{uw}={d}^{\tilde{{T}}}_{vw}+1={d}^{{T}}_{vw}+1> {t}_v+1 > {t}_u$$ where $d_{vw}^{T}>t_v$ because ${T}\in {\cal K}({\cal O}, v, t_v).$
As a summary, for any ${T}\in {\cal K}({{\cal O}},v,{t}_v),$ we have $$\forall w \in {{\cal I}}, {d}^{T}_{uw}\leq {t}_u, \forall w \in {{\cal H}}, {d}^{T}_{uw}>{t}_u$$ Therefore, ${T}\in {\cal K}({{\cal O}},u,{t}_u).$
Hence, we proved $${\cal K}({{\cal O}},v,{t}_v = t+1)\subset{\cal K}({{\cal O}},u,{t}_u = t)$$ which implies $$\begin{aligned}
\Pr({{\cal O}}|v,{t}_v = t+1)= & \sum_{{T}\in {\cal K}({{\cal O}},v,{t}_v = t+1)}\Pr({T})\\
\leq & \sum_{{T}\in {\cal K}({{\cal O}},u,{t}_u = t)}\Pr({T})\\
= & \Pr({{\cal O}}|u,{t}_u = t).\end{aligned}$$ Hence, we proved Inequality (\[eqn:neighborInequality\]).
Denote by $\Pr(v)$ the probability that $v$ is the source, $\Pr({t})$ the probability that the observation time is ${t},$ and $\Pr({{\cal O}}|v,{t})$ is the probability of snapshot ${{\cal O}}$ given $v$ is the source and ${t}$ is the observation time. Since the observation time ${t}$ is independent of the source node, we obtain $$\begin{aligned}
\Pr(v|{{\cal O}})&=\frac{1}{\Pr({{\cal O}})}\Pr(v,{{\cal O}})\\
&=\frac{1}{\Pr({{\cal O}})}\sum_{{t}\geq {e}(v,{{\cal I}})} \Pr(v,{t},{{\cal O}})\\
&=\frac{1}{\Pr({{\cal O}})}\sum_{{t}\geq {e}(v,{{\cal I}})}\Pr({{\cal O}}|v,{t})\Pr(v,{t})\\
&=\frac{\Pr(v)}{\Pr({{\cal O}})} \sum_{{t}\geq {e}(v,{{\cal I}})}\Pr({{\cal O}}|v,{t})\Pr({t})\\
&\leq_{\hbox{(a)}} \frac{\Pr(v)}{\Pr({{\cal O}})} \sum_{{t}\geq {e}(v,{{\cal I}})}\Pr({{\cal O}}|u,{t}-1)\Pr({t})\\
&=_{\hbox{(b)}}\frac{\Pr(u)}{\Pr({{\cal O}})} \sum_{{t}\geq {e}(u,{{\cal I}})}\Pr({{\cal O}}|u,{t})\Pr({t}+1)\\
&\leq_{\hbox{(c)}} \frac{\Pr(u)}{\Pr({{\cal O}})} \sum_{{t}\geq {e}(u,{{\cal I}})}\Pr({{\cal O}}|u,{t})\Pr({t}) \\
&=\Pr(u|{{\cal O}})\end{aligned}$$ (a) is due to Inequality (\[eqn:neighborInequality\]), (b) is based on $\Pr(u)=\Pr(v)$ and $e(v, {\cal I})=e(u, {\cal I})+1,$ and (c) is based on $\Pr({t})\geq \Pr({t}+1).$
Based on the proof of Theorem 4 in [@ZhuYin_14_2], there exists a path from any node to a Jordan infection center in the tree network such that the infection eccentricity strictly decreases along the path. By repeatedly applying Lemma \[cor:MAPneighbors\], we conclude that a MAP estimator must be a Jordan infection center.
Recall ${\cal J}$ is the set of Jordan infection centers. Next, we will show that the MAP estimator , say node $v,$ has the maximum $\sum_{(v,w)\in{{\cal F}'_u}}|\log(1-q_{vw})|$ among all nodes in ${\cal J}.$
Define an edge set $${\cal F}=\{(v,w)|(v,w)\in{{\cal E}}(\tilde{{T}}),v\in {{\cal I}},w\in {{\cal H}}\}.$$ We call the edges in ${\cal F}$ the frontier edges since they are the edges between ${{\cal I}}$ and ${{\cal H}}.$
Define another edge set $${\cal B}=\{(v,w)|(v,w)\in {{\cal E}}(\tilde{{T}}), v,w\in {{\cal I}}\}.$$ The edges in ${\cal B}$ are the edges between infected nodes.
In addition, for any $u\in{\cal J}$ define $${\cal F}_u({t}_u)=\{(v,w)|(v,w)\in{{\cal E}}(\tilde{{T}}),v\in {{\cal I}},w\in {{\cal H}},{d}_{uw}\leq {t}_u\}.$$ ${\cal F}_u({t}_u)$ is set of edges which cannot be live edges when $u$ is the source and ${t}_u$ is the observation time.
For a complete observation, we have $$\begin{aligned}
\Pr({{\cal O}}|u,{t}_u)=\prod_{(v,w)\in {\cal B}}q_{vw}\prod_{(v,w)\in{\cal F}_u({t}_u)}(1-q_{vw})\label{eqn:AP}\end{aligned}$$
Denote by $e^*$ the minimum infection eccentricity, i.e., $$\forall v \in {\cal J}, {e}(v,{{\cal I}}) = e^*$$ Intuitively, when $t_u>e^*,$ none of frontier edges should be a live edge in a feasible live edge graph to make sure healthy nodes are not infected. So when ${t}_{u}>e^*,$ we have $${\cal F}_u({t}_u)={\cal F}.$$ Hence, $$\Pr({{\cal O}}|u,{t}_u)=\prod_{(v,w)\in {\cal B}}q_{vw}\prod_{(v,w)\in{\cal F}}(1-q_{vw})\triangleq C,$$ which is not a function of either $t_{u}$ or $u.$ Substituting into Equation (\[eqn:AP\]), we have $$\begin{aligned}
\Pr({{\cal O}}|u,e^*)=\frac{C}{\prod_{(v,w)\in{\cal F}\backslash{\cal F}_u(e^*)}(1-q_{vw})}\label{eqn:newAP}\end{aligned}$$
Follow a similar procedure in Lemma \[cor:MAPneighbors\], for an Jordan infection center $u,$ we have $$\begin{aligned}
\Pr(u|{{\cal O}})&=\frac{\Pr(u)}{\Pr({{\cal O}})} \sum_{{t}}\Pr({{\cal O}}|u,{t})\Pr({t})\\
&=\frac{\Pr(u)}{\Pr({{\cal O}})}\bigg( \Pr({{\cal O}}|u,e^*)\Pr(t = e^*)\\
&+\sum_{{t}>e^*}\Pr({{\cal O}}|u,{t})\Pr({t})\bigg)\\
&=\frac{\Pr(u)}{\Pr({{\cal O}})}\left( \Pr({{\cal O}}|u,e^*)\Pr(t = e^*)+\Pr({t}>e^*)C\right)\end{aligned}$$ Therefore, $$\begin{aligned}
&\arg\max_{u}\Pr(u|{{\cal O}})\\
&=\arg\max_{u\in {\cal J}}\Pr(u|{{\cal O}})\\
&=\arg\max_{u\in {\cal J}}\frac{\Pr(u)}{\Pr({{\cal O}})}\big( \Pr({{\cal O}}|u,e^*)\Pr(t = e^*)\\
&+\Pr({t}>e^*)C\big)\\
&=\arg\max_{u\in {\cal J}}\Pr({{\cal O}}|u,e^*)\label{eqn:MAP}\\
&=\arg \min_{u\in {\cal J}}\prod_{(v,w)\in{\cal F}\backslash{\cal F}_u(e^*)}(1-q_{vw}).\label{eqn:MLE}\end{aligned}$$
Note we have $$\begin{aligned}
&{\cal F}\backslash{\cal F}_u(e^*)\\
=&\{(v,w)|(v,w)\in{{\cal E}}(\tilde{{T}}),v\in {{\cal I}},w\in {{\cal H}},{d}_{vw}>e^*\}\end{aligned}$$ Since $e^*$ is the minimum eccentricity and $u$ is the Jordan infection center, we have ${d}_{uv}\leq e^*$ for all $v\in {{\cal I}}.$ Hence for all $w\in {{\cal H}}$ which have at least one edge to the infected nodes, we have ${d}_{uw}\leq e^*+1.$ Therefore, we have $$\begin{aligned}
&{\cal F}\backslash{\cal F}_u(e^*)\\
=&\{(v,w)|(v,w)\in{{\cal E}}(\tilde{{T}}),v\in {{\cal I}},w\in {{\cal H}},{d}_{vw}=e^*+1\}\\
=&{\cal F}'_u.\end{aligned}$$ Based on equations \[eqn:MAP\] and \[eqn:MLE\], we conclude $$\begin{aligned}
&\arg\min_{u\in{\cal J}} \prod_{(v,w)\in{{\cal F}'_u}}(1-q_{vw})\\
=&\arg\max_{u\in{\cal J}} \sum_{(v,w)\in{{\cal F}'_u}}|\log(1-q_{vw})|\\
=&\arg\max_u\Pr(u|{{\cal O}}).\end{aligned}$$
[**Remark:**]{} Theorem \[thm:treepartialMAP\] contains two important properties for the MAP estimator on tree networks: 1) the MAP estimator is a Jordan infection center; 2) the Jordan infection center with minimum $\prod_{(v,w)\in{{\cal F}'_u}}(1-q_{vw})$ is the MAP estimator. The short-fat tree algorithm is designed based on these properties, which identifies the Jordan infection centers first and then selects the one with maximum $\sum_{(v,w)\in{{\cal F}'_u}}|\log(1-q_{vw})|.$
Proof of Theorem \[thm:sourceIsJordanER\] {#sec:ERJordanIsSource}
=========================================
We first introduce and recall some necessary notations. Consider an ER random graph ${g}.$
- Denote by ${s}$ the actual source.
- A node $v$ is said to locate on level $k$ if ${d}_{{s}v}=k.$ Denote by ${{\cal L}}_k$ the set of nodes from level $0$ to level $k$ and ${l}_k=|{{\cal L}}_k|.$
- The *descendants* of node $v$ in a tree are all the nodes in the subtree rooted at $v$ and $v$ is the *ancestor* of these nodes.
- The offsprings of a node on level $k$ (say $v$) the set of the nodes which are on level $k+1$ and have edges to $v.$ Denote by ${\Phi}(v)$ the offspring set of $v$ and ${\phi}(v)=|{\Phi}(v)|.$
- Denote by $p$ the wiring probability in the ER random graph.
- Denote by $n$ the total number of nodes.
- Denote by $\mu=np.$
- Recall that $\hbox{Bi}(n,p)$ is the binomial distribution with $n$ number of trials and each trial succeeds with probability $p.$
- Denote by $q$ the minimum infection probability of all the edges, i.e., $q=\min_{e\in{{\cal E}}({g})} q_e.$
For simplicity, we use ${d}_{vu}={d}^{g}_{vu}.$
We first elaborate the construction of the BFS tree. Denote by $v_{ij}$ the $j$th saturated node on level $i$ of the BFS tree from the source. $v_{01} = {s}$ is the first node on level zero. Denote by $b_i$ the number of nodes on level $i$ of the BFS tree starting from the source. We start with an empty graph ${T}^\dag.$ Initially, we add $v_{01}$ to the tree. Starting from $v_{01},$ we explore all neighbors $v_{11},v_{12},\cdots,v_{1b_1}$ of $v_{01},$ mark $v_{01}$ as saturated and add the edges from $v_{01}$ to $v_{11},v_{12},\cdots,v_{1b_1}$ to ${T}^\dag$. Then we explore all neighbors $v_{21},v_{22},\cdots,v_{2r_1}$ of $v_{11}$ in the set ${{\cal V}}(g)\backslash\{v_{01},v_{11},\cdots,v_{1b_1}\},$ mark $v_{11}$ as saturated and add the edges from $v_{11}$ to $v_{21},v_{22},\cdots,v_{2r_1}$ to ${T}^\dag$. Then we explore all neighbors $v_{2r_1+1},v_{2r_1+2},\cdots,v_{2r_1+r_2}$ of $v_{12}$ in the set ${{\cal V}}(g)\backslash\{v_{01},v_{11},\cdots,v_{1b_1},v_{21},v_{22},\cdots,v_{2r_1}\}$ and add the corresponding edges. Only after all nodes on level $i$ are saturated, we explore nodes on level $i+1.$ The exploration terminates after all nodes on level $t-1$ are saturated. The resulting tree ${T}^\dag$ is the BFS tree.
We further introduce some notations for the BFS tree.
- Denote by ${\Phi}'(v)$ the set of offsprings of node $v$ on ${T}^\dag$ and ${\phi}'(v)=|{\Phi}'(v)|.$
- Denote by ${g}_{t}$ the subgraph induced by all nodes within ${t}$ hops from $s$ on the ER graph. The *collision edges* are the edges which are not in ${T}^\dag$ but in ${g}_{t},$ i.e., $e\in {{\cal E}}({g}_{t})\backslash{{\cal E}}({T}^\dag).$ A node who is an end node of a collision edge is called a *collision node*. Denote by ${{\cal R}}_k$ the set of collision edges whose end nodes are on level $0$ to level $k$ and ${R}_k=|{{\cal R}}_k|.$
- Denote by ${\cal Z}^{i}_j(v)$ the set of nodes that are infected at time slot $i,$ on level $j$ and the descendants of node $v$ in the BFS tree ${T}^\dag$. In addition, denote by $Z^i_j(v)=|{\cal Z}^i_j(v)|$. We often use ${\cal Z}^{i}_j={\cal Z}^{i}_j({s})$ and $Z^i_j=Z^{i}_j({s})$ for simplicity.
We first define the probability space of the problem. Define the sample space $\Omega$ to be the set of live edge subgraphs of all ER graphs. The probability measure of a live-edge graph is defined by edge generations. Edge $(v,w)$ exists in a live edge subgraph with probability $pq_{vw}.$
To prove that ${s}$ is the *only* Jordan infection center, we consider the following asymptotically high probability events.
- [**Offsprings of each node.**]{} Define $$E_1=\{\forall v \in {{\cal L}}_{{t}-1}, \phi'(v)\in((1-\delta)\mu,(1+\delta)\mu)\}.$$ $E_1$, when occurs, provides upper and lower bounds for the number of offsprings of each node in ${{\cal L}}_{{t}-1}.$
- [**Collision edges.**]{} We define event $E_2$ when the following upper bound on the collision edges holds $${R}_j
\begin{cases} =0 & \mbox{if } 0<j\leq \lfloor m^-\rfloor, \\
\leq 8\mu & \mbox{if } \lfloor m^-\rfloor<j<\lceil m^+\rceil,\\
\leq \frac{4[(1+\delta)\mu]^{2j+1}}{n} & \mbox{if } \lceil m^+\rceil\leq j\leq \frac{\log n}{(1+\alpha)\log \mu}.
\end{cases}$$ where $m^+=\frac{\log n}{2\log[(1+\delta)\mu]}$ and $m^-=\frac{\log n-2\log \mu-\log 8}{2\log[(1+\delta)\mu]}.$ $E_2$ provides the upper bounds for collision edges at different levels. Note that a subgraph with diameter $\leq m^-$ is a tree with high probability since there is no collision edge.
- [**Infected nodes.**]{} Define $$\begin{aligned}
E_3&=\{Z^1_1\geq (1-\delta)^2\mu q\}\\
&\cap \{\forall v\in{\cal Z}^1_1, \cap_{i=2}^t Z^{i}_i(v)\geq (1-\delta)^2\mu qZ^{i-1}_{i-1}(v)\}.\end{aligned}$$ Level $1$ has at least $(1-\delta)^2\mu q$ infected nodes and the number of nodes grows exponentially by each level with a factor of $(1-\delta)^2\mu q.$ One immediate consequence of event $E_3$ is that $$\forall v \in {\cal Z}^1_1, Z^{{t}}_{{t}}(v)\geq [(1-\delta)^2\mu q]^{{t}-1},$$ i.e., there are at least $[(1-\delta)^2\mu q]^{{t}-1}$ infected descendants on level $t$ in ${T}^\dag$ for each infected node on level $1.$
Based on Lemma \[lem:ERAllOffspring\], \[lem:E1\] and \[lem:E3\], for any $\epsilon>0,$ with the union bound, we have that when $t\leq \frac{\log n}{(1+\alpha)\log \mu}$ and $n$ is sufficiently large, $$\begin{aligned}
&\Pr(E_1\cap E_2\cap E_3)\\
\geq & \Pr(E_1)\left(1-\Pr(\bar{E}_2|E_1)-\Pr(\bar{E}_3|E_1)\right)\\
\geq &1-\epsilon
\end{aligned}$$
Next, we show that ${s}$ is the only Jordan infection center when $E_1,E_2,E_3$ occur.
For $t\leq\lfloor m^-\rfloor,$ the nodes within $t$ hops from the source form a tree because there is no collision edge (due to event $E_2$). When event $E_3$ occurs, we have $\forall v \in {\cal Z}^1_1, Z^{{t}}_{{t}}(v)\geq [(1-\delta)^2\mu q]^{{t}-1}$ which means there exists at least one observed infected node on $t$ level for each subtree rooted on level $1.$ Consider infected node ${s}'.$ Recall that $a({s}')$ is the ancestor of ${s}'$ on level $1$ of ${T}^\dag.$ Consider node $u\in {\cal Z}^1_1$ such that $u\neq a({s}')$ and node $w\in {\cal Z}^t_t(u).$ We have $d^{{T}^\dag}_{{s}' w} = d^{{T}^\dag}_{{s}' {s}}+d^{{T}^\dag}_{{s}w}> t.$ Hence the infection eccentricity of ${s}'$ is larger than $t.$ Therefore, ${s}$ is the only Jordan infection center. The positions of ${s},{s}',a({s}'),u$ and $w$ are illustrated in Figure \[fig:onebyoneexploration\].
Consider the case $t>\lfloor m^-\rfloor$ and an infected node ${s}'$ on level $k\in[1,t].$ In the rest of the proof, we show that there exists node $v\in{{\cal I}}$ such that ${d}_{{s}' v}>t,$ which means that $s'$ cannot be the Jordan infection center.
![A pictorial example of ${\cal Z}^{t}_{t}(u)$ in BFS tree ${T}^\dag$[]{data-label="fig:onebyoneexplorationCopy"}](ER_lower_level_tree){width="\columnwidth"}
Consider node $u\in {\cal Z}_1^1, u\neq a({s}')$ (the existence of $u$ is guaranteed since $Z^1_1\geq (1-\delta)\mu q\geq 2$). For the convenience of the reader, we copied Figure \[fig:onebyoneexploration\] as Figure \[fig:onebyoneexplorationCopy\] which shows the relative positions of ${s}',a({s}'),u,$ and ${\cal Z}^{t}_{t}(u).$ The distance between a node in ${\cal Z}^{t}_{t}(u)$ and ${s}'$ on the tree ${T}^\dag$ is $k+t.$ Therefore, if ${s}'$ is the Jordan infection center, there exists at least one collision node on the path between ${s}'$ and each node in ${\cal Z}^{t}_{t}(u)$ to make the distance $\leq t.$
Define $H$ to be the total number of nodes each of which has the shortest path to ${s}'$ within $t$ hops and containing at least one collision node on $g_t$. If $H< Z^{t}_{t}(u),$ there exists a node $v\in{\cal Z}^{t}_{t}(u)$ such that ${d}_{{s}' v}>t.$ Therefore, ${s}'$ can not be the Jordan infection center and the theorem is proved.
In the rest of the proof, we will show that $H< Z^{t}_{t}(u).$ We first have the lower bound on $Z^{t}_{t}(u)$ according to $E_3,$ $$\begin{aligned}
Z^{t}_{t}(u)\geq [(1-\delta)^2\mu q]^{t-1}\label{eqn:ztt_upperbound}\end{aligned}$$ The upper bound of $H$ is computed in Lemma \[prop:collisionNodesUpperBounds\]. $$H\leq c[(1+\delta)\mu]^{\frac{3}{4}t+\frac{1}{2}}+c[(1+\delta)\mu]^{(\frac{5}{4}-\frac{\alpha}{2})t+2},$$
Since $\frac{1}{2}<\alpha<1,$ we have $\alpha=\frac{1}{2}+\alpha'$ where $0<\alpha'<\frac{1}{2}$ is a constant. Based on Lemma \[prop:collisionNodesUpperBounds\], we have $$\begin{aligned}
\frac{H}{Z^t_t(u)}&\leq
\frac{c[(1+\delta)\mu]^{\frac{3}{4}t+\frac{1}{2}}+c[(1+\delta)\mu]^{(\frac{3}{2}-\alpha)t+2}}{ [(1-\delta)^2\mu q]^{t-1}}\\
&\leq \frac{c[(1+\delta)\mu]^{\frac{3}{4}t+\frac{1}{2}}}{ [(1-\delta)^2\mu q]^{t-1}}+\frac{c[(1+\delta)\mu]^{(\frac{5}{4}-\frac{\alpha}{2})t+2}}{ [(1-\delta)^2\mu q]^{t-1}}\\
&= \frac{c}{\mu}\left(\frac{(1+\delta)^{\frac{3}{4}+\frac{1}{2t}}}{[(1-\delta)^2q]^{1-\frac{1}{t}}\mu^{\frac{1}{4}-\frac{5}{2t}}}\right)^t\\
&+\frac{c}{\mu}\left(\frac{(1+\delta)^{\frac{5}{4}-\frac{\alpha}{2}+\frac{2}{t}}}{[(1-\delta)^2q]^{1-\frac{1}{t}}\mu^{\frac{\alpha}{2}-\frac{1}{4}-\frac{4}{t}}}\right)^t\\
&\leq \frac{c}{\mu}\left(\frac{(1+\delta)^{\frac{3}{4}+\frac{1}{2t}}}{[(1-\delta)^2q]^{1-\frac{1}{t}}\mu^{\frac{1}{4}-\frac{4}{t}}}\right)^t\\
&+\frac{c}{\mu}\left(\frac{(1+\delta)^{1-\frac{\alpha'}{2}+\frac{2}{t}}}{[(1-\delta)^2q]^{1-\frac{1}{t}}\mu^{\frac{\alpha'}{2}-\frac{4}{t}}}\right)^t\end{aligned}$$ For $t>16/\alpha'$ we have $$\frac{H}{Z^t_t(u)}\leq \frac{2c}{\mu}\left(\frac{(1+\delta)}{[(1-\delta)^2q]\mu^{\frac{\alpha'}{4}}}\right)^t.$$ Since $\mu>3\log n$ and $\delta,q,\alpha'$ are constants, we have $$\frac{(1+\delta)}{[(1-\delta)^2q]\mu^{\frac{\alpha'}{4}}}<1$$ when $$n> \exp\left(\frac{1}{2}\left(\frac{(1+\delta)}{(1-\delta)^2q}\right)^{\frac{4}{\alpha'}}\right).$$ Therefore, we have $$\frac{H}{Z^t_t(u)}\leq \frac{2c}{\mu}\leq \epsilon',$$ where $\epsilon'\in(0,1)$ is a constant and the inequality holds for sufficiently large $n.$ Therefore, there are at least $(1-\epsilon')Z^t_t(u)$ nodes which cannot be reached from ${s}'$ on level $k$ with time $t.$ Hence we have ${e}({s}',{{\cal O}})>t,\forall {s}'\neq {s}.$
Bounds on the Number of Offsprings of Each Node
-----------------------------------------------
\[lem:ERAllOffspring\] Assume the conditions in Theorem \[thm:sourceIsJordanER\] hold, for any $\epsilon>0,$ we have $$\Pr\left(E_1\right)\geq 1-\epsilon$$ for sufficient large $n.$
Consider $\delta\in(0,1).$ Since ${t}\leq\frac{\log n}{(1+\alpha)\log\mu},$ we have for sufficiently large $n,$ $$\sum_{i=0}^{t}[(1+\delta)\mu]^i\leq 2[(1+\delta)\mu]^t\leq \delta'n,$$ where $\delta'\in(0,1)$ is a constant which can be arbitrarily close to $0$. This condition shows that the ${t}$ hop neighborhood of node ${s}$ includes at most a constant fraction of the total number of nodes.
Denote by ${{\cal E}}({{\cal V}}_1, {{\cal V}}_2)$ the set of edges between node set ${{\cal V}}_1$ and ${{\cal V}}_2.$ Recall that $v_{ij}$ is the $j$th nodes on level $i$ to be explored in the BFS tree starting from the source and $b_i$ is the number of nodes on level $i.$
Define the edge set from $v_{01}$ to all other nodes in the ER graph to be $$\Psi(v_{01}) = {{\cal E}}(\{v_{01}\}, {{\cal V}}(g)\backslash\{v_{01}\}),$$ which is the set of edges between $v_{01}$ and all other nodes in the graph.
Define $$\Phi'(v_{01})=\{v|(v,v_{01})\in \Psi(v_{01})\}.$$
Define $$\Psi(v_{01},v_{11}) = {{\cal E}}(\{v_{11}\}, {{\cal V}}(g)\backslash(\Phi'(v_{01})\cup \{v_{01}\})),$$ which is the set of edges from node $v_{11}$ to all nodes that are not already included in the BFS tree and $$\Phi'(v_{01},v_{11}) = \{v|(v,v_{11})\in \Psi(v_{01},v_{11})\},$$ which is the set of offsprings of $v_{11}.$
For simplicity, we use $\Psi(v_{ij})$ to denote $$\Psi(v_{01},v_{11},\cdots, v_{1 b_1},\cdots, v_{i1},\cdots,v_{ij}).$$ and use $\Phi'(v_{ij})$ to denote $$\Phi'(v_{01},v_{11},\cdots, v_{1 b_1},\cdots, v_{i1},\cdots,v_{ij}).$$
Iteratively, we define $$\begin{aligned}
&\Psi(v_{ij})\\
\triangleq &{{\cal E}}(\{v_{ij}\}, {{\cal V}}(g)\backslash(\{v_{01}\}\cup \Phi'(v_{01})\cup \cdots \cup \Phi'(v_{i j-1})))\end{aligned}$$ and $$\begin{aligned}
&\Phi'(v_{ij}) \triangleq\{v|(v,v_{ij})\in\Psi(v_{ij})\}\end{aligned}$$ which is the set of offsprings of node $v_{ij}$ in the BFS tree from the source. Define $\phi'(v_{ij}) = |\Phi'(v_{ij})|$ and $\psi(v_{ij}) = |\Psi(v_{ij})|.$
Note that $\Psi(v_{ij})$ uniquely determines $\Phi'(v_{ij})$ and vice versa. In addition, according to the definition, $\Psi(v_{ij})$ for any $i$ and $j$ are pairwise disjoint.
Define $$\Lambda(v_{ij}) = \{\Phi'(v_{ij})|\phi'(v_{ij})\in(\mu(1-\delta),\mu(1+\delta))\}.$$ which is the set of $\Phi'(v_{ij})$ which satisfies the given bounds on the number of offsprings.
Therefore, we have $$\begin{aligned}
&\Pr(E_1)\\
&=\Pr\left(\forall v \in {{\cal L}}_{{t}-1}, \phi'(v)\in(\mu(1-\delta),\mu(1+\delta))\right)\\
&= \sum_{\Phi'(v_{01})\in \Lambda(v_{01})}\Pr\Big(\Phi'(v_{01}), \\
& \phi'(v)\in(\mu(1-\delta),\mu(1+\delta)),\forall v \in {{\cal L}}_{{t}-1}\backslash\{v_{01}\}\Big)\\
& = \sum_{\Phi'(v_{01})\in \Lambda(v_{01})}\Pr(\Phi'(v_{01}))\Pr\big(\forall v \in {{\cal L}}_{{t}-1}\backslash\{v_{01}\},\\
& \phi'(v)\in(\mu(1-\delta),\mu(1+\delta))|\Phi'(v_{01})\big)\end{aligned}$$ Given $\Phi'(v_{01}),$ the order of the nodes to be explored during the construction of the BFS tree on the next level is determined and we have $$\begin{aligned}
&\Pr\left(\forall v \in {{\cal L}}_{{t}-1}, \phi'(v)\in(\mu(1-\delta),\mu(1+\delta))\right)\\
& = \sum_{\Phi'(v_{01})\in \Lambda(v_{01})}\Pr(\Phi'(v_{01}),\Phi'(v_{11}))\\
&\times\Pr\big(\forall v \in {{\cal L}}_{{t}-1}\backslash\{v_{01} , v_{11}\},\\
& \phi'(v)\in(\mu(1-\delta),\mu(1+\delta))|\Phi'(v_{01}) ,\Phi'(v_{11}) \big)\end{aligned}$$ Iteratively, we have $$\begin{aligned}
&\Pr\left(\forall v \in {{\cal L}}_{{t}-1}, \phi'(v)\in(\mu(1-\delta),\mu(1+\delta))\right)\\
& = \sum_{\Phi'(v_{01})\in \Lambda(v_{01})}\cdots \sum_{\Phi'(v_{t-1 b_{t-1}-1})\in \Lambda(v_{t-1 b_{t-1}-1})}\label{eqn:iterativeForNumberOfOffsprings1}\\
&\Pr(\Phi'(v_{01}),\cdots,\Phi'(v_{t-1 b_{t-1}-1}))\\
&\times\Pr\Big(\phi'(v_{t-1 b_{t-1}})\in(\mu(1-\delta),\mu(1+\delta))\\
&|\Phi'(v_{01}),\cdots,\Phi'(v_{t-1 b_{t-1}-1})\Big)\label{eqn:iterativeForNumberOfOffsprings2}\end{aligned}$$ Next, we focus on the last term in Equation (\[eqn:iterativeForNumberOfOffsprings2\]). Note, $\Psi(v_{ij})$ uniquely determines $\Phi'(v_{ij})$ and vice versa. Therefore, $$\begin{aligned}
\Pr\Big(&\phi'(v_{t-1 b_{t-1}})\in(\mu(1-\delta),\mu(1+\delta))\\
&|\Phi'(v_{01}),\cdots,\Phi'(v_{t-1 b_{t-1}-1})\Big)\\
= \Pr\Big(&\phi'(v_{t-1 b_{t-1}})\in(\mu(1-\delta),\mu(1+\delta))\\
&|\Psi(v_{01}),\cdots,\Psi(v_{t-1 b_{t-1}-1})\Big)\end{aligned}$$ Since $\Psi(v_{t-1 b_{t-1}})$ is disjoint with $\Psi(v_{01}),\cdots,\Psi(v_{t-1 b_{t-1}-1})$ and each edge is generated independently in the ER graph. Therefore, conditioned on $\Psi(v_{01}),\cdots,\Psi(v_{t-1 b_{t-1}-1}),$ we have $\phi'(v_{t-1 b_{t-1}})$ follows $$\hbox{Bi}(n - \sum_{i=0}^{t-2}\sum_{j=1}^{b_{i}}\phi'(v_{ij})-\sum_{j=1}^{b_{t-1}-1}\phi'(v_{t-1j})-1, p).$$
Note, $\phi(v_{01}),\cdots,\phi(v_{t-1 b_{t-1}-1})$ are in $(\mu(1-\delta),\mu(1+\delta))$ according to the condition in Equation (\[eqn:iterativeForNumberOfOffsprings1\]). Hence $$\sum_{i=0}^{t-2}\sum_{j=1}^{b_{i}}\phi'(v_{ij})+\sum_{j=1}^{b_{t-1}-1}\phi'(v_{t-1j})+1\leq\sum_{i = 0}^{t}[\mu(1+\delta))]^i$$
Therefore, $\phi'(v_{t-1 b_{t-1}})$ stochastically dominates $\hbox{Bi}(n - \sum_{i = 0}^{t}[\mu(1+\delta))]^i, p)$ and is stochastically dominated by $\hbox{Bi}(n,p)$ which implies
$$\begin{aligned}
&\Pr\big(\phi'(v_{t-1 b_{t-1}})\in(\mu(1-\delta),\mu(1+\delta))\\
&|\Phi'(v_{01}),\cdots,\Phi'(v_{t-1 b_{t-1}-1})\big)\\
&\geq 1-\Pr\left(\hbox{Bi}\left(n-\sum_{i = 0}^{t}[\mu(1+\delta))]^i,p\right)\leq (1-\delta)\mu\right)\\
&-\Pr\left(\hbox{Bi}\left(n,p\right)\geq \mu(1+\delta )\right)\end{aligned}$$
Note $\sum_{i=0}^{t}[(1+\delta )\mu]^i\leq \delta' n.$ Therefore, we have $$\begin{aligned}
&\Pr\big(\phi'(v_{t-1 b_{t-1}})\in(\mu(1-\delta),\mu(1+\delta))\\
&|\Phi'(v_{01}),\cdots,\Phi'(v_{t-1 b_{t-1}-1})\big)\\
&\geq 1-\Pr\left(\hbox{Bi}\left((1-\delta')n,p\right)\leq (1-\delta)\mu\right)\\
&-\Pr\left(\hbox{Bi}\left(n,p\right)\geq \mu(1+\delta )\right)\label{eqn:singleCase}\end{aligned}$$
By using the Chernoff bound in Lemma \[lem:chernoff\], we have $$\Pr\left(\hbox{Bi}\left((1-\delta')n,p\right)\leq \mu(1-\delta)\right)\leq \exp\left(-\frac{(\delta-\delta')^2\mu}{2(1-\delta')}\right),$$ and $$\Pr\left(\hbox{Bi}\left(n,p\right)\geq \mu(1+\delta)\right)\leq \exp\left(-\frac{\delta ^2\mu}{2+\delta }\right)$$ Substitute into Inequality (\[eqn:singleCase\]), we obtain $$\begin{aligned}
&\Pr\big(\phi'(v_{t-1 b_{t-1}})\in(\mu(1-\delta),\mu(1+\delta))\\
&|\Phi'(v_{01}),\cdots,\Phi'(v_{t-1 b_{t-1}-1})\big)\\
&\geq 1-\exp\left(-\frac{(\delta-\delta')^2\mu}{2(1-\delta')}\right) -\exp\left(-\frac{\delta ^2\mu}{2+\delta }\right)\triangleq \Delta\label{eqn:singleCaseFinal}\end{aligned}$$ Substitute Inequality (\[eqn:singleCaseFinal\]) into Equation (\[eqn:iterativeForNumberOfOffsprings2\]), we obtain $$\begin{aligned}
&\Pr\left(\forall v \in {{\cal L}}_{{t}-1}, \phi'(v)\in(\mu(1-\delta),\mu(1+\delta))\right)\\
&\geq \sum_{\Phi'(v_{01})\in \Lambda(v_{01})}\cdots \sum_{\Phi'(v_{t-1 b_{t-1}-1})\in \Lambda(v_{t-1 b_{t-1}-1})}\\
&\Pr(\Phi'(v_{01}),\cdots,\Phi'(v_{t-1 b_{t-1}-1}))\times\Delta\\
& = \Delta\sum_{\Phi'(v_{01})\in \Lambda(v_{01})}\cdots \sum_{\Phi'(v_{t-1 b_{t-1}-2})\in \Lambda(v_{t-1 b_{t-1}-2})}\\
&\Big( \sum_{\Phi'(v_{t-1 b_{t-1}-1})\in \Lambda(v_{t-1 b_{t-1}-1})}\Pr(\Phi'(v_{01}),\cdots,\Phi'(v_{t-1 b_{t-1}-1})) \Big)\\
& = \Delta\sum_{\Phi'(v_{01})\in \Lambda(v_{01})}\cdots \sum_{\Phi'(v_{t-1 b_{t-1}-2})\in \Lambda(v_{t-1 b_{t-1}-2})}\\
&\Pr(\Phi'(v_{01}),\cdots,\Phi'(v_{t-1 b_{t-1}-2}))\\
& \Pr\big(\phi'(v_{t-1 b_{t-1}-1})\in(\mu(1-\delta),\mu(1+\delta))\\
&|\Phi'(v_{01}),\cdots,\Phi'(v_{t-1 b_{t-1}-2})\big)\\
& = \Delta^2\sum_{\Phi'(v_{01})\in \Lambda(v_{01})}\cdots \sum_{\Phi'(v_{t-1 b_{t-1}-2})\in \Lambda(v_{t-1 b_{t-1}-2})}\\
&\Pr(\Phi'(v_{01}),\cdots,\Phi'(v_{t-1 b_{t-1}-2}))\label{eqn:IterativeApply}\end{aligned}$$ Applying Equation (\[eqn:IterativeApply\]) iteratively, we have $$\begin{aligned}
&\Pr\left(\forall v \in {{\cal L}}_{{t}-1}, \phi'(v)\in(\mu(1-\delta),\mu(1+\delta))\right)\\
\geq & \Delta^{\sum_{i=0}^{{t}-1}[(1+\delta )\mu]^i}\\
\geq & \bigg(1-\exp\left(-\frac{(\delta-\delta')^2\mu}{2(1-\delta')}\right)-\exp\left(-\frac{\delta ^2\mu}{2+\delta }\right)\bigg)^{\sum_{i=0}^{{t}-1}[(1+\delta )\mu]^i}\\\end{aligned}$$ When $\delta'\rightarrow 0,$ we have $$\frac{(\delta-\delta')^2\mu}{2(1-\delta')}\rightarrow \frac{\delta^2}{2}>\frac{\delta^2}{2+\delta}$$ Therefore, we can choose a sufficiently small $\delta'$ such that $$\begin{aligned}
&\Pr\left(\forall v \in {{\cal L}}_{{t}-1}, \phi'(v)\in(\mu(1-\delta),\mu(1+\delta))\right)\\
\geq & \left(1-2\exp\left(-\frac{\delta^2\mu}{2+\delta}\right)\right)^{\sum_{i=0}^{{t}-1}[(1+\delta )\mu]^i}\\
\geq & \left(1-2\exp\left(-\frac{\delta^2\mu}{2+\delta}\right)\right)^{2[(1+\delta )\mu]^{{t}-1}}\\
\geq_{(a)}& \exp\left(-8[(1+\delta )\mu]^{{t}-1}\exp\left(-\frac{\delta^2\mu}{2+\delta}\right)\right)\\
\geq & \exp\left(-8\exp\left(-\frac{\delta^2\mu}{2+\delta}+({t}-1)\log[(1+\delta)\mu]\right)\right),\end{aligned}$$ where $(a)$ is based on Lemma \[lem:exponentialBounds\] and holds when $\mu$ is sufficiently large (i.e., when $n$ is sufficiently large). To make the above bound greater than $1-\epsilon,$ we need $${t}\leq \frac{\frac{\delta^2\mu}{2+\delta}-\log8+\log\log\left(\frac{1}{1-\epsilon}\right)}{\log(1+\delta)+\log \mu}+1.$$ When $\mu> \frac{2+\delta}{\delta^2}\log n$, we have $$\begin{aligned}
{t}&\leq\frac{\log n}{(1+\alpha)\log\mu}\\
&<\frac{\log n-\log8+\log\log\left(\frac{1}{1-\epsilon}\right)}{\log(1+\delta)+\log\mu}+1.\end{aligned}$$ for sufficiently large $n.$
Note $ \frac{2+\delta}{\delta^2}\rightarrow 3$ when $\delta \rightarrow 1$ which matches the condition that $\mu>3\log n.$ Therefore, we prove the lemma.
Bounds on the Number of Collision Edges
---------------------------------------
Next, we analyze the number of collision edges on different levels. We have the following lemma.
\[lem:E1\] If the conditions in Theorem \[thm:sourceIsJordanER\] hold, for any $\epsilon>0,$ $$\Pr(E_2|E_1)\geq 1-\epsilon$$ for sufficiently large $n.$
We have $$\begin{aligned}
\Pr(E_2|E_1)&\geq 1-\Pr(R_{\lfloor m^-\rfloor}\neq 0|E_1)\\
&-\sum_{j=\lfloor m^-\rfloor+1}^{\lceil m^+\rceil-1}\Pr(R_j>8\mu|E_1)\\
&-\sum_{j=\lceil m^+\rceil}^t\Pr\left(R_j>\frac{4[(1+\delta)\mu]^{2j+1}}{n}|E_1\right)\end{aligned}$$
- [**No collision edge at the first $\lfloor m^- \rfloor$ levels.**]{}
We will show that $$\Pr\left({R}_{\lfloor m^-\rfloor}\neq 0|E_1\right)\leq 1-\exp\left(-\frac{1}{\mu}\right)\leq \frac{1}{\mu}$$ when $n$ is sufficiently large.
Conditioning on $E_1,$ we have ${R}_j$ is stochastically dominated by $\hbox{Bi}(l_j^2,p).$ Since $l_j\leq 2[(1+\delta)\mu]^j,$ ${R}_j$ is stochastically dominated by $\hbox{Bi}(4[(1+\delta)\mu]^{2j},p).$ We have for sufficiently large $n,$ $$\begin{aligned}
\Pr\left({R}_j=0\big|E_1\right)\geq& \left(1-p\right)^{\left(2 [(1+\delta)\mu]^j\right)^2}\\
= & \left(1-\frac{\mu}{n}\right)^{4 [(1+\delta)\mu]^{2j}}\\
\geq_{(a)} & \exp\left(-8[(1+\delta)\mu]^{2j}\frac{\mu}{n}\right)\\
\geq_{(b)}&\exp\left(-\frac{1}{\mu}\right)\end{aligned}$$ Inequality $(a)$ is based on Lemma \[lem:exponentialBounds\]. To obtain Inequality $(b),$ note $$j\leq m^-=\frac{\log n-2\log \mu-\log 8}{2\log[(1+\delta)\mu]}.$$ We have $$8[(1+\delta)\mu]^{2j}\frac{\mu}{n}\leq \frac{1}{\mu}$$ which explains $(b).$
- [**The number of collision edges at levels between $\lfloor m^-\rfloor+1$ and $\lceil m^+\rceil -1$.**]{}
We will show $$\Pr\left({R}_j> \frac{4[(1+\delta)\mu]^{2j+1}}{n}\bigg|E_1 \right)\leq\exp\left(-\frac{4\delta^2}{2+\delta}\mu\right)$$ when $n$ is sufficiently large.
Let $$\delta'=\frac{2n}{[(1+\delta)\mu]^{2j}}-1$$ Since $j\leq m^+ = \frac{\log n}{2\log[(1+\delta)\mu]},$ we have $n\geq[(1+\delta)\mu]^{2j}.$ Hence $$\delta'\geq 1$$ Conditioned on event $E_1,$ ${R}_j$ is stochastically dominated by $\hbox{Bi}(4[(1+\delta)\mu]^{2j},p).$ Using the Chernoff bound in Lemma \[lem:chernoff\], we have, $$\begin{aligned}
&\Pr\left({R}_j\leq (1+\delta')4[(1+\delta)\mu]^{2j}p \big|E_1\right)\\
&\geq \Pr\left(\hbox{Bi}(4[(1+\delta)\mu]^{2j},p)\leq (1+\delta')4[(1+\delta)\mu]^{2j}p\right)\\
&\geq 1-\exp\left(-\frac{\delta'^2}{2+\delta'} 4[(1+\delta)\mu]^{2j}\frac{\mu}{n}\right)\\
&\geq 1-\exp\left(-\frac{\delta'}{2+\delta'} 4(2n-[(1+\delta)\mu]^{2j})\frac{\mu}{n}\right)\\
&\geq_{(a)}1-\exp\left(-\frac{\delta'}{2+\delta'} 4\mu\right)\\
&\geq_{(b)}1-\exp\left(-\frac{4}{3}\mu\right)\\\end{aligned}$$ Note $(a)$ is due to $n>[(1+\delta)\mu]^{2j}$ and $(b)$ is due to $\delta'\geq 1.$ Note, $$\begin{aligned}
&(1+\delta')4[(1+\delta)\mu]^{2j}p\\
=&\left(1+\frac{2n}{[(1+\delta)\mu]^{2j}}-1\right)4[(1+\delta)\mu]^{2j}p\\
=&\frac{2n}{[(1+\delta)\mu]^{2j}}4[(1+\delta)\mu]^{2j}\frac{\mu}{n}\\
=& 8 \mu\end{aligned}$$
- [**The number of collision edges at levels between $\lceil m^+\rceil$ and $\frac{\log n}{(1+\alpha) \log \mu}.$** ]{}
We will show that $$\Pr\left({R}_{j}> 8\mu|E_1\right)\leq \exp\left(-\frac{4}{3}\mu\right)$$ when $n$ is sufficiently large.
Conditioned on event $E_1,$ ${R}_j$ is stochastically dominated by $\hbox{Bi}(l_j^2,p).$ Since $l_j\leq 2[(1+\delta)\mu]^j,$ ${R}_j$ is stochastically dominated by $\hbox{Bi}(4[(1+\delta)\mu]^{2j},p).$ Then $$\begin{aligned}
&\Pr\left({R}_j\leq \frac{4 [(1+\delta)\mu]^{2j+1}}{n} \big|E_1\right)\\
&\geq \Pr\left(\hbox{Bi}(4[(1+\delta)\mu]^{2j},p)\leq \frac{4 [(1+\delta)\mu]^{2j+1}}{n}\right)\\
&\geq \Pr\left(\hbox{Bi}(4[(1+\delta)\mu]^{2j},p)\leq (1+\delta)4 [(1+\delta)\mu]^{2j}p\right)\\
&\geq 1-\exp\left(-\frac{\delta^2}{2+\delta} 4[(1+\delta)\mu]^{2j}\frac{\mu}{n}\right)\\
&\geq_{(a)} 1-\exp\left(-\frac{\delta^2}{2+\delta}4\mu\right)\end{aligned}$$ From $j\geq \lceil m^+\rceil\geq \frac{\log n}{2\log[(1+\delta)\mu]},$ we obtain $$\begin{aligned}
n\leq [(1+\delta)\mu]^{2j}.\label{eqn:LowLevelCollision}\end{aligned}$$ we obtain Inequality $(a)$ by substituting Inequality (\[eqn:LowLevelCollision\]).
Since $m^+-m^-< 2$ we have $$\begin{aligned}
\Pr(E_2|E_1)\geq & 1-\frac{1}{\mu}-(m^+-m^-)\exp\left(-\frac{4}{3}\mu\right)\\
&-\sum_{j=\lceil m ^+\rceil}^t\exp\left(-\frac{4\delta^2}{2+\delta}\mu\right)\\
\geq & 1-\frac{1}{\mu}-2\exp\left(-\frac{4}{3}\mu\right)-t\exp\left(-\frac{4\delta^2}{2+\delta}\mu\right)\end{aligned}$$
Note we have $\mu\geq 3\log n$ and ${t}\leq\frac{\log n}{(1+\alpha)\log\mu},$ therefore, for $n$ sufficiently large, we have $$\Pr(E_2|E_1)\geq 1-\epsilon.$$
Bounds on the Number of Infected Nodes
--------------------------------------
\[lem:E3\] Assume the conditions in Theorem \[thm:sourceIsJordanER\] hold, for any $\epsilon>0,$ we have $$\Pr\left(E_3|E_1\right)\geq 1-\epsilon$$ for sufficiently large $n.$
- We first show that for any $\epsilon>0,$ $$\Pr(Z^1_1\geq(1-\delta)^2\mu q|E_1)\geq 1-\epsilon$$ for sufficient large $n.$ $Z^1_1$ is lower bounded by a binomial distribution $\hbox{Bi}((1-\delta)\mu, q).$ Hence using Chernoff bound, we have $$\Pr(Z^1_1\geq(1-\delta)(1-\delta)\mu q|E_1)\geq 1-\exp\left(-\frac{\delta^2}{2}(1-\delta)\mu q\right)$$ Note $\mu\rightarrow \infty$ while all other parameters are constants, for any $\epsilon>0,$ we have $$\Pr(Z^1_1\geq(1-\delta)^2\mu q|E_1)\geq 1-\epsilon$$
- We show that for any $\epsilon>0,$ $$\Pr\left(\{\forall v \in {\cal Z}^1_1, Z^{{t}}_{{t}}(v)\geq [(1-\delta)^2\mu q]^{{t}-1} \}|E_1\right)\geq 1-\epsilon$$ for sufficiently large $n.$ Define $$E_4=\{(1-\delta)^2\mu q\leq Z^1_1\leq(1+\delta)\mu\}$$ Note when $n$ is sufficiently large, the following holds $$\Pr(Z^1_1\geq(1-\delta)^2\mu q)\geq 1-\frac{\epsilon}{4}.$$ When $E_1$ occurs, we have $Z^1_1\leq(1+\delta)\mu.$ Therefore, we have $$\Pr(E_4|E_1)\geq 1-\frac{\epsilon}{4}$$
Define $$\begin{aligned}
&{\cal S}^t_2(v)=\{({\cal Z}^2_2(v),\cdots,{\cal Z}^t_t(v))| \\
&\cap_{i=2}^t Z^i_i(v)\geq (1-\tilde{\delta} )(1-\delta)\mu qZ^{i-1}_{i-1}(v).\}\end{aligned}$$ We have $$\begin{aligned}
&\Pr\left(\cap_{i=2}^t Z^{i}_i(v)\geq (1-\tilde{\delta} )(1-\delta)\mu qZ^{i-1}_{i-1}(v)|E_4,E_1\right)\\
=&\Pr\big( Z^{t}_t(v)\geq (1-\tilde{\delta} )(1-\delta)\mu qZ^{t-1}_{t-1}(v),\\
&\cap_{i=2}^{t-1} Z^{i}_i(v)\geq (1-\tilde{\delta} )(1-\delta)\mu qZ^{i-1}_{i-1}(v)|E_4,E_1\big)\\
&= \sum_{{\cal Z}^2_2(v),\cdots,{\cal Z}^{t-1}_{t-1}(v)\in {\cal S}^{t-1}_2(v)}\Pr\big( Z^{t}_t(v)\geq \\
&(1-\tilde{\delta} )(1-\delta)\mu qZ^{t-1}_{t-1}(v)|{\cal Z}^{2}_2(v),\cdots, {\cal Z}^{t-1}_{t-1}(v), E_4,E_1\big)\\
&\times \Pr({\cal Z}^{2}_2(v),\cdots, {\cal Z}^{t-1}_{t-1}(v)| E_4,E_1)\label{eqn:ZiiIterative}\end{aligned}$$
Conditioned on $E_1$ and $E_4,$ we have $Z_1^1\neq 0.$ For any $v\in {\cal Z}^1_1,$ $Z^i_i(v)$ stochastically dominates $\hbox{Bi}((1-\delta)\mu Z^{i-1}_{i-1}(v),q)$ given ${\cal Z}^{i-1}_{i-1}(v).$ Therefore, denote by $\tilde{\delta} \in(0,1),$ we have $$\begin{aligned}
&\Pr\big( Z^{t}_t(v)\geq (1-\tilde{\delta} )(1-\delta)\mu qZ^{t-1}_{t-1}(v)\\
&|{\cal Z}^{2}_2(v),\cdots, {\cal Z}^{t-1}_{t-1}(v), E_4,E_1\big)\\
\geq &\Pr(Z^t_t(v)\geq (1-\tilde{\delta} )(1-\delta)\mu qZ^{t-1}_{t-1}(v)|{\cal Z}^{t-1}_{t-1}(v),E_4,E_1)\\
\geq & 1-\exp\left(-\frac{\tilde{\delta} ^2(1-\delta)\mu q Z^{t-1}_{t-1}(v)}{2+\tilde{\delta} }\right)\\
\geq & \exp\left(-2\exp\left(-\frac{\tilde{\delta} ^2(1-\delta)\mu q Z^{t-1}_{t-1}(v)}{2+\tilde{\delta} }\right)\right)\end{aligned}$$ Since we have ${\cal Z}^2_2(v),\cdots,{\cal Z}^{t-1}_{t-1}(v)\in {\cal S}^{t-1}_2(v),$ therefore, $$Z^{t-1}_{t-1}(v)\geq [(1-\tilde{\delta} )(1-\delta)\mu q]^{t-2}$$ Hence, we have $$\begin{aligned}
&\Pr\big( Z^{t}_t(v)\geq (1-\tilde{\delta} )(1-\delta)\mu qZ^{t-1}_{t-1}(v)\\
&|{\cal Z}^{2}_2(v),\cdots, {\cal Z}^{t-1}_{t-1}(v), E_4,E_1\big)\\
\geq & \exp\left(-2\exp\left(-\frac{\tilde{\delta} ^2(1-\delta)\mu q [(1-\tilde{\delta} )(1-\delta)\mu q]^{t-2}}{2+\tilde{\delta} }\right)\right)\\
\geq & \exp\left(-2\exp\left(-\frac{\tilde{\delta} ^2[(1-\tilde{\delta} )(1-\delta)\mu q]^{t-1}}{(2+\tilde{\delta}) (1-\tilde{\delta} )}\right)\right)\end{aligned}$$ Substituting back to Equation (\[eqn:ZiiIterative\]), we obtain $$\begin{aligned}
&\Pr\left(\cap_{i=2}^t Z^{i}_i(v)\geq (1-\tilde{\delta} )(1-\delta)\mu qZ^{i-1}_{i-1}(v)|E_4,E_1\right)\\
&\geq \sum_{{\cal Z}^2_2(v),\cdots,{\cal Z}^{t-1}_{t-1}(v)\in {\cal S}^{t-1}_2(v)}\exp\Big(-2\exp\big(\\
&-\frac{\tilde{\delta} ^2[(1-\tilde{\delta} )(1-\delta)\mu q]^{t-1}}{(2+\tilde{\delta}) (1-\tilde{\delta} )}\big)\Big)\\
&\times \Pr({\cal Z}^{2}_2(v),\cdots, {\cal Z}^{t-1}_{t-1}(v)| E_4,E_1)\\
&=\exp\left(-2\exp\left(-\frac{\tilde{\delta} ^2[(1-\tilde{\delta} )(1-\delta)\mu q]^{t-1}}{(2+\tilde{\delta}) (1-\tilde{\delta} )}\right)\right)\\
&\times\sum_{{\cal Z}^2_2(v),\cdots,{\cal Z}^{t-1}_{t-1}(v)\in {\cal S}^{t-1}_2(v)}\Pr({\cal Z}^{2}_2(v),\cdots, {\cal Z}^{t-1}_{t-1}(v)| E_4,E_1)\\
&=\exp\left(-2\exp\left(-\frac{\tilde{\delta} ^2[(1-\tilde{\delta} )(1-\delta)\mu q]^{t-1}}{(2+\tilde{\delta}) (1-\tilde{\delta} )}\right)\right)\\
&\times \Pr(\cap_{i=2}^{t-1} Z^{i}_i(v)\geq (1-\tilde{\delta} )(1-\delta)\mu qZ^{i-1}_{i-1}(v)| E_4,E_1)
\label{eqn:ZiiIterative2}\end{aligned}$$ Use Equation (\[eqn:ZiiIterative2\]) iteratively on all levels, we obtain $$\begin{aligned}
&\Pr\left(\cap_{i=2}^t Z^{i}_i(v)\geq (1-\tilde{\delta} )(1-\delta)\mu qZ^{i-1}_{i-1}(v)|E_4,E_1\right)\\
\geq & \prod_{i=2}^t\exp\left(-2\exp\left(-\frac{\tilde{\delta} ^2[(1-\tilde{\delta} )(1-\delta)\mu q]^{i-1}}{(2+\tilde{\delta}) (1-\tilde{\delta} )}\right)\right)\\
=& \exp\left(-2\sum_{i=2}^t\exp\left(-\frac{\tilde{\delta} ^2[(1-\tilde{\delta} )(1-\delta)\mu q]^{i-1}}{(2+\tilde{\delta}) (1-\tilde{\delta} )}\right)\right)\\
\geq& \exp\left(-2(t-1)\exp\left(-\frac{\tilde{\delta} ^2(1-\delta )\mu q}{2+\tilde{\delta} }\right)\right)\\
\geq & 1-2(t-1)\exp\left(-\frac{\tilde{\delta} ^2(1-\delta )\mu q}{2+\tilde{\delta} }\right)\end{aligned}$$ Using the union bound for all $v\in{\cal Z}^1_1,$ we have $$\begin{aligned}
&\Pr\bigg(\forall v\in{\cal Z}^1_1, \cap_{i=2}^t Z^{i}_i(v)\\
&\geq (1-\tilde{\delta} )(1-\delta)\mu qZ^{i-1}_{i-1}(v)|E_4,E_1\bigg)\\
\geq &1-2(1+\delta)\mu t\exp\left(-\frac{\tilde{\delta} ^2(1-\delta )\mu q}{2+\tilde{\delta} }\right)\end{aligned}$$ Note $t\leq \frac{\log n }{(1+\alpha)\log \mu}$ and $\mu>3\log n.$ We have $t\leq \log n \leq \mu,$ and
$$\begin{aligned}
&\Pr\bigg(\forall v\in{\cal Z}^1_1, \cap_{i=1}^t Z^{i}_i(v)\\
&\geq (1-\tilde{\delta} )(1-\delta)\mu qZ^{i-1}_{i-1}(v)|E_4,E_1\bigg)\\
&\geq 1-2(1+\delta)\mu^2\exp\left(-\frac{\tilde{\delta} ^2(1-\delta )\mu q}{2+\tilde{\delta} }\right)\\
&\geq 1-\frac{\epsilon}{2}\end{aligned}$$
for sufficiently large $n.$
Define $$E_5 = \{\forall v\in{\cal Z}^1_1, \cap_{i=2}^t Z^{i}_i(v)\geq (1-\delta)^2\mu qZ^{i-1}_{i-1}(v)\}$$ We further have, $$\begin{aligned}
&\Pr\left(E_5|E_1\right)\\
\geq &\Pr\left(E_5|E_4,E_1\right)\\
\times & \Pr(E_4|E_1)\\
\geq & (1-\frac{\epsilon}{4})(1-\frac{\epsilon}{4})\\
\geq & 1-\frac{\epsilon}{2}.\end{aligned}$$ Choosing $\tilde{\delta}=\delta,$ we have $$\begin{aligned}
&\Pr\left(\forall v\in{\cal Z}^1_1, \cap_{i=2}^t Z^{i}_i(v)\geq (1-\delta)^2\mu qZ^{i-1}_{i-1}(v)|E_1\right)\\
\geq & 1-\frac{\epsilon}{2}.\end{aligned}$$
Note $E_3 = E_4\cap E_5.$ We have $$\begin{aligned}
\Pr(E_3|E_1) &= \Pr( E_4\cap E_5|E_1)\\
&\geq 1- \Pr( \bar{E}_4|E_1)- \Pr(\bar{E}_5|E_1) \\
&\geq 1-\epsilon.\end{aligned}$$ where $\bar{E}_4, \bar{E}_5$ are the complement of event $E_4,E_5.$
\[prop:collisionNodesUpperBounds\] When $E_1$ and $E_2$ occurs, if $\lfloor m^-\rfloor<t\leq\frac{\log n}{(1+\alpha)\log \mu},$ we have $$H\leq c[(1+\delta)\mu]^{\frac{3}{4}t+\frac{1}{2}}+c[(1+\delta)\mu]^{(\frac{5}{4}-\frac{\alpha}{2})t+2},$$ where $c$ is a constant.
![A pictorial example of upper bounds of $H$ []{data-label="fig:Hupperbound"}](ER_collision_node_tree_new){width="\columnwidth"}
Define a *collision removed* breadth-first search tree to be a BFS tree on the graph with all collision nodes removed. Denote by ${\cal U}^{h}(v)$ the set of nodes in the [**collision removed**]{} BFS tree from node $v$ with $h$ hops in $g_t$ and $U^h(v)=|{\cal U}^{h}(v)|.$ Denote by $\tilde{{\cal U}}^{h}(v)$ the set of nodes that are within $h$ hops from node $v$ on $g_t.$ Recall that a node $v$ is said to locate on level $j$ if ${d}_{{s}v}=j.$ Denote by ${\cal C}_j$ the set of collision nodes on level $j$ in $g_t.$ Therefore, we have $$H\leq \left|\cup_{j=0}^t\cup_{v\in {\cal C}_j} \tilde{{\cal U}}^{t-d_{{s}' v}}(v)\right|,$$ where $\tilde{\cal U}^{t-d_{s'v}}(v)$ is the set of nodes that can be reached from $s'$ within $t$ hops via the collision node $v.$ Next, we will prove that $$\cup_{j=0}^t\cup_{v\in {\cal C}_j} \tilde{{\cal U}}^{t-d_{{s}' v}}(v) = \cup_{j=0}^t\cup_{v\in {\cal C}_j} {\cal U}^{t-d_{{s}' v}}(v)$$ Since ${\cal U}^{t-d_{{s}' v}}(v)\subset \tilde{{\cal U}}^{t-d_{{s}' v}}(v),$ we have $$\cup_{j=0}^t\cup_{v\in {\cal C}_j} \tilde{{\cal U}}^{t-d_{{s}' v}}(v) \supseteq \cup_{j=0}^t\cup_{v\in {\cal C}_j} {\cal U}^{t-d_{{s}' v}}(v).$$ We only need to show that $$\cup_{j=0}^t\cup_{v\in {\cal C}_j} \tilde{{\cal U}}^{t-d_{{s}' v}}(v) \subseteq \cup_{j=0}^t\cup_{v\in {\cal C}_j} {\cal U}^{t-d_{{s}' v}}(v).$$ For any node $w\in \cup_{j=0}^t\cup_{v\in {\cal C}_j} \tilde{{\cal U}}^{t-d_{{s}' v}}(v),$ we consider the following cases.
- If $w$ is a collision node, we have $w\in {\cal U}^{t-d_{{s}' w}}(w).$ Hence $w\in \cup_{j=0}^t\cup_{v\in {\cal C}_j} {\cal U}^{t-d_{{s}' v}}(v).$
- If $w$ is not a collision node, there exists $v'$ such that $w \in \tilde{{\cal U}}^{t-d_{{s}' v'}}(v').$
- If the shortest path from $w$ to $v'$ does not contain any other collision nodes, we have $w\in {\cal U}^{t-d_{{s}' v'}}(v').$
- If the shortest path from $w$ to $v'$ contains other collision nodes, denote by $u$ the collision node on that path which is the closest to node $w.$ Therefore, there is no collision node on the shortest path from node $u$ to node $w.$ We have $$d_{uw}=d_{wv'}-d_{uv'}$$ Note we have $w\in \tilde{{\cal U}}^{t-d_{{s}' v'}}(v'),$ therefore $d_{wv'}\leq t-d_{{s}' v'}.$ Hence, we have $$d_{uw}=d_{wv'}-d_{uv'}\leq t-d_{{s}' v'}-d_{uv'}\leq_{(a)} t-d_{{s}'u},$$ where $(a)$ is due to the triangle inequality. Therefore, we have $$d_{uw}\leq t-d_{{s}'u},$$ and the shortest path from node $u$ to node $w$ contains no collision nodes. Hence, $$w\in {\cal U}^{t-d_{{s}' u}}(u)$$
As a summary, we proved $$\cup_{j=0}^t\cup_{v\in {\cal C}_j} \tilde{{\cal U}}^{t-d_{{s}' v}}(v) = \cup_{j=0}^t\cup_{v\in {\cal C}_j} {\cal U}^{t-d_{{s}' v}}(v)$$ Now we can use the collision removed BFS tree to bound $H$ since the branch of collision nodes of traditional BFS tree are already counted in the collision removed BFS tree rooted at these collision nodes. For example, consider the collision removed BFS tree from node $w$ in Figure \[fig:Hupperbound\]. We ignore the presence of node $u$ since the branch of node $u$ are already considered in the collision removed BFS tree rooted at node $u.$
Hence, we have $$\begin{aligned}
H&\leq \left|\cup_{j=0}^t\cup_{v\in {\cal C}_j} \tilde{{\cal U}}^{t-d_{{s}' v}}(v)\right| \\
&=\left|\cup_{j=0}^t\cup_{v\in {\cal C}_j} {\cal U}^{t-d_{{s}' v}}(v)\right|\\
&\leq \sum_{j=0}^t\sum_{v \in {\cal C}_j}U^{t-d_{{s}' v}}(v)\end{aligned}$$
Since $U^{\lambda}(u)$ is an increasing function of $\lambda$ and $d_{{s}'v}\geq |k-j|$ for node $v$ on level $j,$ we have $$H\leq \sum_{j=0}^t\sum_{v \in {\cal C}_j}U^{t-|k-j|}(v).$$ We next establish the lemma using the following steps.
- [**Step 1: Upper bound on $U^{t-|k-j|}(w).$**]{} Denote by $\hbox{par}(w)$ the parent of $w$ on the BFS tree from the source and denote by $\hbox{par}^i(w)$ the $i$th ancestor of $w$ on the BFS tree from the source. For example, $v$ is the first ancestor of $w$ ($\hbox{par}(w) = v$) and $\hbox{par}(v)$ is the second ancestor of $w$ ($\hbox{par}^2(w)= \hbox{par}(v)$) as shown in Figure \[fig:Hupperbound\]. Denote by $\sigma^h(v)$ the number of nodes in the collision removed BFS subtree rooted in node $v$ with height $h$ without branch of $\hbox{par}(v).$
Consider node $w$ in Figure \[fig:Hupperbound\] and ignore the presence of the collision nodes. Firstly, we remove the branch of the parent of $w.$ The remaining nodes on the tree are below the level of $w$. This is because the level of a node $w'$s neighbor can differ from node $w'$s level by at most one and those neighbors that are at the same or higher levels must be collision nodes. The height of the tree is no larger than the total number of hops $\lambda.$ On the other hand, the tree only contains the nodes within $t$ hops from the actual source ${s}$ since the tree is based on the infection subnetwork. Since $w$ locates on level $j,$ the height of the tree must be no larger than $t-j.$ Therefore, the maximum height of the tree is $\min(t-j,\lambda)$ and denote by $\sigma^{\min(t-j,\lambda)}(w)$ the total number of nodes in the tree as shown in Figure \[fig:Hupperbound\].
Next, we consider the branch of the parent of $w$ ($v=\hbox{par}(w)$) in Figure \[fig:Hupperbound\]. Note $w$ has only one parent $v$ (all other parents are collision nodes thus removed). Since we considered the $\lambda$ hops of the removed collision BFS tree rooted at $w$ and it takes one hop from $w$ to $v,$ the branch of node $v$ in the collision removed BFS tree is contained in ${\cal U}^{\lambda-1}(v).$
Therefore, we have $$\begin{aligned}
U^{\lambda}(w) \leq& \sigma^{\min(t-j,\lambda)}(w)+U^{\lambda-1}(v)\\
=&\sigma^{\min(t-j,\lambda)}(w)+U^{\lambda-1}(\hbox{par}(w))\label{eqn:Urecursive}\end{aligned}$$ Repeatedly using Equation (\[eqn:Urecursive\]), we have $$\begin{aligned}
U^{\lambda}(w)\leq\sum_{i=0}^{\min(\lambda,j)}\sigma^{\min(t-(j-i),\lambda-i)}(\hbox{par}^i(w)).\label{eqn:Urepeat}\end{aligned}$$ Note the maximum number of hops upward is no larger than $\lambda$ and the total number of levels above $w$ is no larger than $j.$ Therefore, we only need to consider $\min(\lambda,j)$ levels above $w$ in Equation (\[eqn:Urepeat\]).
Intuitively, the upper bound on $U^{\lambda}(w)$ is a collection of trees rooted at level $j-i$ with height $\min(t-(j-i),\lambda-i),\forall i\leq j.$ For example, in Figure \[fig:Hupperbound\], the blue area shows the tree rooted in level $j$ with height $\min(t-j,\lambda)$ and the green area shows in tree rooted in level $j-1$ with height $\min(t-(j-1),\lambda-1).$ In this example, we consider the removed collision BFS tree rooted at $w.$ The blue area is the collision removed BFS tree from $w$ after further removing the branch from $v$ and the green area is the collision removed BFS tree from $v$ by further removing the branch from $\hbox{par}(v).$ The height is no larger than $t-(j-1)$ since node $v$ locates on level $j-1$ and we consider the $t$ hop neighborhood of $s.$ In addition, the height is no larger than $\lambda-1$ since it takes one hop from node $w$ to node $v$ and the total number of possible hops from node $w$ is $\lambda.$
According to $E_1,$ we have $$\sigma^{l}(v)\leq\sum_{h=0}^{l}[(1+\delta)\mu]^h,\forall v \in {{\cal V}}(g_t).$$ Hence, $$\begin{aligned}
U^{\lambda}(w)\leq
&\sum_{i=0}^{\min(\lambda,j)}\sum_{h=0}^{\min(\lambda-i,t-(j-i))}[(1+\delta)\mu]^{h}.\end{aligned}$$ Consider $\lambda\in(0,t)$ and $j\in[1,t].$ We obtain an upper bound on $U^{\lambda}(w)$ by analyzing different ranges of $\lambda.$
- $\lambda < j.$ We have $\min(\lambda,j)=\lambda.$
- $\lambda < t-j.$ We have $\min(\lambda-i,t-(j-i))=\lambda-i.$ Hence, $$\begin{aligned}
U^{\lambda}(w)\leq
&\sum_{i=0}^{\lambda}\sum_{h=0}^{\lambda-i}[(1+\delta)\mu]^{h}\\
\leq & \sum_{i=0}^{\lambda}2[(1+\delta)\mu]^{\lambda-i}\\
\leq & 4[(1+\delta)\mu]^{\lambda}.\end{aligned}$$
- $\lambda\geq t-j.$ When $i\leq \frac{\lambda-t+j}{2},$ we have $\lambda-i>t-(j-i).$ Therefore, $\min(\lambda-i,t-(j-i))=t-(j-i).$ Hence, $$\begin{aligned}
&U^{\lambda}(w)\\
\leq
&\sum_{i=0}^{\left\lfloor\frac{\lambda-t+j}{2}\right\rfloor}\sum_{h=0}^{t-j+i}[(1+\delta)\mu]^{h} \\
&+ \sum_{i=\left\lceil\frac{\lambda-t+j}{2}\right\rceil}^{\lambda}\sum_{h=0}^{\lambda -i}[(1+\delta)\mu]^{h}\\
\leq & \sum_{i=0}^{\left\lfloor\frac{\lambda-t+j}{2}\right\rfloor}2[(1+\delta)\mu]^{t-j+i} \\
&+ \sum_{i=\left\lceil\frac{\lambda-t+j}{2}\right\rceil}^{\lambda}2[(1+\delta)\mu]^{\lambda -i }\\
\leq & 4[(1+\delta)\mu]^{t-j+\left\lfloor\frac{\lambda-t+j}{2}\right\rfloor}\\
&+4[(1+\delta)\mu]^{\lambda -\left\lceil\frac{\lambda-t+j}{2}\right\rceil }\\
\leq & 4[(1+\delta)\mu]^{t-j+\frac{\lambda-t+j}{2}}+4[(1+\delta)\mu]^{\lambda -\frac{\lambda-t+j}{2} }\\
\leq & 8[(1+\delta)\mu]^{\frac{\lambda+t-j}{2}}\end{aligned}$$
- $\lambda \geq j.$ We have $\min(\lambda,j)=j.$
- $\lambda < t-j.$ We have $\min(\lambda-i,t-(j-i))=\lambda-i.$ Hence, $$\begin{aligned}
U^{\lambda}(w)\leq
&\sum_{i=0}^{j}\sum_{h=0}^{\lambda-i}[(1+\delta)\mu]^{h}\\
\leq & \sum_{i=0}^{j}2[(1+\delta)\mu]^{\lambda-i}\\
\leq & 4[(1+\delta)\mu]^{j}.\end{aligned}$$
- $\lambda\geq t-j.$ When $i\leq \frac{\lambda-t+j}{2},$ we have $\lambda-i>t-(j-i).$ Hence, $$\begin{aligned}
&U^{\lambda}(w)\\
\leq
&\sum_{i=0}^{\left\lfloor\frac{\lambda-t+j}{2}\right\rfloor}\sum_{h=0}^{t-j+i}[(1+\delta)\mu]^{h} \\
&+ \sum_{i=\left\lceil\frac{\lambda-t+j}{2}\right\rceil}^{j}\sum_{h=0}^{\lambda -i}[(1+\delta)\mu]^{h}\\
\leq & \sum_{i=0}^{\left\lfloor\frac{\lambda-t+j}{2}\right\rfloor}2[(1+\delta)\mu]^{t-j+i} \\
&+ \sum_{i=\left\lceil\frac{\lambda-t+j}{2}\right\rceil}^{j}2[(1+\delta)\mu]^{\lambda -i }\\
\leq & 4[(1+\delta)\mu]^{t-j+\left\lfloor\frac{\lambda-t+j}{2}\right\rfloor}+4[(1+\delta)\mu]^{\lambda -\left\lceil\frac{\lambda-t+j}{2}\right\rceil }\\
\leq & 4[(1+\delta)\mu]^{t-j+\frac{\lambda-t+j}{2}}+4[(1+\delta)\mu]^{\lambda -\frac{\lambda-t+j}{2} }\\
\leq & 8[(1+\delta)\mu]^{\frac{\lambda+t-j}{2}}\end{aligned}$$
As a summary, we have $$U^{\lambda}(w)\leq \begin{cases} 4[(1+\delta)\mu]^{\min(\lambda,j)}. & \mbox{if } \lambda< t-j \\
8[(1+\delta)\mu]^{\frac{\lambda+t-j}{2}}& \mbox{if } \lambda\geq t-j.
\end{cases}$$ For simplicity, we define $U^{\lambda}_j$ to be the upper bound on $U^{\lambda}(w)$ for $w$ on level $j.$ We have $$\begin{aligned}
U^{\lambda}_j= \begin{cases} 4[(1+\delta)\mu]^{\min(\lambda,j)}. & \mbox{if } \lambda< t-j \\
8[(1+\delta)\mu]^{\frac{\lambda+t-j}{2}}& \mbox{if } \lambda\geq t-j.
\end{cases}\label{eqn:upperboundU}\end{aligned}$$ and $U^{\lambda}(w)\leq U^{\lambda}_j$ where the subscript means the level of the nodes and the superscript means the number of hops.
Hence, we have $$H\leq \sum_{j=0}^t\sum_{v \in {\cal C}_j}U^{t-|k-j|}(v)\leq \sum_{j=0}^t|{\cal C}_j| U^{t-|k-j|}_j$$
- [**Step 2: Upper bound on $|{\cal C}_j|.$**]{} Recall that ${\cal C}_j$ is the set of collision nodes on level $j.$ Note one collision edge may connect two nodes on the same level or connect one node on level $j$ and one node on level $j-1.$ Therefore, we have $$|{\cal C}_j|\leq 2{R}_{j+1},\quad \forall j\leq t-1$$ For $j=t,$ since we only consider the $t$ hop neighborhood of the actual source $s,$ we have $$|{\cal C}_t|\leq 2{R}_{t}$$ Therefore, we have $$\begin{aligned}
H&\leq 2{R}_{t}U^{t-|k-t|}_{t}+\sum_{j=0}^{t-1}2{R}_{j+1}U^{t-|k-j|}_{j}\\
&=\underbrace{2{R}_{t}U^{k}_{t}}_{(a)}+\underbrace{ \sum_{j=1}^{t}2{R}_{j}U^{t-|k-(j-1)|}_{j-1}}_{(b)}\label{eqn:Hupperbound2}\end{aligned}$$
- [**Step 3**]{} We analyze part $(a)$ and part $(b)$ in equation (\[eqn:Hupperbound2\]) separately.
- [**Step 3.a: Upper bound on part $(a)$ in Equation (\[eqn:Hupperbound2\])**]{}
Define $$\alpha' = \frac{\alpha}{2}+\frac{1}{4}.$$ and we have $\alpha'\in(1/2, 3/4).$ Since $t\leq \frac{\log n}{(1+\alpha)\log \mu},$ $\alpha<1$ and $\delta<1,$ we have when $n$ is sufficiently large, $$\begin{aligned}
&t\leq \frac{\log n}{(1+\alpha)\log \mu}\\
&t \leq \frac{\log n}{(1+\alpha')\log [(1+\delta)\mu]}\\
&[(1+\delta)\mu]^{(1+\alpha')t}\leq n,\label{eqn:lowerBoundOnN}\end{aligned}$$
According to Equation (\[eqn:upperboundU\]), since $k\geq t-t =0,$ we have $$U_{t}^k =8[(1+\delta)\mu]^{\frac{k+t-t}{2}} = 8[(1+\delta)\mu]^{\frac{k}{2}}$$
Based on event $E_2,$ we have $$\begin{aligned}
2{R}_{t}U^{k}_{t}\leq& \frac{64}{n}[(1+\delta)\mu]^{2t+1+\frac{k}{2}}\end{aligned}$$ Since $[(1+\delta)\mu]^{(1+\alpha')t}\leq n,$ $$2{R}_{t}U^{k}_{t}\leq 64[(1+\delta)\mu]^{(1-\alpha')t+1+\frac{k}{2}}$$ Since $k\leq t,$ we have $$\begin{aligned}
2{R}_{t}U^{k}_{t}\leq 64[(1+\delta)\mu]^{(\frac{3}{2}-\alpha')t+1} \label{eqn:HupperboundpartA}\end{aligned}$$
- [**Step 3.b: Upper bound on part $(b)$ in Equation (\[eqn:Hupperbound2\])**]{}
Based on event $E_3,$ we have $$\sum_{j=1}^{t}2{R}_jU_{j-1}^{t-|k-(j-1)|} = \sum_{j=\lfloor m^-\rfloor+1}^{t}2{R}_jU_{j-1}^{t-|k-(j-1)|}$$
Therefore, we only consider the cases when $j\geq \lfloor m^-\rfloor+1.$ We will show that $$t-|k-(j-1)| \geq t-(j-1),$$ when $j\geq \lfloor m^-\rfloor+1.$ As a consequence, we have $$\begin{aligned}
U_{j-1}^{t-|k-(j-1)|}&=8[(1+\delta)\mu]^{\frac{t-|k-(j-1)|+t-(j-1)}{2}}\\
&=8[(1+\delta)\mu]^{t-\frac{(j-1)+|k-(j-1)|}{2}}\end{aligned}$$
Based on $t\leq \frac{\log n}{(1+\alpha)\log \mu},$ we have $$\begin{aligned}
\frac{t}{2}&\leq \frac{\log n}{2(1+\alpha)\log \mu}\\
&\leq \lfloor m^-\rfloor \label{eqn:m-lowerbound}\end{aligned}$$ for sufficiently large $n.$ Therefore, we have $j-1\geq \lfloor m^-\rfloor>\frac{t}{2}.$
When $j-1\leq k\leq t,$ we have $|k-(j-1)|\leq \frac{t}{2}\leq j-1.$ When $0\leq k< j-1,$ we have $|k-(j-1)|=(j-1)-k\leq j-1.$ We have $$|k-(j-1)| \leq j-1, \forall k\in[0,t].$$ Hence, $$t-|k-(j-1)| \geq t-(j-1).$$ Therefore, for all the discussions in Step 3.b, based on Equation \[eqn:Hupperbound2\], we have $$U_{j-1}^{t-|k-(j-1)|}=8[(1+\delta)\mu]^{t-\frac{(j-1)+|k-(j-1)|}{2}}$$
Based on event $E_2,$ we have $$\begin{aligned}
&\sum_{j=1}^{t}2{R}_jU_{j-1}^{t-|k-(j-1)|}\\
\leq&128\mu\sum_{j= \lfloor m^-\rfloor+1}^{\lceil m^+\rceil-1} [(1+\delta)\mu]^{t-\frac{j-1+|k-(j-1)|}{2}}\\
+& \frac{64}{n}\sum_{j=\lceil m^+\rceil}^{t}[(1+\delta)\mu]^{2j+1+t-\frac{j-1+|k-(j-1)|}{2}}\end{aligned}$$ Since $[(1+\delta)\mu]^{(1+\alpha')t}\leq n,$ $$\begin{aligned}
&\sum_{j=1}^{t}2{R}_jU_{j-1}^{t-|k-(j-1)|}\\
\leq & 128\mu\sum_{j= \lfloor m^-\rfloor+1}^{\lceil m^+\rceil-1} [(1+\delta)\mu]^{t-\frac{j-1+|k-(j-1)|}{2}} \\
&+ 64\sum_{j=\lceil m^+\rceil}^{t}[(1+\delta)\mu]^{2j+1-\alpha' t-\frac{j-1+|k-(j-1)|}{2}}\\\end{aligned}$$ Next, we discuss the upper bounds for different $k$ values. We first show several necessary inequalities. We have $$\begin{aligned}
m^+-m^-=\frac{2\log \mu+\log 8}{2\log[(1+\delta)\mu]}\leq 1+\frac{\log 8}{2\log \mu}< 2.\label{eqn:m+-inequality}\end{aligned}$$ Recall, we consider the case where $t>\lfloor m^-\rfloor.$ As shown in \[eqn:m-lowerbound\], we have $$\begin{aligned}
\lfloor m^-\rfloor\in\left[\frac{t}{2},t\right)\label{eqn:m-allbounds}\end{aligned}$$ Hence, we have $$\begin{aligned}
\lceil m^+\rceil \in\left[\frac{t}{2},t+3\right)\label{eqn:m+allbounds}\end{aligned}$$ Then, we consider the following cases for different $k$ values. Recall that $k$ is the level of node $s'.$
- $k\leq \lfloor m^-\rfloor.$ We have $$\begin{aligned}
&\sum_{j=1}^{t}2{R}_jU_{j-1}^{t-|k-(j-1)|}\\
\leq & 128\mu\sum_{j= \lfloor m^-\rfloor+1}^{\lceil m^+\rceil-1} [(1+\delta)\mu]^{t-\frac{j-1+|k-(j-1)|}{2}} \\
&+ 64\sum_{j=\lceil m^+\rceil}^{t}[(1+\delta)\mu]^{2j+1-\alpha' t-\frac{j-1+|k-(j-1)|}{2}}\\
= & 128\mu\sum_{j= \lfloor m^-\rfloor+1}^{\lceil m^+\rceil-1} [(1+\delta)\mu]^{t-\frac{j-1-(k-(j-1))}{2}} \\
&+ 64\sum_{j=\lceil m^+\rceil}^{t}[(1+\delta)\mu]^{2j+1-\alpha' t-\frac{j-1-(k-(j-1))}{2}}\\
= & 128\mu\sum_{j= \lfloor m^-\rfloor+1}^{\lceil m^+\rceil-1} [(1+\delta)\mu]^{t-j+1+\frac{k}{2}} \\
&+ 64\sum_{j=\lceil m^+\rceil}^{t}[(1+\delta)\mu]^{-\alpha' t+j+2+\frac{k}{2}}\\
\leq & 256\mu [(1+\delta)\mu]^{t-\lfloor m^-\rfloor+\frac{k}{2}} \\
&+ 128[(1+\delta)\mu]^{(1-\alpha')t+2+\frac{k}{2}}\\
\leq_{(a)} & 256 [(1+\delta)\mu]^{\frac{3}{4}t}+ 128[(1+\delta)\mu]^{(\frac{3}{2}-\alpha')t+2}\end{aligned}$$ where $(a)$ is due to $k\leq \lfloor m^-\rfloor$ and $ \frac{t}{2}\leq\lfloor m^-\rfloor< t.$
- $\lfloor m^-\rfloor+1\leq k\leq \lceil m^+\rceil -2.$ In this case, we have $$k\in[\frac{t}{2}+1,t+1),$$ according to Inequalties (\[eqn:m+allbounds\]) and (\[eqn:m-allbounds\]). Hence, we have $$\begin{aligned}
&\sum_{j=1}^{t}2{R}_jU_{j-1}^{t-|k-(j-1)|}\\
\leq & 128\mu\sum_{j=\lfloor m^-\rfloor+1}^{k} [(1+\delta)\mu]^{t-\frac{j-1+|k-(j-1)|}{2}}\\
&+128\mu\sum_{j=k+1}^{\lceil m^+\rceil -1} [(1+\delta)\mu]^{t-\frac{j-1+|k-(j-1)|}{2}}\\
& + 64\sum_{j=\lceil m^+\rceil}^{t}[(1+\delta)\mu]^{2j+1-\alpha' t-\frac{j-1+|k-(j-1)|}{2}}\\
= & 128\mu\sum_{j=\lfloor m^-\rfloor+1}^{k} [(1+\delta)\mu]^{t-\frac{j-1+(k-(j-1))}{2}}\\
&+128\mu\sum_{j=k+1}^{\lceil m^+\rceil -1} [(1+\delta)\mu]^{t-\frac{j-1-(k-(j-1))}{2}} \\
&+ 64\sum_{j=\lceil m^+\rceil}^{t}[(1+\delta)\mu]^{2j+1-\alpha' t-\frac{j-1-(k-(j-1))}{2}}\\
= & 128\mu\sum_{j=\lfloor m^-\rfloor+1}^{k} [(1+\delta)\mu]^{t-\frac{k}{2}}\\
&+128\mu\sum_{j=k+1}^{\lceil m^+\rceil -1} [(1+\delta)\mu]^{t-j+1+\frac{k}{2}}\\
& + 64\sum_{j=\lceil m^+\rceil}^{t}[(1+\delta)\mu]^{-\alpha' t+j+2+\frac{k}{2}}\\
\leq_{(a)} & 256\mu[(1+\delta)\mu]^{t-\frac{k}{2}}\\
&+128[(1+\delta)\mu]^{(1-\alpha')t+2+\frac{k}{2}}\\
\leq & 256 [(1+\delta)\mu]^{\frac{3}{4}t+\frac{1}{2}}+ 128[(1+\delta)\mu]^{(\frac{3}{2}-\alpha')t+2},\end{aligned}$$ where $(a)$ is due to $m^+-m^-<2$ which we proved in Inequality (\[eqn:m+-inequality\]) and $k\in[\frac{t}{2}+1,t+1).$
- $k\geq \lceil m^+\rceil-1.$ In this case, we have $$k\in[\frac{t}{2}-1,t],$$ according to Inequalties (\[eqn:m+allbounds\]) and (\[eqn:m-allbounds\]). Hence, we have $$\begin{aligned}
&\sum_{j=1}^{t}2{R}_jU_{j-1}^{t-|k-(j-1)|}\\
\leq & 128\mu\sum_{j= \lfloor m^-\rfloor+1}^{\lceil m^+\rceil-1} [(1+\delta)\mu]^{t-\frac{j-1+|k-(j-1)|}{2}} \\
&+ 64\sum_{j=\lceil m^+\rceil}^{k}[(1+\delta)\mu]^{2j+1-\alpha' t-\frac{j-1+|k-(j-1)|}{2}}\\
&+64\sum_{j=k+1}^{t}[(1+\delta)\mu]^{2j+1-\alpha' t-\frac{j-1+|k-(j-1)|}{2}}\\
= & 128\mu\sum_{j= \lfloor m^-\rfloor+1}^{\lceil m^+\rceil-1} [(1+\delta)\mu]^{t-\frac{j-1+(k-(j-1))}{2}} \\
&+ 64\sum_{j=\lceil m^+\rceil}^{k}[(1+\delta)\mu]^{2j+1-\alpha' t-\frac{j-1+(k-(j-1))}{2}}\\
&+64\sum_{j=k+1}^{t}[(1+\delta)\mu]^{2j+1-\alpha' t-\frac{j-1-(k-(j-1))}{2}}\\
= & 128\mu\sum_{j= \lfloor m^-\rfloor+1}^{\lceil m^+\rceil-1} [(1+\delta)\mu]^{t-\frac{k}{2}} \\
&+ 64\sum_{j=\lceil m^+\rceil}^{k}[(1+\delta)\mu]^{2j+1-\alpha' t-\frac{k}{2}}\\
&+64\sum_{j=k+1}^{t}[(1+\delta)\mu]^{-\alpha' t+j+2+\frac{k}{2}}\\
\leq_{(a)} & 256\mu[(1+\delta)\mu]^{t-\frac{k}{2}}+128[(1+\delta)\mu]^{\frac{3k}{2}-\alpha' t+1}\\
&+128[(1+\delta)\mu]^{(1-\alpha')t+2+\frac{k}{2}}\\
\leq & 256[(1+\delta)\mu]^{\frac{3}{4}t+\frac{1}{2}}+128[(1+\delta)\mu]^{(\frac{3}{2}-\alpha')t+1}\\
&+128[(1+\delta)\mu]^{(\frac{3}{2}-\alpha')t+2}\\
\leq & 256[(1+\delta)\mu]^{\frac{3}{4}t+\frac{1}{2}}+256[(1+\delta)\mu]^{(\frac{3}{2}-\alpha')t+2}\\\end{aligned}$$ $(a)$ is due to $m^+-m^-<2$ which we proved in Inequality \[eqn:m+-inequality\] and $k\in[\frac{t}{2}-1,t].$
Therefore, we obtain a universal bound for different $k.$ $$\begin{aligned}
\sum_{j=1}^{t}2{R}_jU_{j-1}^{t-|k-(j-1)|}&\leq c'[(1+\delta)\mu]^{\frac{3}{4}t+\frac{1}{2}}\\
&+c'[(1+\delta)\mu]^{(\frac{3}{2}-\alpha')t+2},\label{eqn:HupperboundpartB}\end{aligned}$$ for all $k\in[1,t],$ where $c'\geq 256.$
As a summary, based on Equations (\[eqn:Hupperbound2\]),(\[eqn:HupperboundpartA\]) and (\[eqn:HupperboundpartB\]) $$\begin{aligned}
H&\leq c[(1+\delta)\mu]^{\frac{3}{4}t+\frac{1}{2}}+c[(1+\delta)\mu]^{(\frac{3}{2}-\alpha')t+2}\\
&= c[(1+\delta)\mu]^{\frac{3}{4}t+\frac{1}{2}}+c[(1+\delta)\mu]^{(\frac{5}{4}-\frac{\alpha}{2})t+2}\end{aligned}$$ for all $k\in[1,t],$ where $c\geq 257.$
Proof of Theorem \[thm:approximateRatio\] {#sec:approximateRate}
=========================================
Similar to the proof of Theorem \[thm:sourceIsJordanER\], we assume $E_1,E_2$ and $E_3$ occur. The BND one node can have is bounded by the number of all infected nodes. Therefore, the upper bound on BND is the sum of degree of all infected nodes. The edges of one infected node compose three parts: (1) the edge which infects the node; (2) the edges between the node and its offsprings; (3) the collision edges attaching to the node. Therefore, the total degree of all the infected nodes is upper bounded by $$\sum_{i=0}^t Z^{\leq t}_i+\sum_{i=0}^t Z^{\leq t}_i[(1+\delta)\mu]+2{R}_{t+1}$$ To use $\sum_{i=0}^t Z^{\leq t}_i[(1+\delta)\mu]$ above as the upper bound on offsprings, we need to extend $E_1$ to the range of ${{\cal L}}_{t}.$ It is easy to check that $$E'_1=\{\forall v \in {{\cal L}}_{{t}}, \phi'(v)\in((1-\delta)\mu,(1+\delta)\mu)\}.$$ happens with a high probability with the same proof of Lemma \[lem:ERAllOffspring\].
A lower bound on BND of the actual source is $$Z^t_t[(1-\delta)\mu]$$ according to $E'_1.$ Therefore, we have $$\begin{aligned}
&\frac{Z^t_t[(1-\delta)\mu]}{\sum_{i=0}^t Z^{\leq t}_i+\sum_{i=0}^t Z^{\leq t}_i[(1+\delta)\mu]+2{R}_{t+1}}\\
&= \frac{(1-\delta)\mu}{\frac{\sum_{i=0}^t Z^{\leq t}_i}{Z^t_t}(1+(1+\delta)\mu)+\frac{2{R}_{t+1}}{Z^t_t}}\\
&\geq_{(a)} \frac{(1-\delta)\mu}{\frac{1+(1+\delta)\mu}{1-\epsilon'}+\frac{2{R}_{t+1}}{Z^t_t}}.\\
&\geq \frac{(1-\delta)}{\frac{\frac{1}{\mu}+(1+\delta)}{1-\epsilon'}+\frac{2{R}_{t+1}}{Z^t_t\mu}}\\
&\geq_{(b)}\frac{(1-\delta)}{\frac{\delta''+(1+\delta)}{1-\epsilon'}+\delta'}\\
&\geq \frac{1-\delta}{1+1.1\delta},\label{eqn:lowerBoundOnDegreeCount}\end{aligned}$$ where inequality (a) holds due to Lemma \[lem:infectedNodeRatio1\], inequality (b) is based on Lemma \[lem:infectedNodeRatio2\] and $\delta'',\delta',\epsilon'$ can be arbitrarily small when $n\rightarrow \infty$.
In the proof of Lemma \[lem:ERAllOffspring\], we need $\mu>\frac{2+\delta}{\delta^2}\log n$. Hence, we have $$\frac{Z^t_t[(1-\delta)\mu]}{\sum_{i=0}^t Z^{\leq t}_i+\sum_{i=0}^t Z^{\leq t}_i[(1+\delta)\mu]+2{R}_{t+1}}\geq \frac{1-\delta}{1+1.1\delta}$$
when $\mu> \frac{2+\delta}{\delta^2}\log n.$
Assume we want the ratio to be $\geq 1-x,$ we have $$\frac{1-\delta}{1+1.1\delta} = 1-x.$$ Therefore $$\delta = \frac{x}{2.1-1.1x}$$ Hence, $$\frac{2+\delta}{\delta^2}=\frac{1.32x^2-7.14x+8.82}{x^2}$$ Therefore, when $$\mu > \frac{9}{x^2}\log n>\frac{1.32x^2-7.14x+8.82}{x^2}\log n$$ we have the ratio $\geq 1-x.$
Note, the condition that $\alpha>\frac{1}{2}$ is not used in the high probability result of $E_1,E_2,E_3.$ Therefore, we only need $\alpha\in(0,1)$ for this theorem. Therefore, the theorem is proved.
\[lem:infectedNodeRatio1\] If the conditions in Theorem \[thm:approximateRatio\] hold and events $E_1,$ $E_2$ and $E_3$ occur, we have given any $\epsilon>0$, for sufficiently large $n,$ the following inequality holds $$\frac{Z^t_t}{\sum_{i=0}^t Z^{\leq t}_i}\geq 1-\epsilon.$$
For any $\epsilon>0,$ we use induction and assume that $$Z_{m-1}^{m-1}\geq (1-\epsilon)\sum_{i=0}^{m-1} Z^{\leq m-1}_i$$ Consider time slot $m,$ we have $$\begin{aligned}
\frac{Z^m_m}{\sum_{i=0}^m Z^{\leq m}_i}&=\frac{Z^m_m}{\sum_{i=0}^{m} Z^{\leq m-1}_i+\sum_{i=0}^m Z^m_i}\\
&=_{(a)}\frac{Z^m_m}{\sum_{i=0}^{m-1} Z^{\leq m-1}_i+\sum_{i=0}^{m-1} Z^m_i+Z_m^m}\\
&=\frac{1}{\underbrace{\frac{\sum_{i=0}^{m-1} Z^{\leq m-1}_i}{Z_m^m}}_{\hbox{A}}+\underbrace{\frac{\sum_{i=0}^{m-1}Z_i^m}{Z^m_m}}_{\hbox{B}}+1}\end{aligned}$$ In (a) we use that $Z_m^{\leq m-1}=0$ and $Z_i^{\leq m}=Z_i^{\leq m-1}+Z_i^m.$
Use induction assumption for part A, we have $$\begin{aligned}
\frac{Z^m_m}{\sum_{i=0}^m Z^{\leq m}_i}\geq & \frac{1}{\frac{Z^{m-1}_{m-1}}{(1-\epsilon) Z_m^m}+\underbrace{\frac{\sum_{i=0}^{m-1}Z_i^m}{Z^m_m}}_{\hbox{B}}+1}\label{eqn:induction}\end{aligned}$$ Based on event $E_3$, we have $$\frac{Z^{m-1}_{m-1}}{Z^m_m}\leq \frac{1}{(1-\delta)^2\mu q}$$ and $$Z^m_m\geq [(1-\delta)^2\mu q]^m.$$ Next, we establish an upper bound on $\sum_{i=0}^{m-1}Z_i^m.$ Note $\sum_{i=0}^{m-1}Z_i^m$ represents the number of nodes which are from level $0$ to level $m-1$ and are infected at time $m.$ Denote by $C({\cal V})$ the set of offsprings of node set ${\cal V}$ who are not collision nodes and were infected by node ${\cal V}.$ Define $C^2({\cal V}) = C(C({\cal V})).$ Recall the number of collsion nodes from level $0$ to level $m-1$ is upper bounded by $2{R}_m.$ We establish an upper bound as following $$\sum_{i=0}^{m-1}Z_i^m\leq 2{R}_m+\left| C\left(\cup_{i=0}^{m-2}{\cal Z}_i^{m-1}\right)\right|.$$ Similarly, we have $$\left|\cup_{i=0}^{m-2}{\cal Z}_i^{m-1}\right|\leq 2{R}_{m-1}+\left| C\left(\cup_{i=0}^{m-3}{\cal Z}_i^{m-2}\right)\right|.$$ Based on $E_1,$ we have $$\begin{aligned}
\sum_{i=0}^{m-1}Z_i^m\leq 2{R}_m+ 2{R}_{m-1}[(1+\delta)\mu]+ \left| C^2\left(\cup_{i=0}^{m-3}{\cal Z}_i^{m-2}\right)\right|\end{aligned}$$ Repeating the step above, we have
$$\sum_{i=0}^{m-1}Z_i^m\leq\sum_{j=0}^{m}2{R}_{j} [(1+\delta)\mu]^{m-j}$$ Based on $E_2,$ we evaluate the upper bound for different values of $m.$
- $0<m\leq \lfloor m^-\rfloor.$ We have $$\sum_{i=0}^{m-1}Z_i^m=0.$$ Hence, $$\begin{aligned}
\frac{Z^m_m}{\sum_{i=0}^m Z^{\leq m}_i}\geq & \frac{1}{\frac{Z^{m-1}_{m-1}}{(1-\epsilon) Z_m^m}+1}\\
\geq &\frac{1}{\frac{1}{(1-\epsilon) (1-\delta)^2\mu q}+1}\end{aligned}$$ For any $\epsilon>0,$ we have $$\frac{1}{(1-\epsilon) (1-\delta)^2\mu q}\leq \epsilon$$ for sufficiently large $n.$
Therefore, we have $$\frac{Z^m_m}{\sum_{i=0}^m Z^{\leq m}_i}\geq (1-\epsilon).$$
- $\lfloor m^-\rfloor<m<\lceil m^+\rceil.$ We have $$\begin{aligned}
\sum_{i=0}^{m-1}Z_i^m\leq& \sum_{j=\lfloor m^-\rfloor+1}^{\lceil m^+\rceil-1} 2{R}_j[(1+\delta)\mu]^{m-j}\\
\leq_{(a)}& 32\mu [(1+\delta)\mu]^{m-\lfloor m^-\rfloor-1}\end{aligned}$$ $(a)$ is based on the fact that ${R}_j\leq 8\mu$ and $\lceil m^+\rceil-\lfloor m^-\rfloor\leq 2.$
Hence, we have $$\begin{aligned}
&\frac{\sum_{i=0}^{m-1}Z_i^m}{Z^m_m}\\
\leq &\frac{32\mu [(1+\delta)\mu]^{m-\lfloor m^-\rfloor-1}}{[(1-\delta)^2\mu q]^m}\\
\leq & \frac{32}{\mu} \frac{(1+\delta)^{m-\lfloor m^-\rfloor-1}}{(1-\delta)^{2m}q^m\mu^{\lfloor m^-\rfloor-1}}\\
=& \frac{32}{\mu} \left(\frac{(1+\delta)^{1-\frac{\lfloor m^-\rfloor-1}{m}}}{(1-\delta)^2q\mu^{\frac{\lfloor m^-\rfloor-1}{m}}}\right)^m\\
\leq & \frac{32}{\mu} \left(\frac{(1+\delta)^{1-\frac{\lfloor m^-\rfloor-1}{\lceil m^+\rceil}}}{(1-\delta)^2q\mu^{\frac{\lfloor m^-\rfloor-1}{\lceil m^+\rceil}}}\right)^m\end{aligned}$$ Note $\lceil m^+\rceil<\lfloor m^-\rfloor+2.$ For sufficiently large $n,$ we have $$\frac{\lfloor m^-\rfloor-1}{\lceil m^+\rceil}\geq 1-\frac{3}{\lfloor m^-\rfloor}\geq \frac{1}{2}.$$ Hence we have $$\frac{\sum_{i=0}^{m-1}Z_i^m}{Z^m_m}\leq \frac{32}{\mu} \left(\frac{(1+\delta)^{\frac{1}{2}}}{(1-\delta)^2q\mu^{\frac{1}{2}}}\right)^m\leq \frac{\epsilon}{2}$$ for sufficiently large $n.$
In addition, we have $$\frac{1}{(1-\delta)^2q\mu}\leq \frac{\epsilon}{2}$$ for sufficiently large $n.$ $$\begin{aligned}
\frac{Z^m_m}{\sum_{i=0}^m Z^{\leq m}_i}\geq & \frac{1}{\frac{Z^{m-1}_{m-1}}{(1-\epsilon) Z_m^m}+\frac{\sum_{i=0}^{m-1}Z_i^m}{Z^m_m}+1}\\
\geq &\frac{1}{
\frac{\epsilon}{2(1-\epsilon)}+\frac{\epsilon}{2}+1}\geq(1-\epsilon).\end{aligned}$$
- $\lceil m^+\rceil\leq m\leq t.$ Define $$\alpha' = \frac{\alpha}{2}+\frac{1}{4}.$$ and we have $\alpha'\in(1/4, 3/4).$ Follow the same argument in Equation (\[eqn:lowerBoundOnN\]), we obtain that $$\begin{aligned}
[(1+\delta)\mu]^{(1+\alpha')t}\leq n.\label{eqn:LowerBoundOnN2}\end{aligned}$$ We have $$\begin{aligned}
\sum_{i=0}^{m-1}Z_i^m\leq& \sum_{j=\lfloor m^-\rfloor+1}^{\lceil m^+\rceil-1} 2{R}_j[(1+\delta)\mu]^{m-j}\\
&+\sum_{j=\lceil m^+\rceil}^{m-1} 2{R}_j[(1+\delta)\mu]^{m-j}\\
\leq& 32\mu [(1+\delta)\mu]^{m-\lfloor m^-\rfloor-1}\\
&+\frac{8}{n}\sum_{j=\lceil m^+\rceil}^{m-1}[(1+\delta)\mu]^{m+j+1}\\
\leq & 32\mu [(1+\delta)\mu]^{m-\lfloor m^-\rfloor-1}+\frac{16}{n}[(1+\delta)\mu]^{2m}\\
\leq & 32\mu [(1+\delta)\mu]^{m-\lfloor m^-\rfloor-1}\\
&+16[(1+\delta)\mu]^{2m-(1+\alpha')t}\end{aligned}$$ The last inequality holds based on Inequality (\[eqn:LowerBoundOnN2\]).
Hence, we have $$\begin{aligned}
&\frac{\sum_{i=0}^{m-1}Z_i^m}{Z^m_m}\\
\leq &\underbrace{\frac{32\mu [(1+\delta)\mu]^{m-\lfloor m^-\rfloor-1}}{[(1-\delta)^2 q \mu]^m}}_{(A)}+\underbrace{\frac{16[(1+\delta)\mu]^{2m-(1+\alpha')t}}{[(1-\delta)^2 q \mu]^m}}_{(B)}\end{aligned}$$ $(A)$ has been handled in the previous case. For sufficiently large $n$ we have $(A)\leq \frac{\epsilon}{4}.$
Next, we focus on $(B).$ Since $m\leq t$, for sufficiently large $n,$ we have $$\frac{m+1}{t}\leq 1+\frac{\alpha'}{2}.$$ Hence, for sufficiently large $n$ $$\begin{aligned}
&\frac{16[(1+\delta)\mu]^{2m-(1+\alpha')t}}{[(1-\delta)^2 q \mu]^m}\\
= & \frac{16}{\mu}\left(\frac{(1+\delta)^{\frac{2m}{t}-1-\alpha'}}{(1-\delta)^{2\frac{m}{t}}q^{\frac{m}{t}}}\mu^{\frac{m+1}{t}-1-\alpha'}\right)^t\\
\leq & \frac{16}{\mu}\left(\frac{(1+\delta)^{2-\alpha'}}{(1-\delta)^2 q}\mu^{-\alpha'/2}\right)^t\\
\leq & \frac{\epsilon}{4}.\end{aligned}$$ Hence, we have $$\frac{\sum_{i=0}^{m-1}Z_i^m}{Z^m_m}\leq \frac{\epsilon}{2}$$ Following the analysis in the previous case, we have $$\frac{Z^m_m}{\sum_{i=0}^m Z^{\leq m}_i}\geq 1-\epsilon.$$
As a summary, we proved that $$\frac{Z^t_t}{\sum_{i=0}^t Z^{\leq t}_i}\geq 1-\epsilon.$$
\[lem:infectedNodeRatio2\] If the conditions in Theorem \[thm:approximateRatio\] hold and events $E_1,$ $E_2$ and $E_3$ occur, we have given any $\epsilon>0$, for sufficiently large $n,$ the following inequality holds $$\begin{aligned}
\frac{2{R}_{t+1}}{Z^t_t\mu}\leq \epsilon \label{eqn:ratio}\end{aligned}$$
Note the upper bound of ${R}_{t+1}$ can be obtained by a same proof of Lemma \[lem:E1\] and the conclusions are the same when we extend the range from $t$ to $t+1$. Based on Lemma \[lem:E3\], we have $$Z^t_t\geq [(1-\delta)^2\mu]^t.$$
When $t< \lceil m^+\rceil,$ Inequality \[eqn:ratio\] trivially holds.
For $t\geq \lceil m^+\rceil,$ we have $$\begin{aligned}
&\frac{2{R}_{t+1}}{Z^t_t\mu}\\
\leq & \frac{8[(1+\delta)\mu]^{2t+3}}{n\mu[(1-\delta)^2\mu]^t}\\
= & \frac{8(1+\delta)^{2t+3}}{(1-\delta)^{2t}}\frac{1}{\mu^{\alpha t -2}}\\
= & \frac{8}{\mu}\left(\frac{1+\delta}{(1-\delta)^2} \times\frac{1}{\mu^{\frac{\alpha t -3}{2t+3}}}\right)^{2t+3}\end{aligned}$$ Note $\frac{\alpha t -3}{2t+3}>0$ for sufficiently large $t$. Therefore, we have $$\frac{2{R}_{t+1}}{Z^t_t\mu}\leq \epsilon.$$ For sufficiently large $n.$
Proof of Lemma \[thm:diameterER\] {#sec:proofImpossibility}
=================================
We present the proof of Theorem 4.2 from [@DraMas_10] with some minor changes to provide a more specific lower bound on $\mu q.$ This proof is included for the sake of completeness and is not a contribution of this paper.
Given some $\epsilon > 0,$ define $$d_{j}^{\pm}=\begin{cases} (1\pm\epsilon)^j\mu^j &\hbox{if } j =1,2, \\
(1\pm\epsilon)^2(1\pm \frac{\epsilon}{\mu})^{j-2}\mu^j & \hbox{if } j=3,\cdots,D'.
\end{cases}$$ where $D'= \left\lceil\frac{\log n}{ 2\log \mu}\right\rceil.$ Define $$\Gamma_i(u) = \{v: d^g_{uv} = i\},$$ and $$d_i(u)=|\Gamma_i(u)|.$$ We first prove the following lemma.
Let $\epsilon>0$ be fixed. Define for all $u\in\{1,\cdots,n\}$ and all $i=1,\cdots,D',$ the event $E_i(u)$ by $$E_i(u)=\{d_i^-\leq d_i(u)\leq d_i^+\}.$$ Assumes that $\gamma_l\log n\leq\mu<< \sqrt{n},$ for large enough $n$,we have $$\Pr(E_i(u))\geq 1-D'n^{-\frac{\gamma_l\epsilon^2}{2+\epsilon}},~ u\in\{1,\cdots,n\},~i=1,\cdots, D'.$$
Let $u\in\{1,\cdots,n\}$ and $i\in\{1,\cdots,D'\}$ be fixed. Note that, conditional on $d_1(u),\cdots,d_{i-1}(u),d_i(u)$ admits a binomial distribution with parameters $$\begin{aligned}
&{\cal L}(d_i(u)|d_1(u),\cdots,d_{i-1}(u))\\
=&\hbox{Bi}(n-1-d_1(u)-\cdots-d_{i-1}(u),1-(1-p)^{d_{i-1}(u)})\end{aligned}$$ where ${\cal L}(X|{\cal F})$ is the distribution of the random variable $X$ conditional on the event ${\cal F}.$ Denote by $\bar{E}_i(u)$ the complement of ${E}_i(u).$ It readily follows that $$\begin{aligned}
&\Pr(\bar{E}_i(u)|{E}_1(u),\cdots,{E}_{i-1}(u))\\
\leq&\Pr(\hbox{Bi}(n,1-(1-p)^{d_{i-1}^+})\geq d_{i}^+)\\
+&\Pr(\hbox{Bi}(n-1-d_1^+-\cdots-d_{i-1}^+,1-(1-p)^{d_{i-1}^-})\leq d_{i}^-).\end{aligned}$$ Note that, for all $j<D',$ one has $$\begin{aligned}
& d_j^-\leq d_j^+\leq d_{D'-1}^+ \\
&= (1+\epsilon)^2\left(1+\frac{\epsilon}{\mu}\right)^{D'-3}\mu^{D'-1}\\
&=\left(\frac{\mu(1+\epsilon)}{\mu+\epsilon}\right)^2(\mu+\epsilon)^{D'-1}\\
&\leq \left(\frac{\mu(1+\epsilon)}{\mu+\epsilon}\right)^2(\mu+\epsilon)^{\frac{\log n}{2\log \mu}}\\
& = \left(\frac{\mu(1+\epsilon)}{\mu+\epsilon}\right)^2\exp\left(\frac{\log n}{2}\frac{\log(\mu+\epsilon)}{\log\mu}\right)\\
& = \left(\frac{\mu(1+\epsilon)}{\mu+\epsilon}\right)^2\exp\left(\frac{\log n}{2}\right)\\
&\times \exp\left(\frac{\log n}{2}\left(\frac{\log(\mu+\epsilon)}{\log\mu}-1\right)\right)\\
&= \left(\frac{\mu(1+\epsilon)}{\mu+\epsilon}\right)^2\exp\left(\frac{\log n}{2}\right)\\
&\times \exp\left(\frac{\log n}{2}\frac{\log(\mu+\epsilon)-\log\mu}{\log\mu}\right)\\
&= \left(\frac{\mu(1+\epsilon)}{\mu+\epsilon}\right)^2\\
\exp\left(\frac{\log n}{2}\right)&\times \exp\left(\frac{\log n}{2}\frac{\log(1+\epsilon/\mu)}{\log\mu}\right)\end{aligned}$$ Note $\log(1+x)\leq x$ for $x\geq 0.$ We have $$\begin{aligned}
& \leq \left(\frac{\mu(1+\epsilon)}{\mu+\epsilon}\right)^2\sqrt{n}\exp\left(\frac{\log n}{2}\frac{\epsilon}{\mu\log\mu}\right)\\
&\leq (1+\epsilon)^3\sqrt{n}\end{aligned}$$ Since $\mu\geq \gamma_1\log n$, we have $$\begin{aligned}
&\leq \left(\frac{\mu(1+\epsilon)}{\mu+\epsilon}\right)^2\sqrt{n}\exp\left(\frac{1}{2}\frac{\epsilon}{\gamma_l\log\mu}\right)\\
&\leq (1+\epsilon)^2\sqrt{n}\exp\left(\frac{1}{2}\frac{\epsilon}{\gamma_l\log\mu}\right)\end{aligned}$$ For sufficiently large $n,$ we have $\mu$ is sufficiently large and $\exp\left(\frac{1}{2}\frac{\epsilon}{\gamma_l\log\mu}\right)\rightarrow 1$ as $n\rightarrow \infty.$ Hence, we have $$=(1+\epsilon)^2(1+o(1))\sqrt{n}$$ Next, we compute the mean of $\hbox{Bi}(n,1-(1-p)^{d_{i-1}^+})$ and $\hbox{Bi}(n-1-d_1^+-\cdots-d_{i-1}^+,1-(1-p)^{d_{i-1}^-}).$
Since $i-1\leq D'-1,$ we have $d^+_{i-1}p=d^+_{i-1}\frac{\mu}{n}\rightarrow 0$ as $n\rightarrow \infty.$ Based on Taylor expansion, we have we have $$(1-p)^{d^+_{i-1}} = 1-d^+_{i-1}p + o(d^+_{i-1}p)$$ Hence, $$\begin{aligned}
&n(1-(1-p)^{d^+_{i-1}}) \\
=& d^+_{i-1}pn-o(d^+_{i-1}pn) = (1-o(1))d^+_{i-1}\mu\end{aligned}$$ Note $$\begin{aligned}
&n-1-d^+_1-\cdots-d^+_{i-1} \geq n-D'(1+\epsilon)^2(1+o(1))\sqrt{n} \\
\geq &n-(1+\epsilon)^2(1+o(1))\log n\sqrt{n}\end{aligned}$$ Therefore $$\begin{aligned}
&(n-1-d^+_1-\cdots-d^+_{i-1})(1-(1-p)^{d^-_{i-1}})\\
&\geq (n-(1+\epsilon)^2(1+o(1)\log n\sqrt{n})(d^-_{i-1}p - o(d^-_{i-1}p))\\
&= d^-_{i-1}pn-o(d^-_{i-1}pn)-d^-_{i-1}p(1+\epsilon)^2(1+o(1)\log n\sqrt{n}\\
&+o(d^-_{i-1}p(1+\epsilon)^2(1+o(1)\log n\sqrt{n})\\
&\geq (1-o(1))d^-_{i-1}\mu\end{aligned}$$ Using the Chernoff bound, we have $$\begin{aligned}
&\Pr\left(\hbox{Bi}(n,1-(1-p)^{d^+_{i-1}})\geq d^+_i\right)\\
&\leq \exp\left(-\frac{\xi^2}{2+\xi}n\left(1-(1-p)^{d^+_{i-1}}\right)\right)\end{aligned}$$ where $$(1+\xi)n\left(1-(1-p)^{d^+_{i-1}}\right) = d^+_{i}$$ Therefore, $$\xi = \frac{d^+_{i}}{(1-o(1))d^+_{i-1}\mu}-1 \geq \begin{cases} \epsilon &\hbox{if } j =1,2, \\
\frac{\epsilon}{\mu} & \hbox{if } j=3,\cdots,D'.
\end{cases}$$ Therefore, when $i=1,2$ we have $$\begin{aligned}
&\Pr\left(\hbox{Bi}(n,1-(1-p)^{d^+_{i-1}})\geq d^+_i\right)\\
&\leq \exp\left(-\frac{\xi^2}{2+\xi}(1-o(1))d^+_{i-1}\mu\right)\\
&\leq \exp\left(-\frac{\xi^2}{2+\xi}(1-o(1))\mu\right)\\
&\leq \exp\left(-\frac{\xi^2}{2+\xi}(1-o(1))\gamma_l \log n\right)\\
& \leq n^{-\gamma_l(1-o(1))\frac{\epsilon^2}{2+\epsilon}}\end{aligned}$$ when $i>2,$ we have $$\begin{aligned}
&\Pr\left(\hbox{Bi}(n,1-(1-p)^{d^+_{i-1}})\geq d^+_i\right)\\
&\leq \exp\left(-\frac{\xi^2}{2+\xi}(1-o(1))d^+_{i-1}\mu\right)\\\end{aligned}$$ Note, since $i>2,$ we have $d^+_{i-1}\geq (1+\epsilon)^2\mu^2.$ Hence, we have $$\begin{aligned}
&\leq \exp\left(-\frac{\xi^2}{2+\xi}(1-o(1)(1+\epsilon)^2\mu^3\right)\\
&\leq \exp\left(-\epsilon^2(1-o(1))(1+\epsilon)\mu\right)\\
&\leq n^{-\epsilon^2(1+\epsilon)(1-o(1)\gamma_l}\end{aligned}$$ Therefore, we have for all $i\leq D',$ $$\Pr\left(\hbox{Bi}(n,1-(1-p)^{d^+_{i-1}})\geq d^+_i\right)\leq n^{-\gamma_l(1-o(1))\frac{\epsilon^2}{2+\epsilon}}$$
Similarly, using the Chernoff bound, we have $$\begin{aligned}
&\Pr(\hbox{Bi}(n-1-d_1^+-\cdots-d_{i-1}^+,1-(1-p)^{d_{i-1}^-})\leq d_{i}^-)\\
&\leq \exp\left(-\frac{\xi'^2}{2}(n-1-d_1^+-\cdots-d_{i-1}^+)(1-(1-p)^{d_{i-1}^-})\right)\end{aligned}$$ where $$(1-\xi')(n-1-d_1^+-\cdots-d_{i-1}^+)(1-(1-p)^{d_{i-1}^-}) = d^-_{i}$$ Therefore, $$\xi' = 1-\frac{d^-_{i}}{(1-o(1))d^-_{i-1}\delta}\geq \begin{cases} (1-\delta)\epsilon &\hbox{if } j =1,2, \\
(1-\delta)\frac{\epsilon}{\mu} & \hbox{if } j=3,\cdots,D'.
\end{cases}$$ for any fixed $\delta \in(0,1)$ when $n$ is sufficiently large. Therefore, when $i=1,2$ we have $$\begin{aligned}
&\Pr(\hbox{Bi}(n-1-d_1^+-\cdots-d_{i-1}^+,1-(1-p)^{d_{i-1}^-})\\
&\leq \exp\left(-\frac{\xi'^2}{2}(1-o(1)d^-_{i-1}\mu\right)\\
&\leq \exp\left(-\frac{\xi'^2}{2}(1-o(1)\mu\right)\\
&\leq \exp\left(-\frac{\xi'^2}{2}(1-o(1)\gamma_l \log n\right)\\
& \leq n^{-\gamma_l(1-o(1))\frac{(1-\delta)^2\epsilon^2}{2}}\end{aligned}$$ when $i>2,$ we have $$\begin{aligned}
&\Pr(\hbox{Bi}(n-1-d_1^+-\cdots-d_{i-1}^+,1-(1-p)^{d_{i-1}^-})\\
&\leq \exp\left(-\frac{\xi'^2}{2}(1-o(1))d^-_{i-1}\mu\right)\\\end{aligned}$$ Note, since $i>2,$ we have $d^-_{i-1}\geq (1-\epsilon)^2\mu^2.$ Hence, we have $$\begin{aligned}
&\leq \exp\left(-\frac{\xi'^2}{2}(1-o(1)(1-\epsilon)^2\mu^3\right)\\
&\leq \exp\left(-\frac{1}{2}(1-\delta)^2\epsilon^2(1-o(1))(1+\epsilon)\mu\right)\\
&\leq n^{-\frac{1}{2}(1-\delta)^2\epsilon^2(1-o(1))(1+\epsilon)\gamma_l}\end{aligned}$$ Therefore, we have for all $i\leq D',$ $$\begin{aligned}
&\Pr(\hbox{Bi}(n-1-d_1^+-\cdots-d_{i-1}^+,1-(1-p)^{d_{i-1}^-})\\
&\leq n^{-\frac{\gamma_l(1-o(1))(1-\delta)^2\epsilon^2}{2}}\end{aligned}$$ Next, using union bounds, we have $$\begin{aligned}
&\Pr(E_i(u))\\
&\geq \Pr(E_1(u),\cdots,E_i(u))\\
&\geq \Pr(E_1(u),\cdots,E_{i-1}(u))\\
&-\Pr(\bar{E}_i(u)|E_1(u),\cdots,E_{i-1}(u))\\
&\geq 1-\sum_{j=1}^{i}\Pr(\bar{E}_j(u)|E_1(u),\cdots,E_{j-1}(u))\\
&\geq 1-D'n^{-\frac{\gamma_l(1-o(1))(1-\delta)^2\epsilon^2}{2}}-D'n^{-\gamma_l(1-o(1))\frac{\epsilon^2}{2+\epsilon}}\\
& \geq 1-D'n^{-\frac{\gamma_l\epsilon^2}{2+\epsilon}}\end{aligned}$$ for sufficiently large $n.$
Next, we consider the upper bound of the diameter.
For any arbitrary nodes $u,v$, note that $$\begin{aligned}
&\Pr(d^g_{uv}>2D'+1|\Gamma_1(u),\cdots,\Gamma_{D'}(u),\Gamma_1(v),\cdots,\Gamma_{D'}(v))\\
&\leq (1-p)^{d_{D'}(u)d_{D'}(v)}\end{aligned}$$ Note if their $D'$ neighborhood has non-empty intersection, we have $d^g_{uv}\leq 2D'.$ Therefore, we obtain that $$\Pr(d^g_{uv}>2D'+1)\leq \Pr(\bar{E}_{D'}(u))+\Pr(\bar{E}_{D'}(v))+(1-p)^{(d^-_{D'})^2}$$ The last term is evaluated as follows: $$\begin{aligned}
&(1-p)^{(d_{D'}^-)^2}\\
&\leq \exp(-p(d_{D'}^-)^2)\\
&=\exp\left(-p\left((1-\epsilon)^2(1-\frac{\epsilon}{\mu})^{D'-2}\mu^{D'}\right)^2\right)\\
&=\exp\left(-p\left(\frac{1-\epsilon}{1-\frac{\epsilon}{\mu}}\right)^4(\mu -\epsilon)^{2D'}\right)\\
&\leq \exp\left(-p\left(\frac{1-\epsilon}{1-\frac{\epsilon}{\mu}}\right)^4(\mu -\epsilon)^{\log n /\log \mu}\right)\\
& = \exp\left(-pn^{\frac{\log(\mu -\epsilon)}{\log \mu}}\left(\frac{1-\epsilon}{1-\frac{\epsilon}{\mu}}\right)^4\right)\\
& = \exp\left(-pn^{1+\frac{\log(1-\epsilon/\mu)}{\log \mu}}\left(\frac{1-\epsilon}{1-\frac{\epsilon}{\mu}}\right)^4\right)\\
& = \exp\left(-\mu e^{\log n\frac{\log(1-\epsilon/\mu)}{\log \mu}}\left(\frac{1-\epsilon}{1-\frac{\epsilon}{\mu}}\right)^4\right)\\
&\leq \exp\left(-\mu(1-\frac{\epsilon}{\mu})^{-2}\left(1-\epsilon\right)^4\right)\\
&\leq\exp\left(-\mu \left(1-\epsilon\right)^4\right)\\
&\leq n^{-\gamma_l\left(1-\epsilon\right)^4}\end{aligned}$$
Therefore, we have $$\Pr(d^g_{uv}>2D'+1)\leq n^{-\gamma_l\left(1-\epsilon\right)^4}+2D'n^{-\frac{\gamma_l\epsilon^2}{2+\epsilon}}$$ Finally, we have $$\begin{aligned}
&\Pr(\hbox{Diamter}>2D'+1)\leq \sum_{u\neq v}\Pr(d^g_{uv}>2D'+1)\\
&\leq n^2 \times\left(n^{-\gamma_l\left(1-\epsilon\right)^4}+2D'n^{-\frac{\gamma_l\epsilon^2}{2+\epsilon}}\right)\end{aligned}$$ Therefore, we have $$\gamma_l> \max\left(\frac{2}{(1-\epsilon)^4},\frac{2(2+\epsilon)}{\epsilon^2}\right)$$ Note $\frac{2}{(1-\epsilon)^4}-\frac{2(2+\epsilon)}{\epsilon^2}$ is a increasing function for $\epsilon(0,1)$ and $\max\left(\frac{2}{(1-\epsilon)^4},\frac{2(2+\epsilon)}{\epsilon^2}\right) \geq 23.35.$ The optimal value is when $\epsilon = 0.459.$ Therefore, when $$\gamma_l>24.$$ we have $$\begin{aligned}
\Pr(\hbox{Diamter}>2D'+1)&\leq \sum_{u\neq v}\Pr(d^g_{uv}>2D'+1)\\
&\leq n^{-\delta_2}+2D'n^{-\delta_1}\end{aligned}$$ where $\delta_1$ and $\delta_2$ is fixed positive constant. Note $D'\leq \log n.$ We have $$\lim_{n}\Pr(\hbox{Diamter}>2D'+1) = 0.$$ Note $2\lceil x/2 \rceil\leq \lceil x\rceil.$ Hence $$2D'+1\leq D+2$$ Hence, we have $$\lim_{n}\Pr(\hbox{Diamter}\leq D+2) = 1.$$ The theorem is proved.
Necessary Inequalities
======================
We use the following Chernoff bounds.
\[lem:chernoff\] Let $X_1,X_2,\cdots,X_n$ be i.i.d Poisson trials such that $\Pr(X_i)=p_i$. Let $X=\sum_{i=1}^n X_i$ and $E(X)=\mu.$ For any $\delta>0,$ we have $$\Pr(X\geq(1+\delta)\mu)\leq \left(\frac{e^\delta}{(1+\delta)^{1+\delta}}\right)^\mu\leq \exp\left(-\frac{\delta^2\mu}{2+\delta}\right)$$ and for $\delta\in(0,1)$ $$\Pr(X\leq(1-\delta)\mu)\leq \exp\left(-\frac{\delta^2\mu}{2}\right)$$
We only need to prove $\left(\frac{e^\delta}{(1+\delta)^{1+\delta}}\right)^\mu\leq \exp\left(-\frac{\delta^2\mu}{2+\delta}\right).$ All other bounds are proved in [@MitUpf_05]. We need to show $$\begin{aligned}
\left(\frac{e^\delta}{(1+\delta)^{1+\delta}}\right)^\mu&\leq \exp\left(-\frac{\delta^2\mu}{2+\delta}\right)\\
\mu\left(\delta-(1+\delta)\log(1+\delta)\right)&\leq -\frac{\delta^2\mu}{2+\delta}\\
(2+\delta)\delta-(1+\delta)(2+\delta)\log(1+\delta)+\delta^2&\leq 0\\
(2\delta-(2+\delta)\log(1+\delta))(1+\delta)&\leq 0\\
2\delta-(2+\delta)\log(1+\delta)&\leq 0\end{aligned}$$ Denote by $f(\delta)=2\delta-(2+\delta)\log(1+\delta).$ We have $$f'(\delta)=2-\log(1+\delta)-\frac{2+\delta}{1+\delta}=1-\log(1+\delta)-\frac{1}{1+\delta}$$ $$f''(\delta)=-\frac{1}{1+\delta}+\frac{1}{(1+\delta)^2}=\frac{1}{1+\delta}(\frac{1}{1+\delta}-1)\leq 0$$ Hence, $f'(\delta)\leq f'(0)=0.$ Therefore, we have $$f(\delta)\leq f(0)=0$$ Hence we prove the lemma.
We need the following bounds
\[lem:exponentialBounds\] When $x>0,$ we have $$1-x\leq e^{-x}$$ and when $x\in(0,\frac{\log 2}{2}),$ $$1-x\geq e^{-2x}.$$
Let $f_1(x)=1-x-e^{-x}.$ We have $$f'_1(x)=-1+e^{-x}<0$$ when $x>0.$ Hence, $f_1(x)\leq f_1(0)=0.$ Therefore, we have $1-x\leq e^{-x}.$
Let $f_2(x)=1-x-e^{-2x}.$ We have $$f'_2(x)=-1+2e^{-2x}.$$ When $x<\frac{\log 2}{2},$ we have $f'_2(x)>0.$ Therefore $f_2(x)\geq f_2(0)=0.$ We have $1-x\geq e^{-2x}.$
We obtain the following bound using the similar proof procedures. $\forall x>0, 1-\frac{1}{x}\leq \log(x)\leq x-1.$
For $x\geq 2$ and integer $n\geq 0$ we have $$x^n \leq \sum_{i=0}^n x^i\leq 2x^n$$
$$\begin{aligned}
\sum_{i=0}^n x^n-2x^n&=\frac{x^{n+1}-1}{x-1}-2x^n\\
&=\frac{x^{n+1}-1-2x^{n+1}+2x^n}{x-1}\\
&=\frac{2x^n-1-x^{n+1}}{x-1}\\
&=\frac{x^n(2-\frac{1}{x^n}-x)}{x-1}\\
&\leq \frac{x^n(2-\frac{1}{x^n}-2)}{x-1}\\
&\leq 0\end{aligned}$$
Hence, we obtain the inequality in the lemma.
[^1]: Available at <http://snap.stanford.edu/data/index.html>
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this brief survey, we will remark the interaction among the Hessian tensor on a semi-Riemannian manifold and some of the several questions in Lorentzian (and also in semi-Riemannian) geometry where this $2-$covariant tensor is involved. In particular, we deal with the characterization of Killing vector fields and the study of a set of consequences of energy conditions in the framework of standard static space-times.'
address:
- ' Dipartimento di Matematica e Informatica, Università degli Studi di Trieste, Via Valerio 12/B, I-34127 Trieste, Italy'
- 'Department of Mathematics, Bilkent University, Bilkent, 06800 Ankara, Turkey'
author:
- Fernando Dobarro
- Bülent Ünal
date: 'February 26, 2008'
title: 'Hessian Tensor and Standard Static Space-times'
---
Introduction
============
The two central concepts in this note will be the Hessian tensor on a semi-Riemannian manifold and the warped product of semi-Riemannnian manifolds (specially standard static space-times). There are many arguments in mathematical-physics where these concepts interact.
We briefly recall some basic definitions. Let $(F,g_F)$ be a semi-Riemannian manifold and $\varphi \in C^\infty(F)$ be a smooth function on $F.$ Then the *Hessian of $\varphi$* is the $(0,2)-$tensor defined by $$\label{eq:def-hessian}
{\rm H}_F^\varphi(X,Y) =
g_F(\nabla_X^F \operatorname{grad}_F \varphi,Y)= \nabla^F \nabla^F \varphi (X,Y) ,$$ for any vector fields $X,Y \in \mathfrak{X} (F)$ where $\nabla^F$ is the Levi-Civita connection and $\operatorname{grad}_F$ is the $g_F-$gradient operator. The $g_F-$trace of ${\rm
H}_F^\varphi$ is the Laplace-Beltrami operator, $\Delta_{F}\varphi$. Notice that $\Delta_{F}$ is elliptic if $(F,g_F)$ is Riemannian.
Let $(B,g_B)$ and $(F,g_F)$ be pseudo-Riemannian manifolds and also let $b \colon B \to (0,\infty)$ be a smooth function. Then the (singly) *warped product*, $B \times_b F$ is the product manifold $B \times F$ furnished with the metric tensor $g=g_B \oplus b^{2}g_F$ defined by $$\label{eq:wp-metric}
g=\pi^{\ast}(g_B) \oplus (b \circ \pi)^2 \sigma^{\ast}(g_F),$$ where $\pi \colon B \times F \to B$ and $\sigma \colon B \times F
\to F$ are the usual projection maps and ${}^\ast$ denotes the pull-back operator on tensors. Here, the function $b$ is called the warping function. Warped product manifolds were introduced in general relativity as a method to find general solutions to Einstein’s field equations [@B; @BEE; @ON]. Two important examples include generalized Robertson-Walker space-times and standard static space-times (a generalization of the Einstein static universe). Precisely a *standard static space-time* is a Lorentzian warped product where the warping function is defined on a Riemannian manifold called the base and acting on the negative definite metric on an open interval of real numbers, called the fiber. More precisely, a standard static space-time, denoted by $I_f \times F$, is a Lorentzian warped product furnished with the metric $g=-f^2{\rm d}t^2 \oplus g_F,$ where $(F,g_F)$ is a Riemannian manifold, $f \colon F \to (0,\infty)$ is smooth and $I=(t_1,t_2)$ with $-\infty \leq t_1 < t_2 \leq \infty$. In [@ON], it was shown that any static space-time is locally isometric to a standard static space-time.
There are many subjects in semi-Riemannian geometry and physics where all these ingredients interact and play a central role. For instance in the study of concircular scalar fields [@Obata1962; @T65]; in recent studies of Hessian manifolds [@Shima2007]; in several questions of curvature of warped products and the construction of Einstein manifolds [@B; @DU04a; @DU07a; @DU07b] and in the characterization of Killing vector fields on Robertson Walker space-times [@S99], among many others. We will concentrate our attention to the study of Killing vector fields and energy conditions on standard static space-times, where the ingredients mentioned above are involved (see [@DU08a] and [@DU08b]). The references mentioned above are mere indications in which the reader can find more specific references for each argument as well as links to alternative approaches and much more.
Preliminaries and Notation
==========================
Throughout the paper $I$ will denote an open real interval $I=(t_1,t_2)$, where $-\infty \leq t_1 < t_2 \leq \infty.$ Moreover, $(F,g_F)$ will be a connected Riemannian manifold without boundary with $\dim F=s$. Finally, on an arbitrary differentiable manifold $N$, $C^{\infty}_{>0}(N)$ denotes the set of all strictly positive $C^{\infty}$ functions defined on $N$ and $\mathfrak X(N)$ will denote the $C^{\infty}(N)-$module of smooth vector fields on $N$.
Suppose that $U \in \mathfrak X (I)$ and $V,W \in \mathfrak
X (F)$. If ${\rm Ric}$ and ${\rm Ric}_F$ denote the Ricci tensors of $I _f\times F$ and $(F,g_F)$, respectively, then $$\label{eq:ricci-sss}
{\rm Ric} \left(U+V, U+W \right)={\rm Ric}_F(V,W)+ f \Delta_F f \,
{\rm d}t^2(U,U) - \frac{1}{f}{\rm H}^f_F(V,W).$$
If $\tau$ and $\tau_F$ denote the scalar curvatures of $I _f\times
F$ and $F$, respectively, then $$\label{eq:sc-sss} \tau = \tau_F -2
\frac{1}{f}\Delta_F f.$$
From now on, for any given $f \in C^{\infty}_{>0}(F)$, $Q_F^f$ will denote the $2$ covariant tensor $$\label{eq: quadratic form 1}
Q_F^f:=\Delta_F f \, g_F- H_F^f.$$
$\mathcal{R}ic_F$(respectively, $\mathcal{Q}^f_F$) denotes the quadratic form associated to $\operatorname{Ric}_F$ (respectively, $Q^f_F$).
Notice that and imply that for any $U \in \mathfrak X (I)$ and $V,W \in \mathfrak
X (F)$ is $$\label{eq:Qricci-sss}
{\rm Ric} (U+V,U+W)= {\rm Ric}_F(V,W) + \frac{1}{f}
Q_F^f(V,W)- g(U+V,U+W) \frac{1}{f}\Delta_F f.$$
Killing Vector Fields
=====================
To begin with, we recall the concepts of Killing and conformal-Killing vector fields on pseudo-Riemannian manifolds. Let $(N, g_N)$ be a pseudo-Riemannian manifold and $X \in \mathfrak X(N)$. Then
- $X$ is said to be Killing if ${\rm L}_X g_N = 0$,
- $X$ is said to be conformal-Killing if $\exists \sigma \in
C^{\infty}(N)$ such that ${\rm L}_X g_N = 2 \sigma g_N$,
where ${\rm L}_X$ denotes the Lie derivative with respect to $X$. Moreover, for any $Y$ and $Z$ in $\mathfrak X(N),$ we have the following identity (see [@ON p.250 and p.61]) $$\label{eq:Lie deriv 1}
{\rm L}_X g_N(Y,Z) = g_N(\nabla_Y X,Z) + g_N(Y,\nabla_Z X).$$
Notice that, any vector field on $(I,g_I=\pm {\rm
d}t^2)$ is conformal Killing. Indeed, if $X$ is a vector field on $(I,g_I)$, then $X$ can be expressed as $X = h \partial_t$ for some smooth function $h \in \mathcal{C}^\infty(I)$.
For the rest of the paper, let $M=I_f \times F$ be a standard static space-time with the metric $g=f^2 g_I \oplus g_F$, where $g_I=-{\rm d}t^2$. Suppose that $X,Y,Z \in \mathfrak X(I)$ and $V,W,U \in \mathfrak
X(F)$, then (see [@U01]) $$\label{eq:Lie deriv 1a}
{\rm L}_{X+V}g(Y+W,Z+U) = f^2 {\rm L}^I_X g_I(Y,Z) + 2f
V(f)g_I(Y,Z)
+ {\rm L}^F_V g_F(W,U).$$ Moreover, we also have $$\label{eq:Lie deriv 2}
{\rm L}_{h \partial_t} g_I(Y,Z) = Y(h)g_I(Z,\partial_t) + Z(h) g_I
(Y, \partial_t).$$ By combining (\[eq:Lie deriv 1a\]) and (\[eq:Lie deriv 2\]), we can state the following result.
\[con-1\] [@DU08a] Let $M=I_f \times F$ be a standard static space-time with the metric $g=-f^2{\rm d}t^2 \oplus g_F.$ Suppose that $h \in C^{\infty}(I)$ and $V \in \mathfrak X(F)$. Then $h
\partial_t + V$ is a conformal-Killing vector field on $M$ with $\sigma \in C^{\infty}(M)$ if and only if the following properties are satisfied:
1. $V$ is conformal-Killing on $F$ with associated $\sigma \in
C^{\infty}(F),$
2. $h$ is affine, i.e, there exist $\mu, \nu \in \mathbb{R}$ such that $h(t)=\mu t + \nu$ for any $t \in I,$
3. $V(f) = (\sigma-\mu) f.$
Consequently, $h \partial_t + V$ is a Killing vector field on $M$ if and only if the following properties are satisfied:
1. $V$ is Killing on $F$,
2. there exist $\mu, \nu \in \mathbb{R}$ such that $h(t)=\mu t + \nu$ for any $t \in I,$
3. $V(f) = - \mu f$.
In [@DU08a], to provide a characterization of Killing vector fields on standard static space-times, we modify the procedure used in [@S99] (see also [@CC93]) to study the structure of Killing and conformal-Killing vector fields on warped products. In [@S99], the author obtains full characterizations of the Killing and conformal-Killing vector fields on generalized Robertson-Walker space-times. Here, we will state some of the main results about the characterization of Killing vector fields obtained in [@DU08a].
Let $(F,g_F)$ be a Riemannian manifold of dimension $s$ admitting at least one *nonzero* Killing vector field. Thus, there exists a basis $\{K_{\overline b} \in \mathfrak X(F) | \,
\overline b = 1,\cdots, \overline s \}$ for the set of Killing vector fields on $F$. At this point, we would like to emphasize that the dimension of the set of conformal Killing vector fields on $(I,-{\rm d}t^2)$ is infinite, so that one cannot apply directly the procedure in [@S99] before observing that the form of conformal Killing vector fields on $(I,-{\rm d}t^2)$ is trivial (i.e, any vector field on $(I,-{\rm d}t^2)$ is conformal). Adapting the Sánchez technique to $M=I _f\times F$, a vector field $K \in \mathfrak X(M)$ is a Killing vector field if and only if $K$ can be written in the form $$\label{eq:Killing structure}
K= \psi h \partial_t +
\phi^{\overline b} K_{\overline b},$$ where $h$, $\phi^{\overline b} \in C^\infty(I)$ for any $\overline
b \in\{1, \cdots, \overline m\}$ and $\psi \in C^\infty(F)$ satisfy $$\label{2eqnst}
\left\{
\begin{array}{rcl}
h^\prime \psi+ \phi^{\overline
b} K_{\overline b}(\ln f)& = & 0 \\
{\rm d} \phi^{\overline b} \otimes g_F(K_{\overline b},\cdot) +
g_I(h \partial_t, \cdot) \otimes f^2 {\rm d} \psi & = & 0.
\end{array}
\right.$$ Since $\displaystyle{{\rm d} \phi^{\overline b} = (\phi^{\overline
b})^\prime {\rm d}t}$ with $\phi^{\overline b} \in \mathcal
C^\infty(I)$ and $g_I(h \partial_t, \cdot)=-h {\rm d}t$, is equivalent to $$\label{3eqnst}
\left\{
\begin{array}{rcl}
h^\prime \psi+ \phi^{\overline
b} K_{\overline b}(\ln f)& = & 0 \\
(\phi^{\overline b})^\prime {\rm d}t \otimes g_F(K_{\overline
b},\cdot) & = & h {\rm d}t \otimes f^2 {\rm d} \psi .
\end{array}
\right.$$
The following notation will be useful. Let $h$ be a continuous function defined on a real interval $I$. If there exists a point $t_0 \in I$ such that $h(t_0)\neq 0$, then $I_{t_0}$ denotes the connected component of $\{t \in I: h(t)\neq
0\}$ such that $t_0 \in I_{t_0}$.
By the method of separation of variables and a detailed analysis of system , one can state the following result.
\[thm:Killing ssst\] [@DU08a] Let $(F,g_F)$ be a Riemannian manifold, $f \in C^\infty_{>0}(F)$ and $\{K_{\overline b}\}_{1 \le \overline b \le \overline m}$ a basis of Killing vector fields on $(F,g_F)$. Let also $I$ be an open interval of the form $I=(t_1,t_2)$ in $\mathbb R,$ where $-\infty \leq t_1 < t_2 \leq \infty.$ Consider the standard static space-time $I _f\times F$ with the metric $g=-f^2{\rm d}t^2 \oplus g_F$.
Then, any Killing vector field on $I _f\times F$ admits the structure $$\label{eq:eq:Killing 2-s no 0}
K = \psi h \partial_t + \phi^{\overline b} K_{\overline b}$$ where $h$ and $\phi^{\overline b} \in C^\infty(I)$ for any $\overline b \in\{1, \cdots, \overline m\}$ and $\psi \in
C^\infty(F).$
Furthermore, assume that $K$ is a vector field on $I
_f\times F$ with the structure as in . Hence,
- if $h \equiv 0,$ then the vector field $K=\phi^{\overline b} K_{\overline b}$ is Killing on $I _f\times
F$ if and only if the functions $\phi^{\overline b}$ are constant and $\phi^{\overline b} K_{\overline b}(\ln f)=0$.
- if $h \equiv h_0 \neq 0$ is constant, then the vector field $K= \psi h_0 \partial_t + \phi^{\overline b}
K_{\overline b}$ is Killing on $I_f \times F$ if and only if is satisfied $$\label{3eqnst vector field h=h0 2} \left\{
\begin{array}{l}
f^{2} {\rm grad}_F \psi \textrm{ is a Killing vector field on } (F,g_F)
\textrm{ with }\\
\textrm{ coefficients } \{\tau_{\overline b}\}_{1 \le {\overline
b} \le {\overline m}} \textrm{ relative to the basis }
\{K_{\overline b}\}_{1 \le
{\overline b} \le {\overline m}};\\
(f^{2} {\rm grad}_F \psi)(\ln f)=0 \,(i.e, \,{\rm grad}_F \psi(f)=0); \\
\forall {\overline b}: \phi^{\overline b}(t)=h_0 \tau^{\overline
b}t+\omega^{\overline b} \textrm{ with } \omega^{\overline b}\in
\mathbb{R} : \omega^{\overline b}
K_{\overline b}(\ln f)=0.\\
\end{array}
\right.$$
- if $K$ is a Killing vector field on $I _f\times F$ with the nonconstant function $h$, then the set of functions $h$, $\psi$ and $\{\phi_{\overline b}\}_{1 \le \overline b \le \overline m}$ satisfy $$\label{eq:3eqnst 2 h no 0}
\left\{
\begin{array}{l}
(a)\left\{
\begin{array}{l}
\psi \equiv 0; \\
\phi^{\overline b}(t)=\omega^{\overline b} \textrm{ on } I_{t_0}
\textrm{ where }\omega^{\overline b} \in
\mathbb{R}: \omega^{\overline b} K_{\overline b}(\ln f)=0\\
\end{array}
\right.
\\
\textrm{or }\\
(b)\left\{
\begin{array}{l}
f^{2} {\rm grad}_F \psi \textrm{ is a Killing vector field on }
(F,g_F)\\
\textrm{with } \textrm{coefficients } \{\tau_{\overline b}\}_{1
\le {\overline b} \le {\overline m}} \, \textrm{ relative to the } \\
\textrm{basis } \{K_{\overline b}\}_{1 \le
{\overline b} \le {\overline m}};\\
(f^{2} {\rm grad}_F \psi)(\ln f) = \nu \psi \textrm{ where }
\nu \, \textrm{is constant} ;\\
h(t)= \left\{
\begin{array}{lcl}
a e^{\sqrt{-\nu}\,t} + b e^{-\sqrt{-\nu}\,t}\, \,
& \textrm{ if } & \nu \neq 0 \\
a t + b \, \, &\textrm{ if }& \nu = 0,\\
\end{array}
\right.\\
\textrm{with } a, b \in \mathbb{R};\\
\forall {\overline b}: \phi^{\overline b}(t)=\tau^{\overline
b}\displaystyle \int_{t_0}^t h(s)ds +\omega^{\overline b} \textrm{
with }
\omega^{\overline b}\in \mathbb{R}:\\
\displaystyle h^\prime (t_0) \psi + \omega^{\overline
b}K_{\overline b}(\ln f)=0 \textrm{ on }
I_{t_0}\\
\end{array}
\right.
\\
\end{array}
\right.$$ for any $t_0 \in I$ with $h(t_0)\neq 0.$
Conversely, if a set of functions $h$, $\psi$ and $\{\phi_{\overline b}\}_{1 \le \overline b \le \overline m}$, satisfy with an arbitrary $t_0$ in $I$ and the entire interval $I$ (instead of $I_{t_0}$) and $\psi \in
C^\infty(F)$, then the vector field $\tilde{K}$ on the standard static space-time $I _f\times F$ associated to the set of functions as in is Killing on $I
_f\times F$.
For clarity, we also state the following lemma which covers the case where the Riemannian manifold $(F,g_F)$ admits no nonidentical zero Killing vector field.
\[lem:0 Killing\] Let $(F,g_F)$ be a Riemannian manifold of dimension $s$ and $f \in
C^\infty_{>0}(F)$. Let also $I$ be an open interval of the form $I=(t_1,t_2)$ in $\mathbb R,$ where $-\infty \leq t_1 < t_2 \leq
\infty$. Suppose that the only Killing vector field on $(F,g_F)$ is the zero vector field. Then all the Killing vector fields on the standard static space-time $I_f\times F$ are given by $h_0 \partial_t$ where $h_0$ is a constant.
*Theorem \[thm:Killing ssst\]* is relevant to the problem given by:
$$\label{eq:pb Killing 1} \left\{
\begin{array}{l}
f \in C^\infty_{>0}(F), \psi \in C^\infty(F);\\
f^{2} {\rm grad}_F \psi \textrm{ is a Killing vector field on }
(F,g_F);\\
(f^{2} {\rm grad}_F \psi)(\ln f) = \nu \psi, \nu \in \mathbb{R} .\\
\end{array}
\right.$$
We are interested in the existence of nontrivial solutions for . To study this, for any $Z \in \mathfrak
{X}(F)$ and $\varphi \in C^\infty(F)$ we define the (0,2)-tensor on $(F,g_F)$ given by $$\label{eq:special Killing 2}
B_Z^\varphi(\cdot,\cdot):=\textrm{d} \varphi (\cdot)
\otimes g_F(Z,\cdot) + g_F(\cdot,Z) \otimes \textrm{d}
\varphi (\cdot).$$
A central role in our study of is played by the next proposition which also shows up the relevance of the Hessian tensor in all these questions.
\[prp:special Killing\] [@DU08a] Let $(F,g_F)$ be a Riemannian manifold, $f \in C^\infty_{>0}(F)$ and $\psi \in C^\infty(F)$. Then the vector field $f^{2} {\rm
grad}_F \psi$ is Killing on $(F,g_F)$ if and only if $$\label{eq:special Killing 1}
{\rm H}_F^\psi + \frac{1}{f} B_{{\rm grad}_F \psi}^{f}=0.$$
The latter proposition and the identity $fg_F({\rm grad}_F
\psi,{\rm grad}_F f)=(f {\rm grad}_F \psi)(f)$, allow to express in the equivalent form $$\label{eq:3eqnst 4}
\left\{
\begin{array}{l}
f \in C^\infty_{>0}(F), \psi \in C^\infty(F);\\
\displaystyle {\rm H}_F^\psi + \frac{1}{f}
B_{{\rm grad}_F \psi}^{f} = 0 ;\\
fg_F({\rm grad}_F \psi,{\rm grad}_F f) = \nu \psi, \nu \in
\mathbb{R}.
\end{array}
\right.$$
By *Proposition \[prp:special Killing\]*, if the dimension of the Lie algebra of Killing vector fields of $(F,g_F)$ is zero, then the system has only the trivial solution given by a constant $\psi $ (this constant is not $0$ only if $\nu =0$). This happens, for instance when $(F,g_F)$ is a compact Riemannian manifold of negative-definite Ricci curvature without boundary, indeed it is sufficient to apply the vanishing theorem due to Bochner (see for instance [@B46], [@B Theorem 1.84]). The next *Lemma \[lem: laplace spectrum 1\]* allows to prove that the system is still *equivalent* to $$\label{eq:3eqnst 5}
\left\{
\begin{array}{l}
f \in C^\infty_{>0}(F), \psi \in C^\infty(F);\\
\displaystyle {\rm H}_F^\psi + \frac{1}{f}
B_{{\rm grad}_F \psi}^{f} = 0 ;\\
-\Delta_{g_F} \psi = \displaystyle \nu \frac{2}{f^2}
\psi \textrm{ where } \nu
\textrm{ is a constant}.
\end{array}
\right.$$
\[lem: laplace spectrum 1\][@DU08a] Let $(F,g_F)$ be a Riemannian manifold and $f \in
C^\infty_{>0}(F)$. If $(\nu,\psi)$ satisfies , then $\nu$ is an eigenvalue and $\psi$ is an associated $\nu-$eigenfunction of the elliptic problem: $$\label{eq:weight Laplace-Beltrami 1}
-\Delta_{g_F} \psi = \nu \frac{2}{f^2}
\psi \, \textrm{ on } (F,g_F).$$
Thus, by arguments of critical points and maximum principle, we obtain the following characterization results.
\[prp:laplace spectrum 1\] Let $(F,g_F)$ be a compact Riemannian manifold and $f \in C^\infty_{>0}(F)$. Then $(\nu,\psi)$ satisfies if and only if $\nu =0$ and $\psi$ is constant.
\[thm:killing compact fiber\] Let $M=I _f\times F$ be a standard static space-time with the metric $g=-f^2{\rm d}t^2 \oplus g_F.$ If $(F,g_F)$ is compact then, the set of all Killing vector fields on the standard static space-time $(M,g)$ is given by $$\{a
\partial_t + \tilde{K}| \, a \in \mathbb{R}, \tilde{K} \textrm{ is a
Killing vector field on } (F,g_F) \textrm{ and } \tilde{K}(f)=0
\}.$$
\[rem:sharipov2007\] *(Killing vector fields in the Einstein static universe)* In [@Sh07], the author studied Killing vector fields of a closed homogeneous and isotropic universe (for related questions in quantum field theory and cosmology see [@F87; @LL]). Theorem 6.1 of [@Sh07] corresponds to Theorem \[thm:killing compact fiber\] for the spherical universe $\mathbb{R} \times
\mathbb{S}^3$ with the pseudo-metric $\displaystyle -( R^2 {\rm
d}t^2 -R^2 h_0)$, where the sphere $\mathbb{S}^3$ endowed with the usual metric $h_0$ induced by the canonical Euclidean metric of $\mathbb{R}^4$ and $R$ is a real constant (i.e., a stable universe).
As we have already mentioned, any Killing vector field of a compact Riemannian manifold of negative-definite Ricci tensor is equal to zero. Thus, one can easily state the following result.
\[cor:killing compact fiber of neggative definite Ricci tensor\] Let $M=I _f\times F$ be a standard static space-time with the metric $g=-f^2{\rm d}t^2 \oplus g_F.$ Suppose that $(F,g_F)$ is a compact Riemannian manifold of negative-definite Ricci tensor. Then, any Killing vector field on the standard static space-time $(M,g)$ is given by $a \partial_t$ where $a \in \mathbb
R.$
In [@S07 Theorem 5], it is shown that the decomposition of a space-time as a standard static one is essentially unique when the fiber $F$ is compact. We observe that [*Corollary \[cor:killing compact fiber of neggative definite Ricci tensor\]*]{} enables us to establish a stronger conclusion (i.e., nonexistence of a nontrivial strictly stationary [^1] field) under a stronger assumption involving the definiteness of the Ricci tensor.
At this point, we would like to make some comments about the case where the Riemannian part of a standard static space-time is not compact. While the Theorem \[thm:Killing ssst\] does not require the compactness of the Riemannian manifold $(F,g_F)$, this assumption is the central idea for a complete characterization similar to the one in Theorem \[thm:killing compact fiber\]. The key question in our approach is the full characterization of the solutions of (or the equivalent problems and ) which is reached if $(F,g_F)$ is compact. In the noncompact case, the latter question is more difficult. It is possible to obtain partial nonexistence results for , but the global question is still open. However, there are particular situations, like Example \[rem:sharipov2007\], where the application of Theorem \[thm:Killing ssst\] is sufficient for a complete classification.
Other relevant and related problem is the full classification of the conformal Killing vector fields of a standard static space-time. There are partial recent results in this direction (see for instance [@AC; @Sha-Iq] and the references therein).
Energy Conditions
=================
Recall that a space-time is said to satisfy the *strong energy condition*, briefly SEC, if ${\rm Ric}(X,X) \geq 0$ for all causal tangent vectors $X$ and the *time-like*(respectively, *null* ) *convergence condition*, briefly TCC (respectively, NCC ), if ${\rm Ric}(X,X) \geq 0$ for all time-like (respectively, null ) tangent vectors $X$. Notice that the SEC implies the NCC. Furthermore the TCC is equivalent to the SEC, by continuity. The actual difference between TCC and SEC follows from the fact that while TCC is just a geometric condition imposed on the Ricci tensor, SEC is a condition on the stress-energy tensor. They can be considered equivalent due to the Einstein equation (see below ).
Moreover, a space-time is called to satisfy the *weak energy condition*, briefly WEC, if ${\rm T}(X,X) \geq 0$ for all time-like vectors, where ${\rm T}$ is the energy-momentum tensor, which is determinated by physical considerations.
Along this article, when we consider the energy-momentum tensor, we will assume that the Einstein equation holds (see [@HE; @ON]). More explicitly, $$\label{eq: Einstein eq} {\rm Ric}-\frac{1}{2} \tau g
= 8 \pi {\rm T}.$$
The WEC has many applications in general relativity theory such as nonexistence of closed time-like curve (see [@ChoPark]) and the problem of causality violation ([@OriSoen]). But its fundamental usage still lies in Penrose’s Singularity theorem (see [@Pen1]).
Let $M=I _f\times F$ be a standard static space-time with the metric $g=-f^2{\rm d}t^2 \oplus g_F$. By , ${\rm Ric} (\partial_t,\partial_t)= f \Delta_F f$. So, since $g(\partial_t,\partial_t)= -f^2 <0$, the warping function $f$ is necessarily subharmonic, i.e. $\Delta_F f \geq 0$, if the standard static space-time satisfies the SEC (or equivalently, the TCC)[@DA1]. On the other hand, it is well known that there is no nonconstant subharmonic functions on compact Riemannian manifolds [@B46], and hence $f$ is a positive constant if $(F,g_F)$ is compact. Furthermore, applying a family of Liouville type results of Li, Schoen and Yau, in [@DU08b] we give a set of sufficient conditions implying the warped function is a positive constant under the hypothesis that $(F,g_F)$ is complete and noncompact.
Below, we state a set of necessary conditions for a standard static space-time to satisfy the NCC and other Ricci curvature conditions which are useful to study conformal hyperbolicity through the studies of Markowitz, more precisely Theorems 5.1 and 5.8 in [@MM1]. We also observe that there are more accurately analogous results for a Generalized Robertson-Walker space-time given in [@ES00 Proposition 4.2] (see also [@DU08b]).
\[ec-t\] [@DU08b; @DA1] Let $M=I _f\times F$ be a standard static space-time with the metric $g=-f^2{\rm d}t^2 \oplus g_F$, where $s = \dim F \ge 2$.
1. If $\mathcal{R}ic_F$ and $\mathcal{Q}^f_F$ are positive semi-definite, then $M$ satisfies the TCC and the NCC.
2. If $\mathcal{R}ic_F$ and $\mathcal{Q}^f_F$ are negative semi-definite, then ${\rm Ric}(w,w) \leq 0$ for any causal vector $w \in \mathfrak X(M)$.
3. If $(F,g_F)$ is Ricci flat, then $\mathcal{Q}^f_F$ is positive semi-definite if and only if $M$ satisfies the NCC.
Now we state a small results about energy conditions in terms of the energy-momentum tensor $T$. It is easy to obtain from , and that for any $U \in \mathfrak X (I)$ and $V \in
\mathfrak X (F)$ is $$\label{eq: em tensor}
8\pi T(U+V,U+V)= \mathcal{R}ic_F(V) + \frac{1}{f}
\mathcal{Q}_F^f(V) - \frac{1}{2} \tau_F g(U+V,U+V).$$ So, as above, since $g(\partial_t,\partial_t)= -f^2 <0$ results that if a standard static space-time satisfies the WEC, then $\tau_F \geq 0$ and as consequence
\[ec-w\] [@DU08b] Let $M=I _f\times F$ be a standard static space-time with the metric $g=-f^2{\rm d}t^2 \oplus g_F$, where $s = \dim F \ge 2$.
1. If $\mathcal{R}ic_F$ and $\mathcal{Q}^f_F$ are positive (respectively negative) semi-definite, then ${\rm T}(w,w) \geq 0$ (respectively $\leq 0$) for any causal vector $w \in \mathfrak X(M).$
2. If $(F,g_F)$ is Ricci flat, then for any $u \in \mathfrak X (I)$ and $v \in \mathfrak X (F)$ is $8\pi
T(u+v,u+v)=\mathcal{Q}^f_F(v)$. Thus, $\mathcal{Q}^f_F$is positive semi-definite if and only if ${\rm T}(w,w) \geq 0$ for any vector $w \in \mathfrak X(M)$.
In [@MM1] the intrinsic Lorentzian pseudo-distance $d_M \colon M \times M \to [0,\infty)$ was defined by $$\label{eq:pseudo-distance}
d_M(p,q)=\inf_{\alpha}L(\alpha),$$ where the infimum is taken over all the chains of null geodesic segments joining $p$ and $q$ and $L(\alpha)$ means the length of the chain $\alpha$. Such a chain $\alpha$ is a sequence of points $p=p_0,p_1,\dots,p_k=q$ in $M,$ pairs of points $(a_1,b_1), \dots,
(a_k,b_k)$ in $(-1,1)$ and projective maps (i.e., a projective map is simply a null geodesics with the projective parameter as the natural parameter) $f_1,\dots,$ $f_k$ from $(-1,1)$ into $M$ such that $f_i(a_i)=p_{i-1}$ and $f_i(b_i)=p_i$ for $i=1,\cdots,k$. Besides, the length of $\alpha $ is $$L(\alpha)=\sum_{i=1}^{k} \rho(a_i,b_i),$$ where $\rho$ is the Poincaré distance in $(-1,1)$ (see [@MM1; @MM2] for details). Notice that $d_M$ is really a pseudo-distance, i.e., it is non-negative, symmetric and satisfies the triangle inequality. A Lorentzian manifold $(M,g)$ where $d_M$ is a distance is called *conformally hyperbolic*.
In [@MM1 Theorem 5.1], it is proved that if $(M,g)$ is a null geodesically complete Lorentz manifold satisfying the reverse NCC condition, i.e. ${\rm Ric}(X,X) \leq 0$ for all null vectors $X$, then it has a trivial Lorentzian pseudo-distance, i.e., $d_M \equiv 0$. Moreover, in [@MM1 Theorem 5.8], it is obtained that if $(M,g)$ is an $n(\geq 3)$-dimensional Lorentzian manifold satisfying the NCC and the null generic condition, briefly NGC, (i.e., ${\rm Ric}(\gamma^\prime,\gamma^\prime) \neq
0,$ for at least one point of each inextendible null geodesic $\gamma$) then, it is conformally hyperbolic.
Under the light of these theorems, one can easily conclude that
- Complete Einstein space-times (in particular, Minkowski, de-Sitter and the anti-de Sitter space-times) have all trivial Lorentzian pseudo-distances because of Theorem 5.1 of [@MM1].
- The Einstein static universe has also trivial Lorentzian pseudo-distance since the space-times in the previous item can be conformally imbedded in the Einstein static universe.
- A Robertson-Walker space-time (i.e., an isotropic homogeneous space-time) is conformally hyperbolic due to Theorem 5.9 of [@MM1].
- The Einstein-de Sitter space $M$ is conformally hyperbolic and (see Theorem 5 in [@MM2] for details and a precise formula for the Lorentzian pseudo-distance on this class of space-time).
Applying *Theorem \[ec-t\]*, and the previous Markowitz results, we obtain the theorems that follow.
\[main-12\] [@DU08b] Let $M=\mathbb R _f\times F$ be a standard static space-time with the metric $g=-f^2{\rm d}t^2 \oplus g_F.$ Suppose that $\mathcal{R}ic_F$ and $\mathcal{Q}^f_F$ are negative semi-definite.
1. If $(F,g_F)$ is compact, then the Lorentzian pseudo-distance $d_M$ on the standard static space-time $(M,g)$ is trivial, i.e., $d_M \equiv 0.$
2. If $(F,g_F)$ is complete and $0 < \inf f $, then the Lorentzian pseudo-distance $d_M$ on the standard static space-time $(M,g)$ is trivial, i.e., $d_M \equiv 0.$
In the Theorem \[main-12\], the general hypothesis ensure the *reversed* NCC, in order to apply [@MM1 <span style="font-variant:small-caps;">Theorem 5.1</span>]. The additional hypothesis in item (2) of the same theorem implies the null geodesic completeness of $M$ by [@ADt Thoerem 3.12]. We observe that there are more general hypotheses which imply null geodesic completeness, see for instance [@RS Th. 3.9(ii b)].
\[main-3\] [@DU08b] Let $M=I _f\times F$ be a standard static space-time with the metric $g=-f^2{\rm d}t^2 \oplus g_F$. Suppose that $\mathcal{R}ic_F$ is positive semi-definite and $\mathcal{Q}^f_F$ is positive definite. Then the standard static space-time $(M,g)$ is conformally hyperbolic.
Now we state some results joining the conformal hyperbolicity and causal conjugate points of a standard static space-time by using [@BJ1; @BE1; @BE2; @CE] and also [@BEE]. In [@CE Theorem 2.3], it was shown that if the line integral of the Ricci tensor along a complete causal geodesic in a Lorentzian manifold is positive, then the complete causal geodesic contains a pair of conjugate points.
Assume that $\gamma=(\alpha, \beta)$ is a complete causal geodesic in a standard static space-time of the form $M=I _f\times F$ with the metric $g=-f^2{\rm d}t^2 \oplus g_F.$ Then by using $g(\gamma^\prime, \gamma^\prime) \leq 0$ and we have, $$\label{} {\rm Ric}(\gamma^\prime,
\gamma^\prime)= {\rm Ric}_F(\beta^\prime, \beta^\prime) +
\frac{1}{f}Q^f_F(\beta^\prime,\beta^\prime)-
\underbrace{g(\gamma^\prime,\gamma^\prime)}_{\leq 0}
\frac{1}{f}\Delta_F f.$$
We can easily state the following existence result for conjugate points of complete causal geodesics in a conformally hyperbolic standard static space-time by [*Theorem \[main-3\]*]{} and [@CE Theorem 2.3].
\[cor-1\] Let $M=I _f\times F$ be a standard static space-time with the metric $g=-f^2{\rm d}t^2 \oplus g_F$. Suppose that $\mathcal{R}ic_F$ is positive semi-definite. If $\mathcal{Q}^f_F$ is positive definite, then $(M,g)$ is conformally hyperbolic and any complete causal geodesic in $(M,g)$ has a pair of conjugate points.
By using , [@BEE Propositions 11.7, 11.8 and Theorem 11.9] and [@ADt Corollary 3.17], we can establish an existence result for conjugate points of time-like geodesics in a standard static space-time which by *Theorem \[cor-1\]* is also conformally hyperbolic. In the next theorem, $\mathbf{L}$ denotes the usual time-like Lorentzian length and ${\rm diam}_\mathbf{L}$ denotes the corresponding time-like diameter (see [@BEE Chapters 4 and 11]).
\[cor-3\] Let $M=I _f\times F$ be a standard static space-time with the metric $g=-f^2{\rm d}t^2 \oplus g_F$. Suppose that $\mathcal{R}ic_F$ and $\mathcal{Q}^f_F$ are positive semi-definite. If there exists a constant $c$ such that $\displaystyle\frac{1}{f}\Delta_F f \geq c >0$, then
1. any time-like geodesic $\gamma \colon [r_1,r_2] \to M$ in $(M,g)$ with $\mathbf{L}(\gamma) \geq \pi \sqrt{\frac{n-1}{c}}$ has a pair of conjugate points,
2. for any time-like geodesic $\gamma \colon [r_1,r_2] \to M$ in $(M,g)$ with $\mathbf{L}(\gamma)
> \pi\sqrt{\frac{n-1}{c}}$, $r=r_1$ is conjugate along $\gamma$ to some $r_0 \in (r_1,r_2),$ and as consequence $\gamma$ is not maximal,
3. if $I=\mathbb{R}$, $(F,g_F)$ is complete and $\sup f<\infty$, then $\displaystyle{\rm diam}_\mathbf{L}(M,g) \leq
\pi\sqrt{\frac{n-1}{c}}.$
In the final part of [@DU08b] we show some examples and results connecting the tensor $Q_F^f$, conformal hyperbolicity, concircular scalar fields and Hessian manifolds, where the role of the Hessian tensor is central.
<span style="font-variant:small-caps;">Acknowledgements</span>
The authors wish to thank the referee for the useful and constructive suggestions. F. D. thanks The Abdus Salam International Centre of Theoretical Physics for their warm hospitality where part of this work has been done.
[A]{}
D. E. Allison, *Energy conditions in standard static space-times*, General Relativity and Gravitation [**20**]{}(2) (1988), 115–122.
D. E. Allison, *Lorentzian warped products and static space-times*, Ph.D. Thesis, University of Missouri-Columbia, 1985.
P. S. Apostolopoulos and J. G. Carot, *Conformal symmetries in warped manifolds*, Journal of Physics: Conference Series 8 (2005) 28–33.
J. K. Beem, *Lorentzian geometry in the large*, Mathematics of gravitation, Part I [**41**]{}, Polish Acad. Sci., Warsaw, (1997), 11–20.
J. K. Beem and P. Ehrlich, *Cut points, conjugate points and Lorentzian comparison theorems*, Math. Proc. Cambridge Philos. Soc. [**86**]{} (1979), 365–384.
J. K. Beem and P. Ehrlich, *Singularities, incompleteness and the Lorentzian distance function*, Math. Proc. Cambridge Philos. Soc. [**85**]{} (1979), 161–178.
J. K. Beem, P. E. Ehrlich and K. L. Easley, *Global Lorentzian Geometry*, (2nd Ed.), Marcel Dekker, New York, 1996.
A. L. Besse, *Einstein Manifolds*, Springer-Verlag, Heidelberg, 1987.
S. Bochner, *Vector fields and Ricci curvature*, Bull. Am. Math. Sc. [**52**]{} (1946), 776–797.
J. Carot J. and J. da Costa, *On the geometry of warped spacetimes*, Class. Quantum Grav. [**10**]{} (1993), 461–482.
C. Chicone and P. Ehrlich, *Line integration of Ricci curvature and conjugate points in Lorentzian and Riemannian manifolds*, Manuscripta Math. [**31**]{} (1980), 297–316.
Y. M. Cho and D. H. Park, *Closed time-like curves and weak energy condition*, Phys. Lett. B [**402**]{}, (1997), 18–24.
F. Dobarro and B. Ünal, *Special standard static spacetimes*, Nonlinear Analysis: Theory, Methods & Applications [**59**]{}(5) (2004), 759–770.
F. Dobarro and B. Ünal, *Curvature in Special Base Conformal Warped Products*, arXiv:math/0412436v3\[math.DG\].
F. Dobarro and B. Ünal, *About curvature, conformal metrics and warped products*, J. Phys. A: Math. Theor. [**40**]{} (2007) 13907–13930, arXiv:0704.0595v1\[math.DG\].
F. Dobarro and B. Ünal, *Killing vector fields on standard static space-times*, arXiv:0801.4692v1\[math.DG\].
F. Dobarro and B. Ünal, *Implications of Energy Conditions on Standard Static Space-times*, work in progress. P. Ehrlich, M. Sánchez, *Some semi-Riemannian volume comparison theorems*, Tohôku Math. J. [**52**]{} (2000) 285–314.
S. A. Fulling, *Aspects of quantum field theory in curved space-time*, Cambridge University Press, UK, 1987.
S. W. Hawking and G. F. R. Ellis, *The Large Scale Structure of Space-time*, Cambridge University Press, UK, 1973.
L. D. Landau and E. M. Lifshits, *Field Theory*, Vol. II of *Theoretical Physics*, Nauka publishers, Moscow, 1988.
M. J. Markowitz, *An intrinsic conformal Lorentz pseudodistance*, Math. Proc. Cambridge Philos. Soc. [**89**]{}(2) (1981), 359–371.
M. J. Markowitz, *Conformal hyperbolicity of Lorentzian warped products*, Gen. Relativity Gravitation [**14**]{}(2) (1982), 1095–1105.
M. Obata, *Certain conditions for a Riemannian manifold isometric with a sphere*, J.Math. Soc. Japan, [**14**]{}(1962), 333–340.
B. O’Neill, *Semi-Riemannian Geometry With Applications to Relativity*, Academic Press, New York, 1983.
A. Ori and Y. Soen, *Causality violation and the weak energy condition*, Phys. Rev. D [**49**]{}, (1994), 3990–3997.
R. Penrose, *Techniques of Differential Topology in Relativity*, Regional Conference Series in Applied Mathematics, SIAM, PA, 1972.
A. Romero and M. Sánchez, *On completeness of certain families of semi-Riemannian manifolds*, Geometriae Dedicata [**53**]{} (1994), 103–117.
M. Sánchez, *On the geometry of generalized Robertson-Walker spacetimes: curvature and Killing fields*, J. Geom. Phys. [**31**]{} (1999), 1–15.
M. Sánchez and José M. M. Senovilla, *A note on the uniqueness of global static decompositions*, Class. Quantum Grav. [**24**]{} (2007), 6121–6126.
G. Shabbir and S. Iqbal, *A note on proper conformal vector fields in cylinderically symmetric static space-times*, arXiv:0711.1207v1 \[gr-qc\].
R. A. Sharipov, *On Killing vector fields of a homogeneous and isotropic universe in closed model*, arXiv:0708.2508v1\[math.DG\].
Shima H., *The Geometry of Hessian Structures*, World Scientific, 2007.
Y. Tashiro Y, *Complete Riemannian manifolds and some vector fields*, Trans. Amer. Math. Soc. **117** (1965), 251–275.
B. Ünal, *Doubly Warped Products*, Differential Geometry and Its Applications [**15**]{}(3), (2001), 253–263.
[^1]: Here, a stationary field means that it is Killing and time-like at the same time (see [@S07]).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The effect of derivative nonlinearity and parity-time- (${\mathcal{PT}}$-) symmetric potentials on the wave propagation dynamics is investigated in the derivative nonlinear Schrödinger equation, where the physically interesting Scarff-II and hamonic-Hermite-Gaussian potentials are chosen. We study numerically the regions of unbroken/broken linear ${\mathcal{PT}}$-symmetric phases and find some stable bright solitons of this model in a wide range of potential parameters even though the corresponding linear ${\mathcal{PT}}$-symmetric phases are broken. The semi-elastic interactions between exact bright solitons and exotic incident waves are illustrated such that we find that exact nonlinear modes almost keep their shapes after interactions even if the exotic incident waves have evidently been changed. Moreover, we exert the adiabatic switching on ${\mathcal{PT}}$-symmetric potential parameters such that a stable nonlinear mode with the unbroken linear ${\mathcal{PT}}$-symmetric phase can be excited to another stable nonlinear mode belonging to the broken linear ${\mathcal{PT}}$-symmetric phase.'
author:
- Yong Chen
- 'Zhenya Yan[^1]'
date: '21 May 2016, Phys. Rev. E [**95**]{}, 012205 (2017)'
title: |
Stable parity-time-symmetric nonlinear modes and excitations\
in a derivative nonlinear Schrödinger equation
---
Introduction
============
The derivative nonlinear Schrödinger (DNLS) equation \[dnls0\] i\_t+\_[xx]{}+ig(||\^2)\_x=0, g>0 where $g$ represents its relative magnitude (the space-reflection transformation $x\to -x$ can make $g<0$) and the derivative nonlinearity term is also called the nonlinear dispersion term [@keer]. In fact, Eq. (\[dnls0\]) has a close relation with the modified nonlinear Schrödinger (MNLS) equation [@phap4; @phap5] \[dnls0g\] iq\_+q\_+|q|\^2q+i(|q|\^2q)\_=0, where $\alpha$ denotes the group velocity dispersion coefficient, the Kerr nonlinear coefficient $\lambda$ and derivative nonlinear coefficient $\gamma$ both depend on nonlinear refractive index $n_2$. Eq. (\[dnls0g\]) can be transformed into Eq. (\[dnls0\]) by using the similarity transformation [@cl04] $q(\tau, \xi)=\psi(x, t)e^{i(kx+k^2t)}$ with $x=\frac{\gamma}{\alpha g}\tau-\frac{2\lambda}{g}\xi$, $t=\frac{\gamma^2}{\alpha g^2}\xi$, and $k=\frac{\alpha g\lambda}{\gamma^2}$. Eq. (\[dnls0\]) (or the similarity Eq. (\[dnls0g\])) can be used to describe many nonlinear wave phenomena in some physical applications such as the propagations of small-amplitude nonlinear Alfvén waves in a low-$\beta$ plasma [@dnlsp], large-amplitude magnetohydrodynamic waves propagating in an arbitrary direction with respect to the magnetic field in a high-$\beta$ plasma [@dnls-02], the filamentation of lower-hybrid waves [@dnls-77], and the sub-picosecond or femtosecond pulses in single-mode optical fiber [@phap4; @phap5]. Eq. (\[dnls0\]) can be solved using the inverse scattering method [@kn78]. Moreover, its some modified versions (e.g., Eq. (\[dnls0g\]) had also been studied such as the Chen-Li-Liu equation [@cll] and the modified NLS equation [@cll; @cons79; @cl04].
The NLS equation describing light propagation in optics [@opso] with real external potentials or/and gain-and-loss distributions has been investigated [@simi0; @yanpla10; @yanpre12; @1dnlsv; @1dnlsv2; @yanpre15; @mu08; @vvk12; @pt1; @pt2; @pt3; @exp9; @exp10; @exp11; @exp12; @exp13; @exp14] since the refractive index of the optical waveguide can be complex [@complex1; @complex2]. It is surprising to find that if the complex refractive index satisfies the property of the parity-time (${\mathcal{PT}}$) symmetry [@bender], that is, if the real and imaginary parts of the refractive index are the even and odd functions of spatial position, respectively, then the propagation constant of the light can still be in all-real spectrum range, hence admitting stationary beam transmission [@real1; @real2; @real3; @real4]. Moreover, the complex ${\mathcal{PT}}$-symmetric potentials can also support continuous families of stable solitons [@yanpre15; @mu08; @vvk12; @pt1; @pt2; @pt3; @exp9; @exp10; @exp11; @exp12; @exp13; @exp14] even if the solitons appear in the range of the broken linear ${\mathcal{PT}}$-symmetric phases (see, e.g., Ref. [@yanpre15]). More recently, the stable nonlinear modes were found in the third-order NLS equation with ${\mathcal{PT}}$-symmetric potentials [@yan16]. Other interesting ${\mathcal{PT}}$-symmetric phenomena or properties can be found in the relevant experimental studies [@real2; @real3; @exp3; @exp4].
It is still a significant subject to study whether stable nonlinear modes exist in other models with ${\mathcal{PT}}$-symmetric potentials. To the best of our knowledge, soliton dynamics of the DNLS equation (\[dnls0\]) (it can be regarded as the extension of the NLS equation) in the ${\mathcal{PT}}$-symmetric potentials was not studied before. Our main goal in this paper is to find stable solitons and study their dynamical behaviors of the DNLS equation (\[dnls0\]) in two kinds of physically interesting ${\mathcal{PT}}$-symmetric potentials (i.e., ${\mathcal{PT}}$-symmetric Scarff-II and hamonic-Hermite-Gaussian potentials).
The rest of this paper is arranged as follows. We firstly present the broken/unbroken regions of the linear spectral problem with ${\mathcal{PT}}$-symmetric potentials. And then we analysis the effect of the ${\mathcal{PT}}$-symmetric potentials and derivative nonlinearity on the stability, wave propagations, interactions, transverse power-flow density of solitons in detail. Finally, based on the adiabatic change technique we also perform some types of stable excitations belonging to the broken linear ${\mathcal{PT}}$-symmetric phases from the nonlinear modes.
Nonlinear physical model with ${\mathcal{PT}}$-symmetric potentials
===================================================================
The nonlinear model
-------------------
We begin our investigation by considering the wave propagations in the derivative nonlinearity and ${\mathcal{PT}}$-symmetric potentials, which can be modelled by the following normalized derivative nonlinear Schrödinger-like equation with ${\mathcal{PT}}$-symmetric potentials \[dnls\] i\_t+\_[xx]{}-\[V(x)+iW(x)\]+ig(||\^2)\_x=0, where $\psi=\psi(x,t)$ is a complex wave function of $x,t$, which is proportional to the electric field envelope, $t$ denotes the scaled propagation time or distance, $x$ represents the normalized transverse coordinate, and $g$ is a positive nonlinear coefficient (without loss of generality we can choose $g=1$). When we make the transformation $t\rightarrow z$ (propagation distance) and $x\rightarrow t$ (propagation time), the above-mentioned model may be used to describe the evolution of pulses inside a single-mode fiber [@keer; @keer2]. The ${\mathcal{PT}}$-symmetric potential $V(x)+iW(x)$ requires that its real and imaginary components satisfy $V(-x)=V(x)$ and $W(-x)=-W(x)$ describing the real-valued external potential and gain-and-loss distribution, respectively. It is easy to show that Eq. (\[dnls\]) is invariant under the ${\mathcal{PT}}$-symmetric transformation if the complex potential $[V(x)+iW(x)]$ is ${\mathcal{PT}}$-symmetric, where ${\mathcal P}$ and ${\mathcal T}$ operators are defined by ${\mathcal P}:\, x\to -x$; ${\mathcal T}:\, i\to-i,\, t\to-t$. Eq. (\[dnls\]) can be rewritten as the form $\psi_t=-\frac{\partial}{\partial x}\frac{\delta\mathcal{H}}{\delta\psi^{*}}$ with the Hamiltonian $\nonumber\mathcal{H}=\int_{-\infty}^{+\infty}\{-i\psi_x\psi^{*}+\psi^{*}\int_0^x[iV(x)-W(x)]\psi dx+\frac{g}2|\psi|^4\} dx$, where the asterisk stands for the complex conjugate. The power and quasi-power of Eq. (\[dnls\]) are given by $P(t)=\int_{-\infty}^{+\infty}|\psi(x,t)|^2dx$ and $Q(t)=\int_{-\infty}^{+\infty}\psi(x,t)\psi^{*}(-x,t)dx$, respectively. One can immediately obtain that $P_t=2\int_{-\infty}^{+\infty}W(x)|\psi(x,t)|^2dx$ and $Q_t=-\int_{-\infty}^{+\infty}g\psi(x,t)\psi^{*}(-x,t)[(|\psi(x,t)|^2)_x-(|\psi(-x,t)|^2)_x+\psi^{*}(x,t)\psi_x(x,t)-\psi(-x,t)\psi^*_x(-x,t)]dx$.
General theory
--------------
The stationary solutions of Eq. (\[dnls\]) are considered in the form $\psi(x,t)=\phi(x) e^{i\mu t}$, where $\mu$ is the real propagation constant and the nonlinear localized eigenmode ($\lim_{|x|\rightarrow\infty}\phi(x)=0$) satisfies \[ode\] \_[xx]{}-\[V(x)+iW(x)\]+ig(||\^2)\_x=(x). For Eq. (\[ode\]) with some functions $V(x)$ and $W(x)$, there exist two cases for the study of solutions of Eq. (\[ode\]): (i) if $\phi(x)$ is a real-valued function, then we have the solution of Eq. (\[ode\]) \[solua\] \^2(x)=\_x\^[-1]{}W(x), with the condition linking the potential and gain-and-loss distribution being W\^2(x)-2W\_x(x)\_x\^[-1]{}W(x)+4\[V(x)+\](\_x\^[-1]{}W(x))\^2=0, where $\partial_x^{-1}W(x)=\int_0^xW(s)ds$.
\(ii) if the function $\phi(x)$ is complex in the form \[solub\] (x)=(x), where $\rho(x)$ is the real amplitude, and the real function $v(x)$ is the hydrodynamic velocity, then we substitute Eq. (\[solub\]) into Eq. (\[ode\]) to yield the relations linking the hydrodynamic velocity \[ode1\] v(x)=\^[-2]{}(x)\^x\_0W(s)\^2(s)ds-\^2(x), and the amplitude satisfying the second-order ordinary differential equation with varying coefficients \[ode2\] \_[xx]{}(x)=\[V(x)+v\^[2]{}(x)+\](x)+gv(x)\^3(x).
In order to further study the linear stability of such nonlinear localized mode $\psi(x,t)=\phi(x) e^{i\mu t}$, we consider the perturbed solutions of Eq. (\[dnls\]) as follow \[pert\] (x,t)={(x)+}e\^[it]{}, where $\epsilon\ll 1$, $F(x)$ and $G(x)$ are the perturbation eigenfunctions of the linearized eigenvalue problem and $\delta$ measures the growth rate of the perturbation instability. Substituting Eq. (\[pert\]) into Eq. (\[dnls\]) and linearizing with respect to $\epsilon$, we obtain the following linear eigenvalue problem for the perturbation modes \[st\] (
[cc]{} \_1 & \_2\
-\_2\^\* & -\_1\^\*\
) (
[c]{} F(x)\
G(x)\
) =(
[c]{} F(x)\
G(x)\
), \[stable\] where $\hat{L}_1=\partial^2_x+2ig[|\phi|^2\partial_x+(|\phi|^2)_x]-[V(x)+iW(x)]-\mu$ and $\hat{L}_2=ig[\phi^2 \partial_x+(\phi^2)_x]$. Obviously, the ${\mathcal{PT}}$-symmetric nonlinear modes are linearly stable if $\delta$ is purely real, otherwise they are linearly unstable.
In what follows we study Eqs. (\[dnls\]) and (\[ode\]) analytically and numerically in detail for two distinct physically interesting ${\mathcal{PT}}$-symmetric potentials.
Nonlinear modes in the ${\mathcal{PT}}$-symmetric Scarff-II potential
=====================================================================
The first potential to consider is the celebrated ${\mathcal{PT}}$-symmetric Scarff-II potential [@real4] \[ps\] V(x)=V\_0[sech]{}\^2x, W(x)=W\_0[sech]{}xx, with the real parameters $V_0<0$ and $W_0$ modulating the amplitudes of the reflectionless potential $V(x)$ and gain-and-loss distribution $W(x)$, respectively. For the case $W_0>0$, $W(x)$ represents the gain (loss) action in the domain of $x\geq 0$ ($x\leq 0$), respectively, whereas $W_0<0$, $W(x)$ represents the gain (loss) action in the domain of $x\leq 0$ ($x\geq 0$), respectively. Evidently, both $V(x)$ and $W(x)$ are bounded and vanish as $|x|\to \infty$. Moreover, the gain-and-loss distribution $W(x)$ always has a global balance in Eq. (\[dnls\]) since $\int^{+\infty}_{-\infty}W(x)dx=0$.
Linear spectral problem
-----------------------
In the absence of the derivative nonlinearity ($g=0$), Eq. (\[ode\]) becomes the following linear eigenvalue problem with the Scarff-II potential (\[ps\]) \[lp\] L(x)=(x),L=-\_x\^2+ V(x)+ iW(x), with $\lambda$ and $\Phi(x)$ being the eigenvalue and localized eigenfunction, respectively. By virtue of the spectral method, we numerically find its symmetry-breaking line in $(V_0, W_0)$-space, which coincides well with the theoretical result that Eq. (\[lp\]) with Eq. (\[ps\]) enjoys entirely real spectra provided that $|W_0|\leq -V_0+1/4$ [@real4] (see Fig. \[fig-sp\]a). Therefore, for a fixed $W_0$ satisfying $|W_0|>1/4$, there always exists a threshold of the potential amplitude $V_0$, beyond which a phase transition occurs and the corresponding spectra become complex in the meantime (see Figs. \[fig-sp\]b, c).
However, more interestingly, even though the phase transition occurs in the linear spectral problem (i.e., Eq. (\[lp\]) has the complex spectra), nonlinear modes can still exist with entirely real eigenvalues, since the beam itself can have a strong influence on the amplitude of the potential through the derivative nonlinearity. Thus for the same parameter $W_0$, the new effective potential with stronger derivative nonlinearity may alter the linear ${\mathcal{PT}}$-symmetric threshold with the result that nonlinear eigenmodes can be found with real eigenvalues. But the broken ${\mathcal{PT}}$ symmetry cannot be nonlinearly restored at the lower power levels subject to the weaker derivative nonlinearity. Thus in what follows we turn to investigate nonlinear modes of Eq. (\[dnls\]) with ${\mathcal{PT}}$-symmetric Scarff-II potential (\[ps\]) analytically and numerically.
Stability and dynamics of nonlinear modes
-----------------------------------------
Without loss of generality, we consider $g=1$. The exact bright solitons of Eq. (\[ode\]) with the Scarff-II potential (\[ps\]) can be found in the form \[sol-s\] (x)=, where $\phi_0=W_0\pm\sqrt{4W_0^2+12V_0+9}>0$ (‘$+$’ denotes Scarff-II-Case-1 and ‘$-$’ Scarff-II-Case-2, hereafter), the propagation constant is $\mu=0.25$, the nontrivial phase is $\varphi(x)=-\frac{(W_0+\phi_0)}2 {\rm tan}^{-1}[{\rm sinh}x]$ .
The existence conditions for the bright solitons (\[sol-s\]) are V\_0>-14(W\_0\^2+3) [for]{} W\_0<0\
[or]{}\
V\_0>-(+34) [for]{} W\_0>0 for the Scarff-II-Case-1 and -(+34)V\_0<-14(W\_0\^2+3) [for]{} W\_0>0 for the Scarff-II-Case-2. Apparently, the nonlinear localized modes (\[sol-s\]) are also ${\mathcal{PT}}$-symmetric. It is easy to see that for the same ${\mathcal{PT}}$-symmetric potential, the solutions (\[sol-s\]) of the DNLS equation and ones of NLS equation (see Refs. [@yanpre15; @mu08]) have the distinct properties.
It is easy to see from Fig. \[fig-sp\]a that except for the only one tangent point $(V_0, W_0)=(-0.75, 0)$, the dashed parabola $V_0=-(W_0^2/3+0.75)$ is completely contained in the solid parabola $V_0=-0.25(W_0^2+3)$, which is tangent with the two linear ${\mathcal{PT}}$-symmetric breaking lines $\pm W_0=0.25-V_0$ with two tangent points being $(V_0, W_0)=(-1.75,\pm2)$. Thus the existence region of bright solitons (\[sol-s\]) for Scaff-II-Case-1 contains both entire region of broken ${\mathcal{PT}}$-symmetric phase and partial region of unbroken ${\mathcal{PT}}$-symmetric phase (see Fig. \[stability-s-1\]b), whereas the existence region of bright solitons (\[sol-s\]) for Scaff-II-Case-2 is only located between the two parabolas in upper half plane, utterly located in the region of unbroken linear ${\mathcal{PT}}$-symmetric phase (see Fig. \[stability-s-2\]b). Moreover, we find that the strength $V_0$ and $W_0$ of the potential (\[ps\]) can modulate not only amplitudes of bright solitons (\[sol-s\]) but also the corresponding power $P=\int_{-\infty}^{+\infty}{|\psi(x,t)|^2}dx=2\pi\phi_0/3$, which is conserved.
In the following we investigate numerically the linear stability of bright solitons (\[sol-s\]) for the Scarff-II-Case-1 and Scarff-II-Case-2 through the direct wave propagation of initially stationary modes (\[sol-s\]) with some $2\%$ noise perturbation. Fig. \[stability-s-1\]a for Scarff-II-Case-1 and Fig. \[stability-s-2\]a for Scarff-II-Case-2 exhibit the stable (blue) and unstable (red) regions of nonlinear localized modes (\[sol-s\]), respectively, which are determined by the maximum absolute value of imaginary parts of the linearized eigenvalue $\delta$ in Eq. (\[st\]) in $(V_0, W_0)$-space. For Scarff-II-Case-1 with $V_0=-1,W_0=-1.1$, belonging to the region of unbroken linear ${\mathcal{PT}}$-symmetric phase (see Fig. \[fig-sp\]a), the corresponding nonlinear localized mode is stable (see Figs. \[stability-s-1\]d). If we fix $V_0=-1$ and change $W_0=-1.4$ (it in fact holds for $W_0\in (-1.25, -1.4]$), in spite of belonging to the region of broken linear ${\mathcal{PT}}$-symmetric phase, the corresponding nonlinear localized mode can still keep stable (see Fig. \[stability-s-1\]g), that is, the derivative nonlinearity can excite the broken linear ${\mathcal{PT}}$-symmetric phase to the unbroken nonlinear ${\mathcal{PT}}$-symmetric phase. If we further increase $W_0$ a little bit to $W_0=-1.5$ (broken ${\mathcal{PT}}$-symmetric phase), the corresponding nonlinear mode begins to grow to become unstable (see Fig. \[stability-s-1\]h).
For the Scarff-II-Case-2, the bright solitons (\[sol-s\]) only exist in the extremely narrow region between those two parabolas contained in the domain of unbroken ${\mathcal{PT}}$-symmetric phase (Fig. \[stability-s-2\]b). We find the stable nonlinear mode for $V_0=-0.9,\, W_0=-0.74$ (Fig. \[stability-s-2\]d). When we fix $V_0=-0.9$ and increase $W_0$ to $W_0=0.78$, in which the solution becomes $\phi_{in}(x)=ia \sqrt{{\rm sech}x}\, {\rm exp}[-ib {\rm tan}^{-1}({\rm sinh}(x))]$ with $a=0.1032471136, b=0.3820050252$. The stationary function $\phi_{in}(x)$ does not solve Eq. (\[ode\]) and its real (imaginary) part is an odd (even) function differing from the former cases, but we surprisedly find it can be stable through the direct evolution using the inexact solution $\phi_{in}(x)$ as an initial solution with some $2\%$ noise perturbation (see Fig. \[stability-s-2\]f). When we fix $W_0=0.78$ and decrease $V_0$ a little bit to $V_0=-0.91$, in which the solution satisfies Eq. (\[dnls\]) and the linear ${\mathcal{PT}}$-symmetric phase is unbroken, a stable nonlinear localized mode is found again (see Fig. \[stability-s-2\]h).
We now investigate the interaction between two solitary waves in the ${\mathcal{PT}}$-symmetric Scarff-II potential. For the Scarff-II-Case-1 and $V_0=-1,\, W_0=-1.1$, we consider the initial condition $\psi(x,0)=\phi(x)+\sqrt{\frac23 \phi_0 {\rm sech}(x+20)} e^{4i x}$ with $\phi(x)$ determined by Eq. (\[sol-s\]), as a result, the semi-elastic interaction is generated in which exact nonlinear mode does not change its shape whereas the exotic incident wave becomes damped before and after interaction (see Fig. \[collision-s-12\]a). When $W_0$ becomes a little bit to $W_0=-1.4$, we consider the initial condition $\psi(x,0)=\phi(x)+\sqrt{\frac23 \phi_0 {\rm sech}(x+40)} e^{10i x}$ with $\phi(x)$ determined by Eq. (\[sol-s\]), then a novel phenomenon occurs in collision that there exists a reflected wave when exotic incident wave interacts with the exact soliton (\[sol-s\]) (see Fig. \[collision-s-12\]d). Through repeated numerical tests, we find the reflected wave is probably related to the simultaneously increasing amplitude of the exact soliton and exotic incident wave. As $W_0$ decreases from $-1.1$ to $-1.4$, it is easy to verify that the amplitude (determined by $\phi_0$) of the exact soliton or exotic incident wave increases and in the meantime the reflected wave begins to occur and then becomes larger and larger (see Figs. \[collision-s-12\](a, b, c, d)). However, the exact nonlinear mode still does not change its shape before and after interaction. Similarly, for the Scarff-II-Case-2, we successively consider the initial condition $\psi(x,0)=\phi(x)+\sqrt{\frac32 \phi_0 {\rm sech}(x+40)} e^{4ix}$ for $V_0=-0.9,W_0=0.74$ and $V_0=-0.9,W_0=0.78$ with $\phi(x)$ determined by Eq. (\[sol-s\]), the similar semi-elastic interactions to Fig. \[collision-s-12\]a are generated (see Figs. \[collision-s-12\](e, f)).
In order to better understand the properties of the nonlinear localized modes (\[sol-s\]), we check its corresponding transverse power flow (Poynting vector), which derives from the nontrivial phase structure of the nonlinear localized modes and is given by $S(x)=\frac{i}2(\psi\psi_x^{\ast}-\psi^{\ast}\psi_x)=-\frac13\phi_0 (W_0+\phi_0){\rm sech}^2x$ with $\phi_0>0$. Signs and directions of the transverse power flow $S(x)$ are discussed and summarized in detail in Fig. \[pf-s\].
Excitations of nonlinear modes
------------------------------
Finally, we discuss the excitation of nonlinear localized modes by means of changing the potential amplitudes as the functions of time, $V_0\rightarrow V_0(t)$ or $W_0\rightarrow W_0(t)$ \[cf. Ref. [@yanpre15]\]. It means that we focus on the simultaneous adiabatic switching on the Scarff-II potential, governed by \[tdnls\] i\_t+\_[xx]{}-\[V(x,t)+iW(x,t)\]+ig(||\^2)\_x=0, where $V(x,t), W(x,t)$ are given by Eq. (\[ps\]) with $V_0\rightarrow V_0(t)$ and $W_0\rightarrow W_0(t)$, and $V_0(t)$, $W_0(t)$ are both choose as the following form \[excite-s\](t)=
(\_2-\_1) (t/2000)+\_1, & ,\
\_2, &
where $\epsilon_{1,2}$ are real constants. It is easy to verify that nonlinear localized modes (\[sol-s\]) with $V_0\rightarrow V_0(t)$ or $W_0\rightarrow W_0(t)$ do not satisfy Eq. (\[tdnls\]) any more, whereas the modes (\[sol-s\]) do satisfy Eq. (\[tdnls\]) for both the initial state $t=0$ and excited states $t\geq 1000$.
For the Scarff-II-Case-1, Fig. \[excited-s-12\]a exhibits the wave propagation of the nonlinear modes $\psi(x,t)$ of Eq. (\[tdnls\]) via the initial condition given by Eq. (\[sol-s\]) with $W_0\rightarrow W_0(t)$ given by Eq. (\[excite-s\]), which excite an initially stable nonlinear localized mode given by Eq. (\[sol-s\]) for $(V_0, W_{01})=(-1, -1.1)$ with the unbroken linear ${\mathcal{PT}}$-symmetric phase to another stable nonlinear localized mode given by Eq. (\[sol-s\]) for $(V_0, W_{02})=(-1, -1.4)$, though with broken linear ${\mathcal{PT}}$-symmetric phase. It also indicates fully that bright solitons (\[sol-s\]) have extremely strong capacity of resisting disturbance.
For the Scarff-II-Case-2, we successively perform three types of excitations by changing potential amplitudes $V_0\rightarrow V_0(t)$ or $W_0\rightarrow W_0(t)$ singly or simultaneously. Similarly, Fig. \[excited-s-12\]b displays the wave propagation of nonlinear modes $\psi(x,t)$ of Eq. (\[tdnls\]) using the initial condition given by Eq. (\[sol-s\]) with $W_0\rightarrow W_0(t)$ given by Eq. (\[excite-s\]), which excites a stable and inexact nonlinear localized mode given by Eq. (\[sol-s\]) for $(V_0, W_{01})=(-0.9, 0.78)$ to another stable and exact nonlinear localized mode given by Eq. (\[sol-s\]) for $(V_0, W_{02})=(-0.9, 0.74)$, both of which belong to unbroken linear ${\mathcal{PT}}$-symmetric phase. The property is fairly significant to find numerically or experimentally the stable nonlinear mode from an inexact mode due to the stability of excitation. Of course, we can also achieve the same purpose only by tuning $V_0$ appropriately. Fig. \[excited-s-12\]c displays the wave propagation of the nonlinear modes $\psi(x,t)$ of Eq. (\[tdnls\]) via the initial condition given by Eq. (\[sol-s\]) with $V_0\rightarrow V_0(t)$ given by Eq. (\[excite-s\]), which excites a stable and inexact nonlinear localized mode given by Eq. (\[sol-s\]) for $(V_{01}, W_0)=(-0.9, 0.78)$ to another stable and exact nonlinear localized mode given by Eq. (\[sol-s\]) for $(V_{02}, W_0)=(-0.91, 0.78)$, both of which belong to unbroken linear ${\mathcal{PT}}$-symmetric phase. Fig. \[excited-s-12\]d shows the wave propagation of nonlinear modes $\psi(x,t)$ of Eq. (\[tdnls\]) via the initial condition given by Eq. (\[sol-s\]) with $V_0\rightarrow V_0(t), W_0\rightarrow W_0(t)$ given by Eq. (\[excite-s\]), which excites a stable and inexact nonlinear localized mode given by Eq. (\[sol-s\]) for $(V_{01}, W_{01})=(-0.9, 0.78)$ to another stable and exact nonlinear localized mode given by Eq. (\[sol-s\]) for $(V_{02}, W_{02})=(-0.91, 0.74)$, both of which also belong to unbroken linear ${\mathcal{PT}}$-symmetric phase.
Nonlinear modes in the ${\mathcal{PT}}$-symmetric harmonic-Hermite-Gaussian potential
=====================================================================================
Next we consider another physically significant potential, that is, the harmonic potential and gain-and-loss distribution of Hermite-Gaussian type \[phv\] V(x)=\^2 x\^2,\
\[phw\] W\_n(x)=H\_n(x)\[x H\_n(x)\
-2nH\_[n-1]{}(x)\]e\^[-x\^2]{}, where the frequency $\omega>0$ and real constant $\sigma>0$ can adjust amplitudes of the harmonic potential $V(x)$ and gain-and-loss distribution $W_n(x)$, and $H_n(x)=(-1)^n e^{x^2}(d^n e^{-x^2})/(dx^n)$ represents the Hermite polynomial with $n$ being a non-negative integer and $H_n(x)\equiv0$ as $n<0$. It is easy to verify that these complex potentials $V(x)+iW_n(x)$ are all ${\mathcal{PT}}$-symmetric for any non-negative integer $n$, which differ from other ones [@yanpre15]. Without loss of generality, in what follows we mainly focus on the ${\mathcal{PT}}$-symmetric potentials (\[phv\]) and (\[phw\]) for $n=0,1,2$.
Linear spectral problem
-----------------------
Here we investigate the linear operator $L$ in Eq. (\[lp\]) with ${\mathcal{PT}}$-symmetric potentials composed of $V(x)$ (\[phv\]) and $W_{0,1,2}(x)$ (\[phw\]), which are explicitly given by ($n=0, 1,2)$ \[W0\] , \[W1\]\
, \[W2\]\
,
For $n=0,1,2$, the regions of unbroken and broken linear ${\mathcal{PT}}$-symmetric phase on $(\omega, \sigma)$-space are all numerically exhibited in Fig. \[spectra-g-012\]. It can be obviously observed that the ranges of unbroken linear ${\mathcal{PT}}$-symmetric phase gradually shrink with $n$ increasing, which is mainly because the higher amplitude of gain-and-loss distribution $W_n(x)$ can possibly lead to the broken linear ${\mathcal{PT}}$-symmetric phase as $n$ increases. For some fixed $\sigma$, we also illustrate numerically the collisions of the first six lowest discrete energy levels as the frequency $\omega$ decreases (see Figs. \[spectra-g-012\](b-g)). Notice that only the first two lowest energy levels interact with each other for $n=0$ whereas the situations become more and more intricate with $n$ growing.
Nonlinear modes and stability
-----------------------------
For the above-mentioned ${\mathcal{PT}}$-symmetric potential $V(x)+iW_n(x)$ with Eqs. (\[phv\]) and (\[phw\]), we find a series of multi-hump bright solitons of Eq. (\[ode\]) \[sol-h\] \_n(x)= H\_n( x) e\^[-x\^2/2]{} e\^[i(x)]{},>0, where the chemical potential $\mu=-\omega (2n+1)$, and the phase function $\varphi_n(x)=-\sigma \int_0^x{H_n^2(\sqrt{\omega}s)e^{-\omega s^2}} ds$.
For the cases $n=0,1,2$, we first give the regions of linear stability \[cf. Eq. (\[st\])\] of nonlinear localized modes (\[sol-h\]) in the $(\omega, \sigma)$ space (see Figs. \[sta-evo-g-012\](a1, b1, c1)). It is more than evident that the stable regions of linear stability have the similar narrowing behaviors to the corresponding unbroken ${\mathcal{PT}}$-symmetric phase above on account of the rising strength of the gain-and-loss distribution $W_n(x)$ as $n$ increases. Moreover, the stable regions of linear stability are entirely included in the regions of the corresponding unbroken ${\mathcal{PT}}$-symmetric phase, which indicates the derivative nonlinear term makes a negative influence on the corresponding linear ${\mathcal{PT}}$-symmetric phase.
We now study numerically the linear stability of bright solitons (\[sol-h\]) for cases $n=0,1,2$ through the direct wave propagation of initially stationary mode (\[sol-h\]) for several specific amplitude parameters $(\omega, \sigma)$ with some $2\%$ noise perturbation. For $n=0$ and the fixed $\omega=1$, we modulate $\sigma$ from a very small positive number (e.g., $\sigma=0.001$) to $\sigma=1.1$ to perform the direct wave evolution of one-hump nonlinear modes (\[sol-h\]) such that we obtain the stable one-hump solitons (see Figs. \[sta-evo-g-012\](a2, a3)). Whereas we further increase $\sigma$ to $\sigma=1.2$, the one-hump nonlinear mode (\[sol-h\]) begins to become extremely unstable (see Fig. \[sta-evo-g-012\](a4)). The main reason is that the gain-and-loss distribution $W_n(x)$ has a stronger effect on the stability of modes as $\sigma$ increases. For the fixed $\sigma=0.1$, we only change $\omega$ from $1$ to $2$ continuously such that we also find a series of stable one-hump solitons, although there exist some small periodic variation as $\omega$ approaches to $2$ (see Fig. \[sta-evo-g-012\](a5)).
For $n=1$, we can also find a family of stable two-hump solitons (\[sol-h\]) for a fixed $\omega=2$ and $\sigma=0.1\rightarrow 0.8$ (see Figs. \[sta-evo-g-012\](b2, b3)). Whereas we further increase $\sigma$ from $0.8$ to $1$, the two-hump nonlinear mode (\[sol-h\]) begins to become extremely unstable (see Fig. \[sta-evo-g-012\](b4)). For a fixed $\sigma=0.1$, we change $\omega$ from $2$ to $3$ continuously such that we can also find a series of stable two-hump solitons (Fig. \[sta-evo-g-012\](b5)). For $n=2$, we also have the similar results for three-hump solitons (see Figs. \[sta-evo-g-012\](c2-c5))
Next we investigate the interactions of bright solitons (\[sol-h\]) in the ${\mathcal{PT}}$-symmetric potential $V(x)+iW_n(x)$ with (\[phv\]) and (\[phw\]). For $n=0$ and $\omega=1, \sigma=0.1$, we consider the initial condition $\psi(x,0)=\phi(x)+0.8 \sqrt{\sigma} e^{-\omega (x+10)^2/2} e^{i \varphi(x)}$ with $\phi(x)$ determined by Eq. (\[sol-h\]), as a result the elastic interaction is generated in which neither the exact one-hump nonlinear mode nor exotic periodic incident wave change their shapes before and after interaction (see Fig. \[collision-h-012\]a). When $\sigma$ becomes large to $\sigma=1.1$, we consider the initial condition $\psi(x,0)=\phi(x)+0.5 \sqrt{\sigma} e^{-\omega (x+10)^2/2} e^{i \varphi(x)}$ with $\phi(x)$ determined by Eq. (\[sol-h\]), then a novel phenomenon occurs in collision that there exists a weak reflected wave when exotic incident wave interacts with the exact one-hump soliton (\[sol-h\]) (see Fig. \[collision-h-012\]b), which is probably related to the increasing amplitude or strength of the exact one-hump soliton (\[sol-h\]). However, the exact one-hump nonlinear mode still doesn’t change its shape before and after interaction. Similarly, for $n=1$, we successively consider the initial condition $\psi(x,0)=\phi(x)+\sqrt{\sigma\omega} (x+5) e^{-\omega (x+5)^2/2} e^{i \varphi(x)}$ for $\omega=2, \sigma=0.1$ and $\omega=2, \sigma=0.8$ with $\phi(x)$ determined by Eq. (\[sol-h\]), the similar elastic interactions between the exact two-hump nonlinear modes and exotic periodic incident waves to Fig. \[collision-h-012\]a are generated (see Figs. \[collision-h-012\]c, d). For $n=2$, the similar elastic interactions between the exact three-hump nonlinear modes and exotic periodic incident waves are generated (see Figs. \[collision-h-012\]e, f).
In order to further understand the properties of the stationary nonlinear localized modes (\[sol-h\]), we also check its corresponding transverse power flow (Poynting vector) $S_n(x)=-\sigma^2 H_n^4(\sqrt{\omega}x) e^{-2\omega x^2}$ with $\omega>0$ and $\sigma>0$. Notice that the signs of the transverse power flow $S_n(x)$ always keep negative definite for any $n$. For $n=0$, the power always flows in one direction, i.e., from the gain toward the loss domain (see Fig. \[pf-h\]a). However, the directions of the power flow for $n=1$ are so complicated, that is (from the negative infinite to positive infinite), firstly from the gain to loss, then from the loss to gain, finally from the gain to loss domain again (see Fig. \[pf-h\]b, the total direction is from the gain to loss). Similar more complicated results hold for $n=2$ and specifically not repeat them.
Excitations of nonlinear modes
------------------------------
Finally, we investigate the excitation of stable nonlinear modes by means of changing the potential amplitudes $\omega\rightarrow \omega(t)$ or $\sigma\rightarrow \sigma(t)$, which means that we exert simultaneous adiabatic switching on the harmonic potential (\[phv\]) and gain-and-loss distribution (\[phw\]), modelled by Eq. (\[tdnls\]), where $V(x,t), W(x,t)$ are given by Eq. (\[phv\]) and Eq. (\[phw\]) with $\omega\rightarrow \omega(t)$ and $\sigma\rightarrow \sigma(t)$. We assume that $\omega(t)$, $\sigma(t)$ are all taken as the same form as Eq. (\[excite-s\]) (i.e., $\epsilon(t)$ can be replaced with $\omega(t)$ or $\sigma(t)$). We have the similar results that the nonlinear localized modes (\[sol-h\]) with $\omega\rightarrow \omega(t)$ or $\sigma\rightarrow \sigma(t)$ do not satisfy Eq. (\[tdnls\]) any more, whereas the modes (\[sol-h\]) indeed satisfy Eq. (\[tdnls\]) for the initial state $t=0$ and excited states $t\geq 1000$.
For $n=0,1,2$, we numerically perform three distinct types of excitations by changing the potential amplitudes $\omega\rightarrow \omega(t)$, $\sigma\rightarrow \sigma(t)$, or both, respectively. For $n=0$, Fig. \[excited-h-012\](a1) exhibits the wave evolution of nonlinear modes $\psi(x,t)$ of Eq. (\[tdnls\]) via the initial condition given by Eq. (\[sol-h\]) with $\sigma\rightarrow \sigma(t)$ given by Eq. (\[excite-s\]), which excites a stable one-hump nonlinear localized mode given by Eq. (\[sol-h\]) for $(\omega, \sigma_1)=(1, 0.1)$ to another stable one-hump nonlinear localized mode given by Eq. (\[sol-h\]) for $(\omega, \sigma_2)=(1, 1.1)$, which both belong to unbroken linear ${\mathcal{PT}}$-symmetric phase (all mentioned points $(\omega, \sigma)$ enjoy the same property of unbroken linear ${\mathcal{PT}}$-symmetric phase hereafter). Fig. \[excited-h-012\](a2) displays the wave propagation of the nonlinear modes $\psi(x,t)$ of Eq. (\[tdnls\]) via the initial condition given by Eq. (\[sol-h\]) with $\omega\rightarrow \omega(t)$ singly given by Eq. (\[excite-s\]), which excite a initially stable one-hump nonlinear localized mode given by Eq. (\[sol-h\]) for $(\omega_1, \sigma)=(1, 0.1)$ to another stable one-hump nonlinear localized mode given by Eq. (\[sol-h\]) for $(\omega_2, \sigma)=(2, 0.1)$. Fig. \[excited-h-012\](a3) displays the wave propagation of the nonlinear modes $\psi(x,t)$ of Eq. (\[tdnls\]) via the initial condition given by Eq. (\[sol-h\]) with $\omega\rightarrow \omega(t)$ and $\sigma\rightarrow \sigma(t)$ simultaneously given by Eq. (\[excite-s\]), which excite a stable one-hump nonlinear localized mode given by Eq. (\[sol-h\]) for $(\omega_1, \sigma_1)=(1, 0.1)$ to another stable one-hump nonlinear localized mode given by Eq. (\[sol-h\]) for $(\omega_2, \sigma_2)=(2, 1.1)$. Similarly for $n=1$, we excite a stable two-hump nonlinear localized mode given by Eq. (\[sol-h\]) for $(\omega, \sigma)=(2, 0.1)$ to another stable two-hump nonlinear localized mode given by Eq. (\[sol-h\]) for $(\omega, \sigma)=(2, 0.8)$, $(\omega, \sigma)=(3, 0.1)$, and $(\omega, \sigma)=(3, 0.8)$, respectively (see Figs. \[excited-h-012\](b1, b2, b3)). For $n=2$, we also excite a stable three-hump nonlinear localized mode given by Eq. (\[sol-h\]) for $(\omega, \sigma)=(2, 0.1)$ to another stable three-hump nonlinear localized mode given by Eq. (\[sol-h\]) for $(\omega, \sigma)=(2, 0.2)$, $(\omega, \sigma)=(3, 0.1)$, and $(\omega, \sigma)=(3, 0.2)$, respectively (see Figs. \[excited-h-012\](c1, c2, c3)). These stable excitations also shows that bright solitons (\[sol-h\]) have extremely strong capacity of resisting disturbance.
Conclusions and discussions
===========================
In conclusion, some stable bright solitons have been investigated in the derivative nonlinear Schrödinger equation with ${\mathcal{PT}}$-symmetric Scarff-II and hamonic-Hermite-Gaussian potentials. Firstly, the linear ${\mathcal{PT}}$-symmetric breaking curves are numerically exhibited. Secondly, in the presence of derivative nonlinearity, such ${\mathcal{PT}}$-symmetric solitons are shown to be stable through the linear stability analysis and direct wave propagation with some noise perturbation. Moreover, the semi-elastic interactions between exact bright solitons and exotic incident waves are illustrated and the transverse power flows are also checked in detail. Finally, the soliton excitations are also studied including from a stable nonlinear mode with unbroken linear ${\mathcal{PT}}$-symmetric phase to another stable nonlinear mode with broken linear ${\mathcal{PT}}$-symmetric phase and from a stable and inexact nonlinear mode to another stable and exact nonlinear mode. In fact, we may change the nonlinear coefficient $g$ as the function of space such that the stable solitons can also be generated. The idea used in this paper can also be extended to the DNLS equation with other ${\mathcal{PT}}$-symmetric potentials.
The authors would like to thank the referee for the valuable suggestions and comments. This work was supported by the NSFC under Grant No. 11571346 and and the Youth Innovation Promotion Association CAS.
[99]{}
G. P. Agrawal, [*Nonlinear Fibre Optics*]{} (5th ed.), Academic Press, New York, 2014.
N. Tzoar and M. Jain, Phys. Rev. A [**23**]{}, 1266 (1981).
D. Anderson and M. Lisak, Phys. Rev. A [**27**]{}, 1393 (1983).
X. Chen and W. K. Lam, Phys. Rev. E [**69**]{}, 066604 (2004). K. Mio, T. Ogino, K. Minami, and S. Takeda, J. Phys. Soc. Japan [**41**]{}, 265 (1976); E. Mjolhus, J. Plasma Phys. [**16**]{}, 321 (1976).
M. S. Ruderman, J. Plasma Phys. [**67**]{}, 271 (2002).
K. H. Spatchek, P. K. Shukla, and M. Y. Yu, Nucl. Fusion [**18**]{}, 290 (1977); H. Steudel, J. Phys. A [**36**]{}, 1931 (2003).
D. J. Kaup and A. C. Newell, J. Math. Phys. [**19**]{}, 798 (1978).
H. H. Chen, Y. C. Lee, and C. S. Liu, Phys. Scr. [**20**]{}, 490 (1979).
T. Kawata, N. Kobayashi, and H. Inoue, J. Phys. Soc. Jpn. [**46**]{}, 1008 (1979).
Y. S. Kivshar and G. P. Agrawal, [*Optical Solitons: from Fibers to Photonic Crystals*]{}, Academic, San Diego, 2003.
J. Belmonte-Beitia, V. M. Pérez-Garcia, V. Vekslerchik, and P. J. Torres, Phys. Rev. Lett. [**98**]{}, 064102 (2007).
Z. Yan, Phys. Lett. A [**374**]{}, 672 (2010); [*ibid*]{} [**374**]{}, 4838 (2010).
Z. Yan and D. Jiang, Phys. Rev. E [**85**]{}, 056608 (2012); Z. Yan and V. V. Konotop, Phys. Rev. E [**80**]{}, 036607 (2009); Z. Yan, Nonlinear Dyn. [**79**]{}, 2515 (2015); Y. Yang, Z. Yan, and D. Mihalache, J. Math. Phys. [**56**]{}, 053508 (2015).
S. Ponomarenko and G. P. Agrawal, Phys. Rev. Lett. [**97**]{}, 13901 (2006).
V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, Phys. Rev. Lett. [**98**]{}, 074102 (2007).
Z. H. Musslimani, [*et al.,*]{} Phys. Rev. Lett. [**100**]{}, 030402 (2008); Z. H. Musslimani, [*et al.,*]{} J. Phys. A [**41**]{}, 244019 (2008).
Z. Yan, Phil. Trans. R. Soc. A [**371**]{}, 20120059 (2013); Z. Yan, [*et al.,* ]{} arXiv.1009.4023 (2010); Z. Wen and Z. Yan, Phys. Lett. A [**379**]{}, 2025 (2015); Z. Yan, Z. Wen, and C. Hang, Phys. Rev. E [**92**]{}, 022913 (2015); Z. Yan, Z. Wen, and V. V. Konotop, Phys. Rev. A. [**92**]{}, 023821 (2015).
D. A. Zezyulin and V. V. Konotop, Phys. Rev. A [**85**]{}, 043840 (2012); [*ibid*]{} Phys. Rev. Lett. [**108**]{}, 213906 (2012).
C. H. Tsang, B. A. Malomed, and K. W. Chow, Phys. Rev. E [**84**]{}, 066609 (2011).
F. K. Abdullaev, [*et al.,*]{} Phys. Rev. E [**82**]{}, 056606 (2010).
V. E. Lobanov, O. V. Borovkova, and B. A. Malomed, Phys. Rev A [**90**]{}, 053820 (2014); J. Yang, Phys. Rev. E [**91**]{}, 023201 (2015).
H. Wang and J. Wang, Opt. Express [**19**]{}, 4030 (2011).
Z. Lu and Z. Zhang, Opt. Express [**19**]{}, 11457 (2011).
S. Hu, [*et al.,*]{} Phys. Rev. A [**84**]{}, 043818 (2011).
X. Zhu, [*et al.,*]{} Opt. Lett. [**36**]{}, 2680 (2011).
S. Nixon, L. Ge, and J. Yang, Phys. Rev. A [**85**]{}, 023822 (2012).
C. Li, H. Liu, and L. Dong, Opt. Express [**20**]{}, 16823 (2012).
A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A [**38**]{}, L171 (2005).
R. El-Ganainy, [*et al.,*]{} Opt. Lett. 32, 2632 (2007).
C. M. Bender and S. Boettcher, Phys. Rev. Lett. [**80**]{}, 5243 (1998); C. M. Bender, Rep. Prog. Phys. [**70**]{}, 947 (2007).
A. Guo, [*et al.,*]{} Phys. Rev. Lett. [**103**]{}, 093902 (2009).
C. E. Rueter, [*et al.,*]{} Nat. Phys. [**6**]{}, 192 (2010).
A. Regensburger, [*et al.,*]{} Nature [**488**]{}, 167 (2012).
Z. Ahmed, Phys. Lett. A [**282**]{}, 343 (2001).
Y. Chen and Z. Yan, Sci. Rep. [**6**]{}, 23478 (2016).
G. Castaldi, [*et al.,*]{} Phys. Rev. Lett. [**110**]{}, 173901 (2013).
A. Regensburger, [*et al.,*]{} Phys. Rev. Lett. [**110**]{}, 223902 (2013); B. Peng, [*et al.,*]{} Nature Phys. [**10**]{}, 394 (2014).
A. Biswas, et al., [*Mathematical Therory of Dispersion-Managed Optical Solitons*]{}, Higher Education Press, Beijing 2010.
[^1]: Corresponding author,email adress: zyyan@mmrc.iss.ac.cn
| {
"pile_set_name": "ArXiv"
} |
---
author:
- Katharina Bogad
- Manuel Huber
bibliography:
- 'biblio.bib'
title: '[Harzer Roller: Linker-Based Instrumentation for Enhanced Embedded Security Testing]{}'
---
<ccs2012> <concept> <concept\_id>10002978.10003014.10003017</concept\_id> <concept\_desc>Security and privacy Mobile and wireless security</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10002978.10003022.10003023</concept\_id> <concept\_desc>Security and privacy Software security engineering</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002978.10003022.10003465</concept\_id> <concept\_desc>Security and privacy Software reverse engineering</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The polarimetric measurement on the prompt phase of GRB 100826A shows that the polarization angle changes $\sim 90^{\circ}$ between two adjacent time intervals. We will show that this phenomenon can be naturally interpreted in the framework of the magnetic-dominated jet (MDJ) model. The MDJ model suggests that the bulk Lorentz factor of the outflow increases as $\Gamma\propto r^{1/3}$, until reaching a saturated value $\Gamma_{\rm sat}$. Electrons move in the globally ordered magnetic field advected by the jet from the central engine and produce synchrotron photons. The polarized synchrotron photons travel alone the jet direction and then collide with the cold electrons at the front of the jet. After the Compton scattering process, these photons escape from the jet and are detected by the observer locating slightly off-axis. If photons are emitted before the bulk Lorentz factor saturates, the change of polarization angle is a natural result of the acceleration of the outflow.'
date: 'Accepted xxxx; Received xxxx; in original form xxxx'
title: 'The change of GRB polarization angles in the magnetic-dominated jet model'
---
\[firstpage\]
gamma-ray burst: individual (GRB 100826A) – polarization – radiation mechanism: non-thermal – scattering
Introduction {#sec:introduction}
============
Gamma-ray bursts (GRBs) are the most energetic explosions in the universe since the Big Bang (for recent reviews, see, e.g., @Piran:1999 [@Meszaros:2006]). They are divided into two categories (short and long) according to the duration $T_{90}$ – the time interval in which 90% of the photons are counted, $T_{90}>2$ seconds for long GRBs, and $T_{90}<2$ seconds for short GRBs[^1]. The isotropic equivalent energy radiated during the prompt phase of a GRB, which typically lasts a few seconds, is as large as the energy radiated from the sun during its whole life. After decades of extensive research since its discovery in the 1960s, it is still not clear how does the energy release in such a short time. The most accepted assumptions are the core collapse of a massive star [@Woosley:1993; @Paczynski:1998; @Fryer:1999], or the merger of two compact objects, such as neutron star–neutron star or neutron star–black hole binaries [@Paczynski:1986; @Goodman:1986; @Eichler:1989; @Meszaros:1997]. The light curves of GRBs have various types, ranging from smooth, fast-rise and quasi-exponential decay, through curves with several peaks, to highly variable curves with many peaks [@Meszaros:2006]. The spectra can often be fitted well by the Band function, and the $\nu F_{\nu}$ spectra generally peak at $0.1\sim 1$ MeV [@Band:1993]. Photons with energy higher than 1 GeV have also been detected in some brightest GRBs [@Abdo:2009a; @Abdo:2009b; @Abdo:2009c; @Ackermann:2011]. In order to avoid the “compactness problem", it is believed that gamma-rays are emitted from a highly relativistic outflow ejected by the central engine [@Rees:1966; @Piran:1999; @Ackermann:2010]. However, there are still controversies on some basic questions, e.g., is the outflow isotropic or collimated, magnetic-dominated or baryon-dominated? The information extracted from spectrum and light curve observation is not complete enough to distinguish one model from another.
The polarimetric measurement provides a deep insight into the nature of GRBs. Since the first detection of linear polarization in the optical afterglow of GRB 990123 [@Hjorth:1999], linear polarization has been observed in many GRBs in the prompt phase [@Coburn:2003; @Rutledge:2004; @Wigger:2004; @McGlynn:2007; @Kalemci:2007; @Yonetoku:2011; @Yonetoku:2012; @Berger:2011], as well as in the afterglow [@Hjorth:1999; @Covino:1999; @Wijers:1999; @Bersier:2003; @Greiner:2003; @Caldwell:2003; @Steele:2009; @Uehara:2012]. It seems that polarization in the prompt phase is often larger than that in the afterglow. The latter is usually less than $10\%$. The first detection of high linear polarization in the prompt phase was from GRB 021206, with polarization degree $\Pi=80\%\pm 20\%$ at a confidence level of $>5.7\sigma$ [@Coburn:2003], although some independent groups could not conform this result [@Rutledge:2004; @Wigger:2004]. @Yonetoku:2011 reported a significant change of polarization angle between two adjacent time intervals in the prompt phase of GRB 100826A. They divided the most intense pulses of duration 100 seconds into two parts according to the shape of the light curve. The first part contains a single pulse lasting 47 seconds, while the second part contains multiple pulses lasting 53 seconds. They found that polarization angle changes $\sim 90^{\circ}$ between the two time intervals, while polarization degree almost keeps the same. A similar analysis to other two GRBs, GRB 110301A and GRB 110721A, however, shows no evident evolution of neither polarization degree nor polarization angle [@Yonetoku:2012]. @McGlynn:2007 investigated the energy dependence of polarization in GRB 041219A, and found that polarization degree decreases as the photon energy increases, while polarization angle is almost independent of photon energy. @Gotz:2009 also found the temporal variability of polarization degree and polarization angle in the prompt phase of GRB 041219A. As in the afterglow phase, the temporal evolution of polarization has also been observed [@Greiner:2003]. Note that the polarization mentioned here is referred to the linear polarization. The circular polarization is very small, especially in the afterglow phase [@Matsumiya:2003]. Besides, GRBs which have polarimetric observations all belong to the long category. The polarization of short GRBs is difficult to detect due to the short time of photon accumulation.
In the theoretical aspect, polarization may have different origins. It is well-known that synchrotron radiation can produce highly polarized photons. For the anisotropic and power-law ($N(E)dE\propto E^{-p}dE$) electrons, the polarization is known to be $\Pi_{\rm syn}=(p+1)/(p+7/3)$, if the magnetic field is globally uniform [@Rybicki:1979]. On the contrary, if the magnetic field is random on large scales, polarization is much reduced, or even vanishes. In the typical cases, the polarization induced by synchrotron radiation is expected to be less than $75\%$. The most prominent feature of polarization induced by synchrotron radiation is that it is energy-independent. Compton scattering is an alternative mechanism to produce high polarization. The polarization of an initially unpolarized photon scattered by a static electron, in the Thomson approximation, is $\Pi_{\rm comp}= (1-\cos^2\theta)/(1+\cos^2\theta)$, where $\theta$ is the scattering angle of the photon [@Rybicki:1979]. The polarization reaches its maximum at $\theta=90^{\circ}$. However, the probability of high polarization is small since the cross section reaches its minimum at this angle. Given some assumptions on the structure of the outflow, both synchrotron radiation and Compton scattering can produce a wide range of polarization [@Sari:1999; @Gruzinov:1999a; @Granot:2003; @Lazzati:2004; @Lazzati:2006; @Toma:2009; @Zhang:2011; @Mao:2013gha]. Besides the above two intrinsic mechanisms, polarization can also originate from geometric effects [@Waxman:2003; @Granot:2003]. If the jet open angle is small enough and the line-of-sight is close to the jet edge, the polarization signal is not completely averaged out. Thus, even randomly ordered magnetic field can produce high polarization. Nevertheless, all the above models can not naturally interpret the change of polarization angle observed in GRB 100826A. @Yonetoku:2011 pointed out that the jet should be non-axisymmetric in order to cause the change of polarization angle. @Lundman:2013qba considered the polarization properties of photospheric emission originating from a highly relativistic jet, and found that significant degrees of linear polarization can be observed for the observers located at viewing angles larger than the jet opening angle. Particularly, the angle of polarization may shift by $\pi/2$ for the time-variable jets.
Synchrotron photons may suffer from the Compton scattering process in the optically thick region. Thus, both synchrotron radiation and Compton scattering process may contribute to the polarization. Recently, @Chang:2013 [@Chang:2014a; @Chang:2014b] investigated the polarization properties of the synchrotron-Compton process in the framework of the magnetic-dominated jet (MDJ) model. According to the MDJ model, the bulk Lorentz factor of the outflow increases as $\Gamma\propto r^{1/3}$, until reaching a saturated value $\Gamma_{\rm sat}$ [@Drenkhahn:2002a; @Drenkhahn:2002b; @Metzger:2010pp; @Granot:2011; @Meszaros:2011]. For typical long GRBs, $\Gamma_{\rm sat}\approx 250$, while it is much larger for short GRBs [@Chang:2012]. Electrons moving in the magnetic field radiate synchrotron photons. A beam of synchrotron photons with polarization degree $\Pi_0$ and polarization angle $\chi_0$ travels along the jet direction and then collide with cold electrons at the front of the jet. After the Compton scattering process, both polarization degree and polarization angle are changed. The polarization is expressed as a function of photon energy and viewing angle. The synchrotron-Compton model can produce a wide range of polarization, ranging from completely unpolarized, to completely polarized. Interestingly, at a special setup, the polarization angle can be changed $90^{\circ}$ exactly after scattering. We will show, in this paper, that the change of polarization angle observed in GRB 100826A can be naturally interpreted by the synchrotron-Compton model. @Chang:2012 have shown that photons with energy $E \lesssim 100$ MeV are emitted before the bulk Lorentz factor saturates. If this is indeed the case, the change of polarization angle is a natural result of the acceleration of the outflow.
The rest of this paper is organized as follows. In section \[sec:model\], we briefly review the polarization of photons in the synchrotron-Compton process. In section \[sec:evolution\], we show the evolution of polarization angle with the bulk Lorentz factor. The $\sim 90^{\circ}$ change of polarization angle observed in GRB 100826A can be naturally interpreted as the acceleration of the outflow. Finally, discussions and conclusions are given in section \[sec:conclusion\].
The polarization of photons in the synchrotron-Compton process {#sec:model}
==============================================================
Suppose a highly relativistic, magnetic-dominated and baryon-loaded (so electron-loaded) jet ejected from the central engine travels towards the observer with a large Lorentz factor. The magnetic field advected by the jet from the central engine is globally ordered [@Spruit:2001; @Fendt:2004]. The jet consists of shells of slightly different velocities. When a fast shell catches up with a slow one, shocks are produced. The shocks accelerate electrons to power-law distribution, and at the same time magnify the magnetic field. The power-law electrons move in the magnetic field and radiate synchrotron photons. A beam of synchrotron photons, which are initially polarized, travel along the jet direction and then collide with the cold electrons at the front of the jet before escaping from it. The cold electrons here mean that the electrons are static with respect to the jet. After the Compton scattering process, both the polarization degree and polarization angle are changed. Since the jet is magnetic-dominated, the dynamical behavior of the jet seems as if it is consist of pure magnetic field, and the jet opening angle can be as small as $\sim 1/\Gamma$ [@Beloborodov:2010; @Meszaros:2011], where $\Gamma$ is the Lorentz factor of the jet.
Consider a polarized photon with energy $\varepsilon_0$ moves in the jet direction and then collides with a static electron. Define a Cartesian coordinate system such that the $z$-axis is along the jet direction, the $y$-axis is in the scattering plane, and the $xyz$ axes form the right-handed set. The energy of the photon after scattering is given as $$\label{eq:energy}
\varepsilon_1= \frac{\varepsilon_0 }{1+ \frac{\varepsilon_0}{m_e c^2}(1- \cos\theta)},$$ where $\theta$ is the scattering angle of the photon. The differential cross section for the scattering of a polarized photon by a static electron is written as [@Berest:1982; @Chang:2014a; @Chang:2014b] $$\label{eq:cross-section}
d \sigma = \frac{1}{4} r_e^2 d \Omega \left(\frac{\varepsilon_1}{\varepsilon_0}\right)^2 \bigg[ F_0 +F_3(\xi_3+\xi'_3) + F_{11} \xi_1 \xi_1' +F_{22} \xi_2\xi'_2+F_{33} \xi_3\xi'_3\bigg],$$ where $r_e=e^2/m_ec^2$ is the classical electron radius, $d\Omega=\sin \theta d \theta d \varphi$ is the infinitesimal solid angle, $\xi_i$ ($\xi'_i$) are the Stokes parameters of the incident (scattered) photon, and $$\label{eq:F}
\begin{cases}
F_0=\displaystyle\frac{\varepsilon_1}{\varepsilon_0}+\frac{\varepsilon_0}{\varepsilon_1}-\sin^2 \theta,\\
F_3=\sin^2 \theta,\\
F_{11}=2 \cos \theta,\\
F_{22}=\displaystyle\left(\frac{\varepsilon_1}{\varepsilon_0} +\frac{\varepsilon_0}{\varepsilon_1} \right) \cos \theta,\\
F_{33}=1+ \cos^2 \theta.
\end{cases}$$ Especially, if the incident and scattered photons are unpolarized, i.e., $\xi_i=\xi'_i=0$, equation (\[eq:cross-section\]) reduces to the famous Klein-Nishina formula.
The physical meaning of the Stokes parameters is presented in Fig.\[fig:stokes\].
![[]{data-label="fig:stokes"}](Stokes.eps){width="9"}
The Stokes parameters stand for the polarization state of a photon. The positive (or negative) $\xi_3$ describes that the photon is linearly polarized along the $x$-axis (or $y$-axis). The positive (or negative) $\xi_1$ means the linear polarization along the direction with azimuthal angle $+\pi/4$ (or $-\pi/4$) relative to the $x$-axis in the $xy$ plane. The positive (or negative) $\xi_2$ stands for the right-handed (or left-handed) circular polarization. Since the circular polarization is very small in GRBs, we will ignore it in the following discussion. It should be paid specific attention that the Stokes parameters of the scattered photon are defined in a new coordinate system $O'x'y'z'$. The $O'x'y'z'$ system is the $Oxyz$ system rotating an angle $\theta$ relative to the $x$-axis, such that the $z'$-axis is along the moving direction of the scattered photon. This ensures that the polarization direction of the scattered photon is still perpendicular to its wave vector. The polarization degree of the incident photon is related to the three Stokes parameters by $$\label{eq:pi0}
\Pi_0=\sqrt{\xi_1^2+\xi_2^2+\xi_3^2}.$$ On the contrary, for a given photon with polarization degree $\Pi_0$ and polarization angle $\chi_0$, we can conveniently write its Stokes parameters as $$\xi_1=\Pi_0\sin 2\chi_0,~~\xi_2=0,~~\xi_3=\Pi_0\cos 2\chi_0,$$ where $\chi_0\in [-\pi/2,\pi/2]$ is the angle between the polarization direction and the $x$-axis.
Note that the Stokes parameters $\xi_i'$ are secondary quantities which essentially represent the properties of the detector as selecting one or the other polarization of the final photon, not the properties of the scattering process as such. The polarization states of the photon resulting from the scattering process itself are denoted by $\xi_i^{\rm f}$. They are given by the ratios of the coefficients of $\xi'_i$ in equation (\[eq:cross-section\]) to the terms independent of $\xi'_i$ [@Berest:1982; @Chang:2014a; @Chang:2014b], i.e., $$\label{eq:xif}
\begin{cases}
\displaystyle\xi^{\rm f}_1=\frac{ \xi_1 F_{11}}{F_0+\xi_3 F_3}=\frac{2 \Pi_0 \sin 2\chi_0 \cos \theta}{\varepsilon_1/\varepsilon_0+\varepsilon_0/\varepsilon_1-(1-\Pi_0 \cos 2 \chi_0)\sin^2 \theta},\\
\displaystyle\xi^{\rm f}_2=\frac{ \xi_2 F_{22}}{F_0+\xi_3 F_3}=0,\\
\displaystyle\xi^{\rm f}_3=\frac{F_3+ \xi_3 F_{33}}{F_0+\xi_3 F_3}=\frac{\sin^2 \theta+ \Pi_0 \cos 2\chi_0(1+\cos^2 \theta)}{\varepsilon_1/\varepsilon_0+\varepsilon_0/\varepsilon_1-(1-\Pi_0 \cos 2\chi_0)\sin^2 \theta}.
\end{cases}$$ The circular polarization occurs only if the incident photon is circularly polarized ($\xi_2^{\rm f}\neq 0$ only if $\xi_2\neq 0$). After scattering, the polarization degree of the photon becomes to $$\label{eq:pi}
\Pi=\sqrt{(\xi_1^{\rm f})^2+(\xi_2^{\rm f})^2+(\xi_3^{\rm f})^2},$$ and the polarization angle is determined by $$\label{eq:chi}
\tan2\chi=\frac{\xi_1^{\rm f}}{\xi_3^{\rm f}},$$ where $\chi$ is the angle between the polarization direction and the $x'$-axis (which is the same to the $x$-axis). For any incident photon with polarization degree $\Pi_0$ and polarization angle $\chi_0$, we can derive the Stokes parameters of the scattered photon from equations (\[eq:energy\]) and (\[eq:xif\]). Then the polarization degree and polarization angle of the scattered photon can be further calculated from equations (\[eq:pi\]) and (\[eq:chi\]).
So far, we are working in the jet frame, which moves highly relativistically towards the observer. In order to transform to the observer frame, we note that the scattering angle between the two frames are related by [@Rybicki:1979] $$\label{eq:angle-transform}
\cos\theta=\frac{\cos\bar{\theta}-\beta_{\rm jet}}{1-\beta_{\rm jet}\cos\bar{\theta}},$$ where $\beta_{\rm jet}=(1-1/\Gamma^2)^{1/2}$ is the velocity of the jet in the unit of light speed, and $\Gamma$ is the bulk Lorentz factor of the jet. Here and after, symbols with a bar denote the quantities in the observer frame. In addition, the energy of the scattered photon in the jet frame can be Doppler-shifted to that in the observer frame, i.e., $$\label{eq:doppler}
\bar{\varepsilon}_1=\varepsilon_1\Gamma(1+\beta_{\rm jet}\cos\theta).$$ Making use of equation (\[eq:energy\]) and equations (\[eq:xif\] – \[eq:doppler\]), one can express the polarization degree and polarization angle of the scattered photon as functions of $(\bar{\varepsilon}_1, \bar{\theta}, \Gamma, \Pi_0, \chi_0)$. In the next section, we will show how the polarization degree and polarization angle evolve with the bulk Lorentz factor of the jet.
The change of polarization angle in the evolution {#sec:evolution}
=================================================
For an observer near the earth, the viewing angle $\bar{\theta}$ is almost fixed. On the other hand, the polarimeters are designed to detect the polarization of photons in a certain energy band. Thus, the photon energy $\bar{\varepsilon}_1$ is a constant. Besides, $\Pi_0$ and $\chi_0$ do not evolve with time, since the photons produced by synchrotron radiation have unambiguous polarization degree and polarization angle. The only parameter that may evolve with time is the bulk Lorentz factor $\Gamma$. The MDJ model predicts that the outflow accelerates as $\Gamma\propto r^{1/3}$, until reaching a saturated value $\Gamma_{\rm sat}$. Then the outflow coasts with a constant velocity. In Fig.\[fig:PI\_075\], we plot the polarization degree of the scattered photon, $\Pi$, as a function of $\Gamma$ for various viewing angles.
![[]{data-label="fig:PI_075"}](PI_075.eps){width="10"}
Fig.\[fig:PI\_075\] is a numerical result of equation (\[eq:pi\]). In the numerical calculation, we set $\Pi_0=0.75$, which is the maximum polarization degree that synchrotron photons can achieve. We assume that the initial polarization direction is parallel to the scattering plane, i.e., $\chi_0=\pi/2$. For the photon energy, we set it to be $\bar{\varepsilon}_1=100$ keV. The $\nu F_{\nu}$ spectra of most GRBs peak at about $100-1000$ keV, and the polarimetric measurement are often carried out near this energy band. For example, the GAP on board IKAROS is fully designed to measure linear polarization in the prompt emission of GRBs in the energy band of $70-300$ keV [@Yonetoku:2006]. In Fig.\[fig:PI\_075\], The positive and negative values of $\Pi$ mean that the polarization direction is parallel and perpendicular to the scattering plane, respectively.
From Fig.\[fig:PI\_075\], we can see that the scattered photons can be completely polarized at the specific viewing angle $\bar{\theta}\Gamma\approx 1$ (corresponds to the valley of each curve). Most interestingly, the polarization angle can be changed $90^{\circ}$ at some special values of $\Gamma$ (corresponds to the intersection of each curve with the red-dashed horizonal line). This may provide a possible explanation for the change of polarization angle observed in the prompt phase of GRB 100812A [@Yonetoku:2011]. According to the MDJ model, the saturation bulk Lorentz factor of outflow for a typical long GRB is about $\Gamma_{\rm sat}\approx 250$, and low energy photons (say $\bar{\varepsilon}_1\lesssim 1$ MeV) are emitted before the bulk Lorentz factor saturates [@Chang:2012]. At a fixed viewing angle, e.g., $\bar{\theta}=0.005$, the polarization angle is initially parallel to the scattering plane. As $\Gamma$ increasing, the polarization degree gradually decreases to zero, and then increases again, but the polarization direction becomes to perpendicular to the scattering plane. Thus, the change of polarization angle is a natural result of the acceleration of the outflow. At a smaller viewing angle, say $\bar{\theta}\lesssim 0.002$, the acceleration of the outflow does not cause any change of polarization angle, although the polarization degree decreases. This may be the reason why no change of polarization angle was observed in the prompt phase of GRB 110301A and GRB 110721A [@Yonetoku:2012]. At a much larger viewing angle, say $\bar{\theta}\gtrsim 0.01$, the polarization angle may change two times. Initially, the polarization direction is parallel to the scattering plane. As $\Gamma$ increasing, the polarization direction changes to perpendicular to the scattering plane. When $\Gamma$ reaches a certain value, the polarization direction returns back to parallel to the scattering plane. However, this situation is not easily observed, since the flux reduces dramatically at large viewing angles.
Discussions and conclusions {#sec:conclusion}
===========================
There are many explanations for the temporal evolution of polarization. @Yonetoku:2012 pointed out that the change of polarization angle is due to the multiple patches of magnetic field whose characteristic angular size $\theta_p$ is much smaller than the jet opening angle $\theta_j$. If the jet open angle satisfies $\theta_j\sim \Gamma^{-1}$, we can see multiple patches with different magnetic field directions, and then significant change of polarization angle can be observed. On the contrary, if $\theta_j\gg \Gamma^{-1}$, we only see a limited range of the curved magnetic fields, thus no significant change of polarization angle occurs. @Sari:1999 investigated the polarization and proper motion from a beamed GRB ejecta, and found that the polarization direction will change $90^{\circ}$ near the jet breaking time, if the offset of an observer from the center of the beam is large enough. The ICMART model proposed by @Zhang:2011 also allows for the temporal evolution of polarization because the ordered magnetic field can be distorted by internal shock. In this paper, we have given an analytic calculation of the polarization properties of the synchrotron-Compton process in the framework of the MDJ model. We showed that both the polarization degree and polarization angle are changed after the Compton scattering process, regardless that the magnetic field is globally ordered and static. In the MDJ model, low energy photons are emitted before the bulk Lorentz factor saturates. As the jet accelerates, the change of polarization angle occurs naturally. If the initial polarization direction is parallel to the scattering plane, the polarization angle can change $90^{\circ}$ exactly. This provides a possible explanation for the polarization observed in the prompt phase of GRB 100826A. Note that the change of polarization angle occurs only if the viewing angle is larger than $\sim \Gamma^{-1}$. Otherwise, if the viewing angle is much smaller than $\sim \Gamma^{-1}$, no change of polarization angle can be observed. GRB 110301A and GRB 110721A may belong to the later case.
The change of polarization angle is not necessarily to be $90^{\circ}$ exactly. It depends on the initial polarization angle $\chi_0$. If the initial polarization direction is neither parallel nor perpendicular to the scattering plane, the change of polarization angle can be any value between $0^{\circ} \sim 90^{\circ}$. The synchrotron-Compton model also allows for the variability of polarization. The high variability of light curve implies that the outflow may contain many shells with different Lorentz factors. Photons emitted from different shells may have different polarization degree and polarization angle. This is consistent with the temporal evolution of polarization observed in the prompt phase of GRB 041219A [@Gotz:2009]. In addition, the synchrotron-Compton model predicts that polarization degree of high energy photons is smaller than that of low energy photons [@Chang:2013; @Chang:2014a; @Chang:2014b]. Thus, the energy dependence of polarization detected in GRB 041219A can be naturally interpreted [@McGlynn:2007]. The recent analysis to GRB 061122 also shows a similar polarization-energy relation . The synchrotron-Compton model considered in this paper, although very simple, can account for many features of polarization of GRBs. Due to the low significance and limited amount of experiment data, it is still premature to exclude one model and favor another with the present data. More polarimetric measurement are needed to gain deep insight into the mechanism of GRBs. We hope that the future gamma-ray polarimeter POLAR[^2] on board the Chinese Space Laboratory Tian-Gong II can provide more polarimetric data with unprecedented precision.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to X. Li, P. Wang, S. Wang and D. Zhao for useful discussion. This work has been funded by the National Natural Science Fund of China under Grant No. 11375203.
Abdo, A. A., et al. 2009, Science, 323, 1688 Abdo, A. A., et al. 2009, ApJ, 706, L138 Abdo, A. A., et al. 2009, Nature, 462, 331 Ackermann, M., et al. 2010, ApJ, 716, 1178 Ackermann, M., et al. 2011, ApJ, 729, 114 Band, D., et al. 1993, ApJ, 413, 281 Beloborodov, A. M. 2010, MNRAS, 407, 1033 Berestetskii, V. B., Lifshitz, E. M., & Pitaevskii, L. P. 1982, Quantum electrodynamics (New York: Pergamon Press) Berger, E., et al. 2011, GCN Circ. 12193 Bersier, D., et al. 2003, ApJ, 583, L63 Caldwell, N., Garnavich, P., Holland, S., Matheson, T., & Staneket, K. Z. 2003, GCN Circ. 2053 Chang, Z., Lin, H.-N., & Jiang, Y. G. 2012, ApJ, 759, 129 Chang, Z., Jiang, Y. G., & Lin, H.-N. 2013, ApJ, 769, 70 Chang, Z., Jiang, Y. G., & Lin, H.-N. 2014a, ApJ, 780, 68 Chang, Z., Lin, H.-N., & Jiang, Y. G. 2014b, ApJ, 783, 30 Coburn, W., & Boggs, S. E. 2003, Nature, 423, 415 Covino, S., et al. 1999, A&A, 348, L1 Drenkhahn, G. 2002, A&A, 387, 714 Drenkhahn, G., & Spuit, H. C. 2002, A&A, 391, 1141 Eichler, D., Livio, M., Piran, T. & Schramm, D. 1989, Nature, 340, 126 Fendt, C., & Ouyed, R. 2004, ApJ, 608, 378 Fryer, C., Woosley, S. & Hartmann, D. 1999, ApJ, 526, 152 Götz, D., et al. 2009, ApJ, 695, L208 Götz, D., et al. 2013, MNRAS, 431, 3550 Golenetskii, S., Aptekar, R., Frederiks, D., et al. 2010, GCN Circ., 11158 Goodman, J. 1986, ApJ, 308, L47 Granot, J. 2003, ApJ, 596, L17 Granot, J., Komissarov, S. S., & Spitkovsky, A. 2011, MNRAS, 411, 1323 Greiner, J., et al. 2003, Nature, 426, 157 Gruzinov, A., & Waxman, E. 1999, ApJ, 511, 852 Hjorth, J., et al. 1999, Science, 283, 2073 Kalemci, E., Boggs, S. E., Kouveliotou, C., Finger, M., & Baring, M. G. 2007, ApJ, 169, S75 Lazzati, D., Rossi, E., Ghisellini, G., & Rees, M. J. 2004, MNRAS, 347, L1 Lazzati, D., et al. 2006, New J. Physics, 8, 131 Lundman, C., Pe’er, A. and Ryde, F. 2014, MNRAS, 440, 3292 Lü, H.-J., Liang, E.-W., Zhang, B.-B., & Zhang, B. 2010, ApJ, 725, 1965 Matsumiya, M., & Ioka, K. 2003, ApJL, 595, L25 Mao, J., & Wang, J. 2013, ApJ, 776, 17 McGlynn, S., et al. 2007, A&A, 466, 895 Mészáros, P. 2006, Rep. Prog. Phys., 69, 2259 Mészáros, P., & Rees, M. J. 1997, ApJ, 482, L29 Mészáros, P., & Rees, M. J. 2011, ApJ, 733, L40 Metzger, B. D., et al. 2011, MNRAS, 413, 2031 Paczyński, B. 1986, ApJ, 308, L43 Paczyński, B. 1998, ApJ, 494, L45 Piran, T. 1999, Phys. Rep., 314, 575 Rees, M. J. 1966, Nature, 211, 468 Řípa, J., Mészáros, A., Veres, P., & Park, I. H. 2012, ApJ, 756, 44 Rutledge, R. E., & Fox, D. B. 2004, MNRAS, 350, 1288 Rybicki, G. B., & Lightman, A. P. 1979, Radiative Processes in Astrophysics (New York: Wiley) Sari, R. 1999, ApJ, 524, L43 Spruit, H. C., Daigne, F., & Drenkhahn, G. 2001, A&A, 369, 694 Steele, I. A., Mundell, C. G., Smith, R. J., Kobayashi, S., & Guidorzi, C. 2009, Nature, 462, 767 Toma, K., et al. 2009, ApJ, 698, 1042 Uehara, T., et al. 2012, ApJ, 752, L6 Waxman, E. 2003, Nature, 423, 388 Wigger, C., Hajdas, W., Arzner, K., Güdel, M., & Zehnder, A. 2004, ApJ, 613, 1088 Wijers, R. A. M. J., et al. 1999, ApJ, 523, L33 Woosley, S. 1993, ApJ, 405, 273 Yonetoku, D., et al. 2006, Proc. SPIE, 6266, 62662C Yonetoku, D., et al. 2011, ApJ, 743, L30 Yonetoku, D., et al. 2012, ApJ, 758, L1 Zhang, B., & Yan, H. 2011, ApJ, 726, 90
\[lastpage\]
[^1]: The threshold $T_{90}=2$ seconds is not strict, because in some cases a long lasting GRB seems to be a short one, and vice versa (see, e.g., @Lu:2010, and the references therein). Besides, a third group with intermediate duration can also exist (see, e.g., @Ripa:2012 and the references therein).
[^2]: http://polar.ihep.ac.cn/cms/.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper a geometric method based on Grassmann manifolds and matrix Riccati equations to make hermitian matrices diagonal is presented. We call it Riccati Diagonalization.'
author:
- |
Kazuyuki FUJII [^1]and Hiroshi OIKE [^2]\
${}^{*}$Department of Mathematical Sciences\
Yokohama City University\
Yokohama, 236–0027\
Japan\
${}^{\dagger}$Takado 85–5, Yamagata, 990–2464\
Japan\
title: '*Riccati Diagonalization of Hermitian Matrices*'
---
Introduction
============
In this paper we consider a finite dimensional quantum model, so its Hamiltonian is a (finite-dimensional) hermitian matrix. In order to solve the model we want to make the Hamiltonian diagonal.
Although we have a standard method for the purpose, to perform it [**explicitly**]{} is very hard (maybe, almost impossible). Let us make a brief introduction, see [@Sa], [@St].
Let $H$ be a hermitian matrix, namely $$H\in \{H\in M(n;{{\mathbf C}})\ |\ H^{\dagger}=H\}.$$ When we want to make $H$ diagonal Elementary Linear Algebra shows the following diagonalization procedure $[A]\Longrightarrow [B]\Longrightarrow [C]$ :
\[A\] First we calculate eigenvalues of $H$, $$0
=|\lambda E-H|
=\lambda^{n}-{{\rm tr}}{H}\lambda^{n-1}+\cdots +(-1)^{n}\det{H}.$$ There are $n$ real solutions although it is almost impossible to look for exact ones, so let those be $\{\lambda_{1}, \lambda_{2}, \cdots, \lambda_{n}\}$.
\[B\] Next we find eigenvectors $\{{{\vert{\lambda}\rangle}}\}$ corresponding to eigenvalues $$H{{\vert{\lambda_{j}}\rangle}}=\lambda_{j}{{\vert{\lambda_{j}}\rangle}}
\quad \mbox{and} \quad
\langle{\lambda_{i}}|{\lambda_{j}}\rangle=\delta_{ij}$$ for all $1\leq i,\ j\leq n$. It is also almost impossible to carry out.
\[C\] Last by setting $$U=({{\vert{\lambda_{1}}\rangle}},{{\vert{\lambda_{2}}\rangle}},\cdots,{{\vert{\lambda_{n}}\rangle}})$$ we finally obtain $$H=UD_{H}U^{\dagger}$$ where $D_{H}=
\mbox{diag}(\lambda_{1},\lambda_{2},\cdots,\lambda_{n})$ is the diagonal matrix.
The procedure is standard, while to carry out it completely is another problem[^3]. Even if $n=3$ it is very hard. In fact, for the hermitian matrix $$H=
\left(
\begin{array}{ccc}
h_{1} & \bar{\alpha} & \bar{\beta} \\
\alpha & h_{2} & \bar{\gamma} \\
\beta & \gamma & h_{3}
\end{array}
\right)\in H(3;{{\mathbf C}})$$ carry out the diagonalization. As far as we know it has not been given in any textbook on linear algebra.
Note that the characteristic equation $f(\lambda)=|\lambda E-H|$ is given by $$\begin{aligned}
f(\lambda)
&=&
\lambda^{3}-(h_{1}+h_{2}+h_{2})\lambda^{2}+
(h_{1}h_{2}+h_{1}h_{3}+h_{2}h_{3}-|\alpha|^{2}-|\beta|^{2}-|\gamma|^{2})
\lambda+ \\
&&|\gamma|^{2}h_{1}+|\beta|^{2}h_{2}+|\alpha|^{2}h_{3}-h_{1}h_{2}h_{3}
-\alpha\bar{\beta}\gamma-\bar{\alpha}\beta\bar{\gamma}.\end{aligned}$$ To look for exact solutions by use of Cardano formula is not easy (for example, try it by use of MATHEMATICA or MAPLE).
Therefore we present another diagonalization method based on [**Grassmann manifolds**]{} and [**matrix Riccati equations**]{}.
Riccati Diagonalization
=======================
In this section we show a new diagonalization method. Before it let us explain the idea with simple example for beginners. The target is $$H=
\left(
\begin{array}{cc}
h_{1} & \bar{\alpha} \\
\alpha & h_{2}
\end{array}
\right)\in H(2;{{\mathbf C}}).$$ To diagonalize $H$ above we consider a matrix $$U\equiv U(z)=\frac{1}{\sqrt{1+{{\vert z\vert}}^{2}}}
\left(
\begin{array}{cc}
1 & -\bar{z} \\
z & 1
\end{array}
\right)\quad \mbox{for}\quad z\in {{\mathbf C}}.$$ It is easy to check $U^{\dagger}U=UU^{\dagger}=1_{2}$ and $|U|=1$, so $U$ is special unitary ($U\in SU(2)$). Namely, $U$ is a map $$U\ :\ {{\mathbf C}}\ (\subset S^{2})\ \longrightarrow\ SU(2).$$ This map is well–known in Mathematics or Mathematical Physics.
The calculation $U^{\dagger}HU$ gives $$\begin{aligned}
U^{\dagger}HU
&=&
\frac{1}{1+{{\vert z\vert}}^{2}}
\left(
\begin{array}{cc}
1 & \bar{z} \\
-z & 1
\end{array}
\right)
\left(
\begin{array}{cc}
h_{1} & \bar{\alpha} \\
\alpha & h_{2}
\end{array}
\right)
\left(
\begin{array}{cc}
1 & -\bar{z} \\
z & 1
\end{array}
\right) \nonumber \\
&=&
\frac{1}{1+{{\vert z\vert}}^{2}}
\left(
\begin{array}{cc}
h_{1}+\bar{\alpha}z+\alpha\bar{z}+h_{2}{{\vert z\vert}}^{2} &
\bar{\alpha}-(h_{1}-h_{2})\bar{z}-\alpha\bar{z}^{2} \\
\alpha -(h_{1}-h_{2})z-\bar{\alpha}z^{2} &
h_{2}-\bar{\alpha}z-\alpha\bar{z}+h_{1}{{\vert z\vert}}^{2}
\end{array}
\right),\end{aligned}$$ so if we assume the equation $$\label{eq:riccati}
\alpha-(h_{1}-h_{2})z-\bar{\alpha}z^{2} =0
\Longleftrightarrow
\bar{\alpha}z^{2}+(h_{1}-h_{2})z-\alpha =0$$ we have the diagonal matrix $$U^{\dagger}HU
=
\left(
\begin{array}{cc}
\frac{h_{1}+\bar{\alpha}z+\alpha\bar{z}+h_{2}{{\vert z\vert}}^{2}}
{1+{{\vert z\vert}}^{2}} & \\
&
\frac{h_{2}-\bar{\alpha}z-\alpha\bar{z}+h_{1}{{\vert z\vert}}^{2}}
{1+{{\vert z\vert}}^{2}}
\end{array}
\right).$$ From this two eigenvalues are obtained by $$\lambda_{1}=\frac{h_{1}+\bar{\alpha}z+\alpha\bar{z}+h_{2}{{\vert z\vert}}^{2}}
{1+{{\vert z\vert}}^{2}},\ \
\lambda_{2}=\frac{h_{2}-\bar{\alpha}z-\alpha\bar{z}+h_{1}{{\vert z\vert}}^{2}}
{1+{{\vert z\vert}}^{2}}$$ under the equation (\[eq:riccati\]), whose solutions are easily given by $$z=\frac{-(h_{1}-h_{2})\pm\sqrt{(h_{1}-h_{2})^{2}+4|\alpha|^{2}}}{2\bar{\alpha}}.$$ A comment is in order. The equation (\[eq:riccati\]) is a special version of (generalized) Riccati equations.
As shown in the example our diagonalization method is different from usual one (a kind of reverse procedure). Let us state our procedure.
[**Riccati Diagonalization**]{}
\[A\] For $H\in H(n;{{\mathbf C}})$ we prepare a unitary matrix $U=U(Z)\in U(n)$ where $Z$ is a parameter matrix and calculate $$U^{\dagger}HU\equiv W=(w_{ij}).$$ \[B\] We set $$w_{ij}=0\quad \mbox{for}\quad 1\leq j < i \leq n$$ and solve these simultaneous equations (a system of Riccati equations) to determine $Z=(z_{kl})$.
\[C\] We finally obtain the diagonal matrix $$W=\mbox{diag}(w_{11}, w_{22},\cdots, w_{nn})$$ where each component is an eigenvalue of $H$ under \[B\].
General Case
============
In this section we consider a generalization of the example in the preceding section. See [@Oi], [@Fu1] as a general introduction to Grassmann manifolds and also [@Ga] and its references as an application.
Namely, we treat a hermitian matrix $$H=
\left(
\begin{array}{cc}
H_{+} & V^{\dagger} \\
V & H_{-}
\end{array}
\right)\in H(n;{{\mathbf C}})$$ where $$H_{+}\in H(k;{{\mathbf C}}),\quad
H_{-}\in H(n-k;{{\mathbf C}}),\quad
V\in M(n-k,k;{{\mathbf C}})$$ for $1\leq k\leq n-1$. In order to make $H$ a direct sum form we prepare a unitary matrix $$\begin{aligned}
U&=&U(Z)=
\left(
\begin{array}{cc}
1_{k} & -Z^{\dagger} \\
Z & 1_{n-k}
\end{array}
\right)
\left(
\begin{array}{cc}
(1_{k}+Z^{\dagger}Z)^{-1/2} & \\
& (1_{n-k}+ZZ^{\dagger})^{-1/2}
\end{array}
\right) \\
&\equiv&U_{M}U_{D} \nonumber\end{aligned}$$ where $Z\in M(n-k,k;{{\mathbf C}})$ is a parameter matrix. $U$ is a map $$U\ :\ M(n-k,k;{{\mathbf C}})\longrightarrow\ SU(n)$$ and $Z$ is a local coordinate of the Grassmann manifold $G_{k}({{\mathbf C}}^{n})$ defined by $$\begin{aligned}
G_{k}({{\mathbf C}}^{n})
&=&
\{P\in M(n;{{\mathbf C}})\ |\ P^{2}=P,\ P^{\dagger}=P,\ \mbox{tr}P=k\} \\
&=&
\{UP_{0}U^{\dagger}\ |\ U\in U(n)\} \\
&\cong & U(n)/U(k)\times U(n-k)\end{aligned}$$ with $P_{0}$ given by $$P_{0}=
\left(
\begin{array}{cc}
1_{k} & \\
& 0_{n-k}
\end{array}
\right).$$ Note that $\mbox{dim}_{{{\mathbf C}}}G_{k}({{\mathbf C}}^{n})=
k(n-k)=\mbox{dim}_{{{\mathbf C}}}M(n-k,k;{{\mathbf C}})$. The local parametrization of $G_{k}({{\mathbf C}}^{n})$ is more explicitly given by $$P(Z)=U(Z)P_{0}U(Z)^{\dagger}
=
\left(
\begin{array}{cc}
1_{k} & -Z^{\dagger} \\
Z & 1_{n-k}
\end{array}
\right)
\left(
\begin{array}{cc}
1_{k} & \\
& 0_{n-k}
\end{array}
\right)
\left(
\begin{array}{cc}
1_{k} & -Z^{\dagger} \\
Z & 1_{n-k}
\end{array}
\right)^{-1}$$ where we have used the relation $$\left(
\begin{array}{cc}
1_{k} & -Z^{\dagger} \\
Z & 1_{n-k}
\end{array}
\right)^{-1}
=
\left(
\begin{array}{cc}
(1_{k}+Z^{\dagger}Z)^{-1} & \\
& (1_{n-k}+ZZ^{\dagger})^{-1}
\end{array}
\right)
\left(
\begin{array}{cc}
1_{k} & -Z^{\dagger} \\
Z & 1_{n-k}
\end{array}
\right)^{\dagger}.$$ Let us calculate $U_{M}^{\dagger}HU_{M}$ : $$\begin{aligned}
U_{M}^{\dagger}HU_{M}
&=&
\left(
\begin{array}{cc}
1_{k} & Z^{\dagger} \\
-Z & 1_{n-k}
\end{array}
\right)
\left(
\begin{array}{cc}
H_{+} & V^{\dagger} \\
V & H_{-}
\end{array}
\right)
\left(
\begin{array}{cc}
1_{k} & -Z^{\dagger} \\
Z & 1_{n-k}
\end{array}
\right) \nonumber \\
&=&
\left(
\begin{array}{cc}
H_{+}+Z^{\dagger}V+V^{\dagger}Z+Z^{\dagger}H_{-}Z &
V^{\dagger}-H_{+}Z^{\dagger}+Z^{\dagger}H_{-}-Z^{\dagger}VZ^{\dagger} \\
V-ZH_{+}+H_{-}Z-ZV^{\dagger}Z &
H_{-}-ZV^{\dagger}-VZ^{\dagger}+ZH_{+}Z^{\dagger}
\end{array}
\right).\end{aligned}$$ From this we set $$\label{eq:matrix Riccati equation}
V-ZH_{+}+H_{-}Z-ZV^{\dagger}Z=0\ \Longleftrightarrow\
ZV^{\dagger}Z+ZH_{+}-H_{-}Z-V=0.$$ This is just the matrix Riccati equation. Under the condition we obtain the block form $$U^{\dagger}HU
=
\left(
\begin{array}{cc}
(1_{k}+Z^{\dagger}Z)^{-1/2}\widetilde{H}_{+}(1_{k}+Z^{\dagger}Z)^{-1/2} & \\
& (1_{n-k}+ZZ^{\dagger})^{-1/2}\widetilde{H}_{-}(1_{n-k}+ZZ^{\dagger})^{-1/2}
\end{array}
\right)$$ where $$\label{eq:reduced Hamiltonian}
\widetilde{H}_{+}=H_{+}+Z^{\dagger}V+V^{\dagger}Z+Z^{\dagger}H_{-}Z,\quad
\widetilde{H}_{-}=H_{-}-ZV^{\dagger}-VZ^{\dagger}+ZH_{+}Z^{\dagger}.$$ How to solve the Riccati equation (\[eq:matrix Riccati equation\]) is not known as far as we know. In fact, it is very hard, so we must satisfy by finding some approximate solution at the present time.
[**Approximation I**]{}
First, by rejecting the quadratic term we have $$\label{eq:linear Riccati equation}
ZH_{+}-H_{-}Z=V.$$ This solution is well–known to become $$\label{eq:linear solution}
Z=\int_{0}^{\infty}e^{tH_{-}}Ve^{-tH_{+}}dt$$ under some condition on $H_{-}$ and $H_{+}$. See for example [@Ari]. In fact, $$\begin{aligned}
ZH_{+}-H_{-}Z
&=&
\int_{0}^{\infty}
\{e^{tH_{-}}Ve^{-tH_{+}}H_{+}-H_{-}e^{tH_{-}}Ve^{-tH_{+}}\}dt \\
&=&
-\int_{0}^{\infty}\frac{d}{dt}(e^{tH_{-}}Ve^{-tH_{+}})dt \\
&=&
-[e^{tH_{-}}Ve^{-tH_{+}}]_{0}^{\infty} \\
&=&
V\end{aligned}$$ under the condition $$\lim_{t\rightarrow \infty}e^{tH_{-}}Ve^{-tH_{+}}=0.$$
[**Approximation II**]{}
Next, let us consider another approximation. We assume that $n=2m,\ k=m$ and $V$ is invertible ($V\in GL(m;{{\mathbf C}})$). Then, by remembering $$ax^{2}+2bx+c=0
\ \Longrightarrow \
a(x+\frac{b}{a})^{2}=-c+\frac{b^{2}}{a}$$ we have $$\{Z-H_{-}(V^{\dagger})^{-1}\}V^{\dagger}\{Z+(V^{\dagger})^{-1}H_{+}\}
=
V-H_{-}(V^{\dagger})^{-1}H_{+}$$ from (\[eq:matrix Riccati equation\]). Here if we can choose $Z$ as $$Z+(V^{\dagger})^{-1}H_{+}\in GL(m;{{\mathbf C}})$$ then we have a recursive relation $$Z=H_{-}(V^{\dagger})^{-1}+
\{V-H_{-}(V^{\dagger})^{-1}H_{+}\}
\frac{1}{Z+(V^{\dagger})^{-1}H_{+}}(V^{\dagger})^{-1}.$$ Now by inserting an approximate solution (\[eq:linear solution\]) into the equation above we obtain the approximate solution $$Z\approx H_{-}(V^{\dagger})^{-1}+
\{V-H_{-}(V^{\dagger})^{-1}H_{+}\}
\frac{1}{
\int_{0}^{\infty}e^{tH_{-}}Ve^{-tH_{+}}dt+
(V^{\dagger})^{-1}H_{+}}
(V^{\dagger})^{-1}.$$ if $$\int_{0}^{\infty}e^{tH_{-}}Ve^{-tH_{+}}dt+(V^{\dagger})^{-1}H_{+}
\in GL(m;{{\mathbf C}})$$ or $$H_{+}+\int_{0}^{\infty}V^{\dagger}e^{tH_{-}}Ve^{-tH_{+}}dt
\in GL(m;{{\mathbf C}}).$$ A comment is in order. We don’t know at the present time whether our approximate solution is convenient enough or not.
Reduction of Riccati Diagonalization
====================================
In this section we give an explicit procedure of Riccati diagonalization. General Hamiltonian is $$H=
\left(
\begin{array}{ccccccc}
h_{1} & \bar{v}_{21} & \bar{v}_{31} & \cdot & \cdot & \bar{v}_{n-1,1} & \bar{v}_{n1} \\
v_{21} & h_{2} & \bar{v}_{32} & \cdot & \cdot & \bar{v}_{n-1,2} & \bar{v}_{n2} \\
v_{31} & v_{32} & h_{3} & \cdot & \cdot & \bar{v}_{n-1,3} & \bar{v}_{n3} \\
\cdot & \cdot & \cdot & \cdot & & \cdot & \cdot \\
\cdot & \cdot & \cdot & & \cdot & \cdot & \cdot \\
v_{n-1,1} & v_{n-1,2} & v_{n-1,3} & \cdot & \cdot & h_{n-1} & \bar{v}_{n,n-1} \\
v_{n1} & v_{n2} & v_{n3} & \cdot & \cdot & v_{n,n-1} & h_{n}
\end{array}
\right)\ \in\ H(n;{{\mathbf C}})$$ and we write as $$H=
\left(
\begin{array}{cc}
H_{+} & V^{\dagger} \\
V & h_{n}
\end{array}
\right),\quad
V=(v_{n1},v_{n2}, \cdots, v_{n,n-1})$$ for simplicity. We prepare a unitary matrix $$U=
\left(
\begin{array}{cc}
1_{n-1} & -Z^{\dagger} \\
Z & 1
\end{array}
\right)
\left(
\begin{array}{cc}
(1_{n-1}+Z^{\dagger}Z)^{-1/2} & \\
& (1+ZZ^{\dagger})^{-1/2}
\end{array}
\right)$$ where $Z=(z_{1},z_{2},\cdots,z_{n-1})$. Then the Riccati equation is $$\begin{aligned}
ZV^{\dagger}Z+ZH_{+}-h_{n}Z-V=0
&\Longleftrightarrow&
\left(\sum_{j=1}^{n-1}\bar{v}_{nj}z_{j}\right)z_{k}+
\sum_{j=1}^{n-1}(H_{+})_{jk}z_{j}-h_{n}z_{k}-v_{nk}=0 \nonumber \\
&&\quad \mbox{for}\ \ 1\leq k\leq n-1. \end{aligned}$$ Note that to solve the equation(s) above explicitly is very hard, so in general we must satisfy by constructing some approximate solution.
If we can solve the equation(s) then $$\begin{aligned}
U^{\dagger}HU
&=&
\left(
\begin{array}{cc}
(1_{n-1}+Z^{\dagger}Z)^{-1/2}\widetilde{H}_{+}(1_{n-1}+Z^{\dagger}Z)^{-1/2} & \\
& \frac{\widetilde{h}_{n}}{1+\sum_{j=1}^{n-1}|z_{j}|^{2}}
\end{array}
\right) \\
\tilde{h}_{n}
&=&h_{n}-\sum_{j=1}^{n-1}(z_{j}\bar{v}_{nj}+c.c.)+
\sum_{j=1}^{n-1}\sum_{k=1}^{n-1}z_{j}(H_{+})_{jk}\bar{z}_{k} \nonumber\end{aligned}$$ and the procedure is reduced to the calculation of $$(1_{n-1}+Z^{\dagger}Z)^{-1/2}\widetilde{H}_{+}
(1_{n-1}+Z^{\dagger}Z)^{-1/2}$$ , so we must calculate the term $(1_{n-1}+Z^{\dagger}Z)^{-1/2}$ exactly.
We write $$\begin{aligned}
&&Z
=(z_{1},z_{2},\cdots,z_{n-1})
=z_{1}(1,w_{2},\cdots,w_{n-1})
\equiv z_{1}(1,W),
\quad
w_{j}=z_{j}/z_{1} \\
&&ZZ^{\dagger}=|z_{1}|^{2}(1+WW^{\dagger})=\sum_{j=1}^{n-1}|z_{j}|^{2}\end{aligned}$$ for simplicity. Then $$1_{n-1}+Z^{\dagger}Z
=
1_{n-1}+|z_{1}|^{2}
\left(
\begin{array}{cc}
1 & W \\
W^{\dagger} & W^{\dagger}W
\end{array}
\right)$$ and a unitary matrix given by $$\widetilde{U}
=
\left(
\begin{array}{cc}
1 & -W \\
W^{\dagger} & 1_{n-2}
\end{array}
\right)
\left(
\begin{array}{cc}
(1+WW^{\dagger})^{-1/2} & \\
& (1_{n-2}+W^{\dagger}W)^{-1/2}
\end{array}
\right)$$ gives $$\widetilde{U}
\left(
\begin{array}{cc}
1 & \\
& 0_{n-2}
\end{array}
\right)
\widetilde{U}^{\dagger}
=
\frac{1}{1+WW^{\dagger}}
\left(
\begin{array}{cc}
1 & W \\
W^{\dagger} & W^{\dagger}W
\end{array}
\right).$$ Therefore $$\begin{aligned}
1_{n-1}+Z^{\dagger}Z
&=&
1_{n-1}+|z_{1}|^{2}(1+WW^{\dagger})
\widetilde{U}
\left(
\begin{array}{cc}
1 & \\
& 0_{n-2}
\end{array}
\right)
\widetilde{U}^{\dagger} \\
&=&
\widetilde{U}
\left\{
1_{n-1}+\sum_{j=1}^{n-1}|z_{j}|^{2}
\left(
\begin{array}{cc}
1 & \\
& 0_{n-2}
\end{array}
\right)
\right\}
\widetilde{U}^{\dagger} \\
&=&
\widetilde{U}
\left(
\begin{array}{cc}
1+\sum_{j=1}^{n-1}|z_{j}|^{2} & \\
& 1_{n-2}
\end{array}
\right)
\widetilde{U}^{\dagger} \\\end{aligned}$$ and we have $$\left(1_{n-1}+Z^{\dagger}Z\right)^{-1/2}
=
\widetilde{U}
\left(
\begin{array}{cc}
\frac{1}{\sqrt{1+\sum_{j=1}^{n-1}|z_{j}|^{2}}} & \\
& 1_{n-2}
\end{array}
\right)
\widetilde{U}^{\dagger}.$$ As a result the reduced Hamiltonian is $$\begin{aligned}
&&(1_{n-1}+Z^{\dagger}Z)^{-1/2}\widetilde{H}_{+}(1_{n-1}+Z^{\dagger}Z)^{-1/2}
\nonumber \\
&=&
\widetilde{U}
\left(
\begin{array}{cc}
\frac{1}{\sqrt{1+\sum_{j=1}^{n-1}|z_{j}|^{2}}} & \\
& 1_{n-2}
\end{array}
\right)
\widetilde{U}^{\dagger}
\widetilde{H}_{+}
\widetilde{U}
\left(
\begin{array}{cc}
\frac{1}{\sqrt{1+\sum_{j=1}^{n-1}|z_{j}|^{2}}} & \\
& 1_{n-2}
\end{array}
\right)
\widetilde{U}^{\dagger}\end{aligned}$$ and we obtain $$\begin{aligned}
&&
U^{\dagger}HU \nonumber \\
&=&
\left(
\begin{array}{cc}
\widetilde{U}
\left(
\begin{array}{cc}
\frac{1}{\sqrt{1+\sum_{j=1}^{n-1}|z_{j}|^{2}}} & \\
& 1_{n-2}
\end{array}
\right)
\widetilde{U}^{\dagger}
\widetilde{H}_{+}
\widetilde{U}
\left(
\begin{array}{cc}
\frac{1}{\sqrt{1+\sum_{j=1}^{n-1}|z_{j}|^{2}}} & \\
& 1_{n-2}
\end{array}
\right)
\widetilde{U}^{\dagger} & \\
& \frac{\widetilde{h}_{n}}{1+\sum_{j=1}^{n-1}|z_{j}|^{2}}
\end{array}
\right). \nonumber \\
&&{}\end{aligned}$$ We have only to continue the reduction process one after another.
A comment is in order. Note that it is not easy to calculate $(1_{n-k}+Z^{\dagger}Z)^{-1/2}$ for $Z\in M(k,n-k;{{\mathbf C}})$ and $2\leq k\leq n-2$.
[**Note**]{} There is no need to calculate $(1_{n-2}+W^{\dagger}W)^{-1/2}$ in $\widetilde{U}$ because $$\begin{aligned}
&&\widetilde{U}
\left(
\begin{array}{cc}
\frac{1}{\sqrt{1+\sum_{j=1}^{n-1}|z_{j}|^{2}}} & \\
& 1_{n-2}
\end{array}
\right)
\widetilde{U}^{\dagger} \\
&&
=
\left(
\begin{array}{cc}
1 & -W \\
W^{\dagger} & 1_{n-2}
\end{array}
\right)
\left(
\begin{array}{cc}
\frac{1}{(1+WW^{\dagger})\sqrt{1+\sum_{j=1}^{n-1}|z_{j}|^{2}}} & \\
& (1_{n-2}+W^{\dagger}W)^{-1}
\end{array}
\right)
\left(
\begin{array}{cc}
1 & W \\
-W^{\dagger} & 1_{n-2}
\end{array}
\right) \\
&&
=
\left(
\begin{array}{cc}
1 & -W \\
W^{\dagger} & 1_{n-2}
\end{array}
\right)
\left(
\begin{array}{cc}
\frac{1}{(1+WW^{\dagger})\sqrt{1+\sum_{j=1}^{n-1}|z_{j}|^{2}}} & \\
& 1_{n-2}-\frac{1}{1+WW^{\dagger}}W^{\dagger}W
\end{array}
\right)
\left(
\begin{array}{cc}
1 & W \\
-W^{\dagger} & 1_{n-2}
\end{array}
\right)\end{aligned}$$ from the definition of $\widetilde{U}$. This is important.
Last, let us make a comment. If $n=3$, namely $Z=(z_{1},z_{2})$ then we have a direct method (without using $W$). For $$1_{2}+Z^{\dagger}Z
=
\left(
\begin{array}{cc}
1+|z_{1}|^{2} & \bar{z}_{1}z_{2} \\
z_{1}\bar{z}_{2} & 1+|z_{2}|^{2}
\end{array}
\right)$$ we set $$U=
\frac{1}{\sqrt{|z_{1}|^{2}+|z_{2}|^{2}}}
\left(
\begin{array}{cc}
\bar{z}_{1} & -z_{2} \\
\bar{z}_{2} & z_{1}
\end{array}
\right)\ \in\ SU(2).$$ Then we have $$1_{2}+Z^{\dagger}Z
=
U
\left(
\begin{array}{cc}
1+|z_{1}|^{2}+|z_{2}|^{2} & \\
& 1
\end{array}
\right)
U^{\dagger}$$ and $$\begin{aligned}
(1_{2}+Z^{\dagger}Z)^{-1/2}
&=&
U
\left(
\begin{array}{cc}
\frac{1}{\sqrt{1+|z_{1}|^{2}+|z_{2}|^{2}}} & \\
& 1
\end{array}
\right)
U^{\dagger} \nonumber \\
&=&
\frac{1}{|z_{1}|^{2}+|z_{2}|^{2}}
\left(
\begin{array}{cc}
\bar{z}_{1} & -z_{2} \\
\bar{z}_{2} & z_{1}
\end{array}
\right)
\left(
\begin{array}{cc}
\frac{1}{\sqrt{1+|z_{1}|^{2}+|z_{2}|^{2}}} & \\
& 1
\end{array}
\right)
\left(
\begin{array}{cc}
z_{1} & z_{2} \\
-\bar{z}_{2} & \bar{z}_{1}
\end{array}
\right) \nonumber \\
&=&
\frac{1}{|z_{1}|^{2}+|z_{2}|^{2}}
\left(
\begin{array}{cc}
\frac{|z_{1}|^{2}}{\sqrt{1+|z_{1}|^{2}+|z_{2}|^{2}}}+|z_{2}|^{2}
&
\frac{\bar{z}_{1}z_{2}}{\sqrt{1+|z_{1}|^{2}+|z_{2}|^{2}}}-\bar{z}_{1}z_{2} \\
\frac{z_{1}\bar{z}_{2}}{\sqrt{1+|z_{1}|^{2}+|z_{2}|^{2}}}-z_{1}\bar{z}_{2}
&
\frac{|z_{2}|^{2}}{\sqrt{1+|z_{1}|^{2}+|z_{2}|^{2}}}+|z_{1}|^{2}
\end{array}
\right).\end{aligned}$$ This form looks smart and will be used in the next section.
Example
=======
In this section we apply our method to the following important example $$H=
\left(
\begin{array}{ccc}
h_{1} & \bar{\alpha} & \bar{\beta} \\
\alpha & h_{2} & \bar{\gamma} \\
\beta & \gamma & h_{3}
\end{array}
\right).$$ Since $$H_{+}=
\left(
\begin{array}{cc}
h_{1} & \bar{\alpha} \\
\alpha & h_{2}
\end{array}
\right),
\quad
V=(\beta,\gamma),
\quad
Z=(z_{1},z_{2})$$ the Riccati equation is $$\left\{
\begin{array}{ll}
\bar{\beta}z_{1}^{2}+\bar{\gamma}z_{1}z_{2}+(h_{1}z_{1}+\alpha z_{2})
-h_{3}z_{1}-\beta=0, \\
\bar{\gamma}z_{2}^{2}+\bar{\beta}z_{1}z_{2}+(\bar{\alpha}z_{1}+h_{2}z_{2})
-h_{3}z_{2}-\gamma=0
\end{array}
\right.$$ or $$\left\{
\begin{array}{ll}
\bar{\beta}z_{1}^{2}+(h_{1}-h_{3})z_{1}-\beta
=-z_{2}(\alpha +\bar{\gamma}z_{1}), \\
\bar{\gamma}z_{2}^{2}+(h_{2}-h_{3})z_{2}-\gamma
=-z_{1}(\bar{\alpha} +\bar{\beta}z_{2}).
\end{array}
\right.$$ Let us solve the equations. We get $z_{1}$ by solving the equation $$\begin{aligned}
\label{eq:reduced-1-i}
&&\{
\bar{\beta}\bar{\gamma}(h_{1}-h_{2})
-\alpha\bar{\beta}^{2}+\bar{\alpha}\bar{\gamma}^{2}
\}z_{1}^{3}+ \nonumber \\
&&[
\bar{\gamma}
\{
(h_{1}-h_{2})(h_{1}-h_{3})+2|\alpha|^{2}-|\beta|^{2}-|\gamma|^{2}
\}
-\alpha\bar{\beta}(h_{1}+h_{2}-2h_{3})
]z_{1}^{2}+ \nonumber \\
&&[
-\alpha
\{
(h_{1}-h_{3})(h_{2}-h_{3})-|\alpha|^{2}-|\beta|^{2}+2|\gamma|^{2}
\}
+\beta\bar{\gamma}(-2h_{1}+h_{2}+h_{3})
]z_{1}+ \nonumber \\
&&\beta^{2}\bar{\gamma}+\alpha\beta(h_{2}-h_{3})-\alpha^{2}\gamma
=
0\end{aligned}$$ by use of Cardano formula and $z_{2}$ by solving the equation $$\label{eq:reduced-1-ii}
\bar{\gamma}z_{2}^{2}+(h_{2}-h_{3}+\bar{\beta}z_{1})z_{2}
+\bar{\alpha}z_{1}-\gamma=0.$$ If $(z_{1},z_{2})$ is given then the reduced Hamiltonian becomes $$\begin{aligned}
&&(1_{2}+Z^{\dagger}Z)^{-1/2}\widetilde{H}_{+}(1_{2}+Z^{\dagger}Z)^{-1/2}
\nonumber \\
&=&
\frac{1}{(|z_{1}|^{2}+|z_{2}|^{2})^{2}}
\left(
\begin{array}{cc}
\frac{|z_{1}|^{2}}{\sqrt{1+|z_{1}|^{2}+|z_{2}|^{2}}}+|z_{2}|^{2}
&
\frac{\bar{z}_{1}z_{2}}{\sqrt{1+|z_{1}|^{2}+|z_{2}|^{2}}}-\bar{z}_{1}z_{2} \\
\frac{z_{1}\bar{z}_{2}}{\sqrt{1+|z_{1}|^{2}+|z_{2}|^{2}}}-z_{1}\bar{z}_{2}
&
\frac{|z_{2}|^{2}}{\sqrt{1+|z_{1}|^{2}+|z_{2}|^{2}}}+|z_{1}|^{2}
\end{array}
\right)\times \nonumber \\
&&\quad\quad\quad\quad\quad\quad\ \
\left(
\begin{array}{cc}
h_{1}+\bar{\beta}z_{1}+\beta\bar{z}_{1}+h_{3}|z_{1}|^{2} &
\bar{\alpha}+\gamma\bar{z}_{1}+\bar{\beta}z_{2}+h_{3}\bar{z}_{1}z_{2} \\
\alpha+\bar{\gamma}z_{1}+\beta\bar{z}_{2}+h_{3}z_{1}\bar{z}_{2} &
h_{2}+\bar{\gamma}z_{2}+\gamma\bar{z}_{2}+h_{3}|z_{2}|^{2}
\end{array}
\right)\times \nonumber \\
&&\quad\quad\quad\quad\quad\quad\ \
\left(
\begin{array}{cc}
\frac{|z_{1}|^{2}}{\sqrt{1+|z_{1}|^{2}+|z_{2}|^{2}}}+|z_{2}|^{2}
&
\frac{\bar{z}_{1}z_{2}}{\sqrt{1+|z_{1}|^{2}+|z_{2}|^{2}}}-\bar{z}_{1}z_{2} \\
\frac{z_{1}\bar{z}_{2}}{\sqrt{1+|z_{1}|^{2}+|z_{2}|^{2}}}-z_{1}\bar{z}_{2}
&
\frac{|z_{2}|^{2}}{\sqrt{1+|z_{1}|^{2}+|z_{2}|^{2}}}+|z_{1}|^{2}
\end{array}
\right) \nonumber \\
&{\equiv}&
\left(
\begin{array}{cc}
k_{1} & \bar{\zeta} \\
\zeta & k_{2}
\end{array}
\right).\end{aligned}$$ Therefore we have only to solve the equation $$\label{eq:reduced-2}
\bar{\zeta}z^{2}+(k_{1}-k_{2})z-\zeta =0$$ as shown in the example in section 2.
Finally we obtain three eigenvalues $$\begin{aligned}
\lambda_{1}&=&\frac{k_{1}+\bar{\zeta}z+\zeta\bar{z}+k_{2}{{\vert z\vert}}^{2}}
{1+{{\vert z\vert}}^{2}},\ \
\lambda_{2}=\frac{k_{2}-\bar{\zeta}z-\zeta\bar{z}+k_{1}{{\vert z\vert}}^{2}}
{1+{{\vert z\vert}}^{2}}, \nonumber \\
\lambda_{3}&=&
\frac{h_{3}-(\bar{\beta}z_{1}+\beta\bar{z}_{1})-
(\bar{\gamma}z_{2}+\gamma\bar{z}_{2})+
h_{1}|z_{1}|^{2}+\bar{\alpha}z_{1}\bar{z}_{2}+{\alpha}\bar{z}_{1}{z}_{2}+
h_{2}|z_{2}|^{2}}
{1+|z_{1}|^{2}+|z_{2}|^{2}}\end{aligned}$$ under (\[eq:reduced-1-i\]), (\[eq:reduced-1-ii\]) and (\[eq:reduced-2\]).
In the process of calculation MATHEMATICA or MAPLE is indispensable (calculation by force is very hard).
Discussion
==========
In this paper we presented a geometric approach to diagonalization of hermitian matrices. The advantage of our method is quick diagonalization, while to obtain eigenvalues is left in the final step. Our method is in a certain sense reverse process of the standard one, so they are dual each other.
It is not clear at the present time whether our method is convenient enough or not. Further work will be needed.
Last, let us make a comment on “Riccati structure" of Quantum Mechanics, see for example [@Mi]. We consider the harmonic oscillator given by $$H=\frac{1}{2}
\left(
-\frac{d^{2}}{dx^{2}}+x^{2}
\right).$$ In order to solve the model we usually define the annihilation and creation operators like $$a=\frac{1}{\sqrt{2}}
\left(
\frac{d}{dx}+x
\right), \quad
a^{\dagger}=\frac{1}{\sqrt{2}}
\left(
-\frac{d}{dx}+x
\right).$$ Then it is well–known that $$aa^{\dagger}=H+\frac{1}{2},\quad a^{\dagger}a=H-\frac{1}{2}.$$ Next we define another annihilation and creation operators like $$b=\frac{1}{\sqrt{2}}
\left(
\frac{d}{dx}+\beta(x)
\right), \quad
b^{\dagger}=\frac{1}{\sqrt{2}}
\left(
-\frac{d}{dx}+\beta(x)
\right)$$ with unknown $\beta(x)$.
Then, in order to satisfy the relation $bb^{\dagger}=H+\frac{1}{2}$ $\beta(x)$ must satisfy the equation $$\beta^{\prime}+\beta^{2}=1+x^{2}.$$ This is a special type of Riccati differential equation.
[**Appendix**]{}
In the appendix we consider the important example $$H=
\left(
\begin{array}{ccc}
h_{1} & \bar{\alpha} & \bar{\beta} \\
\alpha & h_{2} & \bar{\gamma} \\
\beta & \gamma & h_{3}
\end{array}
\right)$$ once more. We want to diagonalize the matrix [**at a time**]{}.
For the purpose we prepare a unitary matrix coming from the flag manifold of the second type (in our terminology, see [@FO], [@Fu2] and also [@Pi], [@DJ]) $SU(3)/U(1)\times U(1)$ $$\begin{aligned}
U=U(x,y,z)&=&
\left(
\begin{array}{ccc}
1 & -(\bar{x}+\bar{y}z) & \bar{x}\bar{z}-\bar{y} \\
x & \Delta_{1}-x(\bar{x}+\bar{y}z) & -\bar{z} \\
y & z\Delta_{1}-y(\bar{x}+\bar{y}z) & 1
\end{array}
\right)
\left(
\begin{array}{ccc}
\frac{1}{\sqrt{\Delta_{1}}} & & \\
& \frac{1}{\sqrt{\Delta_{1}\Delta_{2}}} & \\
& & \frac{1}{\sqrt{\Delta_{2}}}
\end{array}
\right) \\
&\equiv& U_{M}U_{D} \nonumber\end{aligned}$$ where $\Delta_{1}$ and $\Delta_{2}$ are given by $$\Delta_{1}=1+|x|^{2}+|y|^{2},\quad
\Delta_{1}=1+|z|^{2}+|xz-y|^{2}.$$ Let us calculate $U^{\dagger}HU$. Since $
U^{\dagger}HU=U_{D}U_{M}^{\dagger}HU_{M}U_{D}
$ we have only to calculate $U_{M}^{\dagger}HU_{M}$ because $U_{D}$ is diagonal. Namely, $$\begin{aligned}
U_{M}^{\dagger}HU_{M}
&=&
\left(
\begin{array}{ccc}
1 & \bar{x} & \bar{y} \\
-(x+y\bar{z}) & \Delta_{1}-\bar{x}(x+y\bar{z}) &
\bar{z}\Delta_{1}-\bar{y}(x+y\bar{z}) \\
xz-y & -z & 1
\end{array}
\right)
\left(
\begin{array}{ccc}
h_{1} & \bar{\alpha} & \bar{\beta} \\
\alpha & h_{2} & \bar{\gamma} \\
\beta & \gamma & h_{3}
\end{array}
\right) \times \nonumber \\
&&\left(
\begin{array}{ccc}
1 & -(\bar{x}+\bar{y}z) & \bar{x}\bar{z}-\bar{y} \\
x & \Delta_{1}-x(\bar{x}+\bar{y}z) & -\bar{z} \\
y & z\Delta_{1}-y(\bar{x}+\bar{y}z) & 1
\end{array}
\right) \nonumber \\
&\equiv&
\left(
\begin{array}{ccc}
w_{11} & \bar{w}_{21} & \bar{w}_{31} \\
w_{21} & w_{22} & \bar{w}_{32} \\
w_{31} & w_{32} & w_{33}
\end{array}
\right)\end{aligned}$$ where $$\begin{aligned}
w_{21}
&=&
-(x+y\bar{z})(h_{1}+x\bar{\alpha}+y\bar{\beta})+
\{\Delta_{1}-\bar{x}(x+y\bar{z})\}(\alpha+xh_{2}+y\bar{\gamma})+ \nonumber \\
&&\{\bar{z}\Delta_{1}-\bar{y}(x+y\bar{z})\}(\beta+x\gamma+yh_{3}), \nonumber \\
w_{31}
&=&
(xz-y)h_{1}-z\alpha+\beta+\{(xz-y)\bar{\alpha}-zh_{2}+\gamma\}x+
\{(xz-y)\bar{\beta}-z\bar{\gamma}+h_{3}\}y \nonumber \\
&=&
(xz-y)(h_{1}+x\bar{\alpha}+y\bar{\beta})-
z(\alpha+xh_{2}+y\bar{\gamma})+(\beta+x\gamma+yh_{3}), \\
w_{32}
&=&
-(\bar{x}+\bar{y}z)[\{(xz-y)h_{1}-z\alpha+\beta\}+
\{(xz-y)\bar{\alpha}-zh_{2}+\gamma\}x+ \nonumber \\
&&\{(xz-y)\bar{\beta}-z\bar{\gamma}+h_{3}\}y]+
\Delta_{1}[(xz-y)\bar{\alpha}-zh_{2}+\gamma\}+
\{(xz-y)\bar{\beta}-z\bar{\gamma}+h_{3}\}z] \nonumber \\
&=&
-(\bar{x}+\bar{y}z)w_{31}+
\Delta_{1}[(xz-y)\bar{\alpha}-zh_{2}+\gamma\}+
\{(xz-y)\bar{\beta}-z\bar{\gamma}+h_{3}\}z]. \nonumber\end{aligned}$$ Here by setting $w_{21}=w_{31}=w_{32}=0$, we have $$\begin{aligned}
0
&=&
-(x+y\bar{z})(h_{1}+x\bar{\alpha}+y\bar{\beta})+
\{\Delta_{1}-\bar{x}(x+y\bar{z})\}(\alpha+xh_{2}+y\bar{\gamma})+ \nonumber \\
&&\{\bar{z}\Delta_{1}-\bar{y}(x+y\bar{z})\}(\beta+x\gamma+yh_{3}), \nonumber \\
0
&=&
(xz-y)(h_{1}+x\bar{\alpha}+y\bar{\beta})-
z(\alpha+xh_{2}+y\bar{\gamma})+(\beta+x\gamma+yh_{3}), \\
0
&=&
(xz-y)\bar{\alpha}-zh_{2}+\gamma+
\{(xz-y)\bar{\beta}-z\bar{\gamma}+h_{3}\}z. \nonumber\end{aligned}$$ Some calculation gives $$\begin{aligned}
&&
\bar{z}=-
\frac
{
x(h_{1}+x\bar{\alpha}+y\bar{\beta})-
(1+|y|^{2})(\alpha+xh_{2}+y\bar{\gamma})+
\bar{y}x(\beta+x\gamma+yh_{3})
}
{
y(h_{1}+x\bar{\alpha}+y\bar{\beta})+
\bar{x}y(\alpha+xh_{2}+y\bar{\gamma})-
(1+|x|^{2})(\beta+x\gamma+yh_{3})
}, \nonumber \\
&&
z=
\frac
{
y(h_{1}+x\bar{\alpha}+y\bar{\beta})-(\beta+x\gamma+yh_{3})
}
{
x(h_{1}+x\bar{\alpha}+y\bar{\beta})-(\alpha+xh_{2}+y\bar{\gamma})
}, \\
&&
(x\bar{\beta}-\bar{\gamma})z^{2}+(h_{3}-h_{2}+x\bar{\alpha}-
y\bar{\beta})z-y\bar{\alpha}+\gamma=0 \nonumber\end{aligned}$$ in terms of $\Delta_{1}=1+|x|^{2}+|y|^{2}$.
It is not easy at the present time to solve the equations at a time.
[99]{} I. Satake : Linear Algebra (in Japanese), 1989 (latest), Shokabo, Tokyo.\
As far as I know this is the best book on Elementary Linear Algebra. G. Strang : Linear Algebra and its Applications, 1976, Academic Press. H. Oike : Introduction to Grassmann Manifolds (in Japanese), 1978, Lecture Note (Yamagata University). K. Fujii : Introduction to Grassmann Manifolds and Quantum Computation, J. Appl. Math, 2(2002), 371, quant-ph/0103011. B. Gardas : Riccati equation and the problem of decoherence, to appear in J. Math. Phys, arXiv:1001.3541 \[quant-ph\]. S. Arimoto : Linear System Theory (in Japanese), 1974 (2’nd), Sangyo-Tosho, Tokyo. B. Mielnik : Factorization method and new potentials with the oscillator spectrum, (1984), 3387. K. Fujii and H. Oike : Reduced Dynamics from the Unitary Group to Some Flag Manifolds : Interacting Matrix Riccati Equations, Int. J. Geom. Methods Mod. Phys, [**6**]{} (2009), 573, arXiv:0809.0165 \[math-ph\]. K. Fujii : A Geometric Parametrization of the Cabibbo-Kobayashi-Maskawa Matrix and the Jarlskog Invariant, Int. J. Geom. Methods Mod. Phys, [**6**]{} (2009), 1057, arXiv:0901.2180 \[math-ph\]. R. F. Picken : The Duistermaat–Heckman integration formula on flag manifolds, J. Math. Phys, [**31**]{} (1990), 616. M. Daoud and A. Jellal : Quantum Hall Effect on the Flag Manifold F$_{2}$, Int. J. Mod. Phys. A, [**20**]{} (2008), 3129, hep-th/0610157.
[^1]: E-mail address : fujii@yokohama-cu.ac.jp
[^2]: E-mail address : oike@tea.ocn.ne.jp
[^3]: in Japan it is called a pie in the sky
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In 2009, Borcea and Br[ä]{}nd[é]{}n characterize all linear operators on multivariate polynomials which preserve the property of being non-vanishing (stable) on products of prescribed open circular regions. We give a representation theoretic interpretation of their findings, which generalizes and simplifies their result and leads to a conceptual unification of many related results in polynomial stability theory. At the heart of this unification is a generalized Grace’s theorem which addresses polynomials whose roots are all contained in some real interval or ray. This generalization allows us to extend the Borcea-Br[ä]{}nd[é]{}n result to characterize a certain subclass of the linear operators which preserve such polynomials.'
author:
- Jonathan Leake
bibliography:
- 'rep\_graces\_v2.bib'
title: 'A Representation Theoretic Explanation of the Borcea-Br[ä]{}nd[é]{}n Characterization, and an Extension to Interval-Rooted Polynomials'
---
Introduction
============
In 1914, P[ó]{}lya and Schur [@polya1914zwei] characterized the set of diagonal linear operators on polynomials which preserve real-rootedness. Since this seminal paper, much work has been done in extending this characterization to other classes of linear operators. This program in essence came to a close in 2009 with a paper of Borcea and Br[ä]{}nd[é]{}n [@bb1], which gave a complete characterization of linear operators on polynomials which preserve real-rootedness.
Their real-rootedness preservation characterization is derived from a more general result pertaining to stable polynomials. Given $\Omega \subset {\mathbb{C}}^m$, we say that a polynomial $f \in {\mathbb{C}}[x_1,...,x_m]$ is *$\Omega$-stable* if $f$ does not vanish in $\Omega$. Further, $f$ is *real stable* if it has real coefficients and is ${\mathcal{H}_+}^m$-stable, where ${\mathcal{H}_+}\subset {\mathbb{C}}$ is the open upper half-plane. (We also denote the open lower half-plane by ${\mathcal{H}_-}\subset {\mathbb{C}}$.) We additionally use the terms *weakly* $\Omega$-stable and *weakly* real stable if we allow $f \equiv 0$. Finally, we write $f \in {\mathbb{C}}_\lambda[x_1,...,x_m]$ for $\lambda \equiv (\lambda_1,...,\lambda_m)$ if $f$ is of degree at most $\lambda_k$ in $x_k$. We are then led to the following problems for ${\mathbb{K}}\in \{{\mathbb{C}},{\mathbb{R}}\}$, generalized from the P[ó]{}lya-Schur characterization:
Characterize linear operators $T: {\mathbb{K}}_\lambda[x_1,...,x_m] \to {\mathbb{K}}[x_1,...,x_m]$ that preserve weak $\Omega$-stability.
Characterize linear operators $T: {\mathbb{K}}[x_1,...,x_m] \to {\mathbb{K}}[x_1,...,x_m]$ that preserve weak $\Omega$-stability.
In [@bb1], Borcea and Br[ä]{}nd[é]{}n were able to solve these problems in many cases. In particular, they solved both problems for ${\mathbb{K}}= {\mathbb{R}}$ and $\Omega = {\mathcal{H}_+}^m$, where $m=1$ corresponds to the case of preservation of real-rooted polynomials. For ${\mathbb{K}}= {\mathbb{C}}$, they were able to solve Problem 1 for $\Omega$ that is *any* product of open circular regions in ${\mathbb{C}}$.
In this paper we will only be concerned with Problem 1, for which we now state the solution from [@bb1]. Given a linear operator $T: {\mathbb{K}}_\lambda[x_1,...,x_m] \to {\mathbb{K}}[x_1,...,x_m]$, a polynomial $\operatorname{Symb}_{BB}(T)$ called the *(Borcea-Br[ä]{}nd[é]{}n) symbol* is associated to $T$. Specifically, the symbol is a polynomial in ${\mathbb{K}}_{\lambda \sqcup \lambda}[x_1,...,x_m, z_1,...,z_m]$ (i.e., of $2m$ variables), where $\lambda \sqcup \lambda := (\lambda_1,...,\lambda_m,\lambda_1,...,\lambda_m)$. The crucial feature of the symbol is that it shares certain stability properties with its associated linear operator, which yields the characterizations stated in the following results. (We will express these results in more detail in §\[Cops\_sect\] and §\[Rops\_sect\].)
\[BB\_Cops\_thm\] Fix a linear operator $T: {\mathbb{C}}_\lambda[x_1,...,x_m] \to {\mathbb{C}}[x_1,...,x_m]$ which has image of dimension greater than one. Then, $T$ maps ${\mathcal{H}_+}^m$-stable polynomials to weakly ${\mathcal{H}_+}^m$-stable polynomials if and only if $\operatorname{Symb}_{BB}(T)$ is ${\mathcal{H}_+}^{2m}$-stable.
\[BB\_Rops\_thm\] Fix a linear operator $T: {\mathbb{R}}_\lambda[x_1,...,x_m] \to {\mathbb{R}}[x_1,...,x_m]$ which has image of dimension greater than two. Then, $T$ maps real stable polynomials to weakly real stable polynomials if and only if either $\operatorname{Symb}_{BB}(T)$ or $\operatorname{Symb}_{BB}(T^-)$ is real stable, where $T^-(p) := T(p(-x_1,...,-x_m))$.
To deal with other products of circular regions, one then conjugates $T$ by certain M[ö]{}bius transformations and applies Theorem \[BB\_Cops\_thm\] to the conjugated operator. Unfortunately though, this is a tedious process which has to be done each time a new stability region is to be considered. Additionally, the image dimension restrictions give rise to degeneracy cases which have to be dealt with separately. Both of these issues obscure the connection between an operator and its symbol.
In this paper, we present a new conceptual approach to the Borcea-Br[ä]{}nd[é]{}n characterization via the representation theory of $SL_2({\mathbb{C}})$. In particular, we derive a new symbol (denoted $\operatorname{Symb}$) in a natural way, and our definition eliminates the issues discussed above. This is seen in the following results, which are our simplified and generalized versions of the Borcea-Br[ä]{}nd[é]{}n characterizations. Note that for the sake of simplicity, we have omitted a few details here regarding non-convex circular regions.
Fix a linear operator $T: {\mathbb{C}}_\lambda[x_1,...,x_m] \to {\mathbb{C}}_\alpha[x_1,...,x_l]$, a product of all open or all closed circular regions $\Omega_0 = C_1 \times \cdots \times C_m$, and a product of sets $\Omega_1 := S_1 \times \cdots \times S_m$. Further, denote $\widetilde{\Omega}_0 := ({\mathbb{C}}\setminus C_1) \times \cdots \times ({\mathbb{C}}\setminus C_m)$. Up to certain degree and convexity (of $C_k$) restrictions, we have that $T$ maps $\Omega_0$-stable polynomials to nonzero $\Omega_1$-stable polynomials if and only if $\operatorname{Symb}(T)$ is $(\widetilde{\Omega}_0 \times \Omega_1)$-stable.
Fix a linear operator $T: {\mathbb{R}}_\lambda[x_1,...,x_m] \to {\mathbb{R}}_\alpha[x_1,...,x_l]$. Up to certain degree restrictions, $T$ maps real stable polynomials to nonzero real stable polynomials if and only if $\operatorname{Symb}(T)$ is either $(\overline{{\mathcal{H}_-}}^m \times {\mathcal{H}_+}^l)$-stable or $(\overline{{\mathcal{H}_-}}^m \times {\mathcal{H}_-}^l)$-stable.
We summarize the specific improvements that this and our other related results give over the Borcea-Br[ä]{}nd[é]{}n characterization as follows.
1. *Different stability regions can be considered using the same symbol.* The symbol we define in this paper is universal: for example, it gives stability-preservation information for *any* product of open circular regions. The Borcea-Br[ä]{}nd[é]{}n symbol, on the other hand, required the application of M[ö]{}bius transformations. In addition, our symbol also allows for the output stability region to be chosen independently of the input stability region. While this does not literally improve the result, it does allow for quicker computations. In particular, see Examples \[add\_conv\_ex1\] and \[mult\_conv\_ex1\] where classical polynomial convolution results are easily derived from our framework.
2. *Our characterization does not require any degeneracy condition.* Our results characterize operators which preserve (strong) stability rather than weak stability. As seen above, this slightly stronger notion of stability enables us to eliminate any image dimension degeneracy condition, as required in the Borcea-Br[ä]{}nd[é]{}n characterizations (Theorems \[BB\_Cops\_thm\] and \[BB\_Rops\_thm\]). This demonstrates a cleaner link between an operator and its symbol.
3. *Closed circular regions and projectively convex regions can be considered.* The symbol we define in this paper handles products of open circular regions, as well as products of closed circular regions. (In [@melamud2015ops], Melamud proves a result similar to the Borcea-Br[ä]{}nd[é]{}n characterization for closed circular regions.) Further, we are also able to consider more general *projectively convex* regions (circular regions with portions of their boundary; also called *generalized circular regions*, see [@zervos1960aspects] and [@zaheer1976polar]) in Proposition \[Cops\_prop\]. This allows us to determine stability-preservation information about real intervals and half-lines. It also turns out, somewhat surprisingly, that our symbol can handle products of any *sets* as possible output space stability regions (as seen in Theorem \[Cops\_graces\] above).
In the process of generalizing the Borcea-Br[ä]{}nd[é]{}n characterization we develop a general algebraic framework which also encompasses many of the classical polynomial tools. This framework aims to motivate classical results and provide intuition for the connection between a stability preserving operator and its symbol.
### The Main Idea {#the-main-idea .unnumbered}
A major purpose of this paper is to explain a certain conceptual thread in the history of polynomial stability theory: that it is often possible to determine general stability information from restricted sets of polynomials. For example, the P[ó]{}lya-Schur and Borcea-Br[ä]{}nd[é]{}n characterizations derive from a single polynomial (i.e., the symbol) stability properties of a whole collection of polynomials in the output of a given linear operator. Additionally, the Grace-Walsh-Szeg[ő]{} coincidence theorem says that stability information of any polynomial can be determined from its polarization, which is of degree at most one in each variable.
As it turns out, these sorts of phenomena can be explained using relatively basic algebraic and representation theoretic concepts. Specifically, we can view ${\mathbb{C}}_n[x]$ as a representation of $SL_2({\mathbb{C}})$ via the following action. For $\phi \in SL_2({\mathbb{C}})$ and $f \in {\mathbb{C}}_n[x]$, we define: $$(\phi \cdot f)(x) := f(\phi^{-1}x)$$ Here $\phi^{-1}$ acts on $x \in {\mathbb{C}}$ as a M[ö]{}bius transformation, or equivalently $\phi$ acts on the roots of $f$. (Similarly, ${\mathbb{C}}_\lambda[x_1,...,x_m]$ can be viewed as a representation of $(SL_2({\mathbb{C}}))^m$ via this action in each variable.) Under this interpretation, important maps like polarization, projection, the apolarity form, and even the symbol turn out to be invariant under these $SL_2({\mathbb{C}})$ actions. This leads us to a conceptual thesis:
$SL_2({\mathbb{C}})$-invariant maps transfer stability information.
The goal of this paper is then to explicate and answer the most important question related to this thesis: what does it mean for the symbol map to be $SL_2({\mathbb{C}})$-invariant and how does it transfer stability information? To answer this, we consider the following standard ideas relating spaces of linear operators to tensor products.
Let $W_1,W_2$ be two finite dimensional representations of a group $G$, and let $\operatorname{Hom}(W_1,W_2)$ denote the space of linear maps from $W_1$ to $W_2$. Then, $\operatorname{Hom}(W_1,W_2) \cong W_1^* \boxtimes W_2$ (the outer tensor product) can viewed as a representation of $G \times G$. If we further have a $G$-invariant bilinear form on $W_1$, then we also have $W_1 \cong W_1^*$. This leads to the following identification: $$\operatorname{Hom}(W_1,W_2) \cong W_1^* \boxtimes W_2 \cong W_1 \boxtimes W_2$$ If $W_1$ and $W_2$ are spaces of polynomials, each in $m$ variables, then their tensor product $W_1 \boxtimes W_2$ is isomorphic to a larger space of polynomials in $2m$ variables. That is, a linear operator between polynomial spaces $W_1$ and $W_2$ can be associated to some polynomial in double the variables, via the above identification of representations. This is precisely the idea of the symbol of an operator.
Let’s see how this works in the univariate case. Consider ${\mathbb{C}}_n[x]$ as a representation of the group $SL_2({\mathbb{C}})$, as described above. It is then a standard result that the classical bilinear *apolarity form* is invariant under the action of M[ö]{}bius transformations. That is, the apolarity form is an $SL_2({\mathbb{C}})$-invariant bilinear form on ${\mathbb{C}}_n[x]$. This form, applied to $f,g \in {\mathbb{C}}_n[x]$ with coefficients $f_k,g_k$, is defined as follows: $$\langle f,g \rangle^n := \sum_{k=0}^n \binom{n}{k}^{-1} (-1)^k f_k g_{n-k}$$ With this, we obtain the identification described above: $\operatorname{Hom}({\mathbb{C}}_n[x],{\mathbb{C}}_m[x]) \cong {\mathbb{C}}_n[x] \boxtimes {\mathbb{C}}_m[x] \cong {\mathbb{C}}_{(n,m)}[x,z]$. The final piece of the puzzle is then to find a way to transfer stability information through this identification of representations. The key result to this end is the classical Grace’s theorem:
Let $f,g \in{\mathbb{C}}_n[x]$ be polynomials of degree exactly $n$. Further, let $C$ be some open or closed circular region such that $f$ is $C$-stable and $g$ is $({\mathbb{C}}\setminus C)$-stable. Then, $\langle f,g \rangle^n \neq 0$.
That is, the apolarity form not only provides the link between a linear operator and its symbol, but also captures stability information. So, whatever stability claims we can make about polynomials in ${\mathbb{C}}_{(n,m)}[x,z]$ can then be seamlessly transferred to corresponding linear operators in $\operatorname{Hom}({\mathbb{C}}_n[x],{\mathbb{C}}_m[x])$. From this we are able to recover the Borcea-Br[ä]{}nd[é]{}n characterization. Additionally, all of the theory here relating stability and the representation theory of $SL_2({\mathbb{C}})$ can be generalized to multivariate polynomials in a straightforward manner. The details will be discussed in §\[reps\_sect\].
In a similar fashion, other important maps also have $SL_2({\mathbb{C}})$-invariance properties (e.g., polarization and projection, as used in the Grace-Walsh-Szeg[ő]{} coincidence theorem, explicitly give the isomorphisms of a classical representation theoretic result; see Appendix \[gws\_app\]). A main feature of our conceptual thesis is that it allows for a unification of many seemingly related results in polynomial stability theory. A crucial point to make then is that Grace’s theorem is at the heart of this unification. That said, a significant portion of this paper is devoted to discussing it.
### A Generalized Grace’s Theorem and Interval- and Ray-Rootedness {#a-generalized-graces-theorem-and-interval--and-ray-rootedness .unnumbered}
In [@bb2], Borcea and Br[ä]{}nd[é]{}n are able to prove a multivariate Grace’s theorem using their operator characterization. In this paper we will prove the multivariate version from scratch, and then use it to derive a new characterization of stability-preserving linear operators. In addition, we generalize it to *projectively convex* regions, which consist of an open circular region with a portion of its boundary (see §\[proj\_conv\_subsect\]). We state our new result as follows. Note that this result can be seen as an extension of the generalized Grace’s theorem given in Corollary 4.4 of [@zaheer1976polar].
Fix $\lambda \in {\mathbb{N}}_0^m$ and $f,g \in {\mathbb{C}}_\lambda[x_1,...,x_m]$ such that $f$ and $g$ both have a nonzero term of degree $\lambda$. Also, denote $C := {\mathcal{H}_+}\cup \overline{{\mathbb{R}}_+}$ and $\widetilde{C} := {\mathcal{H}_-}\cup \overline{{\mathbb{R}}_-}$. If $f$ is $C^m$-stable and $g$ is $\widetilde{C}^m$-stable, then $\langle f,g \rangle^\lambda \neq 0$.
This result can, for instance, give stability information about positive- and negative-rooted polynomials. Since the apolarity form is invariant under the action of M[ö]{}bius transformations, we immediately obtain similar statements regarding the union of any open circular region and a portion of its boundary. Notice also that, unlike the classical Grace’s theorem, the stability regions $C$ and $\widetilde{C}$ have non-empty intersection.
In the vein of this extension, we provide a new characterization of a certain class of linear operators which preserve ray- and interval-rootedness. The problem of classifying all such operators is still open in general (see e.g., the end of [@bba]). Here, we solve this problem for a restricted class of operators: namely, operators which both preserve weak real-rootedness and also preserve ray- or interval-rootedness. Our main result in this direction is stated as follows, where a polynomial is called $J$-rooted when all of its roots are in $J$:
Fix a linear operator $T: {\mathbb{R}}_n[x] \to {\mathbb{R}}_m[x]$ which has image of dimension greater than two. Further, let $I,J \subseteq {\mathbb{R}}$ be intervals or rays. Up to certain degree restrictions, $T$ preserves weak real-rootedness and maps $I$-rooted polynomials to nonzero $J$-rooted polynomials if and only if $\operatorname{Symb}(T)$ is either $({\mathcal{H}_-}\cup I) \times (\overline{{\mathcal{H}_+}} \setminus J)$-stable or $({\mathcal{H}_-}\cup I) \times (\overline{{\mathcal{H}_-}} \setminus J)$-stable.
In §\[Jops\_subsect\], this result is stated in a more restricted manner as the degeneracy condition (image dimension) and degree restrictions end up being more tedious than in the other results. Corollary \[Jops\_graces\_cor\] and further explication then give the result as stated here.
As a final note, all of the results given here in the introduction are stated slightly differently in §\[graces\_sect\], §\[Cops\_sect\], §\[Rops\_sect\]. In particular, the notation $V(\lambda)$ is used in place of ${\mathbb{C}}_\lambda[x_1,...,x_m]$, and reference is made to ${\mathbb{CP}}^1$ (i.e., the Riemann sphere). This notation has to do with consideration of “roots at infinity”, which allows us to remove degree restrictions and avoid reference to convex circular regions. We discuss this rigorously in §\[homog\_subsect\].
### A Roadmap {#a-roadmap .unnumbered}
We now describe the content of the remainder of this paper. In §\[prelim\_sect\], we discuss the use of homogeneous polynomials via the notation $V(n)$ and $V(\lambda)$, and we describe the relation of these spaces to the notion of roots at infinity.
In §\[reps\_sect\], we explicate some very basic representation theory of $SL_2({\mathbb{C}})$. We then demonstrate how the apolarity form and the symbol arise as natural constructs in this context. Results like the Symbol Lemma and the $SL_2({\mathbb{C}})$-invariance of the apolarity form are stated here.
In §\[stability\_sect\], we discuss some classical and some new polynomial stability theory results, and their multivariate analogues, in the homogeneous context. We also extend Laguerre’s theorem to projectively convex regions (generalized circular regions), which later allows us to prove results regarding polynomials which have all their roots in a given interval.
In §\[graces\_sect\], we state and prove our generalized Grace’s theorem. We also discuss other stability regions to which the theorem applies, and consider symbols of linear operators given by evaluation at a particular point. We call these polynomials *evaluation symbols*, as they turn out to play a crucial role in the proofs of the operator characterizations.
In §\[Cops\_sect\], we finally state and prove our improved characterizations of stability-preserving operators. We also demonstrate how the Borcea-Br[ä]{}nd[é]{}n characterizations can be seen (with a bit of work) to be corollaries of our characterizations. We then provide examples of the use of our results. In particular, we show how stability results related to classical polynomial convolutions can be immediately recovered.
In §\[Rops\_sect\], we state and prove the analogous characterization of strong real stability-preserving operators. As with complex operators, we show how the Borcea-Br[ä]{}nd[é]{}n characterization can be obtained as a corollary. In this section, we also state and prove our characterization of operators which preserve both weak real-rootedness as well as interval- (or ray-) rootedness.
In Appendix \[gws\_app\], we discuss how polarization and the Grace-Walsh-Szeg[ő]{} coincidence theorem fit in to the framework presented in this paper. We also demonstrate that the classical isomorphism $V(n) \cong \operatorname{Sym}^n(V(1))$ (for representations of $SL_2({\mathbb{C}})$) can be realized as the polarization map and therefore has a stability-theoretic interpretation. While important to the conceptual thesis stated above, we place this discussion in an appendix as it is not utilized elsewhere.
Preliminaries {#prelim_sect}
=============
Here, we discuss basic notation and results related to polynomials and stability. In particular, we discuss in more detail the notation and consequences related to the use of homogeneous polynomials in place of usual univariate and mutlivariate polynomials. Then, we state a number of basic stability results in the language of homogeneous polynomials.
Notation
--------
Let $[n] := \{1,2,...,n\}$ and $(1^m) := (1,...,1) \in {\mathbb{N}}_0^m$. For $\lambda \equiv (\lambda_1,...,\lambda_m) \in {\mathbb{N}}_0^m$, we define: $${\mathbb{C}}_\lambda[x_1,...,x_m] := \left\{f \in {\mathbb{C}}[x_1,...,x_m] : \deg_{x_k}(f) \leq \lambda_k\right\}$$ That is, elements of ${\mathbb{C}}_\lambda[x_1,...,x_m]$ are of degree at most $\lambda_k$ in the variable $x_k$. In particular, we call polynomials in ${\mathbb{C}}_{(1^m)}[x_1,...,x_m]$ *multi-affine*. We will also use the shorthand ${\mathbb{C}}_n[x]$ to refer to univariate polynomials of degree at most $n$.
Now we define similar spaces of polynomials which are homogeneous in pairs of variables. These polynomials should be seen as per-variable homogenizations of polynomials of the spaces defined above. For $\lambda \equiv (\lambda_1,...,\lambda_m) \in {\mathbb{N}}_0^m$ and ${\mathbb{K}}= {\mathbb{C}}$ or ${\mathbb{K}}= {\mathbb{R}}$, we define: $$V_{\mathbb{K}}(\lambda) \equiv V_{\mathbb{K}}(\lambda_1,...,\lambda_m) := \left\{p \in {\mathbb{K}}[x_1,y_1...,x_m,y_m] : p \text{ is homogeneous of degree $\lambda_k$ in $x_k$, $y_k$}\right\}$$ We also use the shorthand $V(\lambda) \equiv V_{\mathbb{C}}(\lambda)$. As above, we call polynomials in $V(1^m)$ *multi-affine*. The notation used here is generalized from what is typically used to denote the irreducible representations of $SL_2({\mathbb{C}})$. As it turns out, spaces of homogeneous polynomials in two variables can be used to define these representations. This will be made more precise in §\[reps\_sect\].
There is also a natural ordering structure on ${\mathbb{N}}_0^m$, along with a few basic operations that will be used throughout. Fix $\lambda \equiv (\lambda_1,...,\lambda_m)$ and $\alpha \equiv (\alpha_1,...,\alpha_m)$ in ${\mathbb{N}}_0^m$, and fix $\beta \equiv (\beta_1,...,\beta_n)$ in ${\mathbb{N}}_0^l$. We say $\alpha \leq \lambda$ whenever $\alpha_k \leq \lambda_k$ for all $k \in [m]$. We define $\lambda + \alpha := (\lambda_1+\alpha_1,...,\lambda_m+\alpha_m)$, $|\lambda| := \lambda_1 + \cdots + \lambda_m$, and $\lambda \sqcup \beta := (\lambda_1,...,\lambda_m,\beta_1,...,\beta_l) \in {\mathbb{N}}_0^{m+l}$.
We also make use of a number of shorthands. Fix $\mu,\lambda \in {\mathbb{N}}_0^m$ such that $\mu \leq \lambda$. We define $x^\mu \in {\mathbb{C}}_\lambda[x_1,...,x_m]$ via $x^\mu := x_1^{\mu_1}x_2^{\mu_2} \cdots x_m^{\mu_m}$. Similarly, we define $\partial_x^\mu := \partial_{x_1}^{\mu_1} \partial_{x_2}^{\mu_2} \cdots \partial_{x_m}^{\mu_m}$. When considering $V(\lambda)$, we define $y^\mu$ and $\partial_y^\mu$ in the same way. Further, we define $\lambda! := \lambda_1! \cdots \lambda_m!$ and $\binom{\lambda}{\mu} := \frac{\lambda!}{\mu!(\lambda-\mu)!} \equiv \binom{\lambda_1}{\mu_1} \cdots \binom{\lambda_m}{\mu_m}$. Finally, we denote $(-1)^\mu := (-1)^{\mu_1} \cdots (-1)^{\mu_m} \equiv (-1)^{|\mu|}$.
Homogenization {#homog_subsect}
--------------
The usual degree-$n$ homogenization of a polynomial $f \in {\mathbb{C}}_n[x]$ is defined on monomials as follows and is extended linearly. $$\begin{split}
\operatorname{Hmg}_n : {\mathbb{C}}_n[x] &\rightarrow V(n) \\
x^k &\mapsto x^ky^{n-k}
\end{split}$$ More generally, for $\lambda \in {\mathbb{N}}_0^m$ the *degree-$\lambda$ homogenization* is defined on monomials as follows and is extended linearly. $$\begin{split}
\operatorname{Hmg}_\lambda: {\mathbb{C}}_\lambda[x_1,...,x_m] &\rightarrow V(\lambda)\\
x^{\mu} &\mapsto x^{\mu}y^{\lambda-\mu}
\end{split}$$ ${\mathbb{C}}_n[x]$ and $V(n)$ are isomorphic as vector spaces via $\operatorname{Hmg}_n$. The difference between these spaces of course is the fact that elements of $V(n)$ are polynomials in two variables, and hence have many more zeros than elements of ${\mathbb{C}}_n[x]$. By homogeneity though, those roots have strong structural properties which essentially mimic the root properties of elements of ${\mathbb{C}}_n[x]$.
What homogeneity gets us is a simplification of a number of issues related to the fact that polynomials in ${\mathbb{C}}_n[x]$ have *at most* $n$ zeros. Specifically, it is more natural to think of the missing zeros (when the number of zeros is less than $n$) as being “at infinity”. Certain results require premises restricting to convex regions or to polynomials of degree exactly $n$ (e.g., the classical Grace-Walsh-Szeg[ő]{} coincidence theorem), and such details vanish when considering homogeneous polynomials with possible roots at infinity.
We now explicitly demonstrate the root connection between ${\mathbb{C}}_n[x]$ and $V(n)$. Let $f \in {\mathbb{C}}_n[x]$ be a monic polynomial of degree $m \leq n$, and let $p := \operatorname{Hmg}_n(f) \in V(n)$ be the degree-$n$ homogenization of $f$. We write $f$ and $p$ in the following ways. (Here, $p_k = 0$ for $k > m$.) $$f(z) = \sum_{k=0}^n p_k x^k = \prod_{k=1}^m (x-\alpha_k)$$ $$p(x,y) = \sum_{k=0}^n p_k x^ky^{n-k} = y^{n-m} \prod_{k=1}^m (x-\alpha_ky)$$ Note also that any element of $V(n)$ can be written in this way (up to scalar). We can also write $p$ in another way: $$p(x,y) = \det\left(
\begin{bmatrix}
x & -1 \\
y & 0
\end{bmatrix}^{n-m}
\begin{bmatrix}
x & \alpha_1 \\
y & 1
\end{bmatrix}
\cdots
\begin{bmatrix}
x & \alpha_m \\
y & 1
\end{bmatrix}
\right)$$ By this expression, the zeros of $p$ are given by the one-dimensional subspaces spanned by the second columns of each of the above $2 \times 2$ matrices (i.e., any vector $(x,y)$ that makes one of the above matrices non-invertible). With this, we can consider the zeros of $p$ to be elements of ${\mathbb{CP}}^1$, the set of one-dimensional complex subspaces of ${\mathbb{C}}^2$. We write elements of ${\mathbb{CP}}^1$ as $(a:b)$, which denotes the subspace spanned by $(a,b)$. In fact, this implies ${\mathbb{CP}}^1 = \{(z:1) ~|~ z \in {\mathbb{C}}\} \cup \{(-1:0)\} \cong {\mathbb{C}}\cup \{\infty\}$ (the Riemann sphere), since each one-dimensional subspace of ${\mathbb{C}}^2$ is spanned by either $(-1,0)$ or an element of the form $(z,1)$.
For the polynomial $p$ given above, $(\alpha_1:1)$, ..., $(\alpha_m:1)$, and $(-1:0)$ (with multiplicity $n-m$) are its $n$ zeros in ${\mathbb{CP}}^1$, where $(-1:0)$ is the point at infinity in ${\mathbb{CP}}^1$. In this way, the fundamental theorem of algebra passes from $f$ to $p$ as follows: homogeneous polynomials in $V(n)$ have *exactly* $n$ zeros in ${\mathbb{CP}}^1$. This unifies notions like “polynomials of degree at most $n$” by allowing us to view all such polynomials as having the same number of zeros.
Of course, this discussion is not possible for polynomials in $V(\lambda)$, as multivariate polynomials do not factor into linear terms in general. Nonetheless, we will still think of polynomials in $V(\lambda)$ as having zeros in $({\mathbb{CP}}^1)^m$, where $\lambda \in {\mathbb{N}}_0^m$. In particular, given $p \in V(\lambda)$, we can write: $$p(x,y) \equiv p(x_1,y_1,x_2,y_2,...,x_m,y_m) = \sum_{\mu \leq \lambda} p_\mu x_1^{\mu_1} \cdots x_m^{\mu_m} y_1^{\lambda_1-\mu_1} \cdots y_m^{\lambda_m-\mu_m} \equiv \sum_{\mu \leq \lambda} p_{\mu} x^{\mu}y^{\lambda-\mu}$$ By homogeneity, scaling any pair of input variables will scale the output. So, for any zero $(z,w) \equiv (z_1,w_1,...,z_m,w_m)$ and any $a \equiv (a_1,...,a_m) \in {\mathbb{C}}^m$, we have that $(az,aw) \equiv (a_1z_1,a_1w_1,...,a_mz_m,a_mw_m)$ will also be a zero of $p$. This validates the interpretation of $p$ as having zeros in $({\mathbb{CP}}^1)^m$, as well as the notation that $(z:w) \equiv (z_1:w_1,...,z_m:w_m)$ is a zero of $p$. With this, we will use the following notation when writing and evaluating homogeneous polynomials: $$p(x:y) \equiv p(x_1:y_1,x_2:y_2,...,x_m:y_m) := \sum_{\mu \leq \lambda} p_{\mu} x^{\mu}y^{\lambda-\mu}$$ Note that this is a bit ambiguous, since $(x:y) = (ax:ay)$ as element of ${\mathbb{CP}}^1$ for any $a \neq 0$. This abuse of notation is mainly used to denote the zeros of $p$ as elements of ${\mathbb{CP}}^1$, and so the ambiguity will be largely irrelevant.
The notation $(a:b) \in {\mathbb{CP}}^1$ is standard, and is meant to give off the connotation of a ratio; that is, $(a:b)$ should feel like $a/b$. This connotation is reasonable given the following equality, where $p := \operatorname{Hmg}_\lambda(f)$ for a given polynomial $f \in {\mathbb{C}}_{\lambda}[x_1,...,x_m]$: $$p(x:y) = p(x_1:y_1,...,x_m:y_m) = \prod_k y_k^{\lambda_k} \cdot f(x_1/y_1,...,x_m/y_m) = y^\lambda \cdot f(x/y)$$
Finally, we give an important definition which will essentially replace the notion of a monic polynomial in the homogeneous world.
Given $p \in V(\lambda)$, we say that $p$ is *top-degree monic* if the coefficient of $x^\lambda$ in $p$ equals 1. In particular, if $p \in V(n)$ is top-degree monic, then $p$ has no roots at infinity.
Homogeneous Polynomials as Representations {#reps_sect}
==========================================
In this section, we will discuss some basic representation theory of $SL_2({\mathbb{C}})$ and show how the apolarity form and the notion of the symbol of an operator arise naturally in the representation theoretic context. Most of the representation theory we use in this section is very basic. There are a number of references which discuss the theory in full detail, albeit with different goals in mind. Typically this is done via the theory of Lie groups and algebras, as in [@fulton2013reptheory] and in [@humphreys2012lietheory].
As a note, most of the content of this section is less relevant to the analytic questions associated to polynomials. Rather, it serves as the foundational structure for a new approach to Grace’s theorem and results concerning stability-preserving operators. For this reason, we believe it worthwhile to explicate key aspects of this foundation and their connection to analytic results. Pushing further into this connection may lead to new results beyond the scope of this paper.
The Action of SL2(C) {#SL_action_subsect}
--------------------
Given $(\alpha:\beta) \in {\mathbb{CP}}^1$ and $\phi \in SL_2({\mathbb{C}})$, we define the usual action of $SL_2({\mathbb{C}})$ on ${\mathbb{CP}}^1$ via $\phi \cdot (\alpha:\beta) := \phi \binom{\alpha}{\beta}$. That is, $\phi$ acts by matrix multiplication on the vector $\binom{\alpha}{\beta}$. Equivalently, $\phi$ acts as its corresponding M[ö]{}bius transformation on ${\mathbb{CP}}^1$. Note that, as with M[ö]{}bius transformations, $SL_2({\mathbb{C}})$ acts transitively on circular regions in ${\mathbb{CP}}^1$.
This then induces an action on $V(n)$ by acting on the roots (in ${\mathbb{CP}}^1$) of polynomials in $V(n)$. Given $p \in V(n)$ and $\phi \in SL_2({\mathbb{C}})$, this action is defined via: $$(\phi \cdot p)(x:y) := p\left(\phi^{-1} \binom{x}{y}\right) \equiv (p \circ \phi^{-1})(x:y)$$ We can define a similar action of $(SL_2({\mathbb{C}}))^m$ on $V(\lambda)$, for $\lambda \in {\mathbb{N}}_0^m$. Specifically, given $p \in V(\lambda)$ and $(\phi_1,...,\phi_m) \in SL_2({\mathbb{C}})^m$, this action is defined via: $$((\phi_1,...,\phi_m) \cdot p)(x_1:y_1,...,x_m:y_m) := p\left(\phi_1^{-1} \binom{x_1}{y_1}, ..., \phi_m^{-1}\binom{x_m}{y_m}\right)$$ These actions turn $V(n)$ and $V(\lambda)$ into representations of $SL_2({\mathbb{C}})$ and $(SL_2({\mathbb{C}}))^m$, respectively. These are precisely the finite dimensional irreducible representations of $SL_2({\mathbb{C}})$ and $(SL_2({\mathbb{C}}))^m$, and so they are the basic building blocks of the $SL_2({\mathbb{C}})$ representation theory. Actions on $V(n)$ and $V(\lambda)$ can be extended to tensor products in the usual way, and in this paper we will make use of both inner and outer tensor products. We now briefly discuss tensor product actions for those less familiar.
The outer tensor product of $V(\lambda_k)$, denoted $V(\lambda_1) \boxtimes \cdots \boxtimes V(\lambda_m)$, is a representation of $(SL_m({\mathbb{C}}))^m$ with action by $(\phi_1,...,\phi_m)$ on simple tensors given as follows: $$(\phi_1, \cdots, \phi_m) \cdot (p_1 \boxtimes \cdots \boxtimes p_m) := (\phi_1 \cdot p_1) \boxtimes (\phi_2 \cdot p_2) \boxtimes \cdots \boxtimes (\phi_m \cdot p_m)$$ This implies that $V(\lambda)$ and $V(\lambda_1) \boxtimes \cdots \boxtimes V(\lambda_m)$ are isomorphic as representations, and this fact will be used when we define the symbol in §\[symb\_subsect\].
The inner tensor product of $V(\lambda_k)$, denoted $V(\lambda_1) \otimes \cdots \otimes V(\lambda_m)$, is a representation of $SL_2({\mathbb{C}})$ with action by $\phi$ on simple tensors given as follows: $$\phi \cdot (p_1 \otimes \cdots \otimes p_m) := (\phi \cdot p_1) \otimes (\phi \cdot p_2) \otimes \cdots \otimes (\phi \cdot p_m)$$ While $V(\lambda)$ and $V(\lambda_1) \otimes \cdots \otimes V(\lambda_m)$ are isomorphic as vector spaces, they are representations of different groups ($(SL_2({\mathbb{C}}))^m$ and $SL_2({\mathbb{C}})$ respectively). The inner tensor product relates to invariants of multiple polynomials with respect to a single $SL_2({\mathbb{C}})$ action. For instance, the apolarity form takes two distinct polynomials as input, and it is a classical result that this form is invariant with respect to a single action by M[ö]{}bius transformation. As it turns out, this form can be viewed as an $SL_2({\mathbb{C}})$-invaiant map on an inner tensor product of polynomial spaces. It will therefore be important for us to understand these inner tensor products in a little more detail.
An Important Invariant Map, and Apolarity {#inv_maps_subsect}
-----------------------------------------
To aide in our investigation of inner tensor products of $SL_2({\mathbb{C}})$ representations, we now define an important $SL_2({\mathbb{C}})$-invariant linear map, denoted by $D$. This map has a long history in invariant theory, and we touch on this below.
\[Dmap\_prop\] The linear map $D \equiv (\partial_x \otimes \partial_y - \partial_y \otimes \partial_x): V(n+1) \otimes V(m+1) \rightarrow V(n) \otimes V(m)$ is $SL_2({\mathbb{C}})$-invariant.
Fix $\phi \equiv \begin{bmatrix} a & b \\ c & d \end{bmatrix} \in SL_2({\mathbb{C}})$, $p \in V(n+1)$, and $q \in V(m+1)$. We compute: $$\begin{split}
(\phi^{-1} \circ D \circ \phi) (p \otimes q) &= (\phi^{-1} \circ D) (p(dx-by, -cx+ay) \otimes q(dx-by, -cx+ay)) \\
&= (d\partial_x-c\partial_y)p \otimes (-b\partial_x+a\partial_y)q - (-b\partial_x+a\partial_y)p \otimes (d\partial_x-c\partial_y)q \\
&= (ad-bc)(\partial_x p \otimes \partial_y q - \partial_y p \otimes \partial_x q) \\
&= D (p \otimes q)
\end{split}$$ That is, $D \circ \phi = \phi \circ D$.
The multiplication map $V(n) \otimes V(m) \xrightarrow{\times} V(n+m)$ is $SL_2({\mathbb{C}})$-invariant.
Trivial.
Powers of the $D$ map actually appear in the literature under a few different names. The first comes from invariant theory, where the application of the map $$V(n) \otimes V(m) \xrightarrow{D^r} V(n-r) \otimes V(m-r) \xrightarrow{\times} V(n+m-2r)$$ to polynomials $p \in V(n)$ and $q \in V(m)$ is called the *$r^\text{th}$ transvectant* of $p$ and $q$. This map is also the result of the $r^\text{th}$ iteration of *Cayley’s $\Omega$ process*. These notions are discussed, for example, in chapters 4 and 5 of [@olver1999invarianttheory], where they are used to explicitly compute invariants and covariants of forms. In particular, the invariance of the Jacobian ($1^\text{st}$ transvectant map applied to $p \otimes q$) and the Hessian ($2^\text{nd}$ transvectant map applied to $p \otimes p$) can be determined in this way.
Additionally, the $n^\text{th}$ transvectant of $p,q \in V(n)$ is used to define a notion of apolarity (see, e.g., [@rota1993apolarity] and [@brennan2007apolarity]), and this notion corresponds to the classical one used in Grace’s theorem. In fact, the original formulation of Grace’s theorem can be found in Grace and Young’s 1903 book, *The Algebra of Invariants* [@grace1903invariants]. This suggests a connection between invariant theory and the analytic consequences of apolarity theory via the $D$ map, and we will indeed see this map play a crucial role in the proof of Grace’s theorem (\[graces\_thm\]).
We are now ready to define the homogeneous apolarity form via the $D$ map. This form and its $SL_2({\mathbb{C}})$-invariance are then the next main step toward the definition of the symbol of an operator. In the next section, we will use this bilinear form to define an important construction called the dual of a representation. This will serve as the link to viewing spaces of linear operators as representations themselves.
\[Uform\_def\] We call the $n^\text{th}$ transvectant $$V(n) \otimes V(n) \xrightarrow{D^n} V(0) \otimes V(0) \xrightarrow{\times} V(0) \cong {\mathbb{C}}$$ the *apolarity form* of $V(n)$. This is the unique (up to scalar) nondegenerate $SL_2({\mathbb{C}})$-invariant bilinear form on $V(n)$, and therefore it is the homogenization of the classical apolarity form.
\[Mform\_def\] We call the map $$V(\lambda) \otimes V(\lambda) \xrightarrow{D^\lambda} V(0^m) \otimes V(0^m) \xrightarrow{\times} V(0^m) \cong {\mathbb{C}}$$ the *apolarity form* of $V(\lambda)$. This is the unique (up to scalar) nondegenerate $(SL_2({\mathbb{C}}))^m$-invariant bilinear form on $V(\lambda)$, and therefore it is the homogenization of the multivariate apolarity form defined by Borcea and Br[ä]{}nd[é]{}n in [@bb2].
Since $V(0) \otimes V(0) \xrightarrow{\times} V(0)$ and $V(0^m) \otimes V(0^m) \xrightarrow{\times} V(0^m)$ are isomorphisms, we will often refer to these maps as $D^n$ and $D^\lambda$, respectively. And as a final note, we do not justify here the claims of uniqueness and nondegeneracy stated above. Proving these claims involves decomposing $V(n) \otimes V(n)$ and $V(\lambda) \otimes V(\lambda)$ into their irreducible components, which is a standard representation theoretic computation.
The Symbol of an Operator {#symb_subsect}
-------------------------
Given representations $V(\lambda)$ and $V(\alpha)$ (for $\lambda \in {\mathbb{N}}_0^m$ and $\alpha \in {\mathbb{N}}_0^l$) of $(SL_2({\mathbb{C}}))^m$ and $(SL_2({\mathbb{C}}))^l$ respectively, the space of linear maps between $V(\lambda)$ and $V(\alpha)$ can be viewed as a representation of $(SL_2({\mathbb{C}}))^{m+l}$ in a standard way. This space of linear maps is denoted $\operatorname{Hom}(V(\lambda),V(\alpha))$. As discussed previously, we will now use the apolarity form defined above to construct a representation isomorphism between $\operatorname{Hom}(V(\lambda),V(\alpha))$ and $V(\lambda \sqcup \alpha)$ (which is a space of polynomials in $m+l$ variables). This will lead us to a natural definition for the symbol of an operator.
The significance of this isomorphism will come from the fact that stability results about $V(\lambda \sqcup \alpha)$ will transfer to $\operatorname{Hom}(V(\lambda),V(\alpha))$ via the Symbol Lemma (\[symb\_lemma\]) stated below. We will see in §\[str\_Cstab\_subsect\] that this lemma and Grace’s theorem almost immediately imply a characterization of stability-preserving operators which is similar to that of Borcea and Br[ä]{}nd[é]{}n.
To this end, consider the standard representation isomorphism $\operatorname{Hom}(V(\lambda),V(\alpha)) \cong V(\lambda)^* \boxtimes V(\alpha)$, given by $T \mapsto \sum_{\mu \leq \lambda} (x^\mu y^{\lambda-\mu})^* \boxtimes T(x^\mu y^{\lambda-\mu})$, where $V(\lambda)^*$ is the dual representation of $V(\lambda)$. We omit here the details regarding explicit definitions of the action of (products of) $SL_2({\mathbb{C}})$ on $\operatorname{Hom}$ and dual representations. Instead, we utilize the fact that the apolarity form provides an $(SL_2({\mathbb{C}}))^m$-invariant isomorphism between $V(\lambda)$ and the dual representation $V(\lambda)^*$, as stated in the following result.
For any $\lambda \in {\mathbb{N}}_0^m$, there is an $(SL_2({\mathbb{C}}))^m$-invariant isomorphism $V(\lambda)^* \rightarrow V(\lambda)$ given by $(x^\mu y^{\lambda-\mu})^* \mapsto \binom{\lambda}{\mu}(-1)^{\mu} x^{\lambda-\mu} y^\mu$.
We use the apolarity form to determine the isomorphism. In particular, up to scalar $(x^\mu y^{\lambda-\mu})^*$ maps to an element $p \in V(\lambda)$ such that $(x^\mu y^{\lambda-\mu})^* \equiv D^\lambda(p \otimes \cdot)$. We compute: $$D^\lambda(p \otimes x^\alpha y^{\lambda-\alpha}) = \alpha!(\lambda-\alpha)! \binom{\lambda}{\alpha} (-1)^{\alpha} \partial_x^{\lambda-\alpha} \partial_y^\alpha p = \lambda! (-1)^{\alpha} \partial_x^{\lambda-\alpha} \partial_y^\alpha p$$ Picking $p(x,y) := (\lambda!)^{-2}\binom{\lambda}{\mu}(-1)^{\mu} x^{\lambda-\mu} y^\mu$ achieves the desired equality exactly, and therefore $(x^\mu y^{\lambda-\mu})^* \mapsto \binom{\lambda}{\mu}(-1)^{\mu} x^{\lambda-\mu} y^\mu$ is an $(SL_2({\mathbb{C}}))^m$-invariant isomorphism.
With this, we consider the following string of $(SL_2({\mathbb{C}}))^{m+l}$-invariant isomorphisms: $$\operatorname{Hom}(V(\lambda),V(\alpha)) \rightarrow V(\lambda)^* \boxtimes V(\alpha) \rightarrow V(\lambda) \boxtimes V(\alpha) \rightarrow V(\lambda \sqcup \alpha)$$ The first map is the standard isomorphism discussed above, the second map is induced by the previous proposition, and the third map is given by the discussion of outer tensor products in §\[SL\_action\_subsect\]. This string of maps is explicitly defined on a given linear operator via: $$\begin{split}
T &\mapsto \sum_{\mu \leq \lambda} (z^\mu w^{\lambda-\mu})^* \boxtimes T(x^\mu y^{\lambda-\mu}) \\
&\mapsto \sum_{\mu \leq \lambda} \binom{\lambda}{\mu} (-1)^{\mu} z^{\lambda-\mu} w^{\mu} \boxtimes T(x^\mu y^{\lambda-\mu}) \\
&\mapsto \sum_{\mu \leq \lambda} \binom{\lambda}{\mu} z^{\lambda-\mu} (-w)^{\mu} \cdot T(x^\mu y^{\lambda-\mu})
\end{split}$$ Here, $T$ acts only on the $x$ and $y$ variables, and $z$ and $w$ are the $\lambda$ variables in $V(\lambda \sqcup \alpha)$. This gives the desired isomorphism between $\operatorname{Hom}(V(\lambda),V(\alpha))$ and $V(\lambda \sqcup \alpha)$, and hence we refer to this map as the $\operatorname{Symb}$ map.
\[symb\_def\] For $\lambda \in {\mathbb{N}}_0^m$ and $\alpha \in {\mathbb{N}}_0^l$, we define the following $(SL_2({\mathbb{C}}))^{m+l}$-invariant isomorphism: $$\operatorname{Symb}: \operatorname{Hom}(V(\lambda),V(\alpha)) \rightarrow V(\lambda \sqcup \alpha)$$ $$T \mapsto T\left[(zy-xw)^\lambda\right] \equiv \operatorname{Hmg}_{(\lambda,\alpha)}\left(T[(z-x)^\lambda]\right) \equiv \sum_{\mu \leq \lambda} \binom{\lambda}{\mu} z^{\lambda-\mu} (-w)^{\mu} \cdot T(x^\mu y^{\lambda-\mu})$$ We call $\operatorname{Symb}(T)$ the *(universal) symbol of $T$*.
This expression bears striking resemblance to the symbol used by Borcea and Br[ä]{}nd[é]{}n in [@bb1], which motivates the use of the name “symbol” here. (In fact, $\operatorname{Symb}$ is almost the homogenization of the Borcea-Br[ä]{}nd[é]{}n symbol.) In §\[Cops\_sect\], $\operatorname{Symb}$ will allow us to reduce the study of $\operatorname{Hom}(V(\lambda),V(\alpha))$ to the study of $V(\lambda \sqcup \alpha)$ via the next lemma. We refer to this next result as the Symbol Lemma, and it demonstrates the fundamental connection between an operator $T$, its symbol, and the apolarity form. Note that the computation done here in the proof of this lemma is in a sense redundant. The operator $\operatorname{Symb}$ was essentially defined such that $\operatorname{Symb}(T)$ acts as $T$ via $D^\lambda$.
\[symb\_lemma\] Fix $\lambda \in {\mathbb{N}}_0^m$, $\alpha \in {\mathbb{N}}_0^l$, and $T \in \operatorname{Hom}(V(\lambda),V(\alpha))$. For $q \in V(\lambda)$ and $r \in V(\alpha)$, we have: $$D^\lambda(\operatorname{Symb}(T) \otimes q \cdot r) = (\lambda!)^2 T(q) \otimes r$$
Letting $q_\mu$ be the coefficient of the $x^\mu y^{\lambda-\mu}$ term of $q$, we compute: $$\begin{split}
D^\lambda(\operatorname{Symb}(T) \otimes q \cdot r) &= D^\lambda \left(\sum_{\mu \leq \lambda} \binom{\lambda}{\mu} x^{\lambda-\mu} (-y)^{\mu} \cdot T(x^\mu y^{\lambda-\mu}) \otimes q \cdot r\right) \\
&= \sum_{\mu \leq \lambda} \binom{\lambda}{\mu}^2 (\lambda-\mu)! \mu! \cdot T(x^\mu y^{\lambda-\mu}) \otimes (\partial_x^\mu \partial_y^{\lambda-\mu} q) r \\
&= (\lambda!)^2 \sum_{\mu \leq \lambda} T(x^\mu y^{\lambda-\mu}) \otimes q_\mu \cdot r \\
&= (\lambda!)^2 T(q) \otimes r \\
\end{split}$$
Polynomial Stability Theory {#stability_sect}
===========================
Given $\lambda \in {\mathbb{N}}_0^m$, a polynomial $p(x_1:y_1,...,x_m:y_m) \in V(\lambda)$ is said to be *stable* if it doesn’t vanish in ${\mathcal{H}_+}^m \subset ({\mathbb{CP}}^1)^m$. More generally, $p$ is said to be *$\Omega$-stable* if it doesn’t vanish in $\Omega$. As above, we say $p$ is *weakly $\Omega$-stable* if possibly $p \equiv 0$. Most all results related to zero location of polynomials then can be translated into statements about stability properties of polynomials and stability preservation properties of operations applied to polynomials.
A linear operator $T$ is said to *preserve weak $\Omega$-stability* if $T(p)$ is $\Omega$-stable or identically zero for all $\Omega$-stable $p$. Further, a real linear operator $T$ *preserves weak real stability* if the same holds for real stable polynomials. In [@bb1], Borcea and Br[ä]{}nd[é]{}n were concerned with classifying such weak stability preserving operators. As seen in their main characterization results (Theorems \[BB\_Cops\_thm\] and \[BB\_Rops\_thm\]), allowing the zero polynomial leads to a degeneracy condition in their characterization.
In order to remove this condition, we define a slightly different notion of stability: we say a linear operator $T$ *preserves (strong) $\Omega$-stability* if $T(p)$ is stable and nonzero for all stable $p$. Similarly, we say a real linear operator $T$ *preserves (strong) real stability* if the same holds for real stable polynomials. Most of the main results of this paper rely on this notion of strong stability preservation, and we will demonstrate how it relates to weak stability preservation in §\[Cops\_deriv\_subsect\] and §\[Rops\_deriv\_subsect\].
Polar Derivatives
-----------------
A crucial tool of classical stability theory is the polar derivative. In particular, this notion leads to Laguerre’s theorem (Proposition \[laguerre\_prop\]), which is the main lemma toward Grace’s theorem. By passing to homogeneous polynomials the polar derivative becomes conceptually simpler, and this in turn sheds further light on the general connection to $SL_2({\mathbb{C}})$-invariance and the $D$ map. One example of this, as we will see below, is that the polar derivative can be defined as the conjugation of $\partial_x$ by some $SL_2({\mathbb{C}})$ action.
Given some “pole” $x_0 \in {\mathbb{C}}$, the *polar derivative* with respect to $x_0$ of $f \in {\mathbb{C}}_n[x]$ is classically defined as follows. $$(d_{x_0}f)(x) := nf(x) - (x-x_0)f'(x)$$ Noticing that the term of degree $n$ cancels out, the resulting polynomial is of degree $n-1$. It is typically said that this operator generalizes the ordinary derivative in the sense that $\lim_{x_0 \rightarrow \infty} x_0^{-1}d_{x_0}f(x) = f'(x)$. However, this operator also generalizes the ordinary derivative in more natural way, which we see by passing to $V(n)$.
Fix any $\phi \equiv \begin{bmatrix} a & b \\ c & d \end{bmatrix} \in SL_2({\mathbb{C}})$. Define the *pole* of $\phi$ to be $(-d:c) \in {\mathbb{CP}}^1$. For $p \in V(n)$, we then define the polar derivative with respect to $\phi$ as follows. $$d_\phi p := (\phi^{-1}\partial_x\phi)p = -(-d\partial_x + c\partial_y)p$$ Notice that $d_\phi$ depends only on $\phi^{-1}\binom{-1}{0} = \binom{-d}{c}$. With this, the pole of $\phi$ should be interpreted as the element of ${\mathbb{CP}}^1$ that $\phi$ sends to $\infty = (-1:0)$.
This definition of the polar derivative with respect to $\phi$ is at very least a natural one, as it can be simply described as the conjugation of $\partial_x$ by the action of $\phi$. The following result then shows that this is actually the correct definition.
Fix $\phi \equiv \begin{bmatrix} a & b \\ c & d \end{bmatrix} \in SL_2({\mathbb{C}})$ with pole $(-d:c)$, and define $x_0 := \frac{-d}{c}$ (for $c \neq 0$). Then: $$d_\phi \circ \operatorname{Hmg}_n = \operatorname{Hmg}_{n-1} \circ (-c \cdot d_{x_0})$$ That is, the polar derivative $d_\phi$ on $V(n)$ is the homogenization of the classical polar derivative $d_{x_0}$ on ${\mathbb{C}}_n[x]$ (up to scalar).
Straightforward computation.
As mentioned above, $d_\phi$ depends only on $(-d:c)$, the pole of $\phi$. So given any pole in ${\mathbb{CP}}^1$, we can actually choose $\phi \in SL_2({\mathbb{C}})$ to be a rotation of the Riemann sphere (i.e., ${\mathbb{CP}}^1$). This then gives the following intuitive description of the polar derivative.
Fix a rotation $\phi \in SL_2({\mathbb{C}})$ with pole $(-d:c)$. The polar derivative $d_\phi$ then acts on $p \in V(n)$ in the following way. First, consider the zeros of $p$ as being placed in the Riemann sphere via stereographic projection. Next, rotate the sphere via $\phi$, which moves $(-d:c)$ to infinity at the top of the sphere. Apply the derivative to the new polynomial given by the new locations of the zeros. Finally, undo the original rotation via $\phi^{-1}$, which moves infinity back to the pole $(-d:c)$.
Projective Convexity and Laguerre’s Theorem {#proj_conv_subsect}
-------------------------------------------
Circular regions play a key role in Grace’s theorem and its corollaries. The main reason for this is Laguerre’s theorem, which essentially says that polar derivatives with respect to points of a circular region preserve stability for that circular region. This theorem in turn relies on the Gauss-Lucas theorem, which deals with convex regions.
The generalization of circular regions to ${\mathbb{CP}}^1$ is the obvious one. A *circular region* in ${\mathbb{CP}}^1$ is defined to be the sets in ${\mathbb{CP}}^1$ for which the stereographic projection is a circular region in ${\mathbb{C}}$. We now state a lemma to Laguerre’s theorem, which gets at the heart of the importance of circular regions.
Let $C \subseteq {\mathbb{CP}}^1$ be a circular region, and let $\phi \in SL_2({\mathbb{C}})$ be such that its pole is not in $C$. Then, the stereographic projection of $\phi \cdot C$ is convex.
Let $(x_0:y_0) \notin C$ be the pole of $\phi$. Then, $\phi$ maps $(x_0:y_0)$ to $\infty \in {\mathbb{CP}}^1$ and maps $C$ to another circular region. Since $(x_0:y_0) \notin C$ implies $\infty \notin \phi \cdot C$, the sterographic projection of $\phi \cdot C$ is either an open half-plane or is bounded away from $\infty$. Since $\phi \cdot C$ is a circular region, it must be convex.
This then leads to a natural extension of the notion of a circular region.
Given $C \subseteq {\mathbb{CP}}^1$, we say that $C$ is *projectively convex* if for every $\phi \in SL_2({\mathbb{C}})$ with pole not in $C$, the stereographic projection of $\phi \cdot C$ is convex.
We now classify all projectively convex sets in ${\mathbb{CP}}^1$ in the following. This result has been demonstrated before in [@zervos1960aspects], where projectively convex regions are referred to as generalized circular regions.
Let $C \subseteq {\mathbb{CP}}^1$ be projectively convex. Then, $C = C^\circ \cup \gamma$, where $C^\circ$ is an open circular region which is the interior of $C$, and $\gamma$ is a connected subset of the boundary of $C^\circ$. In particular, projective convexity is preserved under taking complements.
So, one example of a projectively convex set which is not quite a circular region is ${\mathcal{H}_+}\cup {\mathbb{R}}_+$. Another is ${\mathcal{H}_+}\cup [0,1]$. Yet another (albeit after a bit of consideration) is ${\mathcal{H}_+}\cup (-\infty,0) \cup (1,\infty]$. We now state a homogeneous version of Laguerre’s theorem, extended to projectively convex sets.
\[laguerre\_prop\] Let $C \subseteq {\mathbb{CP}}^1$ be projectively convex, and fix $\phi \in SL_2({\mathbb{C}})$. If the pole of $\phi$ is in $C$, then $d_\phi$ preserves strong $C$-stability.
Gauss-Lucas and the fact that ${\mathbb{CP}}^1 \setminus C$ is projectively convex give the result. Specifically, for $C$-stable $p \in V(n)$ consider $\phi \cdot p$, which is stable in $\phi \cdot C \ni \infty$. Letting $B$ be the complement of $C$, the dehomogenization of this polynomial is then of degree exactly $n$ with all of its roots in the stereographic projection of $\phi \cdot B$. By projective convexity, $\phi \cdot B$ is convex and therefore Gauss-Lucas implies $\partial_x(\phi \cdot p)$ is $\phi \cdot C$-stable and not identically zero. Applying $\phi^{-1}$ then implies $d_\phi p = (\phi^{-1} \partial_x \phi) p$ is $C$-stable.
\[laguerre\_cor\] Let $C_k \subseteq {\mathbb{CP}}^1$ be projectively convex regions for $k \in [m]$, and fix $\phi \in SL_2({\mathbb{C}})$. If the pole of $\phi$ is in $C_{k_0}$, then $d_\phi$ acting on the variables $(x_{k_0}:y_{k_0})$ preserves strong $(C_1 \times \cdots \times C_m)$-stability.
Follows from the fact that taking derivatives in some variables commutes with evaluation in the others. Specifically, $p \in V(\lambda)$ is $(C_1 \times \cdots \times C_m)$-stable iff $p \neq 0$ for all evaluations in $C_1 \times \cdots \times C_m$. So, evaluating $p$ in all variables in that product of sets except $(x_{k_0}:y_{k_0})$ gives us a $C_{k_0}$-stable polynomial in $V(\lambda_{k_0})$. Applying the previous proposition then gives the result.
Real Stable Polynomials {#stab_subsect}
-----------------------
We now give a number of classical real stability results, along with a few results from [@bb1] and [@bb2]. Additionally, we state these results for homogeneous polynomials in $V_{\mathbb{R}}(\lambda)$, taking roots at infinity into account. The results of this section will come in to play mainly in §\[Rops\_sect\], where we discuss real linear operators and operators preserving interval- and ray-rootedness.
The first result we will need for our considerations of $V_{\mathbb{R}}(\lambda)$ is a version of the Hermite-Biehler theorem, often called the Hermite-Kakeya-Obreschkoff theorem. We state here without proof the multivariate version essentially used in Theorem 1.9 of [@bb1] (see also §2.4 of [@wagner2011polys]). First we need a definition.
\[pp\_def\] We say that $p,q \in V_{\mathbb{R}}(\lambda)$ are in *proper position*, denoted $p \ll q$, if $q + ip$ is weakly ${\mathcal{H}_+}^m$-stable (equivalently, if $p + iq$ is weakly ${\mathcal{H}_-}^m$-stable).
For $p,q \in V_{\mathbb{R}}(\lambda)$, $ap + bq$ is weakly real stable for all $a,b \in {\mathbb{R}}$ if and only if either $p \ll q$ or $q \ll p$.
This result will be crucial to our consideration of real polynomials and real stability (as it was in [@bb1]). Its main use for us in this direction is made explicit in the following.
\[rs\_lemma\] Fix $\lambda \in {\mathbb{N}}_0^m$, $\alpha \in {\mathbb{N}}_0^l$, and a linear operator $T: V(\lambda) \to V(\alpha)$ which restricts to a real linear operator from $V_{\mathbb{R}}(\lambda)$ to $V_{\mathbb{R}}(\alpha)$. If $T$ preserves weak real stability and $p \in V(\lambda)$ is stable, then $T(p)$ is either ${\mathcal{H}_+}^l$-stable, ${\mathcal{H}_-}^l$-stable, or identically zero.
By the Hermite-Biehler theorem, there exist $q,r \in V_{\mathbb{R}}(\lambda)$ such that $p = q + ir$ and $a q + b r$ is real stable or zero for all $a,b \in {\mathbb{R}}$. So, $a T(q) + b T(r)$ is real stable or zero for all $a,b \in {\mathbb{R}}$. By Hermite-Biehler again, $T(p) = T(q) + iT(r)$ is either ${\mathcal{H}_+}^l$-stable, ${\mathcal{H}_-}^l$-stable, or identically zero.
The next two results are from [@bb1], the first of which gives an equivalent characterization for a polynomial to be a scalar multiple of a real stable polynomial. This result will be specifically used in §\[Rops\_sect\] to generalize complex operator theoretic stability results to the real stability case.
\[scal\_rs\_lemma\] Let $p \in V_{\mathbb{C}}(\lambda)$ be both ${\mathcal{H}_+}^m$-stable and ${\mathcal{H}_-}^m$-stable. Then, $p$ is a (complex) scalar multiple of a real stable polynomial. In particular, if nonzero $q,r \in V_{\mathbb{R}}(\lambda)$ are such that $q \ll r$ and $r \ll q$, then $r$ is a (real) scalar multiple of $q$.
The next result provides the degeneracy cases in the Borcea-Br[ä]{}nd[é]{}n characterizations (recall the dimension restrictions of Theorems \[BB\_Cops\_thm\] and \[BB\_Rops\_thm\]). We will use this result to explicate the link between our operator characterization and the Borcea-Br[ä]{}nd[é]{}n characterization (see Lemmas \[bb\_Clink\] and \[bb\_Rlink\]).
\[dim\_lemma\] Let $W \subseteq V_{\mathbb{K}}(\lambda)$ be a ${\mathbb{K}}$-vector subspace (for ${\mathbb{K}}= {\mathbb{C}}$ or ${\mathbb{K}}= {\mathbb{R}}$) consisting only of weakly stable (resp. weakly real stable) polynomials. We have:
1. If ${\mathbb{K}}= {\mathbb{C}}$, then $\dim(W) \leq 1$.
2. If ${\mathbb{K}}= {\mathbb{R}}$, then $\dim(W) \leq 2$.
By applying appropriate M[ö]{}bius transformations, note that $(a)$ of the above lemma can be generalized to $(C_1 \times \cdots \times C_m)$-stable polynomials for any open circular regions $C_1,...,C_m \subseteq {\mathbb{CP}}^1$.
We now state the last result of this section, which refines the Hermite-Biehler theorem for top-degree monic polynomials in $V_{\mathbb{R}}(n)$. This refinement comes through the notion of *interlacing polynomials* and is much closer to the original statement of the classical Hermite-Biehler theorem (e.g., see Theorem 6.3.4 in [@rahman2002anthpoly]).
\[total\_order\_lemma\] For top-degree monic $p,q \in V_{\mathbb{R}}(n)$, $p \ll q$ if and only if the roots of $p$ and $q$ (denoted in increasing order by $(\alpha_k:1)$ and $(\beta_k:1)$, respectively) interlace on the real line in the following way: $$\alpha_1 \leq \beta_1 \leq \alpha_2 \leq \beta_2 \leq \cdots \leq \alpha_n \leq \beta_n$$ Further, if these equivalent conditions hold, then $\ll$ gives a total order on the top-degree monic elements of the span of $p$ and $q$ in $V_{\mathbb{R}}(n)$. This order is equivalently defined via the order of the $k^\text{th}$ largest roots, for any $k \in [n]$ such that $\alpha_k \neq \beta_k$.
The fact that $p \ll q$ is equivalent to interlacing roots is the classical univariate Hermite-Biehler theorem. That $q$ has larger roots than $p$ can be obtained by the fact that the $(n-1)^\text{st}$ derivative of $q + ip$ must be ${\mathcal{H}_+}$-stable. Since both polynomials are top-degree monic, this $(n-1)^\text{st}$ derivative will be a complex linear combination of two linear terms. A simple computation determines how the roots of each of the terms compare.
As for the total ordering property, let $r$ and $s$ be two polynomials in the real span of $p$ and $q$. Any real linear combination of these polynomials is then a real linear combination of $p$ and $q$ (and hence is real-rooted), and Hermite-Biehler implies either $r \ll s$ or $s \ll r$. By the above interlacing condition, it is straightforward to see that this total order is given by looking at the order of the $k^\text{th}$ roots, for any $k \in [n]$.
Grace’s Theorem {#graces_sect}
===============
We now prove the multivariate homogeneous Grace’s theorem for some specific projectively convex regions and then derive a few important corollaries. These corollaries will be almost immediate once Grace’s theorem has been proven, and yet will quickly yield stronger results regarding linear operators in the next section.
In the usual proof of the classical univariate Grace’s theorem, reference to linear factors of $f \in {\mathbb{C}}_n[x]$ is necessary. This makes generalization to ${\mathbb{C}}_\lambda[x_1,...,x_m]$ difficult, as multivariate polynomials do not necessarily have any linear factors. In our new proof, we are able avoid reference to linear terms by using particular features of the $D$ map. This means that our proof method works for any $\lambda$.
\[graces\_thm\] Fix $\lambda \in {\mathbb{N}}_0^m$ and $p,q \in V(\lambda)$. Also, denote $C := {\mathcal{H}_+}\cup \overline{{\mathbb{R}}_+}$ and $\widetilde{C} := {\mathcal{H}_-}\cup \overline{{\mathbb{R}}_-}$, where the closures are considered to be in ${\mathbb{CP}}^1$. If $p$ is $C^m$-stable and $q$ is $\widetilde{C}^m$-stable, then $D^\lambda(p \otimes q) \neq 0$.
We prove the theorem by induction on degree. For $\lambda \equiv 0$, the result is obvious. For $|\lambda| \geq 1$, we can assume WLOG that $\lambda_1 \geq 1$ by permuting the variables. Define $\delta_1 := (1,0,0,...,0) \in {\mathbb{N}}_0^m$.
Since $C$ and $\widetilde{C}$ are projectively convex, Corollary \[laguerre\_cor\] implies $(a\partial_{x_1} + b\partial_{y_1})p$ is $C^m$-stable for all $(a:b) \in C$ and $(c\partial_{x_1} + d\partial_{y_1})q$ is $\widetilde{C}^m$-stable for all $(c:d) \in \widetilde{C}$. To obtain a contradiction, we assume $D^\lambda(p \otimes q) = 0$. This gives: $$\begin{split}
D^{\lambda-\delta_1}((\alpha \partial_{x_1} + \partial_{y_1})p \otimes (\alpha \partial_{x_1} - \partial_{y_1})q) &= \alpha^2 D^{\lambda-\delta_1}(\partial_{x_1} p \otimes \partial_{x_1} q) - D^{\lambda-\delta_1}(\partial_{y_1} p \otimes \partial_{y_1} q) - \alpha D^\lambda(p \otimes q) \\
&= \alpha^2 D^{\lambda-\delta_1}(\partial_{x_1} p \otimes \partial_{x_1} q) - D^{\lambda-\delta_1}(\partial_{y_1} p \otimes \partial_{y_1} q)
\end{split}$$ By induction and the stability properties discussed above, we have $D^{\lambda-\delta_1}(\partial_{x_1} p \otimes \partial_{x_1} q) \neq 0$, $D^{\lambda-\delta_1}(\partial_{y_1} p \otimes \partial_{y_1} q) \neq 0$, and $D^{\lambda-\delta_1}((\alpha \partial_{x_1} + \partial_{y_1})p \otimes (\alpha \partial_{x_1} - \partial_{y_1})q) \neq 0$ for $\alpha \in {\mathcal{H}_+}\cup {\mathbb{R}}_+ \subset C$ (equivalently, $-\alpha \in {\mathcal{H}_-}\cup {\mathbb{R}}_- \subset \widetilde{C}$). This implies: $$\alpha^2 D^{\lambda-\delta_1}(\partial_{x_1} p \otimes \partial_{x_1} q) - D^{\lambda-\delta_1}(\partial_{y_1} p \otimes \partial_{y_1} q) \neq 0 \Longrightarrow \alpha^2 \neq \frac{D^{\lambda-\delta_1}(\partial_{y_1} p \otimes \partial_{y_1} q)}{D^{\lambda-\delta_1}(\partial_{x_1} p \otimes \partial_{x_1} q)} \in {\mathbb{C}}\setminus \{0\}$$ However, we can pick $\alpha \in {\mathcal{H}_+}\cup {\mathbb{R}}_+$ such that $\alpha^2$ is any value of ${\mathbb{C}}\setminus \{0\}$ we want, including that of $\frac{D^{\lambda-\delta_1}(\partial_{y_1} p \otimes \partial_{y_1} q)}{D^{\lambda-\delta_1}(\partial_{x_1} p \otimes \partial_{x_1} q)}$. This contradiction gives the result.
Other Regions
-------------
We now generalize the above theorem to other regions via $SL_2({\mathbb{C}})$ action and topological considerations. Theorem \[grace\_pairs\_thm\] can then be considered our most general form of Grace’s theorem. First though, we define two new notions in order to simplify the rest of this section.
Fix $m \in {\mathbb{N}}_0$ and any sets $S_1,S_2 \subseteq ({\mathbb{CP}}^1)^m$. We call $(S_1,S_2)$ a *Grace pair* if: for all $\lambda \in {\mathbb{N}}_0^m$ and $p,q \in V(\lambda)$ such that $p$ is $S_1$-stable and $q$ is $S_2$-stable, we have that $D^\lambda(p \otimes q) \neq 0$. That is, if Grace’s theorem holds for $S_1$ and $S_2$.
We say that a Grace pair is *disjoint* if it is of the form $(C_1 \times \cdots \times C_m, B_1 \times \cdots \times B_m)$ and $C_k$ and $B_k$ are disjoint for all $k \in [m]$.
This yields the following restatement of the above theorem.
For any $m \in {\mathbb{N}}_0$, $(({\mathcal{H}_+}\cup \overline{{\mathbb{R}}_+})^m, ({\mathcal{H}_-}\cup \overline{{\mathbb{R}}_-})^m)$ is a Grace pair.
The sets considered above intersect at 2 points ($0$ and $\infty$), and this ends up being crucial to the proof. So, in order to extend to the full generality of Grace’s theorem, we will need to find such points even when the stability sets of two polynomials $p$ and $q$ do not a priori intersect at all. To this end, we give the following lemmas.
Fix $\lambda \in {\mathbb{N}}_0^m$ and any closed circular regions $C_1,...,C_m \subset {\mathbb{CP}}^1$, and let $p \in V(\lambda)$ be $(C_1 \times \cdots \times C_m)$-stable. There exist open circular regions $U_1,...,U_m$ such that $C_k \subset U_k$ for all $k \in [m]$ and $p$ is $(U_1 \times \cdots \times U_m)$-stable.
Follows from compactness of ${\mathbb{CP}}^1$ and closedness of the set of roots of $p$.
Fix $n \in {\mathbb{N}}_0$ and any projectively convex $C \equiv C^\circ \cup \gamma \subset {\mathbb{CP}}^1$, and let $p \in V(n)$ be $C$-stable. With respect to the subspace topology of the boundary of $C^\circ$, there is an open set $\Gamma$ such that $\gamma \subseteq \Gamma$ and $p$ is $(C^\circ \cup \Gamma)$-stable.
Elements of $V(n)$ have finitely many roots in ${\mathbb{CP}}^1$.
Using these lemmas and the $SL_2({\mathbb{C}})$-invariance of the apolarity form, we obtain the following generalization of Grace’s theorem. Here, $(ii)$ and $(iii)$ give the multivariate Grace’s theorem proven in [@bb2].
\[grace\_pairs\_thm\] For $m \in {\mathbb{N}}_0$ and $C_1,...,C_m,B_1,...,B_m \subseteq {\mathbb{CP}}^1$, we have that $(C_1 \times \cdots \times C_m, B_1 \times \cdots \times B_m)$ is a Grace pair for the following regions.
1. For all $k \in [m]$, $C_k$ and $B_k$ are projectively convex, $C_k \cup B_k = {\mathbb{CP}}^1$, and $C_k \cap B_k$ is exactly two points.
2. For all $k \in [m]$, $C_k$ is a closed circular region, $B_k$ is an open circular region, and $C_k \cup B_k = {\mathbb{CP}}^1$.
3. For all $k \in [m]$, $C_k$ is an open circular region, $B_k$ is a closed circular region, and $C_k \cup B_k = {\mathbb{CP}}^1$.
4. For $m = 1$, $C_1$ and $B_1$ are projectively convex and $C_1 \cup B_1 = {\mathbb{CP}}^1$.
$(i)$. The previous theorem and $SL_2({\mathbb{C}})$-invariance.
$(ii),(iii)$. If $p$ is $(C_1 \times \cdots \times C_m)$-stable and $q$ is $(B_1 \times \cdots \times B_m)$-stable, then the first lemma above gives an annulus of simultaneous stability in each coordinate. So, we can find regions as in $(i)$ for which $p$ and $q$ are stable. The result then follows from $(i)$.
$(iv)$. If $B_1$ and $C_1$ are circular regions, then the result follows from $(ii)$. Otherwise, let $p$ be $C_1$-stable and $q$ be $B_1$-stable. The second lemma above gives at least two points of simultaneous stability, each on the boundary of $C_1$ and $B_1$. The result then follows from $(i)$.
Notice that $(ii)$ and $(iii)$ in this result do not allow for mixed open and closed stability regions. That is, all of the $C_k$ must be open and all of the $B_k$ closed, or vice versa. We show that this particular point cannot be ignored, using the following example.
Let $\lambda = (1,1,1)$, denote $E := {\mathbb{CP}}^1 \setminus \overline{{\mathbb{D}}}$, and consider the polynomial $p := x_1x_2x_3 - y_1y_2y_3 = \operatorname{Hmg}_\lambda(x_1x_2x_3 - 1)$. First, $D^\lambda(p \otimes p) = 0$ by a simple computation. Also, $p$ is ${\mathbb{D}}^3$-stable and $E^3$-stable, but it is not $\overline{{\mathbb{D}}}^3$-stable nor $\overline{E}^3$-stable (zero at $(x_k:y_k) = (1:1)$). That is, the fact that $D^\lambda(p \otimes p) = 0$ does not contradict $(ii)$ or $(iii)$ of the previous theorem.
On the other hand, $p$ is both $(\overline{{\mathbb{D}}} \times {\mathbb{D}}\times {\mathbb{D}})$-stable and $(E \times \overline{E} \times \overline{E})$-stable. This shows that $(\overline{{\mathbb{D}}} \times {\mathbb{D}}\times {\mathbb{D}}, E \times \overline{E} \times \overline{E})$ is not a Grace pair. That is, mixed open and closed stability regions cannot be included in $(ii)$ and $(iii)$ of the previous theorem.
As for whether or not the two-point intersection condition can be removed from $(i)$ seems to be a more subtle point. It would be quite nice if this condition could be removed, but we suspect it is not possible.
Evaluation Symbols
------------------
One way to interpret the stability properties of a given polynomial is via the stability-preservation properties of a particular type of linear operator: the evaluation map. That is, the map which evaluates polynomials at the point $(a:b)$ preserves strong $\{(a:b)\}$-stability. Further, given $\lambda \in {\mathbb{N}}_0^m$ and $(a:b) \equiv (a_1:b_1,...,a_m:b_m) \in ({\mathbb{CP}}^1)^m$, we can define the corresponding evaluation map as an element of $\operatorname{Hom}(V(\lambda),V(0))$ since $V(0) \cong {\mathbb{C}}$. This allows us to obtain symbols for evaluation maps, and these play an important role in our linear operator characterization.
Fix $\lambda \in {\mathbb{N}}_0^m$ and $(a:b) \equiv (a_1:b_1,...,a_m:b_m) \in ({\mathbb{CP}}^1)^m$. Let $\operatorname{ev}_{(a:b)}: V(\lambda) \rightarrow V(0) \cong {\mathbb{C}}$ be the evaluation operator which maps $p$ to $p(a:b) \equiv p(a,b)$. We call $\operatorname{Symb}(\operatorname{ev}_{(a:b)}) \in V(\lambda)$ the *evaluation symbol with root $(a:b)$*. Further: $$\operatorname{Symb}(\operatorname{ev}_{(a:b)}) = \prod_{j=1}^m (b_jx_j - a_jy_j)^{\lambda_j} \equiv (bx - ay)^\lambda$$
The main significance of this notion comes from the following result, which is essentially just a restatement of the symbol lemma (\[symb\_lemma\]) for evaluation symbols.
\[ev\_lemma\] Fix $\lambda \in {\mathbb{N}}_0^m$, $p \in V(\lambda)$, and $(a:b) \in ({\mathbb{CP}}^1)^m$. Considering $(bx-ay)^\lambda$, the evaluation symbol with root $(a:b)$, we have: $$D^\lambda((bx-ay)^\lambda \otimes p) = (\lambda!)^2 p(a:b) \otimes 1$$
It should be noted here the abuse of notation with elements $(a:b)$ of ${\mathbb{CP}}^1$. By definition, $(a:b) = (ca:cb)$ as elements of ${\mathbb{CP}}^1$ for any $0 \neq c \in {\mathbb{C}}$. However, these two different expressions of the same element of ${\mathbb{CP}}^1$ would yield two different evaluation symbols. Note though, that these evaluation symbols would be equal up to scalar. Since Grace’s theorem is about zero vs. nonzero values, we will often be unconcerned with scalar differences, and this is why we allow the abuse of notation.
In what follows, we will extend Grace’s theorem in a number of ways, mainly relying on this lemma and the symbol lemma itself. As we will see, the representation theoretic mentality combined with repeated use of the symbol lemma will yield many of the results of this paper with surprising simplicity.
We now obtain an interesting corollary of Grace’s theorem, making use of the notion of a disjoint Grace pair. This particular formulation of the theorem will serve as a model for our linear operator characterization in §\[str\_Cstab\_subsect\].
\[ev\_graces\] Fix $\lambda \in {\mathbb{N}}_0^m$, $q \in V(\lambda)$, and any disjoint Grace pair $(C_1 \times \cdots \times C_m, B_1 \times \cdots \times B_m)$. Then the following are equivalent.
1. $D^\lambda(p \otimes q) \neq 0$ for all $(C_1 \times \cdots \times C_m)$-stable $p \in V(\lambda)$.
2. $D^\lambda(p \otimes q) \neq 0$ for all $(C_1 \times \cdots \times C_m)$-stable evaluation symbols $p \in V(\lambda)$.
3. $q$ is $(B_1 \times \cdots \times B_m)$-stable.
$(i) \Rightarrow (ii)$ Trivial.
$(ii) \Rightarrow (iii)$ For each $(a:b) \in (B_1 \times \cdots \times B_m)$, $p \equiv \operatorname{Symb}(\operatorname{ev}_{(a:b)})$ is a $(C_1 \times \cdots \times C_m)$-stable evaluation symbol by disjointness. The previous lemma implies: $$0 \neq D^\lambda(p \otimes q) = D^\lambda(\operatorname{Symb}(\operatorname{ev}_{(a:b)}) \otimes q) = (\lambda!)^2 q(a:b) \otimes 1$$
$(iii) \Rightarrow (i)$ Grace’s theorem (\[graces\_thm\]).
Stability Properties of Complex Linear Operators {#Cops_sect}
================================================
In [@bb1], Borcea and Br[ä]{}nd[é]{}n were concerned with classifying the class of weak $\Omega$-stability preserving operators, where $\Omega$ is some product of open circular regions. What they found is that an operator preserves weak $\Omega$-stability if a particular associated polynomial (what they called the symbol) is $\Omega$-stable. However, the “only if” direction does not necessarily hold. In particular, there are some weak $\Omega$-stability preserving operators for which the corresponding symbol is not $\Omega$-stable. They then showed that this could only happen under very specific circumstances: the operator must have image of dimension at most one.
Here, we will characterize all *strong* $\Omega$-stability preserving linear operators (for a bit more general $\Omega$), as well as linear operators which map between different stability regions. And, as it turns out, the extra premise of strong stability preservation is exactly what is needed to have symbol stability be an equivalent condition. In a way, this makes sense: weak $\Omega$-stability preservation counts the zero polynomial as $\Omega$-stable, which in turn corresponds to potential zeros of the symbol in the region of stability. This does not happen with strong stability preservation, allowing for a more straightforward characterization.
First though, let’s take a closer look at the Borcea-Br[ä]{}nd[é]{}n characterization of weak stability-preserving linear operators.
Weak Stability Preservation
---------------------------
Borcea and Br[ä]{}nd[é]{}n define the following symbol: $$\operatorname{Symb}_{BB}(T) := T[(x+z)^\lambda] = \sum_{\mu \leq \lambda} \binom{\lambda}{\mu} z^{\lambda-\mu} T(x^\mu)$$ They then obtain the following characterization of stability-preserving linear operators.
Fix $\lambda \in {\mathbb{N}}_0^m$ and any linear operator $T: {\mathbb{C}}_\lambda[x_1,...,x_m] \to {\mathbb{C}}[x_1,...,x_m]$. The following are equivalent.
1. $T$ maps ${\mathcal{H}_+}^m$-stable polynomials to weakly ${\mathcal{H}_+}^m$-stable polynomials.
2. One of the following holds:
1. $\operatorname{Symb}_{BB}(T)$ is ${\mathcal{H}_+}^{2m}$-stable.
2. $T$ has image of dimension at most one, and is of the form $$T: p \mapsto q \cdot \psi(p)$$ where $q \in {\mathbb{C}}[x_1,...,x_m]$ is ${\mathcal{H}_+}^m$-stable, and $\psi$ is some linear functional.
Using our terminology, this is a characterization of weak stability-preserving linear operators. This fantastic result perhaps has but one unfortunate piece: the degeneracy condition $(ii)(b)$. Its necessity is demonstrated in the following.
Define $T: {\mathbb{C}}_n[x] \to {\mathbb{C}}[x]$ via: $$T: \sum_{k=0}^n \binom{n}{k} a_k x^k \mapsto (a_n + a_{n-2}) x^n$$ This operator obviously preserves weak ${\mathcal{H}_+}$-stability. We then have that $\operatorname{Symb}_{BB}(T) = (z^2 + 1)x^n$, which is not ${\mathcal{H}_+}^2$-stable.
As we will see below, this condition can be removed once we only consider strong stability-preserving operators. So then, maybe strong stability is the more natural notion? However “natural” it is, unfortunately it leaves out operators one might wish to consider. The most fundamental of such operators is the derivative operator $\partial_x$. While $\partial_x$ preserves strong $\overline{{\mathcal{H}_+}}$-stability, it only preserves weak ${\mathcal{H}_+}$-stability. Specifically, $1 \in {\mathbb{C}}_n[x]$ is ${\mathcal{H}_+}$-stable (all its roots are at $\infty$), but $\partial_x 1 \equiv 0$. With this, one obviously wants to be able to include weak stability preserving operators in any characterization of ${\mathcal{H}_+}$-stability preserving operators. We discuss how to use our strong stability preservation characterization to deal with operators like $\partial_x$ in Example \[deriv\_ex\].
Strong Stability Preservation {#str_Cstab_subsect}
-----------------------------
We now state one of our main characterization results, the strong stability preservation characterization. We then derive the Borcea-Br[ä]{}nd[é]{}n characterization as a corollary.
\[Cops\_graces\] Fix $\lambda \in {\mathbb{N}}_0^m$, $\alpha \in {\mathbb{N}}_0^l$, $T \in \operatorname{Hom}(V(\lambda),V(\alpha))$, any disjoint Grace pair $(C_1 \times \cdots \times C_m, B_1 \times \cdots \times B_m)$, and any sets $S_1,...,S_l \subseteq {\mathbb{CP}}^1$. The following are equivalent.
1. $T$ maps $(C_1 \times \cdots \times C_m)$-stable polynomials to nonzero $(S_1 \times \cdots \times S_l)$-stable polynomials.
2. $T$ maps $(C_1 \times \cdots \times C_m)$-stable evaluation symbols to nonzero $(S_1 \times \cdots \times S_l)$-stable polynomials.
3. $\operatorname{Symb}(T)$ is $(B_1 \times \cdots \times B_m) \times (S_1 \times \cdots \times S_l)$-stable.
One should notice the generality of this result in terms of stability regions. First note that any disjoint Grace pair can be considered, without altering the symbol in any way (e.g., via conjugation by M[ö]{}bius transformations). And further, the output sets that can be considered have no restrictions whatsoever. The power of these extra features can be seen in the following examples, which demonstrate classical results regarding polynomial convolutions in a very symbol-oriented way.
\[add\_conv\_ex1\] Fix $q \in V(n)$, so that $(z_j:1)$ are the roots of $q$ for $j \in [n]$. So, $q$ has no roots at $\infty$. The additive (Walsh) convolution of $p$ and $q$ is defined via: $$p *_+^n q := \frac{1}{n!} \sum_{k=0}^n \partial_x^k p \cdot (\partial_x^{n-k} q)(0:1)$$ With this, $T_q(p) := p *_+^n q$ is a linear operator in $\operatorname{Hom}(V(n),V(n))$, and up to scalar we have: $$\operatorname{Symb}(T_q) = \prod_{j=1}^n (xw - (z + z_jw)y) = \operatorname{Hmg}_{(n,n)}\left[\prod_{j=1}^n (x - (z + z_j))\right]$$ Let $C \subset {\mathbb{CP}}^1$ be any projectively convex region, and define $S := \bigcup_j (C + z_j)$. If we order the variables $(z:w,x:y)$, it is straightforward to show that $\operatorname{Symb}(T_q)$ is $C \times ({\mathbb{CP}}^1 \setminus S)$-stable. (First deal with possible $(x:y) = \infty$ or $(z:w) = \infty$ cases, and then assume $y = w = 1$ to simplify the remaining cases.) Applying the previous theorem, this implies $T_q$ maps polynomials with roots in $C$ to polynomials with roots in $S$. (This is Theorem 5.3.1 in [@rahman2002anthpoly].) Picking $C = \overline{{\mathcal{H}_-}}$ and real-rooted $q$ implies $T_q$ maps ${\mathcal{H}_+}$-stable polynomials to ${\mathcal{H}_+}$-stable polynomials. Restricting to $p \in V_{\mathbb{R}}(n)$ then shows that $T_q$ preserves real-rootedness.
\[mult\_conv\_ex1\] Fix $q \in V(n)$, so that $(z_j:1) \neq 0$ are the roots of $q$ for $j \in [n]$. So, $q$ has no roots at $0$ or $\infty$. The multiplicative (Grace-Szeg[ő]{}) convolution of $p$ and $q$ (with coefficients $p_k$ and $q_k$, respectively) is defined via: $$p *_\times^n q := \sum_{k=0}^n \binom{n}{k}^{-1} (-1)^k p_k q_k x^k y^{n-k}$$ With this, $T_q(p) := p *_\times^n q$ is a linear operator in $\operatorname{Hom}(V(n),V(n))$, and up to scalar we have: $$\operatorname{Symb}(T_q) = \prod_{j=1}^n (xw - z_jzy) = \operatorname{Hmg}_{(n,n)}\left[\prod_{j=1}^n (x - z_j z)\right]$$ Let $C \subset {\mathbb{CP}}^1$ be any projectively convex region, and define $S := \bigcup_j (z_j \cdot C)$. If we order the variables $(z:w,x:y)$, it is straightforward to show that $\operatorname{Symb}(T_q)$ is $C \times ({\mathbb{CP}}^1 \setminus S)$-stable. (As above, first deal with possible $(x:y) = \infty$ or $(z:w) = \infty$ cases, and then assume $y = w = 1$ to simplify the remaining cases.) Applying the previous theorem, this implies $T_q$ maps polynomials with roots in $C$ to polynomials with roots in $S$. (This is Theorem 3.4.1d in [@rahman2002anthpoly].) Picking $C = {\mathcal{H}_-}\cup {\mathbb{R}}_+$ and $q$ with only positive roots implies $T_q$ maps $({\mathcal{H}_+}\cup \overline{{\mathbb{R}}_-})$-stable polynomials to $({\mathcal{H}_+}\cup \overline{{\mathbb{R}}_-})$-stable polynomials. Restricting to $p \in V_{\mathbb{R}}(n)$ then shows that $T_q$ preserves positive-rootedness.
In order to prove the above theorem, we need an operator-theoretic corollary to Grace’s theorem. The following result is the main motivation for the symbol lemma (\[symb\_lemma\]), and demonstrates just how closely Grace’s theorem relates to stability properties of linear operators. Further, it gives a slightly stronger result in one direction of the above characterization, as Grace pair disjointness is not a required premise.
\[Cops\_prop\] Fix $\lambda \in {\mathbb{N}}_0^m$, $\alpha \in {\mathbb{N}}_0^l$, $T \in \operatorname{Hom}(V(\lambda),V(\alpha))$, any Grace pair $(C_1 \times \cdots \times C_m, B_1 \times \cdots \times B_m)$, and any sets $S_1,...,S_l \subseteq {\mathbb{CP}}^1$. If $\operatorname{Symb}(T)$ is $(B_1 \times \cdots \times B_m) \times (S_1 \times \cdots \times S_l)$-stable, then $T$ maps $(C_1 \times \cdots \times C_m)$-stable polynomials to nonzero $(S_1 \times \cdots \times S_l)$-stable polynomials.
Fix any $(C_1 \times \cdots \times C_m)$-stable $q \in V(\lambda)$ and any $(c:d) \in (S_1 \times \cdots \times S_l)$. Let $c_{\lambda,\alpha} := (-1)^\alpha(\lambda!)^2(\alpha!)^2$. The evaluation symbol lemma (\[ev\_lemma\]) and the symbol lemma (\[symb\_lemma\]) then give us the following expression of $T(q)$ evaluated at $(c:d)$: $$\begin{split}
c_{\lambda,\alpha} T(q)(c:d) \otimes 1 &= c_{\lambda,0} D^\alpha(T(q) \otimes (dx-cy)^\alpha) \\
&= D^{\lambda \sqcup \alpha}(\operatorname{Symb}(T) \otimes q \cdot (dx-cy)^\alpha) \\
&= c_{0,\alpha} D^\lambda(\operatorname{Symb}(T)(\cdot,c:d) \otimes q)
\end{split}$$ Since $\operatorname{Symb}(T)(\cdot,c:d)$ is $(B_1 \times \cdots B_m)$-stable and $q$ is $(C_1 \times \cdots \times C_m)$-stable, Grace’s theorem then implies this last expression is nonzero.
With this, we now give the proof of Theorem \[Cops\_graces\].
The statement of this result, as well as its proof, is quite similar to that of the evaluation symbol version of Grace’s theorem given in Corollary \[ev\_graces\]. We explicitly give the proof anyway, as it is rather short and straightforward.
$(i) \Rightarrow (ii)$. Trivial.
$(ii) \Rightarrow (iii)$. Fix $(a:b) \in (B_1 \times \cdots \times B_m)$ and $(c:d) \in (S_1 \times \cdots \times S_l)$. Let $c_{\lambda,\alpha} := (-1)^\alpha(\lambda!)^2(\alpha!)^2$. Disjointness of $B_k,C_k$ implies $(bx-ay)^\lambda \in V(\lambda)$ is a $(C_1 \times \cdots \times C_m)$-stable evaluation symbol. Using the evaluation symbol lemma (\[ev\_lemma\]) and the symbol lemma (\[symb\_lemma\]), we compute: $$(-1)^\lambda c_{\lambda,\alpha} \operatorname{Symb}(T)(a:b,c:d) = D^{\lambda \sqcup \alpha}(\operatorname{Symb}(T) \otimes (bx-ay)^\lambda (dx-cy)^\alpha) = c_{\lambda,\alpha} T[(bx-ay)^\lambda](c:d)$$ By $(ii)$, the last expression is nonzero.
$(iii) \Rightarrow (i)$. The previous proposition.
As mentioned above, the previous proposition gives a slightly stronger result in the (symbol stability $\Rightarrow$ operator stability) direction. Using it, we revisit the additive and multiplicative convolutions with a more algebraic/symbolic mentality.
By Definition \[symb\_def\], the $\operatorname{Symb}$ map gives a bijection between certain spaces of linear operators and polynomials. So, we can uniquely define a linear operator by giving its symbol. Using this idea, we specify $T \in \operatorname{Hom}(V(n,n), V(n))$ by defining its symbol in $V(n,n,n)$ (with variables $(z:w)$, $(t:s)$, and $(x:y)$) as follows: $$\operatorname{Symb}(T) := \operatorname{Hmg}_{(n,n,n)}\left[(x - (z+t))^n\right] = (xws - (zs + tw)y)^n$$ Now, let us consider the additive convolution $*_+^n$ as an element of $\operatorname{Hom}(V(n,n),V(n))$ in the following way. Since $V(n,n) \cong V(n) \boxtimes V(n)$, we define $*_+^n$ on elements $p \boxtimes q \in V(n) \boxtimes V(n)$ via $*_+^n(p \boxtimes q) := p *_+^n q$ and extend linearly. We then compute $\operatorname{Symb}(*_+^n)$ as follows: $$\operatorname{Symb}(*_+^n) = *_+^n\left[(zy-xw)^n \boxtimes (ty-xs)^n\right] = (xws - (zs + tw)y)^n$$ That is, $*_+^n$ is the operator that has our desired symbol. Fixing any $a,b,c,d \in {\mathbb{R}}$ such that $a<b$ and $c<d$, we define the sets $C_1 := \overline{{\mathcal{H}_+}} \setminus (a,b)$, $C_2 := \overline{{\mathcal{H}_+}} \setminus (c,d)$, $B_1 := {\mathcal{H}_-}\cup [a,b]$, $B_2 := {\mathcal{H}_-}\cup [c,d])$, and $S := \overline{{\mathcal{H}_+}} \setminus [a+c,b+d]$. The previous proposition then implies $p *_+^n q$ has all its roots in $[a+c,b+d]$ whenever $p,q \in V_{\mathbb{R}}(n)$ have all their roots in $(a,b)$ and $(c,d)$, respectively. For real-rooted $p,q$ of degree $n$, this implies: $$\operatorname{minroot}(p) + \operatorname{minroot}(q) \leq \operatorname{minroot}(p *_+^n q) \leq \operatorname{maxroot}(p *_+^n q) \leq \operatorname{maxroot}(p) + \operatorname{maxroot}(q)$$ Notice that we actually get a bit more. For $(C_1 \times C_2)$-stable $r := \sum_j p_j \boxtimes q_j \in V(n) \boxtimes V(n) \cong V(n,n)$, we have that $*_+^n[r]$ is $S$-stable. That is, $*_+^n$ has stability properties as an operator in $\operatorname{Hom}(V(n,n),V(n))$, not just as a convolution between two polynomials in $V(n)$.
As in the previous example, we can consider the multiplicative convolution $*_\times^n$ as an element of $\operatorname{Hom}(V(n,n),V(n))$ by defining $*_\times^n(p \boxtimes q) := p *_\times^n q$ on elements $p \boxtimes q \in V(n) \boxtimes V(n) \cong V(n,n)$ and extending linearly. We then compute its symbol in $V(n,n,n)$ (with variables $[z;w]$, $[t;s]$, and $[x;y]$) as follows: $$\operatorname{Symb}(*_\times^n) = *_\times^n\left[(zy-xw)^n \boxtimes (ty-xs)^n\right] = (xws - zty)^n = \operatorname{Hmg}_{(n,n,n)}\left[(x-zt)^n\right]$$ Fixing any $a,b,c,d \in {\mathbb{R}}_+$ such that $0<a<b$ and $0<c<d$, we define the sets $C_1$, $C_2$, $B_1$, and $B_2$ as in the previous example. We then define $S := \overline{{\mathbb{R}}} \setminus [ac, bd]$. The previous proposition then implies $p *_\times^n q$ has all its real roots in $[ac,bd]$ whenever $p,q \in V_{\mathbb{R}}(n)$ have all their roots in $(a,b)$ and $(c,d)$, respectively. (Notice that we could not apply the proposition if ${\mathcal{H}_+}\subset S$ or ${\mathcal{H}_-}\subset S$.) Since Example \[mult\_conv\_ex1\] implies $p *_\times^n q$ is positive-rooted (and hence, real-rooted) whenever $p$ and $q$ are, this implies: $$\operatorname{minroot}(p) \cdot \operatorname{minroot}(q) \leq \operatorname{minroot}(p *_\times^n q) \leq \operatorname{maxroot}(p *_\times^n q) \leq \operatorname{maxroot}(p) \cdot \operatorname{maxroot}(q)$$ As in the previous example, we also obtain stability properties for $*_\times^n$ as an operator in $\operatorname{Hom}(V(n,n),V(n))$, and not just as a polynomial convolution.
Using similar techniques, we can also circumvent the issue that arises from the fact that $\partial_x$ only preserves weak stability.
\[deriv\_ex\] For fixed $n \geq 1$, consider the operator $\partial_x \in \operatorname{Hom}(V(n),V(n-1))$. We compute: $$\operatorname{Symb}(\partial_x) = \partial_x[(zy-xw)^n] = -nw(zy-xw)^{n-1} = \operatorname{Hmg}_{(n,n-1)}\left[-n(z-x)^{n-1}\right]$$ For any $a,b \in {\mathbb{R}}$ such that $a<b$, it is straightforward to see that $\operatorname{Symb}(\partial_x)$ is $(C \times B)$-stable for $C := {\mathcal{H}_-}\cup (a,b)$ and $B := \overline{{\mathcal{H}_+}} \setminus (a,b))$, where the variables are ordered $(z:w),(x:y)$. (Notice that this does not hold when $\infty \in C$, due to the $w$ factor in the symbol.) Since $(C,B)$ is a disjoint Grace pair, the previous theorem implies $\partial_x$ preserves strong $B$-stability.
With this, let $f \in {\mathbb{C}}_n[x]$ be a ${\mathcal{H}_+}$-stable polynomial of degree $1 \leq m \leq n$, and let $p \in V(m)$ be its degree-$m$ homogenization. Then $p$ has no roots at infinity, and therefore there exists $a < b$ such that $p$ is $(\overline{{\mathcal{H}_+}} \setminus (a,b))$-stable. The previous discussion implies $\partial_x p$ is $(\overline{{\mathcal{H}_+}} \setminus (a,b))$-stable, and in particular $\partial_x p$ is ${\mathcal{H}_+}$-stable. Since $\partial_x$ commutes with homogenization, this also implies $\partial_x f$ is ${\mathcal{H}_+}$-stable.
Other issues related to weak stability preservation can be dealt with in a similar way, by considering stability regions with small intervals in $\overline{{\mathbb{R}}}$ about $\infty$ attached. More generally though, the Borcea-Br[ä]{}nd[é]{}n characterization ends up being a corollary of Theorem \[Cops\_graces\], which we discuss and demonstrate now.
Deriving the Complex Borcea-Br[ä]{}nd[é]{}n Characterization {#Cops_deriv_subsect}
------------------------------------------------------------
As mentioned above, we hope to obtain the Borcea-Br[ä]{}nd[é]{}n characterization from our strong stability characterization given in Theorem \[Cops\_graces\]. To this end, we state two corollaries to Theorem \[Cops\_graces\], which look (naively) as close to the Borcea-Br[ä]{}nd[é]{}n characterization as possible. Let $C^c$ denote the complement of $C$ in ${\mathbb{CP}}^1$.
\[naive\_CBB\_cor1\] Fix $\lambda,\alpha \in {\mathbb{N}}_0^m$, $T \in \operatorname{Hom}(V(\lambda),V(\alpha))$, and a Grace pair of the form $(C_1 \times \cdots \times C_m, C_1^c \times \cdots \times C_m^c)$. The following are equivalent.
1. $T$ preserves strong $(C_1 \times \cdots \times C_m)$-stability.
2. $\operatorname{Symb}(T)$ is $(C_1^c \times \cdots \times C_m^c) \times (C_1 \times \cdots \times C_m)$-stable.
\[naive\_CBB\_cor2\] Fix $\lambda,\alpha \in {\mathbb{N}}_0^m$ and $T \in \operatorname{Hom}(V(\lambda),V(\alpha))$. $T$ preserves strong stability iff $\operatorname{Symb}(T)$ is $(\overline{{\mathcal{H}_-}}^m \times {\mathcal{H}_+}^m)$-stable.
In Theorem \[BB\_Cops\_thm\], the analogous “if” direction of the previous corollary is paraphrased as follows: *$T$ preserves weak stability if the Borcea-Br[ä]{}nd[é]{}n symbol of $T$ is stable*. To see how this statement relates, we restate the definition of the Borcea-Br[ä]{}nd[é]{}n symbol: $$\operatorname{Symb}_{BB}(T) := T\left[(z+x)^\lambda\right] = \sum_{\mu \leq \lambda} \binom{\lambda}{\mu} z^{\lambda-\mu} T(x^\mu)$$ Notice that by applying $z \mapsto -z$ and homogenizing, we obtain (up to scalar) the universal symbol $\operatorname{Symb}(T)$ defined in this paper. The crucial difference then is the fact that the Borcea-Br[ä]{}nd[é]{}n “if” direction deals only with *open* upper half-planes, whereas the previous corollary requires *closed* half-plane stability of $\operatorname{Symb}(T)$ in the first $m$ pairs of variables. That is, the required premises of the “if” direction of the previous corollary are strictly stronger than that of the Borcea-Br[ä]{}nd[é]{}n result.
These two results can be reconciled, however, which we now demonstrate. The following result provides the main link to the Borcea-Br[ä]{}nd[é]{}n characterization, and it can be intuitively described as follows: with the exception of having a one-dimensional range, a linear operator which maps $(C_1 \times \cdots \times C_m)$-stable polynomials to weak $(B_1 \times \cdots \times B_m)$-stable polynomials can only have zeros on the boundary of the set of $(C_1 \times \cdots \times C_m)$-stable polynomials.
\[bb\_Clink\] Fix $\lambda,\alpha \in {\mathbb{N}}_0^m$, $T \in \operatorname{Hom}(V(\lambda),V(\alpha))$, and any open circular regions $C_1,...,C_m,B_1,...,B_m \subseteq {\mathbb{CP}}^1$. The following are equivalent.
1. $T$ maps $(C_1 \times \cdots \times C_m)$-stable polynomials to weakly $(B_1 \times \cdots \times B_m)$-stable polynomials.
2. One of the following holds:
1. $T$ maps $(\overline{C_1} \times \cdots \times \overline{C_m})$-stable polynomials to nonzero $(B_1 \times \cdots \times B_m)$-stable polynomials.
2. $T \equiv p_0 \cdot \psi$ for some weakly $(B_1 \times \cdots \times B_m)$-stable polynomial $p_0 \in V(\alpha)$ and some linear functional $\psi$.
By appropriate $SL_2({\mathbb{C}})$ action, we can assume WLOG that $C_k = B_k = {\mathbb{D}}$ for all $k \in [m]$.
$(i) \Rightarrow (ii)$. We show that if $(a)$ is not the case, then $(b)$ must hold. It follows from $(i)$ that $T$ maps $\overline{{\mathbb{D}}}^m$-stable polynomials to (possibly identically zero) ${\mathbb{D}}^m$-stable polynomials. So, if $(a)$ is not the case, we have that $T(p) \equiv 0$ for some $\overline{{\mathbb{D}}}^m$-stable polynomial $p \in V(\lambda)$.
The rest of the argument is essentially the proof of necessity found in [@bb1] for Theorem \[BB\_Cops\_thm\]. Since the set of nonzero $\overline{{\mathbb{D}}}^m$-stable polynomials is open in $V(\lambda)$, there is some ball $B(p) \subset V(\lambda)$ centered at $p$ such that $B(p)$ contains only $\overline{{\mathbb{D}}}^m$-stable polynomials. So, $T[B(p)]$ is an open set in the image of $T$ containing 0 and otherwise consisting of ${\mathbb{D}}^m$-stable polynomials. Therefore, the image of $T$ is a vector space consisting of ${\mathbb{D}}^m$-stable polynomials. Lemma \[dim\_lemma\] (and appropriate $SL_2({\mathbb{C}})$ action) then implies the image of $T$ is of dimension $\leq 1$, and $(b)$ follows.
$(ii) \Rightarrow (i)$. If $(b)$ holds, then $(i)$ is trivial. Otherwise, fix $p \in V(\lambda)$ such that $p$ is ${\mathbb{D}}^m$-stable. For all $n \in {\mathbb{N}}$, define: $$p_n := p((1-n^{-1})x_1, y_1, (1-n^{-1})x_2, y_2, ..., (1-n^{-1})x_m, y_m)$$ So, $p_n$ is $\overline{{\mathbb{D}}}^m$-stable for all $n$, and $\lim_{n \rightarrow \infty} p_n = p$ coefficient-wise. By $(ii)$, $T(p_n)$ is ${\mathbb{D}}^m$-stable for all $n$, and by continuity, $T(p) = \lim_{n \rightarrow \infty} T(p_n)$. Hurwitz’s theorem then implies $T(p)$ is either identically zero or ${\mathbb{D}}^m$-stable.
This lemma then yields the following corollaries to Theorem \[Cops\_graces\]. Applying the necessary maps to convert $\operatorname{Symb}$ to $\operatorname{Symb}_{BB}$ as discussed above, these results give precisely the Borcea-Br[ä]{}nd[é]{}n characterization proven in Theorem \[BB\_Cops\_thm\] and more generally in Theorem 6.3 of [@bb1]. In particular, Corollary \[CBB\_cor1\] can be seen as a unification of the complex characterization results of [@bb1].
\[CBB\_cor1\] Fix $\lambda,\alpha \in {\mathbb{N}}_0^m$, $T \in \operatorname{Hom}(V(\lambda),V(\alpha))$, and any open circular regions $C_1, ..., C_m$. The following are equivalent.
1. $T$ preserves weak $(C_1 \times \cdots \times C_m)$-stability.
2. One of the following holds:
1. $\operatorname{Symb}(T)$ is $(\overline{C_1}^c \times \cdots \times \overline{C_m}^c) \times (C_1 \times \cdots \times C_m)$-stable.
2. $T \equiv p_0 \cdot \psi$ for some weakly $(C_1 \times \cdots \times C_m)$-stable polynomial $p_0 \in V(\alpha)$ and some linear functional $\psi$.
The result follows from the previous lemma and Theorem \[Cops\_graces\] applied to an operator $T$ which maps $(\overline{C_1} \times \cdots \times \overline{C_m})$-stable polynomials to nonzero $(C_1 \times \cdots \times C_m)$-stable polynomials.
\[CBB\_cor2\] Fix $\lambda,\alpha \in {\mathbb{N}}_0^m$ and $T \in \operatorname{Hom}(V(\lambda),V(\alpha))$. $T$ preserves weak stability iff one of the following holds:
1. $\operatorname{Symb}(T)$ is $({\mathcal{H}_-}^m \times {\mathcal{H}_+}^m)$-stable.
2. $T \equiv p_0 \cdot \psi$ for some weakly stable polynomial $p_0 \in V(\alpha)$ and some linear functional $\psi$.
Notice that our naive guess at strong stability results which emulate the Borcea-Br[ä]{}nd[é]{}n characterization (Corollaries \[naive\_CBB\_cor1\] and \[naive\_CBB\_cor2\]) was incorrect. We actually needed to consider *closed* circular stability regions $\overline{C_k}$, so that their complements in ${\mathbb{CP}}^1$ would be open (i.e., to ensure Grace pair disjointness, which is required to apply Theorem \[Cops\_graces\]). We see this play out in condition $(ii)(a)$ of Corollary \[CBB\_cor1\].
Stability Properties of Real Linear Operators {#Rops_sect}
=============================================
Borcea and Br[ä]{}nd[é]{}n also classified the class of weak real stability preserving linear operators. As in the complex case, they showed that weak real stability preservation of a linear operator $T$ is almost equivalent to real stability of the associated symbol $\operatorname{Symb}_{BB}(T)$. We have to say “almost equivalent” here because there are certain weak real stability preserving operators for which the corresponding symbol is not real stable. As before, this implies a certain dimension restriction: such operators must have image of dimension at most two.
We will now characterize all strong real stability preserving linear operators. As above, strong real stability preservation will serve to eliminate the degeneracy condition of the Borcea-Br[ä]{}nd[é]{}n characterization. In this section, we duplicate the outline of our previous discussion on complex operators, making use of arguments similar to those found in [@bb1] to fill in the gaps.
Further, we also obtain a characterization of a certain class of operators which preserve ray- and interval-rootedness. The question of a full characterization of such operators is as of yet still an open problem (see [@bba]). Here, we answer this question for operators which preserve both strong ray- or interval-rootedness as well as weak real-rootedness.
Weak Real Stability Preservation
--------------------------------
Borcea and Br[ä]{}nd[é]{}n obtain the following characterization of weak real stability preserving linear operators. Recall the notion of *proper position* (denoted by $\ll$) given in Definition \[pp\_def\].
Fix $\lambda \in {\mathbb{N}}_0^m$ and any linear operator $T: {\mathbb{R}}_\lambda[x_1,...,x_m] \to {\mathbb{R}}[x_1,...,x_m]$. The following are equivalent.
1. $T$ maps real stable polynomials to weakly real stable polynomials.
2. One of the following holds:
1. $\operatorname{Symb}_{BB}(T)$ is ${\mathcal{H}_+}^{2m}$-stable.
2. $\operatorname{Symb}_{BB}(T)$ is $({\mathcal{H}_-}^m \times {\mathcal{H}_+}^m)$-stable.
3. $T$ has image of dimension at most two, and is of the form $$T: p \mapsto q \cdot \psi_1(p) + r \cdot \psi_2(p)$$ where $q,r \in {\mathbb{R}}[x_1,...,x_m]$ are weakly real stable such that $q \ll r$, and $\psi_1,\psi_2$ are real linear functionals.
As in the case of complex operators, the degeneracy condition $(ii)(c)$ is the result of allowing weak real stability preserving operators. We now give an example which demonstrates its necessity.
Define $T: {\mathbb{R}}_n[x] \to {\mathbb{R}}[x]$ via: $$T: \sum_{k=0}^n \binom{n}{k} a_k x^k \mapsto a_nx^n + a_{n-2}x^{n-1} = (a_nx + a_{n-2})x^{n-1}$$ This operator obviously preserves weak real stability. We then have that $\operatorname{Symb}_{BB}(T) = (z^2+x)x^{n-1}$, which is not ${\mathcal{H}_+}^2$-stable nor $({\mathcal{H}_-}\times {\mathcal{H}_+})$-stable.
Again, the degeneracy condition is required for the characterization but obscures the connection between an operator and its symbol. To remove it, we now turn to our characterization of strong real stability preserving operators.
Strong Real Stability Preservation
----------------------------------
We state and prove our strong real stability preservation characterization here, and then derive the Borcea-Br[ä]{}nd[é]{}n characterization as a corollary. The proof here takes a bit more work than in the complex case, and will rely on many of the real stability results discussed in §\[stab\_subsect\]. This extra work is essentially taken from the proof of Theorem \[BB\_Rops\_thm\] found in [@bb1].
\[Rops\_graces\] Fix $\lambda \in {\mathbb{N}}_0^m$, $\alpha \in {\mathbb{N}}_0^l$, and a linear operator $T \in \operatorname{Hom}(V(\lambda),V(\alpha))$ such that $T$ restricts to a real linear operator from $V_{\mathbb{R}}(\lambda)$ to $V_{\mathbb{R}}(\alpha)$. The following are equivalent.
1. $T$ preserves strong real stability.
2. $\operatorname{Symb}(T)$ is either $(\overline{{\mathcal{H}_-}}^m \times {\mathcal{H}_+}^l)$-stable or $(\overline{{\mathcal{H}_-}}^m \times {\mathcal{H}_-}^l)$-stable.
$(i) \Rightarrow (ii)$. Fixing $(z_0:w_0) \in \overline{{\mathcal{H}_-}}^m$, we have that $(w_0x - z_0y)^\lambda$ is ${\mathcal{H}_+}^m$-stable. If $(z_0:w_0) \in \overline{{\mathbb{R}}}^m$, then $T[(w_0x - z_0y)^\lambda]$ is nonzero and real stable by assumption. Combining the symbol lemma and the evaluation symbol lemma, this implies $\operatorname{Symb}(T)(z_0:w_0,x:y) = (-1)^\lambda T[(w_0x - z_0y)^\lambda]$ is both ${\mathcal{H}_+}^l$-stable and ${\mathcal{H}_-}^l$-stable.
On the other hand, suppose $(z_0:w_0) \not\in \overline{{\mathbb{R}}}^m$. By Lemma \[rs\_lemma\], we have that $T[(w_0x - z_0y)^\lambda]$ is either ${\mathcal{H}_+}^l$-stable, ${\mathcal{H}_-}^l$-stable, or zero. Now suppose there are $(z_0:w_0), (z_0':w_0') \in \overline{{\mathcal{H}_-}}^m \setminus \overline{{\mathbb{R}}}^m$ such that $T[(w_0x - z_0y)^\lambda]$ is ${\mathcal{H}_+}^l$-stable and $T[(w_0'x - z_0'y)^\lambda]$ is ${\mathcal{H}_-}^l$-stable. By a homotopy argument, there exists $(z_0'':w_0'') \in \overline{{\mathcal{H}_-}}^m \setminus \overline{{\mathbb{R}}}^m$ such that $T[(w_0''x - z_0''y)^\lambda]$ is $({\mathcal{H}_+}^l \cup {\mathcal{H}_-}^l)$-stable or zero. By Lemma \[scal\_rs\_lemma\], $T[c_0(w_0''x - z_0''y)^\lambda]$ is either real stable or zero for some complex scalar $c_0 \neq 0$.
Let $c_0(w_0''x - z_0''y)^\lambda = q(x:y) + ir(x:y)$ for $q,r \in V_{\mathbb{R}}(\lambda)$, which are both real stable or zero by Hermite-Biehler. Note further that $r \not\equiv 0$ since $(z_0'':w_0'') \not\in \overline{{\mathbb{R}}}^m$. However, since $T(q + ir) = T(q) + iT(r)$ is real stable or zero and $T$ restricts to real linear operator, it must be that $T(r) \equiv 0$. This contradicts the fact that $T$ strongly preserves real stability.
So, $\operatorname{Symb}(T)(z_0:w_0,x:y) = (-1)^\lambda T[(w_0x - z_0y)^\lambda]$ is either ${\mathcal{H}_+}^l$-stable for all $(z_0:w_0) \in \overline{{\mathcal{H}_-}}^m \setminus \overline{{\mathbb{R}}}^m$, or ${\mathcal{H}_-}^l$-stable for all $(z_0:w_0) \in \overline{{\mathcal{H}_-}}^m \setminus \overline{{\mathbb{R}}}^m$. Combining this with the $(z_0:w_0) \in \overline{{\mathbb{R}}}^m$ case, we have that $\operatorname{Symb}(T)$ is either $(\overline{{\mathcal{H}_-}}^m \times {\mathcal{H}_+}^l)$-stable or $(\overline{{\mathcal{H}_-}}^m \times {\mathcal{H}_-}^l)$-stable.
$(ii) \Rightarrow (i)$. By the complex stability characterization (Theorem \[Cops\_graces\]), $T$ maps ${\mathcal{H}_+}^m$-stable polynomials to either nonzero ${\mathcal{H}_+}^l$-stable polynomials or nonzero ${\mathcal{H}_-}^l$-stable polynomials. Since $T$ restricts to a real linear operator on $V_{\mathbb{R}}(\lambda)$, $T$ preserves strong real stability.
As a final note, the “homotopy argument” used in the previous proof is not quite that of the proof found in [@bb1], though it is similar. Here, one just needs to be a bit more careful about the precise homotopy with respect to points at infinity.
Deriving the Real Borcea-Br[ä]{}nd[é]{}n Characterization {#Rops_deriv_subsect}
---------------------------------------------------------
As in the complex case, we now obtain the Borcea-Br[ä]{}nd[é]{}n weak real stability characterization as a corollary to our strong real stability characterization given in Theorem \[Rops\_graces\]. To this end, we start by giving a sort of real stability version of Lemma \[bb\_Clink\]. The proof of this lemma is similar in spirit to that of the strong real stability characterization given above.
\[bb\_Rlink\] Fix $\lambda,\alpha \in {\mathbb{N}}_0^m$ and $T \in \operatorname{Hom}(V(\lambda),V(\alpha))$ such that $T$ restricts to a real linear operator from $V_{\mathbb{R}}(\lambda)$ to $V_{\mathbb{R}}(\alpha)$. The following are equivalent.
1. $T$ preserves weak real stability.
2. One of the following holds:
1. $T$ maps $\overline{{\mathcal{H}_+}}^m$-stable polynomials to nonzero ${\mathcal{H}_+}^m$-stable polynomials.
2. $T$ maps $\overline{{\mathcal{H}_+}}^m$-stable polynomials to nonzero ${\mathcal{H}_-}^m$-stable polynomials.
3. $T$ has image of dimension at most two, and is of the form $$T: p \mapsto q \cdot \psi_1(p) + r \cdot \psi_2(p)$$ where $q,r \in V_{\mathbb{R}}(\alpha)$ are weakly real stable such that $q \ll r$, and $\psi_1,\psi_2$ are real linear functionals.
$(i) \Rightarrow (ii)$. By the complex characterization (Theorem \[Cops\_graces\]), we only need to consider evaluation symbols when demonstrating $(a)$ or $(b)$. For any $(z_0:w_0) \in {\mathcal{H}_-}^m$, Lemma \[rs\_lemma\] then implies $T[(w_0x-z_0y)^\lambda]$ is either ${\mathcal{H}_+}^m$-stable, ${\mathcal{H}_-}^m$-stable, or identically zero. We now show that $(c)$ holds if $(a)$ and $(b)$ do not.
If neither $(a)$ nor $(b)$ holds for evaluation symbols, then there exist $(z_0:w_0),(z_0':w_0') \in {\mathcal{H}_-}^m$ such that $T[(w_0x-z_0y)^\lambda]$ is ${\mathcal{H}_+}^m$-stable or zero and $T[(w_0'x-z_0'y)^\lambda]$ is ${\mathcal{H}_-}^m$-stable or zero. As in the proof of the strong real stability characterization (Theorem \[Rops\_graces\]), a homotopy argument implies there exists $(z_0'':w_0'') \in {\mathcal{H}_-}^m$ such that $T[(w_0''x-z_0''y)^\lambda]$ is $({\mathcal{H}_+}^m \cup {\mathcal{H}_-}^m)$-stable or zero. Lemma \[scal\_rs\_lemma\] then implies $T[c_0(w_0''x-z_0''y)^\lambda]$ is real stable or zero, for some complex scalar $c_0 \neq 0$.
For the sake of simplicity, we denote $q_0 := c_0(w_0''x-z_0''y)^\lambda$. Since the set of $\overline{{\mathcal{H}_+}}^m$-stable polynomials is open in $V(\lambda)$, let $B(0)$ be some open ball centered at $0$ such that $q_0 + iB(0)$ consists of nonzero $\overline{{\mathcal{H}_+}}^m$-stable polynomials. So, for any $r_0 \in B(0)$, Lemma \[rs\_lemma\] implies $T(q_0) + iT(r_0)$ is either ${\mathcal{H}_+}^m$-stable, ${\mathcal{H}_-}^m$-stable, or zero. Hermite-Biehler then implies $T(r_0)$ is real stable or zero whenever $r_0 \in B(0) \cap V_{\mathbb{R}}(\lambda)$. Therefore, $T[V_{\mathbb{R}}(\lambda)]$ consists of real stable polynomials, and $(c)$ follows from Lemma \[dim\_lemma\].
$(ii) \Rightarrow (i)$. If $(c)$ holds, then $(i)$ follows from Hermite-Biehler. Otherwise, suppose WLOG that $(a)$ holds. We can then use an argument similar in spirit to that of Lemma \[bb\_Clink\] to show that $T$ maps ${\mathcal{H}_+}^m$-stable polynomials to weakly ${\mathcal{H}_+}^m$-stable polynomials. Since $T$ restricts to a real operator, this implies $(i)$.
As in Lemma \[bb\_Clink\], we use the previous lemma to link the characterizations of weak and strong stability preserving operators as follows. Applying the necessary maps to convert $\operatorname{Symb}(T)$ to $\operatorname{Symb}_{BB}(T)$ below gives essentially the characterization of weak real stability preserving operators given in Theorem \[BB\_Rops\_thm\].
\[bb\_Rops\] Fix $\lambda,\alpha \in {\mathbb{N}}_0^m$ and $T \in \operatorname{Hom}(V(\lambda),V(\alpha))$ such that $T$ restricts to a real linear operator from $V_{\mathbb{R}}(\lambda)$ to $V_{\mathbb{R}}(\alpha)$. The following are equivalent.
1. $T$ preserves weak real stability.
2. One of the following holds:
1. $\operatorname{Symb}(T)$ is $({\mathcal{H}_-}^m \times {\mathcal{H}_+}^m)$-stable.
2. $\operatorname{Symb}(T)$ is $({\mathcal{H}_-}^m \times {\mathcal{H}_-}^m)$-stable.
3. $T$ has image of dimension at most two, and is of the form $$T: p \mapsto q \cdot \psi_1(p) + r \cdot \psi_2(p)$$ where $q,r \in V_{\mathbb{R}}(\alpha)$ are weakly real stable such that $q \ll r$, and $\psi_1,\psi_2$ are real linear functionals.
Apply the complex characterization (Theorem \[Cops\_graces\]) to conditions $(ii)(a)$ and $(ii)(b)$ of the previous lemma.
Ray and Interval Stability {#Jops_subsect}
--------------------------
We now apply the above results to projectively convex regions of the form ${\mathcal{H}_+}\cup J^c$, where $J \subset \overline{{\mathbb{R}}}$ is some connected set. From this, we obtain a classification of operators which both preserve strong $J$-rootedness and weak real-rootedness (a polynomial $p \in V(n)$ is $J$-rooted if all its roots lie in $J$). This of course does not completely solve the open problem of providing a classification of interval- and ray-stability preserving operators (see, e.g., [@bba]). However, it does seem to be the natural corollary obtained by applying proof methods similar to that of [@bb1].
That said, we now proceed to prove the main result of this subsection, Theorem \[Jops\_graces\]. We first start with a short-hand definition in order to simplify the proof.
Fix $\lambda,\alpha \in {\mathbb{N}}_0^m$ and $T \in \operatorname{Hom}(V(\lambda),V(\alpha))$ such that $T$ restricts to a real linear operator and preserves weak real-stability. We say $T$ is *degenerate* if it satisfies condition $(ii)(c)$ of Corollary \[bb\_Rops\].
We now prove two lemmas. The first is straightforward, but rather interesting in its own right.
\[conv\_lemma\] Fix a closed bounded interval $J \subset {\mathbb{R}}$ and a subspace $W \subseteq V_{\mathbb{R}}(n)$ consisting of weakly real-rooted polynomials. Let $S \subseteq W$ denote the subset of top-degree monic $J$-rooted polynomials. There exist $p,q \in S$ such that $p \ll q$ and $S$ is the convex hull of $p$ and $q$.
Lemma \[dim\_lemma\] implies $W$ is of dimension at most two, and so then Lemma \[total\_order\_lemma\] implies the relation $\ll$ is a total order on $S$. Applying the root ordering property of Lemma \[total\_order\_lemma\], the closedness of $S$ implies there are $p,q \in S$ such that $p \ll q$ and $p \ll r \ll q$ for all $r \in S$. Basic sign arguments and the fact that $S$ is contained in the span of $\{p,q\}$ then imply $S$ is the the convex hull of $\{p,q\}$.
The second lemma is perhaps less straightforward in terms of proof, but follows from the following intuitive idea: an open ball in some complex subspace of polynomials yields, roughly speaking, an open ball of zeros.
Fix $n,m \in {\mathbb{N}}_0$ and a linear operator $T \in \operatorname{Hom}(V(n),V(m))$ which restricts to a real linear operator and preserves weak real-rootedness. If there exist some $\overline{{\mathcal{H}_+}}$-stable $p_0 \in V(n)$ and some $(x_0:y_0) \in \overline{{\mathbb{R}}}$ such that $T(p_0)(x_0:y_0) = 0$, then one of the following holds:
1. $T(p_0)$ is real-rooted or identically zero.
2. $T(p)(x_0:y_0) = 0$ for all $p \in V(n)$.
Let $q_0,r_0 \in V_{\mathbb{R}}(m)$ be such that $T(p_0) = q_0 + ir_0$. Also suppose that $T(p_0) \not\equiv 0$ and that $(b)$ does not hold, and let $p_1$ be such that $T(p_1)(x_0:y_0) \neq 0$. WLOG, we may also assume $p_1 \in V_{\mathbb{R}}(n)$ by considering its real or imaginary part. We will now prove that $T(p_0)$ must be real-rooted.
First, suppose further that $T(p_0)$ has a multiple root at $(x_0:y_0)$. For small fixed $\epsilon$, $p_0 + \epsilon p_1$ is $\overline{{\mathcal{H}_+}}$-stable and so Lemma \[rs\_lemma\] implies $T(p_0 + \epsilon p_1)$ is either ${\mathcal{H}_+}$-stable or ${\mathcal{H}_-}$-stable. Hermite-Biehler then implies $q_0 + \epsilon T(p_1)$ and $r_0$ have interlacing roots. However, since $T$ restricts to a real linear operator, it must be that $q_0$ and $r_0$ both have a multiple root at $(x_0:y_0)$. The fact that $q_0 + \epsilon T(p_1)$ has no root at $(x_0:y_0)$ yields a contradiction, as interlacing is then impossible.
Otherwise, $T(p_0)$ has a simple root at $(x_0:y_0)$. Defining $R \in \operatorname{Hom}(V(n),V(1))$ as $R := d_{(x_0:y_0)}^{n-1} \circ T$, we see that $R(p_0)(x_0:y_0) = 0$, but $R(p_0) \not\equiv 0$ since the root is simple. Further, $R(p_1)(x_0:y_0) \neq 0$, and therefore $R$ is a surjective continuous linear map. By the open mapping theorem, there exists a one-real-dimensional curve $\Gamma \subset V(n)$ through $p_0$, for which $R(\Gamma)$ contains elements with root in ${\mathcal{H}_+}$ on one side of $p_0$ (call this side $\Gamma_+$) and elements with root in ${\mathcal{H}_-}$ on the other side (call it $\Gamma_-$). So by Laguerre’s theorem, elements of $T(\Gamma_+)$ have some roots in ${\mathcal{H}_+}$ and elements of $T(\Gamma_-)$ have some roots in ${\mathcal{H}_-}$. Since polynomials near $p_0$ are $\overline{{\mathcal{H}_+}}$-stable, Lemma \[rs\_lemma\] implies elements of $T(\Gamma \cap B_\epsilon(p_0))$ are all ${\mathcal{H}_+}$-stable or ${\mathcal{H}_-}$-stable for some small ball $B_\epsilon(p_0)$ about $p_0$. So elements of $T(\Gamma_+ \cap B_\epsilon(p_0))$ are ${\mathcal{H}_-}$-stable and elements of $T(\Gamma_- \cap B_\epsilon(p_0))$ are ${\mathcal{H}_+}$-stable, and therefore $T(p_0)$ is real-rooted.
We now prove our main result on ray- and interval-stability preserving operators. First we state the theorem for closed bounded output intervals, as it clarifies the proof quite a bit. We will then extend the result to other connected regions in $\overline{{\mathbb{R}}}$.
\[Jops\_graces\] Fix $n,m \in {\mathbb{N}}_0$ and a linear operator $T \in \operatorname{Hom}(V(n),V(m))$ which restricts to a real linear operator. Further, let $I \subseteq {\mathbb{R}}$ be any interval, and let $J \subset {\mathbb{R}}$ be any closed bounded interval. The following are equivalent.
1. $T$ preserves weak real-rootedness and maps $I$-rooted polynomials to nonzero $J$-rooted polynomials.
2. One of the following holds:
1. $\operatorname{Symb}(T)$ is $({\mathcal{H}_-}\cup I) \times (\overline{{\mathcal{H}_+}} \setminus J)$-stable.
2. $\operatorname{Symb}(T)$ is $({\mathcal{H}_-}\cup I) \times (\overline{{\mathcal{H}_-}} \setminus J)$-stable.
3. $T$ has image of dimension at most two, and is of the form $$T: p \mapsto q \cdot \psi_1(p) + r \cdot \psi_2(p)$$ where $q,r \in V_{\mathbb{R}}(m)$ are top-degree monic and weakly $J$-rooted such that $q \ll r$, and $\psi_1$ and $\psi_2$ are real linear functionals such that $\psi_1(p) \cdot \psi_2(p) \geq 0$ (not both zero) holds for any $I$-rooted $p$.
$(i) \Rightarrow (ii)$. Suppose $T$ is nondegenerate. So, $\operatorname{Symb}(T)$ is either $({\mathcal{H}_-}\times {\mathcal{H}_+})$-stable or $({\mathcal{H}_-}\times {\mathcal{H}_-})$-stable by Corollary \[bb\_Rops\]. By Lemma \[bb\_Rlink\], either $T$ maps $\overline{{\mathcal{H}_+}}$-stable evaluation symbols entirely to nonzero ${\mathcal{H}_+}$-stable polynomials or entirely to nonzero ${\mathcal{H}_-}$-stable polynomials. If for some $(z_0:w_0) \in {\mathcal{H}_-}$, $T[(w_0x-z_0y)^n]$ has a root in $\overline{{\mathbb{R}}}$, we can apply the previous lemma. If condition $(a)$ of the lemma holds, then $T[(w_0x-z_0y)^n]$ is real-rooted or identically zero. The proof of Lemma \[bb\_Rlink\] then implies $T$ is degenerate, a contradiction. Otherwise condition $(b)$ of the lemma holds, and therefore the real roots of $T[(w_0x-z_0y)^n]$ must be in $J$. So in fact, $T$ maps $\overline{{\mathcal{H}_+}}$-stable evaluation symbols entirely to nonzero $(\overline{{\mathcal{H}_+}} \setminus J)$-stable polynomials or entirely to nonzero $(\overline{{\mathcal{H}_-}} \setminus J)$-stable polynomials. Finally, $T$ maps $I$-rooted evaluation symbols to nonzero $(\overline{{\mathcal{H}_+}} \setminus J)$-stable and $(\overline{{\mathcal{H}_-}} \setminus J)$-stable polynomials by assumption. The complex characterization (Theorem \[Cops\_graces\]) then implies $(a)$ or $(b)$.
Otherwise, $T$ is degenerate and $T[V_{\mathbb{R}}(n)]$ consists entirely of real-rooted polynomials. Condition $(c)$ follows from Lemma \[conv\_lemma\].
$(ii) \Rightarrow (i)$. By Corollary \[bb\_Rops\], $T$ preserves weak real-rootedness. If $(a)$ or $(b)$ holds, then the complex characterization (Theorem \[Cops\_graces\]) and the fact that $T$ restricts to a real operator imply $T$ maps $I$-rooted polynomials to nonzero $J$-rooted polynomials.
Otherwise $(c)$ holds. For any real-rooted $p$, let $\lambda(p)$ and $\mu(p)$ denote the largest and smallest roots of $p$, respectively. Since $q,r$ are top-degree monic, every convex combination of $q$ and $r$ has all its roots in the interval $[\mu(q), \lambda(r)] \subseteq J$. Since $\psi_1 \cdot \psi_2 \geq 0$ (not both zero) holds for $I$-rooted polynomials, we have that $T$ maps $I$-rooted polynomials to nonzero $J$-rooted polynomials.
Notice that this result immediately holds for other closed, connected regions $I,J \subset \overline{{\mathbb{R}}}$ by the action of some appropriate $\phi \in SL_2({\mathbb{R}})$. In fact, one can directly apply the action of $\phi$ to conditions $(ii)(a)$ and $(ii)(b)$, due to the fact that our definition of the “universal” symbol works for any projectively convex regions. The only significant change comes when applying $\phi$ to condition $(ii)(c)$. Further, the only issue with $(ii)(c)$ as it is written now is the requirement that $p_1$ and $p_2$ be top-degree monic polynomials. Having zeros at infinity, for instance, means that a polynomial cannot ever be top-degree monic (as the leading homogeneous coefficient is 0). There are ways to rewrite $(ii)(c)$ that avoids this problem, but it is probably more intuitive to state the result as above and apply $\phi \in SL_2({\mathbb{R}})$.
Additionally, the result holds for open and half-open bounded intervals $J \subset {\mathbb{R}}$, with a bit of tweaking to condition $(ii)(c)$. (Again, the universality of the symbol means that $(ii)(a)$ and $(ii)(b)$ remain unchanged.) We state this in the following, where the action of $\phi \in SL_2({\mathbb{R}})$ can be used to obtain similar results regarding open and half-open connected regions in $\overline{{\mathbb{R}}}$.
\[Jops\_graces\_cor\] The previous theorem holds when $J \subset {\mathbb{R}}$ is an open (or half-open) bounded interval, given the following alterations to condition $(ii)(c)$: if the image of $T$ is of dimension exactly two, then $p_1,p_2 \in V_{\mathbb{R}}(m)$ are top-degree monic $\overline{J}$-rooted polynomials such that the largest root of $p_1$ and the smallest root of $p_2$ are in $J$ (for $p_1 \ll p_2$), and $\psi_1 \neq 0$ (resp. $\psi_2 \neq 0$) whenever $p_2$ (resp. $p_1$) is not $J$-rooted.
The condition that the largest root of $p_1$ and the smallest root of $p_2$ are in $J$ (and the fact that $p_1 \ll p_2$) implies that $\alpha p_1 + \beta p_2$ is $J$-rooted for all $\alpha,\beta > 0$. Applying Lemma \[conv\_lemma\] to $\overline{J}$ completes the proof.
We now give a few examples. The first demonstrates the necessity of the premise that $T$ preserves weak real-rootedness.
Consider the operator $T_n: V(n) \to V(n)$ defined via: $$T_n: x^ky^{n-k} \mapsto \operatorname{Hmg}_n[x(x-1)(x-2)\cdots(x-k+1)]$$ By Proposition 7.31 in [@fisk2006polys], $T_n$ preserves positive-rootedness for all $n$. However, $T_2$ does not preserve real-rootedness, for example. In particular: $$T_2(x^2+2xy+y^2) = x(x-y) + 2xy + y^2 = x^2 + xy + y^2$$ We now compute the symbol of $T_2$: $$\operatorname{Symb}(T_2) = T_2[(xw-zy)^2] = x(x-y)w^2 - 2xzyw + z^2y^2 = \operatorname{Hmg}_{(2,2)}[(x-z)^2 - x]$$ Notice that for $x = -1$, we have that $(-1-z)^2 + 1$ is not real rooted. Therefore, $\operatorname{Symb}(T_2)$ is neither $({\mathcal{H}_-}\cup (0,\infty)) \times (\overline{{\mathcal{H}_+}} \setminus (0,\infty))$-stable nor $({\mathcal{H}_-}\cup (0,\infty)) \times (\overline{{\mathcal{H}_-}} \setminus (0,\infty))$-stable (when the variables are ordered $(z:w),(x:y)$). That is, the operator $T_2$ does not contradict the previous theorem.
In the second example, we demonstrate root preservation properties of $f(\partial_x)$ for real-rooted $f$. These are standard results of the classical theory: see, e.g., Corollary 5.4.1 in [@rahman2002anthpoly].
For any real-rooted $f \in {\mathbb{C}}[x]$, consider the operator $D_f \in \operatorname{Hom}(V(n),V(n))$ defined via $D_f: g \mapsto f(y\partial_x)g$ (i.e., the homogenized version of $f(\partial_x)$). To determine properties of this operator, we first write: $$f(y\partial_x) = c_0 \prod_{j=1}^m (y\partial_x - \alpha_j)$$ Here, the $\alpha_j \in {\mathbb{R}}$ are the roots of $f$. Next, we compute the symbol of $(y\partial_x - \alpha_j) \in \operatorname{Hom}(V(n),V(n))$ for $j \in [m]$: $$\begin{split}
\operatorname{Symb}(y\partial_x - \alpha_j) &= (y\partial_x - \alpha_j)(zy-xw)^n \\
&= -(\alpha_j (zy-xw) + nwy)(zy-xw)^{n-1} \\
&= \operatorname{Hmg}_{(n,n)}\left[-(\alpha_j(z-x)+n)(z-x)^{n-1}\right]
\end{split}$$ We now have three cases, depending on the sign of $\alpha_j$. If $\alpha_j > 0$, we have that $\operatorname{Symb}(y\partial_x - \alpha_j)$ is both $({\mathcal{H}_-}\cup [a,\infty]) \times (\overline{{\mathcal{H}_+}} \setminus [a,\infty])$-stable and $({\mathcal{H}_-}\cup [-\infty,a]) \times (\overline{{\mathcal{H}_+}} \setminus [-\infty,a+\frac{n}{\alpha_j}])$-stable for any $a \in {\mathbb{R}}$. (As usual, we order the variables $(z:w),(x:y)$.) Using Theorem \[Jops\_graces\] and the discussion following the proof, this implies $(y\partial_x - \alpha_j)$ preserves $[a,\infty]$-rootedness and maps $[-\infty,a]$-rooted polynomials to $[-\infty,a+\frac{n}{\alpha_j}]$-rooted polynomials. So, if the (non-infinite) roots of $g$ are contained in the interval $[b,c]$, then the (non-infinite) roots of $(y\partial_x - \alpha_j)g$ are contained in the interval $[b,c+\frac{n}{\alpha_j}]$.
If $\alpha_j < 0$, we have that $\operatorname{Symb}(y\partial_x - \alpha_j)$ is both $({\mathcal{H}_-}\cup [a,\infty]) \times (\overline{{\mathcal{H}_+}} \setminus [a+\frac{n}{\alpha_j},\infty])$-stable and $({\mathcal{H}_-}\cup [-\infty,a]) \times (\overline{{\mathcal{H}_+}} \setminus [-\infty,a])$-stable for any $a \in {\mathbb{R}}$. As above, this implies $(y\partial_x - \alpha_j)$ preserves $[-\infty,a]$-rootedness and maps $[a,\infty]$-rooted polynomials to $[a+\frac{n}{\alpha_j},\infty]$-rooted polynomials. So, if the (non-infinite) roots of $g$ are contained in the interval $[b,c]$, then the (non-infinite) roots of $(y\partial_x - \alpha_j)g$ are contained in the interval $[b+\frac{n}{\alpha_j},c]$.
Finally for $\alpha_j = 0$, the operator $(y\partial_x - \alpha_j) = y\partial_x$ weakly preserves any interval in which the (non-infinite) roots reside. The main difference for this case is that $y\partial_x$ only preserves weak real-rootedness. Combining these three cases, we are lead to the following root preservation property of $f(\partial_x): {\mathbb{C}}_n[x] \to {\mathbb{C}}_n[x]$. Let $\alpha_j^+$ and $\alpha_j^-$ be the positive and negative roots of $f$, respectively. We then have the following, which refers to non-infinite roots: $$f(\partial_x): [b,c]\text{-rooted} \to \left[b + \sum_j \frac{n}{\alpha_j^-}, c + \sum_j \frac{n}{\alpha_j^+}\right]\text{-rooted}$$ If $f$ has zeros at 0, then $f(\partial_x)$ may map some nonzero $[b,c]$-rooted polynomials to 0. Otherwise, $f(\partial_x)$ is invertible on ${\mathbb{C}}_n[x]$.
The Grace-Walsh-Szego Coincidence Theorem {#gws_app}
=========================================
A classical result in the representation theory of $SL_2({\mathbb{C}})$ is the fact that $V(n) \cong \operatorname{Sym}^n(V(1))$. Here, $\operatorname{Sym}^n(V(1))$ denotes the set of symmetric tensors in $V(1)^{\otimes n}$, or alternatively, the set of symmetric elements in $V(1^n)$. That is, there is some $SL_2({\mathbb{C}})$-invariant injection from $V(n)$ to $V(1)^{\otimes n}$, and by our conceptual thesis this map should transfer stability information. In fact, this idea is formalized in the Grace-Walsh-Szeg[ő]{} coincidence theorem, and the injective map is known as the polarization map.
Polarization and Projection
---------------------------
For polynomials of degree $m \leq n$, the degree-$n$ polarization map is defined on monomials as follows and is extended linearly. $$\begin{split}
\Pi_n^\uparrow : {\mathbb{C}}_n[x] &\rightarrow {\mathbb{C}}_{(1^n)}[x_1,...,x_n] \\
x^k &\mapsto \frac{1}{n!} \sum_{\sigma \in S_n} \prod_{j=1}^k x_{\sigma(j)}
\end{split}$$ This definition can be extended to homogeneous polynomials in $V(n)$ by composing with $\operatorname{Hmg}_n^{-1}$ and $\operatorname{Hmg}_{(1^n)}$. The map $\Pi_n^\uparrow$ has a left inverse $\Pi_n^\downarrow$, called the projection map, which we define as follows. $$\begin{split}
\Pi_n^\downarrow : {\mathbb{C}}_{(1^n)}[x_1,...,x_n] &\rightarrow {\mathbb{C}}_{n}[x] \\
f(x_1,x_2,...,x_n) &\mapsto f(x,x,...,x)
\end{split}$$ That is, $\Pi_n^\downarrow \circ \Pi_n^\uparrow$ is the identity map. Similarly, this definition can be extended to homogeneous polynomials by composing with $\operatorname{Hmg}_{(1^n)}^{-1}$ and $\operatorname{Hmg}_n$.
It is well-known that $\Pi_n^\uparrow$ is an injective linear map onto the subspace of symmetric multi-affine polynomials. This fact then extends to homogeneous polynomials, where the terms *symmetric* and *multi-affine* each refer to pairs of homogeneous variables. Further, one can define multivariate polarization and projection maps via composition: $\Pi_\lambda^\uparrow := \Pi_{\lambda_m}^\uparrow \circ \cdots \circ \Pi_{\lambda_1}^\uparrow$ and $\Pi_\lambda^\downarrow := \Pi_{\lambda_m}^\downarrow \circ \cdots \circ \Pi_{\lambda_1}^\downarrow$. Injectivity then automatically extends to $\Pi_\lambda^\uparrow$, and $\Pi_\lambda^\downarrow \circ \Pi_\lambda^\uparrow$ is the identity map.
These two maps arise naturally in the theory of polynomials in general, and play an important role in the theory of stability, via the Grace-Walsh-Szeg[ő]{} coincidence theorem as well as in the proof of the Borcea-Br[ä]{}nd[é]{}n characterization of linear operators. The next result shows they also have represention theoretic importance.
\[pol\_proj\_inv\_prop\] Fix $\lambda \in {\mathbb{N}}_0^m$, and view $V(\lambda) \cong V(\lambda_1) \boxtimes \cdots \boxtimes V(\lambda_m)$ and $V(1^\lambda) \cong V(1)^{\otimes \lambda_1} \boxtimes \cdots \boxtimes V(1)^{\otimes \lambda_m}$ as representations of $(SL_2({\mathbb{C}}))^m$. The maps $\Pi_\lambda^\uparrow: V(\lambda) \rightarrow V(1^\lambda)$ and $\Pi_\lambda^\downarrow: V(1^\lambda) \rightarrow V(\lambda)$ are $(SL_2({\mathbb{C}}))^m$-invariant.
Note that by proving the result for $\Pi_n^\uparrow$ and $\Pi_n^\downarrow$ with $m=1$, the general result follows since $\Pi_\lambda^\uparrow$ and $\Pi_\lambda^\downarrow$ are compositions of such maps. To prove it for $m=1$, note that the set of symmetric elements in $V(1)^{\otimes n} \cong V(1^n)$ is invariant under the diagonal action of $SL_2({\mathbb{C}})$. Further, since $V(n)$ is irreducible of dimension $n+1$ and $V(1)^{\otimes n}$ has a single irreducible component of dimension $n+1$, Schur’s lemma implies the result.
This result then has a few corollaries which will help to shed light on results related to polarization and the apolarity form. The first will be useful in elucidating the representation theoretic ties to the Grace-Walsh-Szeg[ő]{} coincidence theorem below.
\[form\_pol\_commute\_lemma\] Fix $\lambda \in {\mathbb{N}}_0^m$. Then the apolarity form commutes with polarization up to scalar. That is: $$\times \circ D^\lambda = \times \circ D^{(1^\lambda)} \circ (\Pi_\lambda^\uparrow \otimes \Pi_\lambda^\uparrow)$$
The map $\times \circ D^{(1^\lambda)} \circ (\Pi_\lambda^\uparrow \otimes \Pi_\lambda^\uparrow): V(\lambda) \otimes V(\lambda) \rightarrow {\mathbb{C}}$ is an $(SL_2({\mathbb{C}}))^m$-invariant bilinear form on $V(\lambda)$. By uniqueness (see Definition \[Mform\_def\]), this then must equal $\times \circ D^\lambda$ up to scalar.
The content of this result is that fact that we have commutativity even though $\times \circ D^{(1^\lambda)}$ is a priori the apolarity form with respect to a different group action than that of $\times \circ D^\lambda$ (i.e., $(SL_2({\mathbb{C}}))^{|\lambda|}$ instead of $(SL_2({\mathbb{C}}))^m$). That said, it should be noted that the analogous commutativity statement with the projection map $\Pi_\lambda^\downarrow$ does *not* hold (unless of course, one restricts to the image of $\Pi_\lambda^\uparrow$).
The purpose of this result is then to demonstrate the connection between a polynomial and its polarization. In particular, if Grace’s theorem gives stability information via the apolarity form, then the previous result shows that the polarizations of those polynomials will have the same stability information. We prove this rigorously in Corollary \[gws\_cor\].
Proposition \[pol\_proj\_inv\_prop\] also leads to one of the crucial results used in the proof of the Borcea-Br[ä]{}nd[é]{}n characterization of linear operators (Lemma 2.5 in [@bb1]). It relies on the notion of “the polarization of an operator”, given by $T \mapsto \Pi_\alpha^\uparrow \circ T \circ \Pi_\lambda^\downarrow$ (see §2.2 in [@bb1]). We do not make explicit use of this result, but we state it here to demonstrate that operator polarization has a representation theoretic interpretation similar to that of the usual polynomial polarization.
The symbol of the polarization of an operator $T$ is the polarization of the symbol of $T$.
Using Proposition \[pol\_proj\_inv\_prop\] and Definition \[symb\_def\], it is straightforward to see that all the maps involved are injective $(SL_2({\mathbb{C}}))^{2m}$-invariant linear maps (i.e., polarization of polynomials, polarization of operators, the $\operatorname{Symb}$ map). The result then follows from a dimension argument and Schur’s lemma, in a way similar to that of the proof of Proposition \[pol\_proj\_inv\_prop\].
The Coincidence Theorem
-----------------------
The Grace-Walsh-Szeg[ő]{} coincidence theorem has strong ties to Grace’s theorem, and most books and surveys on the subject state the two results side by side. Some books (e.g., [@rahman2002anthpoly]) even go so far as to demonstrate their equivalence, perhaps with other results involving typical polynomial convolutions. Here, we will state and prove the general multivariate version of the theorem in terms of homogeneous polynomials, making use of evaluation symbols and Grace’s theorem.
First though, consider the following corollary to the symbol lemma which is similar in spirit to the evaluation symbol lemma (\[ev\_lemma\]). Note that when applied to $p \in V(n)$ with $m=1$, this result has the following intuitive statement as a corollary: *$D^n(q \otimes p)$ is equal to the evaluation of $\Pi_n^\uparrow p$ at the roots of $q$*.
Fix $\lambda \in {\mathbb{N}}_0^m$, $p \in V(\lambda)$, and any $(a:b) \in ({\mathbb{CP}}^1)^{|\lambda|}$, more explicitly defined as follows: $$(a:b) \equiv (a_{1,1}:b_{1,1}, ..., a_{1,\lambda_1}:b_{1,\lambda_1}, ..., a_{m,1}:b_{m,1}, ..., a_{m,\lambda_m}:b_{m,\lambda_m}) \in ({\mathbb{CP}}^1)^{\lambda_1 + \cdots + \lambda_m}$$ We have the following: $$(\Pi_\lambda^\uparrow p)(a:b) = D^\lambda\left(\Pi_\lambda^\downarrow(\operatorname{Symb}(\operatorname{ev}_{(a:b)})) \otimes p\right) \equiv D^\lambda\left(\prod_{k=1}^m \prod_{j=1}^{\lambda_k} (b_{k,j}x_k - a_{k,j}y_k) \otimes p \right)$$
Lemma \[form\_pol\_commute\_lemma\] implies: $$D^\lambda\left(\Pi_\lambda^\downarrow(\operatorname{Symb}(\operatorname{ev}_{(a:b)})) \otimes p\right) = D^{(1^\lambda)}\left(\Pi_\lambda^\uparrow \circ \Pi_\lambda^\downarrow(\operatorname{Symb}(\operatorname{ev}_{(a:b)})) \otimes \Pi_\lambda^\uparrow p\right)$$ Since $\Pi_\lambda^\uparrow \circ \Pi_\lambda^\downarrow(\operatorname{Symb}(\operatorname{ev}_{(a:b)}))$ is the symmetrization of $\operatorname{Symb}(\operatorname{ev}_{(a:b)})$ in each set of $\lambda_k$ pairs of variables, we can write it as a sum over $S_{\lambda_1} \times \cdots \times S_{\lambda_m}$ (product of symmetric groups) of $\operatorname{Symb}(\operatorname{ev}_{(a:b)})$ with permuted variables. Since $\Pi_\lambda^\uparrow p$ is symmetric in each set of $\lambda_k$ pairs of variables, we then have: $$D^{(1^\lambda)}\left(\Pi_\lambda^\uparrow \circ \Pi_\lambda^\downarrow(\operatorname{Symb}(\operatorname{ev}_{(a:b)})) \otimes \Pi_\lambda^\uparrow p\right) = D^{(1^\lambda)}\left(\operatorname{Symb}(\operatorname{ev}_{(a:b)}) \otimes \Pi_\lambda^\uparrow p\right)$$ The result then follows from the evaluation symbol lemma (\[ev\_lemma\]).
Generally speaking, the above lemma demonstrates the strong connection between the apolarity form and the polarization map. We now utilize this to prove the coincidence theorem.
\[gws\_cor\] Fix $\lambda \in {\mathbb{N}}_0^m$, $p \in V(\lambda)$, and any disjoint Grace pair $(C_1 \times \cdots \times C_m, B_1 \times \cdots \times B_m)$. If $p$ is $(C_1 \times \cdots \times C_m)$-stable, then $\Pi_\lambda^\uparrow p$ is $(C_1^{\lambda_1} \times \cdots \times C_m^{\lambda_m})$-stable.
So as to prove the contrapositive, suppose $\Pi_\lambda^\uparrow p$ is not $(C_1^{\lambda_1} \times \cdots \times C_m^{\lambda_m})$-stable. That is, suppose $(\Pi_\lambda^\uparrow p)(a:b) = 0$ for some $(a:b) \equiv (a_{1,1}:b_{1,1},...,a_{m,\lambda_m}:b_{m,\lambda_m}) \in C_1^{\lambda_1} \times \cdots \times C_m^{\lambda_m}$. By the previous lemma, this implies: $$D^\lambda\left(\Pi_\lambda^\downarrow(\operatorname{Symb}(\operatorname{ev}_{(a:b)})) \otimes p\right) = 0$$ By disjointness, $\operatorname{Symb}(\operatorname{ev}_{(a:b)})$ is $(B_1^{\lambda_1} \times \cdots \times B_m^{\lambda_m})$-stable, which implies $\Pi_\lambda^\downarrow(\operatorname{Symb}(\operatorname{ev}_{(a:b)}))$ is $(B_1 \times \cdots \times B_m)$-stable. Grace’s theorem (\[graces\_thm\]) then implies $p$ is not $(C_1 \times \cdots \times C_m)$-stable.
By Theorem \[grace\_pairs\_thm\], this implies the coincidence theorem for circular regions when $m > 1$ and for any projectively convex regions when $m = 1$. Notice that there is no reference made to degree or convexity restrictions (compare this to Theorems 1.1 and 1.2 in [@bb2]). This is due to the fact that homogeneity and the interpretation of zeros as being in ${\mathbb{CP}}^1$ symmetrizes the Riemann sphere, so to speak. That is, the point $\infty \in {\mathbb{CP}}^1$ can be thought of as a generic point.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Discrete time dynamics on the $SU\left( 1,1\right) $ group is studied. It is shown that a map acting in the dual space is the stroboscopic map for a $SU\left( 1,1\right) $ Bloch equation. Exact solution of the map is used to elucidate the corresponding dynamics. It is shown that dynamics of the $SU\left( 1,1\right) $ Bloch equation in the elliptic case bears close analogy to the $SU\left( 2\right) $ Bloch dynamics.'
author:
- |
A. Okninski\
Physics Division, Politechnika Swietokrzyska,\
Al. 1000-lecia PP 7, 25-314 Kielce, Poland.
title: '$SU(1,1)$ group action and the corresponding Bloch equation'
---
Introduction
============
Discrete-time dynamical systems can be formulated in terms of group actions to exploit the group structure and get a better understanding of the corresponding dynamics. This approach was used to study discrete-time dynamics on some groups, see [@Okninski2009] and references therein. On the other hand, structure of Kleinian groups is naturally studied in the setting of discrete-time dynamical systems, revealing in this way connections with fractals [@Brooks1981; @Gehring1989; @Mumford2002]. For example, the Shimizu-Leutbecher map is a typical tool to study group structure [@Shimizu1963; @Leutbecher1967], see also [@Brooks1981; @Gehring1989; @Beardon1983].
Recently, we have investigated a possibility of relating group actions with stroboscopic maps of ordinary differential equations [@Okninski2009]. More exactly, we have studied the following dynamical system on a Lie group $\mathcal{G}$:$$R_{N+1}=Q_{N}R_{N}Q_{N-1}R_{N-1}Q_{N-1}^{-1}R_{N}^{-1}Q_{N}^{-1},\qquad
N=1,2,\ldots, \label{Q_NR_N}$$ where $Q_{N},\ R_{N}\in\mathcal{G}$. A general solution of the map (\[Q\_NR\_N\]) has been constructed and it was demonstrated that for $\mathcal{G}=SU\left( 2\right) $ and $Q_{N}\equiv Q$ Eq.(\[Q\_NR\_N\]) is a stroboscopic map of the Bloch equation [@Okninski2009]. The latter result is generalized in the present paper for the case $\mathcal{G}=SU\left(
1,1\right) $.
Let us note here that the $SU\left( 1,1\right) $ Bloch equation finds important applications in quantum optics. More exactly, the group of squeezings is generated by the Lie algebra $\mathfrak{su}\left( 1,1\right)
$, the geometry of group manifold is that of Minkowski spaces and time evolution is described by the $SU\left( 1,1\right) $ Bloch equation [@Dattoli1986; @Aravind1988; @King1999; @Puri2001].
The paper is organized as follows. In the next Section discrete time dynamics, defined and solved in [@Okninski2009] for arbitrary group $\mathcal{G}$, is studied in the case $\mathcal{G}=SU\left( 1,1\right) $. It is shown in Section 3 that the map (\[Q\_NR\_N\]), considered in the dual space, is the stroboscopic map for the $SU\left( 1,1\right) $ Bloch equation which is written in the elliptic, parabolic and hyperbolic cases. In Section 4 computations intended to elucidate dynamics of the $SU\left( 1,1\right)
$ Bloch equation are presented for the elliptic case and analogy with the $SU\left( 2\right) $ Bloch equation is stressed. The obtained results are summarized in the last Section.
Discrete-time dynamics on the $SU\left( 1,1\right) $ group
============================================================
Let us recall that the map (\[Q\_NR\_N\]) admits an exact solution:
\[SOL\]$$\begin{aligned}
R_{2K} & =S_{2K}S_{2K-1}\ldots S_{2}R_{0}S_{2}^{-1}\ldots S_{2K-1}^{-1}S_{2K}^{-1},\label{solR}\\
S_{N} & =Q_{N-1}\ldots Q_{1}S_{1}Q_{-1}^{-1}\ldots Q_{N-3}^{-1},
\label{solS}$$ where
$$S_{N}\overset{df}{=}R_{N}Q_{N-1}R_{N-1}Q_{N-2}, \label{defS}$$
and similar equations can be written for $R_{2K+1}$ [@Okninski2009].
We shall consider a special case $Q_{N}\equiv Q$ in (\[Q\_NR\_N\]). In the case $\mathcal{G}=SU(1,1)$ the following parameterization is used [@Chiribella2006]:$$\begin{aligned}
R_{N} & =\exp\left( i\tfrac{\chi_{N}}{2}\overrightarrow{\kappa}\cdot\overrightarrow{r_{N}}\right) ,\qquad\overrightarrow{r_{N}}\cdot\overrightarrow{r_{N}}=\eta,\label{defR}\\
Q & =\exp\left( i\tfrac{\alpha}{2}\overrightarrow{\kappa}\cdot
\overrightarrow{q}\right) ,\hspace{0.4in}\overrightarrow{q}\cdot
\overrightarrow{q}=\eta, \label{defQ}$$ where $i^{2}=-1$, $\eta\in\left\{ +1,0,-1\right\} $ and is fixed, $\overrightarrow{\kappa}\overset{df}{=}\left[ i\sigma^{1},i\sigma^{2},\sigma^{3}\right] $ where $\sigma^{1}$, $\sigma^{2}$, $\sigma^{3}$ are the Pauli matrices and scalar products are defined as
\[DEFSCALAR\]$$\begin{aligned}
& \overrightarrow{\kappa}\cdot\overrightarrow{x}\overset{df}{=}-\kappa
^{1}x^{1}-\kappa^{2}x^{2}+\kappa^{3}x^{3},\label{defkx}\\
& \overrightarrow{x}\cdot\overrightarrow{x}\overset{df}{=}-x^{1}x^{1}-x^{2}x^{2}+x^{3}x^{3}, \label{defxx}$$ where $\overrightarrow{x}\overset{df}{=}\left[ x^{1},x^{2},x^{3}\right] $. The three cases $\eta=+1,0,-1$ are referred to as elliptic, parabolic and hyperbolic, respectively. For an exposition of theory of the $SU(1,1)$ group the reader can consult [@Chiribella2006; @Wawrzynczyk1984].
We obtain from (\[SOL\]) the following solution:
$$R_{2K}=Q^{2K}P^{K}R_{0}P^{-K}Q^{-2K},\label{2k}$$
where $P\overset{df}{=}Q^{-1}S_{1}Q^{-1}=Q^{-1}R_{1}QR_{0}
,\ K=0,\ 1,\ 2,\ \ldots\ $. We still have to impose initial condition $R_{0}$ while $R_{1}$ is computed as $R_{1}=QPR_{0}^{-1}Q^{-1}$.
Matrix $P$ is parameterized in form$$P=\exp\left( i\tfrac{\beta}{2}\overrightarrow{\kappa}\cdot\overrightarrow
{p}\right) ,\qquad\overrightarrow{p}\cdot\overrightarrow{p}=\eta, \label{P}$$ and equation (\[2k\]) can be written as$$\overrightarrow{\kappa}\cdot\overrightarrow{r}_{2K}=\exp\left( iK\alpha
\overrightarrow{\kappa}\cdot\overrightarrow{q}\right) \exp\left(
iK\tfrac{\beta}{2}\overrightarrow{\kappa}\cdot\overrightarrow{p}\right)
\,\overrightarrow{\kappa}\cdot\overrightarrow{r}_{0}\,\exp\left(
-iK\tfrac{\beta}{2}\overrightarrow{\kappa}\cdot\overrightarrow{p}\right)
\exp\left( -iK\alpha\overrightarrow{\kappa}\cdot\overrightarrow{q}\right)
\mathbf{,} \label{RSsol1a}$$ and, after introducing new quantities, $K\alpha=\theta,\ \lambda=\frac{\beta
}{2\alpha},$ reads $$\overrightarrow{\kappa}\cdot\overrightarrow{r}\left( \theta\right)
=\exp\left( i\theta\overrightarrow{\kappa}\cdot\overrightarrow{q}\right)
\exp\left( i\lambda\theta\overrightarrow{\kappa}\cdot\overrightarrow
{p}\right) \,\overrightarrow{\kappa}\cdot\overrightarrow{r}_{0}\,\exp\left(
-i\lambda\theta\overrightarrow{\kappa}\cdot\overrightarrow{p}\right)
\exp\left( -i\theta\overrightarrow{\kappa}\cdot\overrightarrow{q}\right)
\mathbf{,} \label{RSsol1b}$$ where $\overrightarrow{r}\left( \theta\right) \overset{df}{=}\overrightarrow
{r}_{2K}$.
Elliptic case
-------------
To obtain $\overrightarrow{r}\left( \theta\right) $ from Eq.(\[RSsol1b\]) we shall need to compute $\overrightarrow{t}\left( \gamma\right) $ given by:$$\overrightarrow{\kappa}\cdot\overrightarrow{t}\left( \gamma\right)
=S\ \overrightarrow{\kappa}\cdot\overrightarrow{t}\ S^{-1}=\exp\left(
i\tfrac{\gamma}{2}\overrightarrow{\kappa}\cdot\overrightarrow{s}\right)
\,\overrightarrow{\kappa}\cdot\overrightarrow{t}\,\exp\left( -i\tfrac{\gamma
}{2}\overrightarrow{\kappa}\cdot\overrightarrow{s}\right) , \label{deft}$$ in the case $\eta=1$. The exponential map is simplified as$$\exp\left( i\tfrac{\gamma}{2}\overrightarrow{\kappa}\cdot\overrightarrow
{s}\right) =\cos\left( \tfrac{\gamma}{2}\right) \mathbf{1}+i\sin\left(
\tfrac{\gamma}{2}\right) \overrightarrow{\kappa}\cdot\overrightarrow
{s},\qquad\overrightarrow{s}\cdot\overrightarrow{s}=1. \label{expe}$$
Now, due to properties of the Pauli matrices we obtain from (\[deft\]):$$\overrightarrow{t}\left( \gamma\right) =\cos\left( \gamma\right)
\overrightarrow{t}-\sin\left( \gamma\right) \overrightarrow{t}\times\overrightarrow{s}+\left( 1-\cos\left( \gamma\right) \right) \left(
\overrightarrow{t}\cdot\overrightarrow{s}\right) \overrightarrow{s},
\label{te}$$ where$$\overrightarrow{x}\times\overrightarrow{y}\overset{df}{=}\left[ -x_{2}y_{3}+x_{3}y_{2},-x_{3}y_{1}+x_{1}y_{3},x_{1}y_{2}-x_{2}y_{1}\right] .
\label{defxy}$$
Using twice the equation (\[te\]) in (\[RSsol1b\]) we get:
\[EXPLE\]$$\begin{aligned}
\overrightarrow{r}\left( \theta\right) & =\cos\left( 2\theta\right)
\overrightarrow{t}\left( \theta\right) -\sin\left( 2\theta\right)
\overrightarrow{t}\left( \theta\right) \times\overrightarrow{q}+\left(
1-\cos\left( 2\theta\right) \right) \left( \overrightarrow{q}\cdot\overrightarrow{t}\left( \theta\right) \right) \overrightarrow
{q}\mathbf{,}\label{r1e}\\
\overrightarrow{t}\left( \theta\right) & =\cos\left( 2\lambda
\theta\right) \overrightarrow{r}_{0}-\sin\left( 2\lambda\theta\right)
\overrightarrow{r}_{0}\times\overrightarrow{p}+\left( 1-\cos\left(
2\lambda\theta\right) \right) \left( \overrightarrow{p}\cdot\overrightarrow
{r}_{0}\right) \overrightarrow{p}. \label{r2e}$$ For growing $\theta$ the vector $\overrightarrow{r}\left( \theta\right) $ evolves on manifold $\overrightarrow{r}\left( \theta\right) \cdot
\overrightarrow{r}\left( \theta\right) =1$, i.e. on two-sheeted hyperboloid.
Parabolic case
--------------
In the case $\eta=0$ the exponential map reduces to
$$\exp\left( i\tfrac{\gamma}{2}\overrightarrow{\kappa}\cdot\overrightarrow
{s}\right) =\mathbf{1}+i\tfrac{\gamma}{2}\overrightarrow{\kappa}\cdot\overrightarrow{s},\qquad\overrightarrow{s}\cdot\overrightarrow{s}=0.
\label{expp}$$
Now, due to properties of the Pauli matrices we compute from (\[deft\]):$$\overrightarrow{t}\left( \gamma\right) =\overrightarrow{t}-\gamma
\overrightarrow{t}\times\overrightarrow{s}+\left( \overrightarrow{t}\cdot\overrightarrow{s}\right) \overrightarrow{s}. \label{tp}$$ Using twice the equation (\[tp\]) in (\[RSsol1b\]) we obtain:
\[EXPLP\]$$\begin{aligned}
\overrightarrow{r}\left( \theta\right) & =\overrightarrow{t}\left(
\theta\right) -2\theta\overrightarrow{t}\left( \theta\right) \times
\overrightarrow{q}+\left( \overrightarrow{q}\cdot\overrightarrow{t}\left(
\theta\right) \right) \overrightarrow{q}\mathbf{,}\label{r1p}\\
\overrightarrow{t}\left( \theta\right) & =\overrightarrow{r}_{0}-2\lambda\theta\overrightarrow{r}_{0}\times\overrightarrow{p}+\left(
\overrightarrow{p}\cdot\overrightarrow{r}_{0}\right) \overrightarrow{p}.
\label{r2p}$$ The vector $\overrightarrow{r}\left( \theta\right) $ evolves on manifold $\overrightarrow{r}\left( \theta\right) \cdot\overrightarrow{r}\left(
\theta\right) =0$, i.e. on the Minkowski cone.
Hyperbolic case
---------------
In the case $\eta=-1$ we have
$$\exp\left( i\tfrac{\gamma}{2}\overrightarrow{\kappa}\cdot\overrightarrow
{s}\right) =\cosh\left( \tfrac{\gamma}{2}\right) \mathbf{1}+i\sinh\left(
\tfrac{\gamma}{2}\right) \overrightarrow{\kappa}\cdot\overrightarrow
{s},\qquad\overrightarrow{s}\cdot\overrightarrow{s}=-1. \label{exph}$$
Due to properties of the Pauli matrices we obtain from (\[deft\]):$$\overrightarrow{t}\left( \gamma\right) =\cosh\left( \gamma\right)
\overrightarrow{t}-\sinh\left( \gamma\right) \overrightarrow{t}\times\overrightarrow{s}+\left( \cosh\left( \beta\right) -1\right) \left(
\overrightarrow{t}\cdot\overrightarrow{s}\right) \overrightarrow{s}.
\label{th}$$
Using twice the equation (\[th\]) in (\[RSsol1b\]) we get:
\[EXPLH\]$$\begin{aligned}
\overrightarrow{r}\left( \theta\right) & =\cosh\left( 2\theta\right)
\overrightarrow{t}\left( \theta\right) -\sinh\left( 2\theta\right)
\overrightarrow{t}\left( \theta\right) \times\overrightarrow{q}+\left(
\cosh\left( 2\theta\right) -1\right) \left( \overrightarrow{q}\cdot\overrightarrow{t}\left( \theta\right) \right) \overrightarrow
{q}\mathbf{,}\label{r1h}\\
\overrightarrow{t}\left( \theta\right) & =\cosh\left( 2\lambda
\theta\right) \overrightarrow{r}_{0}-\sinh\left( 2\lambda\theta\right)
\overrightarrow{r}_{0}\times\overrightarrow{p}+\left( \cosh\left(
2\lambda\theta\right) -1\right) \left( \overrightarrow{p}\cdot
\overrightarrow{r}_{0}\right) \overrightarrow{p}. \label{r2h}$$ The vector $\overrightarrow{r}\left( \theta\right) $ evolves on manifold $\overrightarrow{r}\left( \theta\right) \cdot\overrightarrow{r}\left(
\theta\right) =-1$, i.e. on one-sheeted hyperboloid.
Symmetry and restrictions of dynamics
-------------------------------------
Dynamical system (\[Q\_NR\_N\]) for $Q_{N}\equiv Q$ has continuous symmetry:
$$R_{N}\rightarrow Q^{\kappa}R_{N}Q^{-\kappa},\qquad\forall\kappa\in\mathbb{R}.
\label{sym}$$
It can be thus expected that dynamics of the quantity $\overrightarrow
{r}\left( \theta\right) \cdot\overrightarrow{q}$ should decouple from other degrees of freedom in (\[Q\_NR\_N\]) [@Okninski2009]. Indeed, it follows from (\[EXPLE\]) that $$\overrightarrow{r}\left( \theta\right) \cdot\overrightarrow{q}=\overrightarrow{t}\left( \theta\right) \cdot\overrightarrow{q}.
\label{decoupling}$$ Since in the elliptic or hyperbolic case $\overrightarrow{t}\cdot
\overrightarrow{t}=\overrightarrow{q}\cdot\overrightarrow{q}=\pm1$ it follows from the Schwartz inequality for the Minkowski metric that $\left(
\overrightarrow{t}\cdot\overrightarrow{q}\right) ^{2}\geq\left(
\overrightarrow{t}\cdot\overrightarrow{t}\right) \left( \overrightarrow
{q}\cdot\overrightarrow{q}\right) =1$. Now, for given $\overrightarrow{p}$, $\overrightarrow{q}$ and $\overrightarrow{r}_{0}$ on the upper sheet of the hyperboloid in the elliptic case we have$$1\leq A_{1}\leq\overrightarrow{t}\left( \theta\right) \cdot\overrightarrow
{q}\leq A_{2}, \label{bounds1}$$ The constants $A_{1,2}$ depending on the parameters $\overrightarrow{p}$, $\overrightarrow{q}$ and the initial condition $\overrightarrow{r}_{0}$ can be computed from (\[r2e\]) by elementary means$$A_{1,2}=c\mp\sqrt{b^{2}+\left( a-c\right) ^{2}}, \label{bounds2}$$ where$$a=\overrightarrow{r}_{0}\cdot\overrightarrow{q}\mathbf{,\quad}b=\left(
\overrightarrow{r}_{0}\times\overrightarrow{p}\right) \cdot\overrightarrow
{q}\mathbf{,\quad}c=\left( \overrightarrow{p}\cdot\overrightarrow{r}_{0}\right) \left( \overrightarrow{p}\cdot\overrightarrow{q}\right) .
\label{bounds3}$$ It thus follows that the motion on the hyperboloid is bounded by two planes:* *$A_{1}\leq\overrightarrow{r}\left( \theta\right)
\cdot\overrightarrow{q}\leq A_{2}$. In the parabolic case we obtain the following simple condition$$\overrightarrow{t}\cdot\overrightarrow{q}=a-2\lambda\theta b+c, \label{cond}$$ i.e. equation of a straight line.
The Bloch equation
==================
The map (\[Q\_NR\_N\]), $Q_{N}\equiv Q$, is the stroboscopic map of a differential equation which is conveniently deduced from the form (\[RSsol1b\]). Since $\alpha$ and $\beta$ are arbitrary we shall treat $\theta$ as a continuous variable. Differentiating Eq.(\[RSsol1b\]) with respect to $\theta$ and using (\[RSsol1b\]) we get $$\dfrac{d\,\overrightarrow{\kappa}\cdot\overrightarrow{r}\left( \theta\right)
}{d\theta}=i\left[ \overrightarrow{\kappa}\cdot\overrightarrow{u}\left(
\theta\right) ,\ \overrightarrow{\kappa}\cdot\overrightarrow{r}\left(
\theta\right) \right] , \label{RSODE}$$ where $\left[ A,\ B\right] \overset{df}{=}AB-BA$ and
\[DEFS\]$$\begin{aligned}
\overrightarrow{u}\left( \theta\right) & =\overrightarrow{q}+\lambda\overrightarrow{p}\left( \theta\right) ,\quad\left( \lambda
=\beta/2\alpha\right) \label{defu}\\
\overrightarrow{\kappa}\cdot\overrightarrow{p}\left( \theta\right) &
=\exp\left( i\theta\overrightarrow{\kappa}\cdot\overrightarrow{q}\right)
\,\overrightarrow{\kappa}\cdot\overrightarrow{p}\,\exp\left( -i\theta
\overrightarrow{\kappa}\cdot\overrightarrow{q}\right) . \label{defp}$$
It follows that the sequence $\overrightarrow{r}_{0},\overrightarrow{r}_{2},\overrightarrow{r}_{4},\ldots,$ generated by $R_{0},R_{2},R_{4},\ldots,$ cf. Eq.(\[2k\]), interpolates flow of Eq.(\[RSODE\]). It turns out that, $\theta$ interpreted as time, Eq.(\[RSODE\]) is the $SU\left( 1,1\right)
$ Bloch equation, cf.[@King1999].
Equations (\[RSODE\]), (\[DEFS\]) can be written in explicit form.
Elliptic case
-------------
Using Eqs. (\[deft\]), (\[te\]) we get from (\[defp\])
$$\overrightarrow{p}\left( \theta\right) =\cos\left( \gamma\right)
\overrightarrow{p}-\sin\left( \gamma\right) \overrightarrow{p}\times\overrightarrow{q}+\left( 1-\cos\left( \gamma\right) \right) \left(
\overrightarrow{p}\cdot\overrightarrow{q}\right) \overrightarrow{q},
\label{defpe}$$
and using properties of the Pauli matrices we obtain from (\[RSODE\]) the Bloch equation in the elliptic case:$$\frac{d\,\overrightarrow{r}}{d\theta}=-2\overrightarrow{r}\left(
\theta\right) \times\overrightarrow{u}\left( \theta\right) , \label{Bloche}$$ with $\overrightarrow{u}\left( \theta\right) $ and $\overrightarrow
{p}\left( \theta\right) $ given by (\[defu\]) and (\[defpe\]), respectively.
Parabolic case
--------------
Applying Eqs. (\[deft\]), (\[tp\]) to (\[defp\]) $$\overrightarrow{p}\left( \theta\right) =\overrightarrow{p}-\gamma
\overrightarrow{p}\times\overrightarrow{q}+\left( \overrightarrow{p}\cdot\overrightarrow{q}\right) \overrightarrow{q}, \label{defpp}$$
and using properties of the Pauli matrices we obtain from (\[RSODE\]) the Bloch equation in the parabolic case:$$\frac{d\,\overrightarrow{r}}{d\theta}=-2\overrightarrow{r}\left(
\theta\right) \times\overrightarrow{u}\left( \theta\right) , \label{Blochp}$$ with $\overrightarrow{u}\left( \theta\right) $ and $\overrightarrow
{p}\left( \theta\right) $ given by (\[defu\]) and (\[defpp\]), respectively.
Hyperbolic case
---------------
Using Eqs. (\[deft\]), (\[th\]) we obtain from (\[defp\]) $$\overrightarrow{p}\left( \theta\right) =\cosh\left( \gamma\right)
\overrightarrow{p}-\sinh\left( \gamma\right) \overrightarrow{p}\times\overrightarrow{q}+\left( \cosh\left( \gamma\right) -1\right)
\left( \overrightarrow{p}\cdot\overrightarrow{q}\right) \overrightarrow{q},
\label{defph}$$
and using properties of the Pauli matrices we obtain from (\[RSODE\]) the Bloch equation in the hyperbolic case:$$\frac{d\,\overrightarrow{r}}{d\theta}=-2\overrightarrow{r}\left(
\theta\right) \times\overrightarrow{u}\left( \theta\right) , \label{Blochh}$$ with $\overrightarrow{u}\left( \theta\right) $ and $\overrightarrow
{p}\left( \theta\right) $ given by (\[defu\]) and (\[defph\]), respectively.
Computational results
=====================
We have performed several computations for the Bloch equation (\[Bloche\]) and the discrete-time dynamical system (\[Q\_NR\_N\]), parameterized as in Eqs. (\[defR\]), (\[defQ\]), $Q_{N}\equiv Q$, $\eta=1$. Exact solutions of the map (\[Q\_NR\_N\]) as well as of the Bloch equation (\[Bloche\]) in the elliptic case are given by (\[2k\]) and (\[EXPLE\]), respectively.
The solution (\[EXPLE\]), of discrete-time dynamical system (\[Q\_NR\_N\]) with $Q$, $P$ given by (\[defQ\]), (\[P\]) has been plotted in Fig. 1 for $\overrightarrow{q}=\left[ 0,0,1\right] $, $\overrightarrow{p}=\left[
1,0,\sqrt{2}\right] $, $\lambda=\beta/\left( 2\alpha\right) =3$ and the initial vector $\overrightarrow{r}_{0}=\left[ \frac{1}{2},\frac{1}{2},\sqrt{\frac{3}{2}}\right] $ on the upper sheet of the hyperboloid. The whole trajectory has three-fold symmetry with respect to the $\overrightarrow{q}$ axis. Circles indicate parallels $A_{1,2}$ given by (\[bounds2\]) confining the dynamics.
In Fig. 2 dynamics of vectors $\overrightarrow{r}_{N}$ obtained from (\[RSsol1a\]) has been plotted for $\alpha=5$, $\beta=20$ $\left(
\lambda=\beta/\left( 2\alpha\right) =2\right) $ and other parameters unchanged. We thus obtain thirty six points marked with dots. The solution (\[EXPLE\]) has been also plotted. The closed curve has two-fold symmetry with respect to the $\overrightarrow{q}$ axis.
[Fig1.eps]{}\
Fig. 1. Exact solution of the Bloch equation (\[Bloche\]), $\lambda=3$.
[Fig2.eps]{}\
Fig. 2. Exact solution of the Bloch equation (\[Bloche\]) (thin line) and discrete-time dynamical system (\[Q\_NR\_N\]) (dots), $\alpha=5$, $\beta=20$, $\lambda=2$.
In Fig. 3 initial stage of dynamics of vectors $\overrightarrow{r}_{N}$ has been plotted for $\lambda=\beta/\left( 2\alpha\right) =1.025$, dot marking the initial vector $\overrightarrow{r}_{N}$.
[Fig3.eps]{}\
Fig. 3. Exact solution of the Bloch equation (\[Bloche\]), $\lambda=1.025$.
Summary and discussion
======================
We have introduced in [@Okninski2009] a class of discrete-time invertible maps (\[Q\_NR\_N\]) on an arbitrary group $\mathcal{G}$ for which the exact solution (\[SOL\]) of this map has been found. Maps of form (\[Q\_NR\_N\]), parameterized on a Lie group, generate points in the dual (parameter) space which sample a trajectory in this space arbitrarily densely. This curve can be generated forward as well as backward from a given initial condition. This suggests that the group action (\[Q\_NR\_N\]) may correspond to a flow of a differential equation. We have demonstrated in the present paper that for $\mathcal{G} =SU\left( 1,1\right) $ the map (\[Q\_NR\_N\]), $Q_{N}\equiv Q$, considered as dynamical system in the dual space, is a stroboscopic map of a $SU\left( 1,1\right) $ Bloch equation. It should be noted that the $SU\left( 1,1\right) $ Bloch equation (\[RSODE\]) is formally analogous to the $SU\left( 2\right) $ Bloch equation, cf. Eq. (4.21) in [@Okninski2009].
Exact solutions of the map constructed in the present paper, (\[EXPLE\]), (\[EXPLP\]), (\[EXPLH\]), lead to a better understanding of the corresponding Bloch equations (\[Bloche\]), (\[Blochp\]), (\[Blochh\]). More exactly, symmetries and restrictions of dynamics have been found explicitely. It is interesting that dynamics of the $SU\left( 1,1\right) $ Bloch equation in the elliptic case bears close analogy to dynamics of the $SU\left( 2\right) $ Bloch equation, compare Figs. 1, 2, 3 from the present paper with analogous figures in [@Okninski2009].
[99]{}
A. Okninski, Acta Phys. Polon. B **40** (2009) 1605-1616; arXiv:0804.2128v3 \[math-ph\].
R. Brooks, J.P. Matelski, in *Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference*, edited by I.Kra, B. Maskit, *Ann. Math. Studies* **97** (Princeton Univ. Press, Princeton, N.J., 1981), p. 65.
F.W. Gehring, G.J. Martin, Bull. Am. Math. Soc. **21** (1989) 57-63.
D. Mumford, C. Series, D. Wright, *Indra’s pearls: The vision of Felix Klein*. Cambridge University Press, Cambridge, 2002.
H. Shimizu, Ann. of Math. **77** (1963) 33-71.
A. Leutbecher, Math. Zeit. **100** (1967) 183-200.
A.F. Beardon, *The Geometry of Discrete Groups*. Springer-Verlag, 1983.
G. Dattoli, A. Dipace, A. Torre, Phys. Rev. A **33** (1986) 4387-4389.
P.K. Aravind, J. Opt. Soc. Am. B **5** (1988) 1545-1553.
B.E. King, *Quantum State Engineering and Information Processing with Trapped Ions*, PhD Thesis, Department of Physics, University of Colorado, 1999 (http://jilawww.colorado.edu/pubs/thesis/king/).
R.R. Puri, *Mathematical Methods of Quantum Optics*. Springer, Berlin 2001.
G. Chiribella, G.M. D’Ariano, P. Perinotti, Laser Physics **16** (2006) 1572-1581; arXiv:0610142v1 \[quant-ph\].
A. Wawrzyńczyk, *Group representations and special functions*. D. Reidel Poblishing Co., Dordrecht/Boston/Lancaster, 1984.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Aharonov’s weak value, which is a physical quantity obtainable by weak measurement, admits amplification and hence is deemed to be useful for precision measurement. We examine the significance of the amplification based on the uncertainty of measurement, and show that the trade-offs among the three (systematic, statistical and nonlinear) components of the uncertainty inherent in the weak measurement will set an upper limit on the usable amplification. Apart from the Gaussian state models employed for demonstration, our argument is completely general; it is free from approximation and valid for arbitrary observables $A$ and couplings $g$.'
author:
- Jaeha Lee
- Izumi Tsutsui
title: Uncertainty of Weak Measurement and Merit of Amplification
---
#### Introduction. {#introduction. .unnumbered}
The novel physical quantity in quantum mechanics called [*weak value*]{}, proposed by Aharonov and co-workers long ago [@Aharonov_1964; @Aharonov_1988], has gained a renewed interest in recent years. One of the reasons for this is that, unlike the standard physical value given by an eigenvalue of an observable $A$, the weak value may be considered meaningful even for a set of non-commutable observables, simultaneously. This inspired a new insight for understanding counter-intuitive phenomena, such as the three-box paradox [@Aharonov_1991] and Hardy’s paradox [@Yokota_2009]. The other, perhaps stronger, motive comes from the realization that the weak value can be amplified by adjusting the process of measurement, [*weak measurement*]{}. Specifically, by choosing properly the initial and the final state (pre and postselection) of the process, the weak value can be made arbitrarily large, and this may be utilized for precision measurement. In fact, it has been reported that a significant amplification is achieved to observe successfully the spin Hall effect of light [@Hosten_2008]. A similar amplification has also been shown to be available for detecting ultrasensitive beam deflection in a Sagnac interferometer [@Dixon_2009].
In view of this, it is quite natural to ask whether there exists a limit on amplification, and if so why. This question was addressed recently in [@Wu_2011], which extended the treatment of [@Josza_2007] to the full order of the coupling $g$ between the system and the measurement device, where the amplification is analyzed based on the average shift of the meter of the device. For the particular case of the observable $A$ fulfilling $A^2 = \mathrm{Id}$ and the Gaussian device states, it has been shown that the amplification rate, as well as the signal-to-noise ratio, has an upper limit [@Koike_2011; @Nakamura_2012]. No such limit arises if the device state can be tuned precisely according to the weak value and the coupling [@Susa_2012].
This Letter presents a completely new approach to the analysis of the weak value amplification. Rather than focusing on the shift of the meter, we consider the uncertainty of the weak measurement and inquire when it is meaningful as measurement for given particular purposes. This should be more important, because during the amplification the obtained amplified shift may well drift away from the intended weak value, rendering the whole measurement meaningless. Our uncertainty is defined from a probabilistic estimation on the deviation of the measured value from the weak value itself, and is shown to be separable into systematic, statistical, and nonlinear components. For the purpose of probing the existence of a physical effect, as done in the experiments [@Hosten_2008; @Dixon_2009], the condition of significant weak measurement is presented by demanding that the uncertainty (with the probability assigned beforehand) be smaller than the weak value to be measured. In the case of the Gaussian state, this explains the appearance of the range of amplification in which the existence of the effect is affirmed with assurance greater than the probability. The case also confirms the anticipation [@Aharonov_2005] that the weak value may be observed with non-small $g$, [*i.e.*]{}, by non-weak measurement. Apart from the Gaussian state analysis, our treatment is completely general; it is valid for an arbitrary dimensional system with arbitrary observables $A$ for all range of couplings $g$, and no approximation is used throughout (detailed discussions with mathematical proofs shall be given in Supplemental Material).
#### Weak Value and Weak Measurement. {#weak-value-and-weak-measurement. .unnumbered}
We begin by recalling the process of weak measurement for obtaining the weak value. Let $\mathcal{H}, \, \mathcal{K}$ be the Hilbert spaces associated with the system and that of the measuring device, respectively. We wish to find the value of an observable $A$ of the system represented by a self-adjoint operator on $\mathcal{H}$. This is done through the measurement of observables $Q$, $P$ of the meter device, which are represented by self-adjoint operators on $\mathcal{K} $ satisfying the canonical commutation relation $[Q, P] = i\hbar$ (we put $\hbar = 1$ hereafter for brevity). Our measurement is assumed to be of von Neumann type, in which the evolution of the composite system $\mathcal{H} \otimes \mathcal{K}$ is described by the unitary operator $e^{-igA \otimes P}$ with a coupling parameter $g \in [0, \infty)$.
Prior to the interaction, the measured system shall be prepared in some state $|\phi_{i}\rangle \in \mathcal{H}$. Along with this *preselection*, the measuring device is also prepared in a state $|\psi_{i}\rangle \in \mathcal{K}$. The state of the composite system evolves after the interaction into $e^{-igA \otimes P} |\phi_{i}\rangle |\psi_{i}\rangle$. We then choose a state $|\phi_{f}\rangle \in \mathcal{H}$ on which a projective measurement in $\mathcal{H}$ is performed. Those that result in $|\phi_{f}\rangle$ shall be kept, otherwise discarded. After this *postselection*, the composite state will be disentangled into $|\phi_{f}\rangle |\psi_{f}\rangle$, where $|\psi_{f}\rangle = \langle\phi_{f}| e^{-igA \otimes P} |\phi_{i}\rangle |\psi_{i}\rangle$.
We intend to extract information of the triplet $(A, |\phi_{i}\rangle, |\phi_{f}\rangle)$ from the above measurement, and to this end we choose an observable $X = Q$ or $P$ of the measuring device and examine its shift in the expectation value $E_{X}(\psi) := \langle\psi |X|\psi\rangle/ \|\psi\|^{2}$ between the two selections: $
\Delta^{w}_{X}(g) := E_{X}(\psi_{f}) - E_{X}(\psi_{i})
$. Imposing certain conditions on $|\phi_{i}\rangle$ and $|\psi_{i}\rangle$, both the functions $g \mapsto \Delta^{w}_{X}(g)$ are proven to be differentiable and well-defined over an open subset of $[0, \infty)$. In particular, the functions $\Delta^{w}_{X}(g)$ are defined at $g=0$ if and only if $\langle\phi_{f} | \phi_{i}\rangle \ne 0$, in which case the derivatives at $g=0$ read [$$\begin{aligned}
&\frac{d}{dg} \Delta^{w}_{Q}(0) = \mathrm{Re}A_{w} \nonumber \\
&\qquad\quad + \left( E_{\{Q, P\}}(\psi_{i}) - 2E_{Q}(\psi_{i})E_{P}(\psi_{i}) \right) \cdot \mathrm{Im}A_{w} , \label{eq:wv_Q}
\\
&\frac{d}{dg} \Delta^{w}_{P}(0) = 2 \mathrm{Var}_{P}(\psi_{i}) \cdot \mathrm{Im}A_{w} ,
\label{eq:wv_P} \end{aligned}$$ ]{}where $\{Q, P\} := QP + PQ$, $\mathrm{Var}_{X}(\psi_{i}) := E_{X^{2}}(\psi_{i}) - \left(E_{X}(\psi_{i})\right)^{2}$ the variance of $X$ on $|\psi_{i}\rangle$ and [$$\begin{aligned}
A_{w} := \frac{\langle \phi_{f} | A | \phi_{i} \rangle}{\langle \phi_{f} | \phi_{i} \rangle},\end{aligned}$$ ]{}a complex valued quantity called the *weak value*.
From the weak measurement described above, the real and imaginary part of $A_{w}$ are obtained, in theory, with arbitrary accuracy by letting $g \to 0$. The fact that $A_{w} $ is obtained from the rate of the shifts $\Delta^{w}_{X}(g)$ at $g=0$ justifies the term weak value. Incidentally, by imposing stricter conditions on the preselected state $|\phi_{i}\rangle$, higher order differentiability of the shifts can also be ensured, and this leads to the notion of higher order weak values, whose formulae are obtained analogously.
Before discussing the complications in actual measurement processes, we note that the weak value $A_{w} $ can take any arbitrary complex value by an appropriate choice of states in the two selections. Indeed, observing that $A|\phi_{i}\rangle = E_{A}(\phi_{i})|\phi_{i}\rangle +\sqrt{\mathrm{Var}_{A}(\phi_{i})}
|\chi\rangle$ holds for any normalized $|\phi_{i}\rangle$ with $|\chi\rangle$ being a normalized vector orthogonal to $|\phi_{i}\rangle$, one can choose the postselection as $|\phi_{f}\rangle = c\, |\phi_{i}\rangle + |\chi\rangle$ with $0 \neq c \in \mathbb{C}$ to find [$$\frac{\langle \phi_{f} | A | \phi_{i} \rangle}{\langle \phi_{f} | \phi_{i} \rangle}
= E_{A}(\phi_{i}) + \frac{\sqrt{\mathrm{Var}_{A}(\phi_{i})}}{c^*}.$$ ]{}Clearly, one can change freely the value of $A_{w} $ by choosing $c$ in $|\phi_{f}\rangle$ appropriately, unless $|\phi_{i}\rangle$ happens to be an eigenstate of $A$ for which $\mathrm{Var}_{A}(\phi_{i}) = 0$. This is a remarkable property of the weak value and can be contrasted to the conventional eigenvalue and expectation value which are always real-valued, and bounded when $A$ is bounded.
#### Conventional Measurement and Uncertainty. {#conventional-measurement-and-uncertainty. .unnumbered}
In order to analyze the merit of weak measurement, we first recall the conventional (indirect) projective measurement, which is obtained by omitting the postselection in the weak measurement process. In this case, defining the shift by $\Delta^c_X(g) := E_{\mathrm{Id} \otimes X}(e^{-igA \otimes P} \phi_{i} \otimes \psi_{i}) - E_{X}(\psi_{i})$, one verifies [$$\label{eq:cm}
\Delta^c_Q(g) = g \cdot E_{A}(\phi_{i}),$$ ]{}while $\Delta^c_P(g) = 0$. Interestingly, for any orthonormal basis $\mathcal{B}$ of the system $\mathcal{H}$, one finds [$$\begin{gathered}
\sum_{|\phi_{f}\rangle \in \mathcal{B}} r(\phi_{i}\to\phi_{f}) \cdot \Delta^{w}_{X}(g) = \Delta^c_X(g),
\label{eq:wqav} \end{gathered}$$ ]{}where [$$\label{eq:sr}
r(\phi_{i}\to\phi_{f}) := \frac{ \left\|\left( {\left| \phi_{f} \right\rangle}\!\!{\left\langle \phi_{f} \right|} \otimes \mathrm{Id} \right) e^{-igA \otimes P} \phi_{i}\otimes\psi_{i} \right\|^{2}}{\left\|e^{-igA \otimes P} \phi_{i}\otimes\psi_{i} \right\|^{2}}$$ ]{}is the survival rate of the postselection process in weak measurement, which tends to $\left|\langle\phi_{f} | \phi_{i}\rangle\right|^{2}$ for $g \to 0$. The relation shows that, even for nonvanishing $g$, the shift of the weak measurement $\Delta^{w}_{X}(g)$ reduces to the shift of the conventional measurement $\Delta^c_{X}(g)$, that is, the effect of postselections disappears completely, when it is averaged over with their corresponding survival rates.
However, the aforementoined idealized measurement processes are not quite possible to implement in practice, due to technical/intrinsic constraints. For instance, in measuring $X$ under the given state $|\psi\rangle$ one has the [*systematic uncertainty*]{} $\delta_{X} \geq 0$ arising from various sources including the finite resolution of the measuring device and its imperfect calibration. Besides, one has the [*statistical uncertainty*]{} arising from the finiteness of the number $N$ of repeated measurements actually performed. To treat these uncertainties explicitly, let $\{ \tilde{X}_{1}, \dots, \tilde{X}_{N} \}$ be the outcomes obtained by the measurements of $X$. The notion of systematic uncertainty implies that there exists a sequence of ideal outcomes $X_{n} \in [\tilde{X}_{n} - \delta_{X}, \tilde{X}_{n} + \delta_{X}]$ for $n = 1, \ldots, N$ whose average approaches the expectation value for large $N$, that is, the error $\kappa^{N}_X(\psi) := \left| \sum_{n=1}^{N} {X}_{n} /N - E_{X}(\psi)\right|$ almost certainly vanishes as $\lim_{N \to \infty} \kappa^{N}_X(\psi) = 0$ (Law of Large Numbers). For the outcomes $\tilde{X}_{n}$ with finite $N$, the triangle inequality yields [$$\label{ineq:uncert}
\left| \frac{\sum_{n=1}^{N} \tilde{X}_{n}}{N} - E_{X}(\psi)\right| \leq \delta_{X} + \kappa^{N}_X(\psi).$$ ]{}Note that $\delta_{X}$ is intrinsic to the measurement setup and is independent of the state $|\psi\rangle$ while $\kappa^{N}_X(\psi)$ is dependent on the statistical ensemble represented by $|\psi\rangle$.
Since the measurement outcomes $\tilde{X}_{n}$ are intrinsically probabilistic, by invoking Chebyshev’s inequality [@Beasley_2004] we learn that the probability of obtaining the error $\kappa^{N}_X(\psi)$ to be less than a value $\kappa$ is bounded as [$${\rm Prob}\left[\kappa^{N}_X(\psi) \le \kappa\right] \ge 1 - \frac{\mathrm{Var}_{X}(\psi)}{N\kappa^{2}}.
\label{prounc}$$ ]{}If one demands that the lower bound (the r.h.s. of ) be a desired value $\eta$, then by solving $\kappa$ in favor of $\eta$, one can rewrite the r.h.s. of as $
\epsilon_{X}(\eta) := \delta_{X} + \sqrt{\mathrm{Var}_{X}(\psi)/N(1-\eta)}
$ which is specified by the probability $\eta$. This gives the inequality the meaning that the deviation of the value estimated from the measured outcomes $\{ \tilde{X}_{1}, \dots, \tilde{X}_{N} \}$ from $E_{X}(\psi)$ is guaranteed to be less than $\epsilon_{X}(\eta)$ with probability greater than $\eta$.
Now, for the conventional measurement, suppose that $N_0$ identical sets of the composite system $|\phi_{i}\rangle|\psi_{i}\rangle$ are prepared by preselection. We collect the outcomes $\tilde{Q}^c_{n}$ of measurements of $Q$ for the meter [*after*]{} the interaction, and thereby obtain $\tilde{\Delta}^c_{Q}(g) := \sum_{n=1}^{N_{0}} \tilde{Q}^c_{n}/N_{0} - E_{Q}(\psi_{i})$. Equation implies that the ratio $
{\tilde{\Delta}^c_{Q}(g)}/{g}
$ can be regarded as the value of $E_{A}(\phi_{i})$ estimated from the experiment, and since $\mathrm{Var}_{\mathrm{Id} \otimes Q}(e^{-igA \otimes P} \phi_{i} \otimes \psi_{i}) = \mathrm{Var}_{Q}(\psi_i) + g^{2} \mathrm{Var}_{A}(\phi_i)$, the accuracy $\vert {\tilde{\Delta}^c_{Q}(g)}/{g} - E_{A}(\phi_{i}) \vert$ of the estimation from the ratio is evaluated by the uncertainty, [$$\label{eq:uncert_CM}
\epsilon^c_Q(\eta ; g, \psi_{i}) := \frac{\delta_Q}{g} + \sqrt{\frac{\mathrm{Var}_{Q}(\psi_i) + g^{2} \mathrm{Var}_{A}(\phi_i)}{g^{2}N_{0}(1-\eta)}}.$$ ]{}Observe that, while the uncertainty $\epsilon^c_Q(\eta ; g, \psi_{i})$ in the conventional measurement is in general dependent on the initial state $|\psi_{i}\rangle$ of the meter, the dependence is washed away in the strong coupling limit $g \to \infty$ where the uncertainty tends to the statistical uncertainty $\sqrt{\mathrm{Var}_{A}(\phi_i) / N_{0}(1-\eta) }$ of the system alone.
#### Weak Measurement and Uncertainty. {#weak-measurement-and-uncertainty. .unnumbered}
Turning to the weak measurement, suppose that $N$ out of $N_0$ identically prepared sets of the composite system $|\phi_{i}\rangle|\psi_{i}\rangle$ survived the postselection process. After the postselection we collect all the outcomes $\tilde{X}^w_{n}$ measured for the final state $|\psi_{f}\rangle$ of the meter and thereby obtain $\tilde{\Delta}^{w}_{X}(g) := \sum_{n=1}^{N} \tilde{X}^w_{n}/N - E_{X}(\psi_{i})$. Specializing to the case $X = Q$ with the initial state $|\psi_{i}\rangle$ of the meter satisfying $E_{\{Q, P\}}(\psi_{i}) - 2E_{Q}(\psi_{i})E_{P}(\psi_{i}) = 0$ for simplicity, the relation and Taylor’s theorem imply that the ratio $
{\tilde{\Delta}^{w}_{Q}(g)}/{g}
$ in the limit $g \to 0$ can be regarded as the estimated value of $\mathrm{Re}A_{w} $ from the experiment. In the actual experiment, however, the coupling constant $g$ should be kept nonvanishing, and hence we have [$$\begin{aligned}
\left| \frac{\tilde{\Delta}^{w}_{Q}(g)}{g} - \mathrm{Re}A_{w} \right|
&= \left| \frac{\tilde{\Delta}^{w}_{Q}(g) - \Delta^{w}_{Q}(g)}{g}
+ \left( \frac{\Delta^{w}_{Q}(g)}{g} - \mathrm{Re}A_{w} \right)\right| \nonumber \\
&\leq \frac{\delta_{Q}}{g} + \frac{\kappa^{N}_Q(\psi_f)}{g}
+ \left| \frac{\Delta^{w}_{Q}(g)}{g} - \mathrm{Re}A_{w} \right|,
\label{ineqp}\end{aligned}$$ ]{}in place of . In addition, since the process of obtaining $N$ out of $N_{0}$ outcomes generally depends on the postselection in relation to the preselection, we must also take the survival rate into account. To discuss the uncertainties along this more realistic line, note that the probability of $N$ survived out of $N_0$ is given by the binomial distribution $\mathrm{Bi}\left[N; N_0,r\right] := \binom{N_0}{N} r^{N} \left( 1 - r \right)^{N_0-N}$ with $r$ given by the survival rate . To each of these $N$ outcomes, inequality holds with the lower bound of the probability . Thus, the average probability that the measurement yields outcomes within the statistical error $\kappa$ is given by the sum over all possible $N$, [$$\begin{aligned}
\label{avpro}
{\Pi}^{N_0}_{Q}(\kappa)
:= \sum_{N=1}^{N_0} \mathrm{Bi}\left[N; N_0, r(\phi_{i}\to\phi_{f})\right] \max \left[ \left( 1-\frac{\mathrm{Var}_{Q}(\psi_{f})}{N\kappa^{2}} \right), 0 \right].\end{aligned}$$ ]{} To ensure the overall uncertainty level by some $\eta > 0$, we may put $\eta = {\Pi}^{N_0}_{Q}(\kappa)$. This relation can be solved for $\kappa$ to obtain the inverse $\kappa_Q^{N_0}(\eta) := [\Pi^{N_0}_{Q}]^{-1}(\eta)$, since each term in the sum is a continuous and monotonically increasing function in $\kappa$. From this, the uncertainty of estimating $\mathrm{Re}A_{w}$ by weak measurement is given by [$$\begin{aligned}
\label{eq:uncert_WM}
\epsilon^{w}_{Q}(\eta; g, \psi_{i}) := \frac{\delta_{Q}}{g} +
\frac{\kappa^{N_{0}}_Q(\eta)}{g} + \left| \frac{\Delta^{w}_{Q}(g)}{g} - \mathrm{Re}A_{w}\right|.\end{aligned}$$ ]{}The third term in , which is absent for the conventional measurement , is due to the nonlinearity of the shift with respect to $g$, which cannot be ignored for nonvanishing $g$ in realistic settings. An analogous argument holds for the estimation of $\mathrm{Im}A_{w} $ with the choice $X = P$.
We thus have obtained a framework for handling both the ideal and realistic measurement in terms of uncertainty, where the ideal case arises in the limit $\delta_{X}\to 0$, $N_{0}\to \infty$ (and $g \to 0$ for weak measurement) for which the uncertainties and vanish for all $\eta$. In passing, we mention that the uncertainties are invariant under translation along the real axis $A \to A + t$ for $t \in \mathbb{R}$. As for scaling $A \to rA,\ g \to g/r$ for $r > 0$, we just have $\epsilon^{c,w}_{X}(\eta; g, \psi_{i}) \to r\epsilon^{c,w}_{X}(\eta; g, \psi_{i})$ as expected.
#### Merit of Weak Measurement and Amplification. {#merit-of-weak-measurement-and-amplification. .unnumbered}
Now we address the question of whether the weak measurement has an advantage over the conventional one for obtaining, [*e.g.*]{}, the expectation value of $A$. Obviously, since the weak measurement requires postselection, one cannot fully exploit all samples prepared prior to the measurement. This yields a larger statistical uncertainty which could quickly become uncontrollably large for higher $\eta$. Even in the ideal limit $N_{0} \to \infty$ where the statistical uncertainty vanishes, there still remains nonlinearity, which is nonexistent in the conventional case. By comparing $\eqref{eq:uncert_CM}$ and $\eqref{eq:uncert_WM}$, we learn that, as long as the uncertainty is concerned, there is no technical merit for adopting the weak measurement. However, the true merit of weak measurement can arise in the situation where the amplification of the weak value outside of the numerical range $W(A) := \{\langle \phi |A| \phi\rangle : \|\phi\| = 1 \}$ is available.
To see this by a simple example, consider a situation where the strength $g$ of interaction cannot be made large enough to curb the systematic error. For instance, if $A$ is bounded and the numerical range $W(A)$ is confined in a much smaller region than $\delta/g$ for available $g$, then the estimated value of any $E_{A}(\phi) \in W(A)$ is completely obscured by $\delta/g$, in which case the projective measurement reveals no meaningful information of the system. In contrast, in the weak measurement the range of the weak value spans the whole complex plane, and one can arrange the selections such that $\mathrm{Re}A_{w}$ is amplified outside of $W(A)$ rendering the systematic error negligible compared to $\mathrm{Re}A_{w}$. To be more specific, suppose that one probes the very existence of a physical effect by looking at the shift of the meter in the measurement, as in the case of the experiments [@Hosten_2008; @Dixon_2009]. We can conclude that the effect exists with confidence $\eta$ when [$$\label{eq:uncert_PMSC}
\epsilon^c_Q(\eta ; g, \psi_{i}) \leq \vert E_{A}(\phi_{i})\vert,$$ ]{}[$$\label{eq:uncert_WMSC}
\epsilon^{w}_Q(\eta ; g, \psi_{i}) \leq \vert \mathrm{Re}A_{w}\vert , \quad \epsilon^{w}_P(\eta ; g, \psi_{i}) \leq \vert \mathrm{Im}A_{w}\vert ,$$ ]{}hold for respective measuremens, which are all sufficient conditions for distinguishing the coupling $g$ from $g=0$. In both of these cases, we may call the measurements *significant with confidence $\eta$*. Clearly, for precision measurement the weak measurement becomes superior when is broken while can be maintained through the amplification of $A_{w}$.
#### Gaussian Model with Two-Point Spectrum. {#gaussian-model-with-two-point-spectrum. .unnumbered}
For illustration, we consider the model in which $A$ has a discrete spectrum consisting of two distinct values $\{\lambda_{1}, \lambda_{2}\}$ and the initial state $\psi_{i}$ of the meter $\mathcal{K} = L^{2}(\mathbb{R})$ is given by normalized Gaussian wave functions $\psi_{i}(x) = ( 1/\pi d^{2})^{\frac{1}{4}} \exp(-\frac{x^{2}}{2d^{2}})$ centered at $x = 0$ with width $d > 0$. Despite its simplicity, this model is sufficiently general in the sense that it covers most of the recent experiments of weak measurement [@Hosten_2008; @Dixon_2009] as well as the recent work [@Wu_2011; @Koike_2011; @Nakamura_2012] where a full order calculation of the shift for $A$ satisfying $A^{2} = \mathrm{Id}$ is performed, which is equivalent to $\{\lambda_{1}, \lambda_{2}\} = \{ -1, 1\}$ under our setting. Identifying the usual position and momentum operators on $L^{2}(\mathbb{R})$ with $Q=\Hat{x}$ and $P=\Hat{p}$, and using the shorthand, $\Lambda_{m} := (\lambda_{1} + \lambda_{2})/{2}$ and $\Lambda_{r} := (\lambda_{2} - \lambda_{1})/{2}$, we find [$$\begin{aligned}
\Delta^{w}_{Q}(g) &= g \cdot \frac{\mathrm{Re} A_{r} }{1 + a\left( 1 - e^{-g^{2}\Lambda_{r}^{2}/d^{2}} \right)} + g \cdot \Lambda_{m}, \label{eq:Gauss_Shift_Q} \\
\Delta^{w}_{P}(g) &= \frac{g}{d^{2}} \cdot \frac{ \mathrm{Im} A_{r} e^{-g^{2}\Lambda_{r}^{2}/d^{2}}}{1 + a \left( 1 - e^{-g^{2}\Lambda_{r}^{2}/d^{2}} \right)} \label{eq:Gauss_Shift_P},\end{aligned}$$ ]{}where $
A_{r} := A_{w} - \Lambda_{m}
$, and $a := \frac{1}{2} \left(\vert A_{r}\vert^{2}/\Lambda_{r}^{2} - 1 \right)$ is a parameter corresponding to the amplification rate. One then verifies that the above functions are of class $C^{\infty}$ with respect to $g$, and the estimated values ${\Delta^{w}_{Q}(g)}/{g}$ and ${\Delta^{w}_{P}(g)}/{(g/d^{2})}$ tend to $\mathrm{Re} A_{w}$ and $\mathrm{Im} A_{w}$, respectively, in the weak limit $g \to 0$.
The uncertainties for this model can be analytically obtained based on and its counterpart for measuring $\mathrm{Im}A_{w}$ (see Supplemental Material). At this point, observe that both the statistical and the nonlinear terms in the overall uncertainties are dependent on $g$ and $d$ only through the combination $g/d$. Thus, instead of considering the weak limit $g \to 0$, one may equally consider the broad limit of the width $d \to \infty$ to obtain the weak value $A_{w}$ from ${\Delta^{w}_{Q}(g)}/{g}$ and ${\Delta^{w}_{P}(g)}/{(g/d^{2})}$. In other words, the aim of weak measurement to obtain the weak value can be achieved with a coupling $g$ which is not weak at all. We also remark that any choice of the spectrum $\{\lambda_{1}, \lambda_{2}\}$ can change into one another by some combination of translation and scaling. In fact, on account of the aforementioned properties of the uncertainties under these transformations, one sees that all these models actually reduce to the simplest case $\{\lambda_{1}, \lambda_{2}\} = \{ -1, 1\}$.
We now return to the problem of fulfillment of the significance condition . With a proper choice of measurement setups, this condition can indeed be fulfilled (while is broken) as shown in Fig. \[fig:Amplification\].
![ Ratio of uncertainty $\epsilon^{w}_{Q}(\eta; g, d)$ to the real part of the weak value of the spin $S_{z} = \sigma_z/2$. By amplifying the weak value out of its numerical range $[-1/2, 1/2]$ to $(S_{z})_{w} \approx 100$, the significance condition is attained with confidence $\eta=0.95$. (Parameters: $\delta_{Q} = 1/2$, $N_{0}= 10^{7}$, $g=1/50$ and $d = 4$.) []{data-label="fig:Amplification"}](fig.eps){width="65mm"}
Note, however, that the advantage of amplification does not come free, because the amplification requires generically a small transition amplitude $\langle\phi_{f} | \phi_{i}\rangle$, which necessitate a much larger number of prepared samples to suppress the statistical uncertainty compared to the conventional one. More importantly, the amplification enhances also the uncertainty coming from the nonlinearity and, together with the statistical uncertainty, eventually ruins the significance condition . In fact, it can be proved that, for an observable $A$ with finite eigenvalues, the shifts $\Delta^{w}_{X}(g)$ for fixed $g$ and $|\psi_{i}\rangle$ are also bounded with respect to a set of pre and postselections of the system and, as a result, the ratio between the amplified weak value and the nonlinear term becomes unity as $|\mathrm{Re}A_{w}|,\ |\mathrm{Im}A_{w}| \to \infty$, suggesting that the qualitative behaviour seen in this specific numerical demonstration is actually universal. Given these trade-offs, for probing a physical effect such as gravitational waves by means of weak measurement, it is vital for us to find a possible range of amplification fulfilling where the measurement is significant.
We thank Prof. A. Hosoya for helpful discussions. This work was supported by JSPS Grant-in-Aid for Scientific Research (C), No. 25400423.
[99]{}
Y. Aharonov, P. G. Bergmann and L. Lebowitz, Phys. Rev. [**134**]{} (1964) B1410.
Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. [**60**]{} (1988) 1351.
Y. Aharonov Y, and L. Vaidman, J. Phys. A: Math. Gen. [**24**]{} (1991) 2315.
K. Yokota, T. Yamamoto, M. Koashi and N. Imoto, New Journ. Phys. [**11**]{} (2009) 033011.
O. Hosten and P. Kwiat, SCIENCE [**319**]{} (2008) 787.
P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Phys. Rev. Lett. [**102**]{} (2009) 173601.
S. Wu, and Y. Li, Phys. Rev. A [**83**]{} (2011) 052106.
R. Josza, Phys. Rev. A [**76**]{} (2007) 044103.
T. Koike and S. Tanaka, Phys. Rev. A 84, 062106 (2011).
K. Nakamura, A. Nishizawa, and M. K. Fujimoto, Phys. Rev. A 85, 012113 (2012).
Y. Susa, Y. Shikano, and A. Hosoya, Phys. Rev. A 85, 052110 (2012).
Y. Aharonov and D. Rohrlich, Quantum Paradoxes: Quantum Theory for the Perplexed, Wiley-VCH, Weinheim (2005).
S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm and A. M. Steinberg, SCIENCE [**332**]{} (2011) 1170.
A. M. Steinberg Phys. Rev. Lett. [**74**]{} (1995) 2405.
J. Dressel, S. Agarwal, and A. N. Jordan, Phys. Rev. Lett. [**104**]{} (2010) 240401; J. Dressel and A. N. Jordan, Phys. Rev. A [**85**]{} (2012) 022123.
B. Reznik and Y. Aharonov, Phys. Rev. A [**52**]{} (1995) 2538.
Y. Aharonov, S. Popescu, J. Tollaksen and L. Vaidman, Phys. Rev. A [**79**]{} (2009) 052110.
T. Mark Beasley et al., Appl. Statist. [**53**]{}, Part 1 (2004), pp. 95-108.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The angle of repose for the flow of granular materials in a half-filled rotating drum is studied by means of experiments and computer simulations. Particles of different material properties are used to investigate the effects of the end caps on the angle of repose. By fitting the numerical results to an exponentially decaying function, we are able to calculate the characteristic range, $\zeta$, of the influence of the wall. We found that $\zeta$ scales with the drum radius but does not depend on either the density or the gravitational constant. For increasing particle diameter, finite size effects are visible.'
address:
- 'Fachbereich Physik, Philipps-Universität, Renthof 6, 35032 Marburg, Germany'
- 'Division of Engineering, Colorado School of Mines, Golden, CO 80401, USA'
author:
- 'Christian M. Dury and Gerald H. Ristow'
- 'Jamie L. Moss and Masami Nakagawa'
bibliography:
- 'habil.bib'
date: 'received 29 August 1997; revised '
title: Boundary Effects on the Angle of Repose in Rotating Cylinders
---
Introduction
============
The behavior of granular materials is of great technological interest [@jaeger96] and its investigation has a history of more than two hundred years. When granular materials are put in a rotating drum, avalanches are observed along the surface of the granular bulk [@evesque88; @rajchenbach90]. In industrial processes, such devices are mostly used for mixing different kinds of particles. However, it is also well known that particles of different sizes tend to segregate in the radial and axial directions .
Recently, the particle dynamics of granular materials in a rotating drum has been described by using quasi two-dimensional systems, tracking individual grains via cameras and computer programs [@cantelaube95]. Extensive numerical studies have also reproduced and predicted many of the experimental findings . The segregation and mixing process depends on many parameters, such as size [@bridgwater76; @dury97], shape [@buchholtz95], mass [@ristow94], frictional forces, angular velocity [@dury97], filling level of the drum [@metcalfe95], etc. The angle of repose of the material also depends on the parameters and it was argued that either the dynamic or static angle difference of the materials in the drum influence the axial segregation process [@donald62; @dasgupta91; @hill94; @hill95].
In this article, we investigate experimentally the dependence of the dynamic angle of repose on the rotation speed of a half-filled drum for particles of different material properties. It is found that the angle is up to 5 degrees higher at the end caps of the drum due to boundary friction. Using a three-dimensional discrete element code, we are able to quantify this boundary effect and discuss its dependence on gravity, particle size and density.
Experimental Results {#experiment}
====================
An acrylic cylinder of diameter 6.9 cm and length 49 cm was placed horizontally on two sets of roller supports and was rotated by a well-regulated electronic motor. The material used was mustard seeds which are relatively round of average diameter about 2.5 mm, and have a coefficient of restitution, estimated from a set of impact experiments, of about 0.75 [@nakagawa93]. A set of experiments were conducted to measure the angle of repose in different flow regimes. For a small rotation speed, $\Omega$, intermittent flow led to a different angle before and after each avalanche occurred, called the starting (maximum) and stopping (minimum) angle, respectively. For a larger rotation speed these intermittent avalanches became a continuous flat surface and thus enabled to define one angle of repose defined as the [*dynamic angle of repose*]{} as shown in Fig. \[fig: sketch\]a. When $\Omega$ increases, the flat surface deforms with increasing rotation speeds and develops a so-called S-shape surface for higher rotation speeds, shown in Fig. \[fig: sketch\]c. The deformation mostly starts from the lower boundary inwards and can be well approximated by two straight lines with different slopes close to this transition, sketched in Fig. \[fig: sketch\]b. For all measurements in this regime, we took the slope of the line to the right which corresponds to the line with the higher slope.
The average maximum and minimum angles of repose for the intermittent avalanches were found to be about 36 and 32 degrees, respectively, see Fig. \[fig: mustard\_all\]. There seems to be a rather sharp transition from intermittent to continuous avalanches, which happens around $\Omega = 4$ rpm. For $\Omega$ greater than 4 rpm where the avalanches are continuous, the mustard seed data indicate a linear dependence of the dynamic angle of repose on the rotation speed which differs from the quadratic dependence found by Rajchenbach [@rajchenbach90].
bound\_0
We also investigated the dynamic angle of repose for different particle diameters and materials in the continuous regime in more detail using a 27 cm long acrylic cylinder of diameter $2R=6.9$ cm. For a given rotation speed, $\Omega$, the dynamic angle of repose was measured four times at one of the acrylic end caps and the average value with an error bar corresponding to a confidence interval of 2$\sigma$, where $\sigma$ is the standard deviation of the data points, was then calculated. First we used mustard seeds of two different diameters, namely 1.7 mm (black) and 2.5 mm (yellow), with a density of 1.3 g/cm$^3$. The latter were the same that were used to produce Fig. \[fig: mustard\_all\]. We varied the rotation speed, $\Omega$, from 5 rpm to 40 rpm and took the higher angle in the S-shaped regime which exists for higher rotation rates, see Fig. \[fig: sketch\]b. Both data sets are shown in Fig. \[fig: mustard\] for black ($\bullet$) and yellow ($\circ$) seeds. The figure also illustrates the transition to the S-shaped regime which occurs at the change of slope, e.g. at around 11 rpm for the smaller seeds and around 16 rpm for the larger seeds. One also notes that the dynamic angle of repose is much higher for the larger particles in the low frequency regime. For values of $\Omega > 15$ rpm in the S-shaped regime, the difference in the dynamic angle of repose for the two different types of mustard seeds decreases with increasing $\Omega$, and both curves cross around 30 rpm giving a slightly higher angle for the smaller seeds with the highest rotation speeds studied.
We applied the same measurements to two sets of glass beads having a density of 2.6 g/cm$^3$. The smaller beads had a diameter of 1.5 mm with no measurable size distribution, whereas the larger beads had a diameter range of 3.0 $\pm$ 0.2 mm. Both data sets are shown in Fig. \[fig: glass\] for small ($\bullet$) and large ($\circ$) beads. It can be seen from this figure that the transition to the S-shaped regime occurs at around 16 rpm for the smaller beads and around 24 rpm for the larger beads. In general, we found that the small particles exhibit the S-shaped surface at lower values of $\Omega$ than the large particles. The angles of repose are, in general, lower for the glass beads compared to the mustard seeds which we attribute to the fact that the mustard seeds are not as round as the glass beads and rotations of the mustard seeds are therefore more suppressed. The coefficient of friction is also higher for mustard seeds.
bound\_2
bound\_1
There are two striking differences when comparing Figs. \[fig: glass\] (glass spheres) and \[fig: mustard\] (mustard seeds). For rotation speeds, $\Omega$, lower than 15 rpm, the small and large glass beads have the same dynamic angle of repose which agrees with the findings in [@zik94], whereas the dynamic angle of repose is significantly higher (3 to 4 degrees) for the larger mustard seeds compared to the smaller ones. For rotation speeds, $\Omega$, higher than 15 rpm, the smaller glass beads show a higher dynamic angle of repose than the larger glass beads, and this angle difference increases with increasing rotation speed. For mustard seeds, the difference in the dynamic angle of repose between the smaller and the larger particles decreases with increasing $\Omega$ and the smaller seeds only show a higher angle for the highest rotation speeds studied. Both Fig. \[fig: mustard\] and Fig. \[fig: glass\] seem to indicate that the increase in the dynamic angle of repose with rotation speed, $\Omega$, in the S-shaped regime is larger for the smaller particles.
All the above angle of repose were measured by looking through one of the acrylic end caps. In order to study the boundary effect of these end caps on the dynamic angle of repose, we performed Magnetic Resonance Imaging (MRI) measurements. This technique of studying non-invasively the flow properties of granular materials was first used by Nakagawa et al. [@nakagawa93] and is in addition explained in more detail in ref. [@nakagawa94]. We used the large mustard seeds, which had an average diameter of 2.5 mm. The dynamic angle of repose was measured based on the concentration data which was averaged in a thin cross-sectional slice in the middle of the cylinder far away from the end caps. It is shown in Fig. \[fig: mri\] as function of the rotation speed by the open circles ($\circ$). We restricted the measurement to the flat surface regime, and all data points then lie approximately on a straight line. On the other hand, non-MRI data were measured at the end caps and are shown as stars ($\star$) in Fig. \[fig: mri\]. The consistently higher dynamic angle of repose at the end cap indicates the significance of the friction between particles and the boundary wall. We also found that the S-shape regime seems to start earlier at the end caps due to the additional wall friction.
bound\_3
Simulation Technique
====================
When measuring the dynamic angle of repose for different materials at the end caps, we found that the angle is always lower in the middle of the drum and the influence of the end caps seems to be rather short range, the angle drops to the value in the middle of the drum within a few centimeters. We are particularly interested in the dependence of this length scale on the particle diameter and density and on the gravitational force. Since we only have a limited number of particle diameters and density available in the experiment, we use three-dimensional discrete element methods, also known as [*granular dynamics*]{}, to overcome this problem.
Each particle $i$ is approximated by a sphere with radius $r_i$. Only contact forces during collisions are considered and the particles are not allowed to rotate. Since the mustard seeds are slightly aspherical, they rotate much less than glass beads. This was the motivation for our non-rotating assumption. The forces acting on particle $i$ during a collision with particle $j$ are $$F_{ij}^n = - \tilde{Y} \ (r_i + r_j - \vec{r}_{ij}\hat{n}) -
\gamma_n \vec{v}_{ij} \hat{n}
\label{eq: fn}$$ in the normal direction ($\hat{n}$) and $$F_{ij}^s = -\min(\gamma_s \vec v_{ij}\cdot\hat{s}(t), \mu|F_{ij}^n|) \ .
\label{eq: fs_p}$$ in the tangential direction ($\hat{s}$) of shearing. In Eqs. (\[eq: fn\]) and (\[eq: fs\_p\]), $\gamma_n$ and $\gamma_s$ represent a dynamic friction force in the normal and tangential direction, respectively, $\vec{r}_{ij}$ represents the vector joining both centers of mass, $\vec{v}_{ij}$ represents the relative motion of the two particles, and $\tilde{Y}$ is related to the Young’s Modulus of the investigated material. Dynamic friction in the model is defined to be proportional to the relative velocity of the particles in the tangential direction.
During particle–wall contacts, the wall is treated as a particle with infinite mass and radius. In the normal direction, Eq. (\[eq: fn\]) is applied, whereas in the tangential direction, the static friction force $$\tilde{F}_{ij}^s = -\min(k_s \int \vec v_{ij}\cdot\hat{s}(t) dt,
\mu|F_{ij}^n|)
\label{eq: fs_w}$$ is used. This was motivated by the observation that when particles flow along the free surface, they dissipate most of their energy in collisions and can come to rest in voids left by other particles. This is not possible at the flat drum boundary. In order to avoid additional artificial particles at the walls which would make the simulations of three-dimensional systems nearly infeasible, we rather use a static friction law to avoid slipping and allowing for a static surface angle when the rotation is stopped. Both tangential forces are limited by the Coulomb criterion which states that the magnitude of the tangential force cannot exceed the magnitude of the normal force multiplied by the friction coefficient $\mu$. For particle–particle collisions we use $\mu=0.2$, and for particle–wall collisions, $\mu_w=0.4$. In order to save computer time, we set $\tilde{Y}$ to which is about one order of magnitude softer than vulcanite but the maximal overlap of two particles is at most $0.3\%$ of the sum of their radii, which is still realistic. This gives a contact time during collisions of $1.1\cdot 10^{-4}$ s. The coefficient of restitution for wall collisions is set to 0.77 which is within the error bar of the experimentally measured value of 0.75, see section \[experiment\]. In experiments with spherical liquid-filled particles, we found only a weak dependence of the restitution coefficient on particle size and therefore used a normal force law, Eq. (\[eq: fn\]), that would make the restitution coefficient independent of particle size. When the same type of force law is applied to particle-particle collisions, it gives a normal restitution coefficient of 0.56. A discussion of the different force laws is given in ref. [@schaefer96] and a review of different applications using granular dynamics is given in [@ristow94b]. Even though detailed experiments for binary collisions of particles were performed, the force relations before and after a collision depend on the material and the aspherity of the particles [@foerster94] and since these two quantities were not available for mustard seeds, we can only take the published values as a guideline. The numerical parameters were fine adjusted by comparing the experimentally determined dynamic angle of repose for 2.5 mm mustard seeds ($\rho = 1.3$ g/cm$^3$) with the simulation results over the $\Omega$–range from 8 to 35 rpm. The radius of the drum was chosen as $R=3.5$ cm. Both data sets are shown in Fig. \[fig: fit\]. Also shown as a solid line in Fig. \[fig: fit\] is the theoretical result based on a model by Zik et al. [@zik94]. They started from the equilibrium condition for the surface flow $j$ in a laminar and thin layer inclined with an angle $\Theta$: $$j = \frac{\rho g}{3\eta} h_0^3 \cos\Theta (\tan\Theta - \tan\Theta_0)
\label{eq: zik1}$$ where $\rho$ denotes the particle density, $g$ gravity, $\eta$ the constant viscosity and $\Theta_0 = \arctan\mu$. The cut-off depth, $h_0$, corresponds to a constant pressure value of $p_0 = h_0 g \cos\Theta$. A second expression for the surface flow in a half-filled drum can be obtained by looking at mass conservation, [@rajchenbach90]: $$j = \rho \frac{\Omega}{2} (R^2 - r^2)
\label{eq: zik2}$$ where $r$ measures distance from the drum center along the free surface. Equating expressions (\[eq: zik1\]) and (\[eq: zik2\]) and using the relation $\tan\Theta=\frac{dy}{dx}=y'$, where $y(x)$ measures the height of the top surface particle along the surface and $(\cos\Theta)^{-1}=\sqrt{1+(y')^2}$, we obtain $$(y')^3 - (y')^2 \tan\Theta_0 + y' + c \Omega (y^2 + x^2 - R^2) = \tan\Theta_0
\label{eq: zik3}$$ with $c = \frac{3\eta g^2}{2 \rho p_0^3}$. Corrections to this model were recently proposed by Khakhar et al. [@khakhar97], but they lead to the same equations for the dynamic angle of repose as above in the case of a half-filled drum due to the symmetry of the thickness of the fluidized layer for shear flow. Solving for $y'$ at the origin (drum center), the only one of the three roots with no imaginary part reads $$y' = \tilde\mu + (B+\sqrt{D})^{1/3} - \frac{1/3 - \tilde{\mu}}
{(B+\sqrt{D})^{1/3}}
\label{eq: angle}$$ where $3\tilde\mu = \tan\Theta_0$, $$B = \tilde\mu(1+\tilde\mu^2) + \frac{1}{2}c \Omega R^2 \quad \mbox{and}$$ $$D = 3 (\tilde\mu^2 + \frac{1}{9})^2 + \tilde\mu(1+\tilde\mu^2) c \Omega R^2 +
(\frac{1}{2}c \Omega R^2)^2 \ .$$ We integrated Eq. (\[eq: zik3\]) numerically and checked that the theoretical profile has a similar shape for different rotation speeds as the numerical data. We adjusted the parameter $c$ using the experimental data points and the best fit was obtained for $c=0.0111 \pm 0.0001$ s/m$^2$ and $\mu = 0.51 \Rightarrow \Theta_0 = 27^\circ$. It is remarkable how well the theoretical curve fits the data points from the experiments and the numerical simulations, as shown in Fig. \[fig: fit\]. The theoretical curve is very close to the arcus-tangent curve proposed by Hager et al. [@hager97] and also found in two-dimensional simulations over a wide $\Omega$-regime [@dury97b]. For low rotation speeds, $\Omega< 8$ rpm, the experiment is very near the discrete avalanche regime, and therefore the simulations where we have used only dynamic friction for the particle-particle interactions and the theory where a steady flow is assumed tend to deviate slightly from the experiment. Rajchenbach experimentally found the relation $\Omega \sim |\Theta - \Theta_c|^m$ with $m=0.5$ leading to $\Theta = \Theta_c
+ \alpha \Omega^2$ [@rajchenbach90] which gives an increasing slope for increasing $\Omega$ in the graph, whereas the experimental data points in our Fig. \[fig: fit\] suggest a decreasing slope with increasing $\Omega$. To illustrate this point further, we replotted all of our available experimental data points of the large mustard seeds measured at the end caps, taken from Fig. \[fig: mustard\] above, in the same fashion as Rajchenbach and obtained a scaling exponent of m=0.87 using $\Theta_c=34.1^\circ$. This has to be compared with m=0.5 found by Rajchenbach and m=0.7 given by the numerical prediction by Tang and Bak [@tang88]. The $\Omega$ range of Rajchenbach is smaller than the one investigated by us and we speculate that his finding is valid close to the transition point to the continuous flow regime where the quadratic fit works rather well. We included this in Fig. \[fig: fit\] as dotted line using a best quadratic fit for the value of $\Omega<20$ rpm.
bound\_5a
Boundary Effect on Surface Angle
================================
As shown in Fig. \[fig: mri\], the dynamic angle of repose of granular material in a rotating drum is significantly higher at the end caps than in the middle. For the experimentally investigated particle sizes of the order of millimeter, the effect was visible up to a few centimeters. But this length scale might depend on the particle diameter and density and also on external parameters like gravity.
Using the above described technique, we simulated extended three-dimensional, half-filled drums. For a particle size of 2.5 mm and a rotation speed of 20 rpm, the time-averaged angle, denoted by $\langle\Theta(z)\rangle$, as function of the position along the rotation axis is shown in Fig. \[fig: profile\]. Each point with corresponding error bar stands for a weighted average over the nearest neighbours. In order to study the characteristic length, $\zeta$, of the boundary effect, we fit all data points by the relation $$\langle\Theta(z)\rangle = \Theta_\infty + \Delta\Theta\left(
e^{-z/\zeta} + e^{-(L-z)/\zeta} \right)
\label{eq: profile}$$ where $L$ stands for the length of the drum, $\Theta_\infty$ for the dynamic angle of repose far away from the boundaries and $\Delta\Theta$ for the angle difference between the value at the boundary and $\Theta_\infty$. For the curve shown in Fig. \[fig: profile\], the corresponding values are $\Delta\Theta\approx 4^\circ$, which is the same value given in Fig. \[fig: mri\] for the MRI experiment, $\zeta = 3.19 \pm 0.25$, $L = 20.6$ cm and $\Theta_\infty = 35^\circ \pm 0.2^\circ$. We tested Eq. (\[eq: profile\]) against the simulation results for different drum lengths of $L/2$, $L/4$ and $L/8$ and found a remarkably good agreement. In the last two cases, the value for $\Theta_\infty$ is never reached in the middle of the drum due to the boundary effects.
bound\_6
Using drums that are at least 64 particle diameters long, we studied the dependence of $\Theta_\infty$ (middle) and the angle at the end caps on the particle diameter $d$ at the same rotation speed of 20 rpm. The simulation values in the tangential direction were chosen in such a way that the normalized tangential velocities before and after impact were [*independent*]{} of the particle diameter $d$. The ration $\tilde{Y}/k_s$ was set to 3.1, a value which gives for acetate spheres a good agreement of simulations [@schaefer96] and experiments [@foerster94]. The value of $\gamma_s$ was chosen sufficiently high to give a similar behavior in particle-particle- and particle-wall-collisions in the sliding regime. The results for the surface angle along the rotation axis are shown in Fig. \[fig: angle\], which illustrates that the angle increases with increasing particle size in agreement with the mustard seed experimental results given in Fig. \[fig: mustard\]. The angle difference of around 4 degrees, which seems to be independent of the particle size, also agrees with the experimental findings. In other experiments, different dependencies were observed: das Gupta et al. [@dasgupta91] mostly found a higher angle for smaller particles using sand grains and Hill and Kakalios [@hill94] measured higher angles for smaller particles when using sand and glass particles although it was also possible to get no angle difference for certain size ratios of glass spheres. The latter was also found by Zik et al. [@zik94], whereas Cantelaube [@cantelaube95b] did not find a clear trend when using discs in a quasi two-dimensional drum. What causes the different behaviour is not clear at the moment and a more detailed analysis would be desirable but is beyond the scope of this article. It is necessary to use appropriate values for the simulation parameters to quantitatively model a desired system, which is why we gathered as much information for the mustard seeds as possible. A arcus-tangent fit which gave a smaller mean deviation than a parabolic fit was added to Fig. \[fig: angle\] to guide the eye. Changing the density, $\rho$, of the particles or the gravitational constant, $g$, has a dramatic effect on the angle of repose: for the latter quantity this is shown in table \[tab: gravity\] and a similar behaviour was seen in recent experiments [@fabi97b]. For both $\rho$ and $g$, an increase in value corresponds to an angle decrease. When a hydrostatic pressure, $p_0 \sim g$, is assumed, the data for $g<30$ m/s$^2$ can be well described by Eq. (\[eq: angle\]). The lower $g$ becomes the more pronounced the S-shaped surface becomes, and in the limit $g \rightarrow 0$, the transition to the centrifugal regime takes place at $$\Omega_c \approx \sqrt{\frac{\sqrt{2}\,g}{R\,\sin\Theta_0}}$$ where $\Theta_0$ denotes the average angle in the limit $\Omega \rightarrow
0$ [@walton93; @dury97b]. In our case, for $\Omega = 20$ rpm the transition to the centrifugal regime occurs at $g \approx 0.45$ m/s$^2$, which is in perfect agreement with the numerical findings. Even though we studied more than one (two) orders of magnitude in $\rho$ ($g$), we could not obtain an accurate infinite value limit.
bound\_8a
$g$ \[m/s$^2$\] 1.62 3.73 9.81 13.6 25.1 274
----------------------- ------- ------- ------- ------- ------- -------
$\Theta$ \[$^\circ$\] 48.2 41.4 35.0 33.3 30.2 18.4
$\zeta/R$ 0.291 0.274 0.277 0.269 0.283 0.260
: Angle of repose, $\Theta$, in the drum middle and dimensionless characteristic length, $\zeta/R$, as function of gravity, $g$ (simulation).[]{data-label="tab: gravity"}
Range of Boundary Effect
========================
In order to study the range of the boundary effect, we extracted figures similar to Fig. \[fig: profile\] from our simulations and varied the drum length, $L$, and radius, $R$, the particle diameter, $d$, and density, $\rho$, the gravitational constant, $g$, and the rotation speed, $\Omega$. The data points for the dynamic angle of repose as a function of position along the rotation axis were fitted by Eq. (\[eq: profile\]) giving the characteristic length, $\zeta$, of this run. As expected, $\zeta$ did not vary when the length of the drum or the rotation speed was changed, but surprisingly, the characteristic length, $\zeta$, did not change when the density of the particles or the gravitational constant was changed by more than one order of magnitude, even though the dynamic angle of repose strongly depends on both as shown in table \[tab: gravity\] for the latter quantity.
bound\_9a
Based on the definition of $\zeta$ in Eq. (\[eq: profile\]), one might speculate that $\zeta \sim R$ since the gradient of the slope along the rotational axis of the surface should be a material property, i.e. it should not depend on the geometry. The angle of repose is independent of the drum radius, $R$, and therefore the height difference between the surface at the end cap and the surface in the middle of the drum must be proportional to $R$. This leads to $\zeta \sim R$ which is indeed the case, and we show in Fig. \[fig: range\] the dimensionless characteristic length, $\zeta/R$, as function of dimensionless particle diameter, $d/R$, for three different drum radii. Below a critical diameter, $d_c$, $\zeta$ seems to be independent of the particle size, and we propose the following relation $$\zeta = \left\{ \begin{array}{ll}
\alpha\, R & \mbox{, if $d \le d_c$} \\
\alpha\, R + \beta (d - d_c) & \mbox{, if $d>d_c$}
\end{array}
\right.$$ where, in our case, $\alpha=0.28$ and $\beta= 3.13$. The critical particle diameter $d_c \approx 0.14\, R$ and it seems to decrease slightly with increasing drum radius. Therefore, particles in the fluidized zone with $d<d_c$ might be describable by a continuum model. For particles with $d>d_c$, we have to take finite size effects into account.
For comparison, we replot in Fig. \[fig: range3\] the data from Fig. \[fig: range\] by showing the characteristic length, $\zeta$, made dimensionless by the average particle diameter, $d$. A strong decrease in $\zeta/d$ and a later saturation is clearly visible for increasing particle diameter. The solid line is an exponential fit which matches all data points very well but it can only serve as a guide to the eye since it does not reproduce the right value in the limit $d \rightarrow 0$.
bound\_9b
Conclusions
===========
We have investigated the dynamic angle of repose, $\Theta$, in a three-dimensional rotating drum in the continuous flow regime. By choosing different materials and particle diameters, we discussed the $\Omega$-dependence of $\Theta$ for glass beads and mustard seeds of two different sizes. In the low rotation speed regime, both types of glass beads showed the same angle of repose, whereas the angle was higher for the [*larger*]{} mustard seeds. Using MRI techniques, we could quantify, for the large mustard seeds, the angle difference between the middle and the end of the drum and its $\Omega$-dependence. In all cases, either a linear or an arcus-tangent dependence of $\Theta$ on $\Omega$ was found. In order to investigate the range, $\zeta$, of the boundary effects, we used a three-dimensional [*discrete element*]{} code and fitted the averaged angle along the rotation axis to two exponentially decaying functions. We found that $\zeta$ scales linearly with the drum radius. On the other hand, it does not depend either on the particle density or the gravitational constant, even though the surface angle changes drastically with these quantities, or on the rotation speed of the drum. A detailed analysis of the dependence of the characteristic length, $\zeta$, on the particle diameter, $d$, revealed that $\zeta$ is independent of $d$ for small particle diameters but shows finite size effects for larger $d$.
CMD and GHR gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft and partial funding for their stay in Golden by the [*Particulate Science & Technology Group*]{} of the Colorado School of Mines. JLM and MN are supported in part by NASA through contract number NAG3-1970. We also would like to thank Susan McCaffery for a critical reading of the manuscript. A generous grant of computer time on the Cray T3E at the Forschungszentrum Jülich made the numerical simulations possible.
$g$ \[m/s$^2$\] 1.62 3.73 9.81 13.6 25.1 274
----------------------- ------- ------- ------- ------- ------- -------
$\Theta$ \[$^\circ$\] 48.2 41.4 35.0 33.3 30.2 18.4
$\zeta/R$ 0.291 0.274 0.277 0.269 0.283 0.260
: Angle of repose, $\Theta$, in the drum middle and dimensionless characteristic length, $\zeta/R$, as function of gravity, $g$ (simulation).[]{data-label="tab: gravity"}
Figure Captions {#figure-captions .unnumbered}
---------------
\(a) Flat surface for low rotation speeds, (b) deformed surface for medium rotation speeds with two straight lines added as approximation and (c) fully developed S-shaped surface for higher rotation speeds.
Experimentally measured starting ($\circ$) and stopping ($\star$) angle and dynamic angle of repose ($\bullet$) for mustard seeds.
Dynamic angle of repose for black ($\bullet$) and yellow ($\circ$) mustard seeds.
Dynamic angle of repose for small ($\bullet$) and large ($\circ$) glass beads.
Comparison of dynamic angle of repose for large mustard seeds taken from MRI ($\circ$) and non-MRI ($\star$) measurements.
Comparison of dynamic angle of repose for large mustard seeds taken from MRI ($\circ$), numerical simulation ($\bullet$) and the theory of Zik et al. [@zik94] (—).
Profile of the dynamic angle of repose along the rotation axis for 2.5 mm spheres: ($\bullet$) simulation, (—) fit.
Dynamic angle of repose as function of sphere diameter for $\Omega=20$ rpm (simulation); ($\star$) end cap, ($\bullet$) drum middle, (—) arcus-tangent fit.
Dimensionless range of boundary effect for spheres with different diameter; ($\star$) $R=3.5$ cm, ($\circ$) $R=7$ cm, ($\bullet$) $R=10.5$ cm, dotted line shows value $\zeta=0.28 \, R$ for radius-independent regime (simulation).
Range of boundary effect measured in sphere diameters for spheres with different diameter; ($\star$) $R=3.5$ cm, ($\circ$) $R=7$ cm, ($\bullet$) $R=10.5$ cm, (—) exponential fit (simulation).
bound\_0
bound\_2
bound\_1
bound\_3
bound\_5a
bound\_6
bound\_8a
bound\_9a
bound\_9b
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Allen Riddell[^1]'
- 'Michael Betancourt[^2]'
bibliography:
- 'references.bib'
date: '23. July 2018, Revision 10'
title: 'Reassembling the English novel, 1789-1919'
---
Introduction {#introduction .unnumbered}
============
Aspirations in the 20th century for sociologically-inclined literary history foundered due to a lack of accessible, trustworthy, and inclusive bibliographies and biographical records. Despite sustained interest, no principled estimates of the number of novelists writing or the number of new novels published during the 19th and early 20th centuries ever materialized [@sutherland1988publishing]. Without a detailed accounting of novelistic production, numerous questions proved impossible to answer. The following three are representative: How many writers made careers as novelists, Are there unacknowledged precursors or forgotten rivals to canonical authors, To what extent is a writer’s critical or commercial success predictable from their social origins? Although material traces of every novel published in Europe and North America survive, gathering particulars required to answer questions such as these proved too time-consuming or too resource intensive.
The lack of credible information about the population of novelists and the population of published novels obstructs research in literary studies, cultural studies, book history, and sociology of literature. Two communities in particular stand to gain from a more detailed accounting of these two populations. The first includes those interested in studying literary form and prose style from below. A characteristic concern of this group is an interest in how the emergence and diffusion of literary morphology reveals information about broader economic, social, and cultural relationships within and across national and linguistic situations (e.g., @escarpit1958sociologie [@moretti1998atlas; @casanova1999republique; @moretti2000conjectures]). The second group includes researchers in cultural studies and sociology of culture interested in uniting literary history with sociological concerns. This group includes those interested in the working conditions facing novelists and those studying the history of occupational gender segregation in the text industry (e.g., @williams1965long, @tuchman1989edging). This group also includes those interested in reassembling an understanding of literary artworks as products of networks of actors whose actions are necessary for works’ existence and whose actions, in turn, shape the art objects [@becker1995introduction xii]. Library digitization and sharing of machine-readable datasets are two developments which support research agendas associated with these communities. More generally, these developments facilitate studying literary works at multiple scales and with a broader range of vocabularies.
To demonstrate the improving prospects for data-intensive, sociologically-inclined literary history, this paper offers two analyses of bibliographic data concerning novels published in the British Isles after 1789. Both of these analyses would not have been possible—or at least not practical—without the availability of digital surrogates of surviving volumes and the sharing of machine-readable bibliographic data. First, we estimate the yearly rates of new novel publication in the British Isles between 1789 and 1919, a period which witnessed, in aggregate, the publication of between [40,000]{} and [63,000]{} previously unpublished novels (“new novels”). Although there has been considerable speculation about this time series, ours are the first principled estimates to be published. The years studied include the rise of mass literacy and one of the more important periods in the history of publishing (1830-1850), a period during which practices and institutional arrangements resembling the modern publishing industry emerge [@raven2007business 328-329]. Second, using the titles of novels published between 1800 and 1829, we resolve a dispute concerning occupational gender segregation in novel subgenres. We show that the remarkable growth in the number of men novelists after 1815 was not concentrated in particular subgenres (military novels, nautical tales, and historical novels).
The analyses presented here are limited to literary production on two islands in the North Atlantic. Although the prospect of comparative research was a primary motivation for this work, a lack of comprehensive bibliographical records outside the British Isles made such research difficult. The exhaustive bibliography of novels published between 1770 and 1836 found in @raven2000english [@garside2000english] (hereafter “RFGS”)—indispensable to the work here—has no real equivalent. (For example, although @brummer1884lexikon is impressive in the number of German-language titles it documents, like @block1961english, it makes no claims to have enumerated all titles published.) Bibliographic work on novels written in languages other than English is, however, ongoing and library digitization makes the work considerably easier. And the estimates presented here provide information about plausible trajectories of literary production elsewhere. For example, because it is hard to imagine per capita novelistic production growing considerably faster than it did in the British Isles during the 1840s, the pace of growth during this decade may be used as an estimate of the upper bound on the pace of growth in established text industries in other geographic regions.
Rise of the Text Industry {#rise-of-the-text-industry .unnumbered}
=========================
No survey of new novels published in the British Isles exists for any year after 1836. There is neither an exhaustive list of new novels published nor principled estimates of the number of new novels published in any given year in any year after 1836. Given the pace of expansion in the publishing industry during the period and the time and resources required to complete RFGS this is understandable.[^3] The absence of information about novels published after 1836 is regrettable because this period witnesses the rise of mass literacy and sees the publishing industry adopt practices and organizational structures characteristic of the modern text industry [@raven2007business 328-329]. What little information we have about the population of literary works published after 1836 relies on inferences drawn from the heterogeneous population of published books (novels and non-novels, new and reissued) [@weedon2003victorian; @eliot1997patterns]. Even here, however, the information is not detailed enough to allow us to estimate the number of novels (new or reprinted), published during any year or decade.
In this section we estimate rates of novelistic production for each year between 1789 and 1919 from five existing data sources using a probabilistic model. In addition to annual publication counts, the data permit us to estimate the proportion of new titles associated with men and women authors. Although we do not directly observe the number of new novels published in any year after 1836—or new novels by author gender after 1829—we infer credible intervals through the use of a model of several correlated time series. Our results make visible, for the first time, a period of particularly intense growth between [1840]{} and [1855]{}.
Background {#background .unnumbered}
----------
There are bibliographies and related resources that purport to provide information about new novels published during specific periods of the 19th century. Most are unusable. Typical are bibliographies of a period or novel subgenre which for one reason or another are not exhaustive. @block1961english is one example. Although it advertises itself as a bibliography of English novels published between 1740 and 1850, it is not clear what novels are included and what novels are missing. Worse, it includes books which are not novels by any prevailing definition @garside2000general [2]. There are, however, a small number of works which are exhaustive for a period or genre and do provide information usable by those interested in an inclusive history of the novel and of novel writing. @bassett2008production, for example, enumerates three-volume editions appearing between 1863 and 1897. RFGS, mentioned earlier, enumerates all novels published between 1770 and 1836. RFGS also helpfully makes clear how they go about the essential task of distinguishing novels from non-novels [@garside2000general].
For those interested in an inventory of new novels published during the 19th century, the most useful information comes from historians of publishing. (With notable exceptions—including @escarpit1958sociologie [@moretti1982anima; @moretti1998atlas; @moretti2000slaughterhouse]—literary historians working after 1950 have not pursued an inclusive history of the novel, one which would include all novels and novelists.) Working with a machine-readable version of the *Nineteenth Century Short Title Catalog* (NTSC), @eliot1997patterns creates a time series which provides information about the number of books published in London, Oxford, Cambridge, Edinburgh, and Dublin each year between 1801 and 1870.[^4] Until an integrated history of the English novel and the book trade is written, this series will be invaluable. It helps us in two specific ways. First, it provides a crude upper bound on the number of new novels published each year as the number of new novels will always be less than the number of books (novels and non-novels) appearing in a given year. Second, because the rate of book production and the rate of new novel production are correlated, the time series gives us considerable insight into how the rate of new novel production likely changed from year to year.
The two most important resources used to estimate the rate of novelistic production are RFGS and a series derived from the *Nineteenth-Century Short Title Catalog* (NSTC). Three other resources used in the model—which tend to cover shorter periods—are introduced in the next section.
Method {#method .unnumbered}
------
We estimate annual rates of novelistic production from five data sources using a probabilistic model. The model assumes that changes in the pace of novelistic production are well described by exponential growth with transitory deviations. Using the model and available data we infer the pace of growth and the character of deviations. Taken together these inferences permit us to estimate the number of novels published each year between 1789 and 1919. In this section we first describe the resources used and then elaborate the model.
#### The English Novel, 1770-1836 (“RFGS”)
The most important source of information is *The English Novel*, an exhaustive survey of novels appearing between 1770 and 1836 [@raven2000english; @garside2000english; @garside2006english]. In this paper we refer to the two-volume printed bibliography, updates, and online database collectively as RFGS.[^5]
RFGS anchors the analysis in this paper in several respects. What RFGS records, counts of new novels—and, for 1800-1829, counts by author gender—is what we wish to infer for the entire period (1789-1919). RFGS provides a principled, descriptive definition of the novel: printed works referred to as novels by readers at the time. The usefulness and specificity of this definition is amplified by the fact that RFGS provides examples of works which meet the definition (the bibliography itself) as well as works which do not meet the definition. RFGS includes detailed records for each title listed in the bibliography. For years 1800–1829, each record includes an indication of the gender of the author. RFGS code author gender as (“Male”,“Female”, “Unknown”). If the title indicates author gender but not author name, the title is associated with the indicated author gender. For example, although the novel *The Castle of Probation* (1802) does not have a named author, it is associated with a “Male” author in RFGS because the novel’s full title includes the words “By a Clergyman”.[^6] As a practical matter, we see RFGS as providing two distinct time series: first, counts of new novels published between 1770 and 1836; and, second, counts of new novels by author gender between 1800 and 1829. We further limit our attention to records associated with 1789 and later years in order to allay concerns about the definitional strategy used. As the 18th century progresses, characteristics associated with works labeled “novels” tend to stabilize. Works published after 1789 which were referred to as novels are very likely to share morphology with works labeled novels published during later decades. This is less often the case for novels published earlier in the 18th century.
To the concern that the definition used by RFGS may be too restrictive, that it may tend to exclude literary works which were not called novels but which are, in all other respects, treated by readers at the time as if they were novels, it is worth noting that different definitions of the novel tend to agree on particulars in more than 85% of cases. Moreover, disagreement is localized. Most disputed cases involve novel-like (didactic) juvenile fiction and novel-like religious fiction (Troy Bassett, personal communication, Nov. 9, 2015). It should, therefore, be straightforward for other researchers to adjust the estimates reported here or to modify the model source code accompanying this paper to accommodate different assumptions about what works count as novels.
**Nineteenth-Century Short Title Catalog (London, Oxford, Cambridge, Edinburgh, or Dublin), 1801-1870 (“LOCED”).** @eliot1997patterns extracts yearly totals of entries (novels and non-novels) listed in the *Nineteenth-Century Short Title Catalog* (NSTC) associated with one of the following places of publication: London, Oxford, Cambridge, Edinburgh, or Dublin. We refer to this time series using Eliot’s abbreviation, “LOCED”. Because RFGS provide an exhaustive survey of new novels between 1801 and 1836, we know what percentage of LOCED titles are new novels for 36 years. During these years there is, therefore, an opportunity to observe how the two time series co-vary.
Our LOCED series differs from Eliot’s in one important respect. The original LOCED series has an unusual feature: undated material is assigned to the nearest half-decade (to a year ending with a “0” or a “5”) [@eliot1997patterns 86]. To deal with this idiosyncrasy, we ignore entirely publication counts from the original series which are associated with years ending in “0” or “5”. Although ignoring counts in these years might appear to bias the counts associated with other years downward (as many works, were their publication years known, “belong” in adjacent years), we have a different view. The original LOCED series mixes two time series, a series recording dated material and a series recording undated material. (New novels, for example, are virtually certain to report publication years on their title pages.) By stripping out counts for years ending with “0” or “5”, we ignore the time series related to undated publications.[^7]
#### Publishers’ Circular, 1843-1919 (“PC”)
The third time series we use records yearly totals of new titles derived from *Publishers’ Circular*, 1843–1919 [@eliot1994patterns] (“PC”). Issues of *Publishers’ Circular* appeared biweekly and listed new books published. The PC time series overlaps with LOCED for 28 years (1843–1870), permitting observation of how these two series co-vary. As one would expect given the similarity in what is being recorded in the two series, the PC series and the LOCED series are highly correlated ($r = 0.72$). Together they give us a guide to year-to-year variation in the rate of book publication over 119 years (1801–1919).
At this point the inference strategy may be growing clearer. We aim to gather several partially overlapping time series which are correlated in order to “triangulate” from observed rates to unobserved rates.
#### *The Athenaeum* Reviews of Novels, 1860, 1865, …, 1900
The fourth and fifth resources are used primarily to improve the estimates of the number of new novels published after 1850. Improving our estimates for this period is important because uncertainty grows as we move further away from the bibliographic terra firma of the early 19th century. The fourth resource appears in @casey1996edging. Casey provides counts for the number of novels reviewed in *The Athenaeum* during nine years: 1860, 1865, 1870, 1875, 1880, 1885, 1890, 1895, and 1900. (*The Athenaeum* was a London literary magazine published from 1828 to 1921.) Casey also breaks down the number of novels reviewed during the nine years by author gender. We make the assumption that every title counted as a novel in this time series meets the definition of a novel used by RFGS.
Counts are taken from Chart 2 in @casey1996edging. In Casey’s series, titles with multiple authors contribute an author fraction to the relevant count. As the model used here is designed to model count data, all non-integer values in Chart 2 are rounded down. As novels with multiple authors are exceedingly rare during the period, we feel that ignoring authors other than the first will not meaningfully change any results presented in our analysis.
*The Athenaeum* does not review all novels published, so these counts are significantly lower than the total number of new novels published. If we knew the percentage of new novels reviewed by the magazine, we could derive the number of new novels published during these nine years. We infer the percentage of novels reviewed by modeling the overlapping time series. This strategy is the same as the one used to infer the percentage of total books published which are novels. In our model, we assume that the percentage of novels reviewed, whatever it turns out to be, is fixed during the period 1860–1900. Supporting this assumption is the observation that novel reviews in *The Athenaeum* increased markedly between 1860 and 1900, suggesting that the periodical enjoyed flexibility in the number of titles it reviewed.
#### Elicited Distributions of New Novel Publications in 1886, 1891, and 1894
The fifth resource, like the fourth, is used to reduce the considerable uncertainty about the number of new novels published in the second half of the 19th century. The fifth resource is a series of three *distributions* over rates of new novels publication in the years 1886, 1891, and 1894. These distributions are elicited from a domain expert, Troy Bassett, editor of *At the Circulating Library: A Database of Victorian Fiction, 1837-1901* (“ATCL”).[^8] We follow the elicitation procedure described in @garthwaite2005statistical. For each year, we asked Bassett to report quartiles of the distribution reflecting his beliefs about the total number of new novels published that year. As editor of ATCL, a database which contains entries for over 15,000 novels published between 1837 and 1901, Bassett is in a position to make accurate estimates of intervals which are likely to contain the total number of new novels published in any year during the Victorian period. Eliciting quartiles of a distribution which describe the likely number of new novels published in a year is roughly equivalent to asking for an interval which contains the true number with probability 0.5. After eliciting quartiles of the distributions for the three years, we find familiar probability distributions which have quartiles as close as possible to those elicited. The three distributions identified in this way are the distributions used in the model. For example, the quartiles elicited for the year 1886 are 394, 482, and 613. A normal distribution with mean 494 and standard deviation 163 has approximately the same quartiles: 384, 494, 604.[^9]
### A Model of Novelistic Production {#a-model-of-novelistic-production .unnumbered}
We view the number of new novels appearing each year as counts generated from a process defined by a year-specific latent rate. The latent rates of new novel publication change from year to year following a simple exponential growth trend with transitory deviations due to disruptions in the book trade— economic depressions, wars, cholera outbreaks, and so forth. We know enough about literary production during the period to safely say that a model failing to account for periodic deviations would be conspicuously inadequate. Since we model latent rates of new novel publication on the log scale our model of growth is a linear model. Such a model is often referred to as a “log-linear model”. Rates are modeled using a Gaussian Process, with the mean of the Gaussian Process capturing the basic trend and the Gaussian Process’s covariance structure capturing transitory deviations. (As Gaussian Processes are covered in detail elsewhere—for example, in @rasmussen2005gaussian or @bishop2007pattern—we do not describe them in any detail here.) In this section we review the most important assumptions we make in our model—exponential growth with transitory deviations—and then describe in detail how the five time series mentioned in the preceding section appear in the full model. To simplify the presentation, we initially describe the model without considering author gender. The minor adjustments required to model author gender are presented at the end of this section.
Seen from a distance, it is obvious that the rate at which new novels appear grows exponentially. We can appreciate this by looking at the rate at which books (novels and non-novels) appear [@eliot1997patterns; @weedon2003victorian]. Additional evidence, if any is needed, is available from @eliot1998patterns which shows nonlinear growth in the number of titles labeled as “Literature” in the NSTC [@eliot1998patterns 85]. The standard approach to modeling this sort of trend is a log-linear model. Taking log publication rates as our estimands, we can describe the trend using a linearly increasing rate of publication. In a log-linear model, the log rate of new novel publication in a given year $t$ is described by a two-parameter expression, $\alpha + \beta t$, where $\beta$ is interpreted as an annual growth rate. (For example, if in year 1800 the annual rate of publication is 100 new novels and the rate grows continuously at a rate of 3%, $\beta$ would be 0.03 and in the year 1900 the annual rate of publication would be roughly 2,000 new novels.) In our model of the log rates of new novel publication, a linear trend appears as the mean function of a Gaussian Process.
Both the linear trend and transitory deviations are modeled by a Gaussian Process. The time series derived from the NSTC (LOCED) and *Publishers’ Circular* (PC) make clear that novel publishing experienced several disruptions between 1789 and 1919. Time series of the number of books published suggest the influence of events including wars, market panics, and epidemics. That the disruptions are transitory is also clear; the text industry always returns to growth. Because Gaussian Processes can model both an underlying trend and periodic or transitory deviations, they are a familiar choice in settings similar to this one. (In particular, Gaussian Processes are more expressive than models with independent and identically distributed errors such as linear regression.) The backbone of our model is a Gaussian Process of the log rate of new novel publication between 1800 and 1919. In symbols, the log rate of new novel appearance for year $t = 1, \ldots, T,\, T = 120$ is given by
$$\begin{aligned}
\lambda_t &= \alpha + \beta t + \epsilon(t),\\
\epsilon(\vec t) &\sim \text{GP}(0, K) \\
k(t, t^\prime) &= \sigma_\lambda^2 \exp\left(-\frac{|t - t^\prime|^2}{l_\lambda^2} \right) \\
\end{aligned}$$ where the year $t = 1$ is associated with 1800, $t = 2$ with 1801, and so on; $\text{GP}(0, K)$ is a zero-mean Gaussian Process with $120 \times 120$ covariance matrix $K$; and the element $(t, t^\prime)$ of $K$ is given by $k(t, t^\prime)$.
To capture the belief that deviations from the trend (parameterized by $\alpha$ and $\beta$) will tend to persist for a bounded number of years, we use an informative prior distribution on the characteristic length-scale $l_\lambda$. This distribution places 90% probability on values between 1 and 10, expressing the prior belief that deviations will tend to persist for between 1 and 10 years. Such a prior distribution is consistent with the belief that, say, a market panic might affect the rate of novel publication in the short term but would likely cease to influence publication rates in years which are more than ten years distant from the event.
The observed annual counts of new novels from RFGS (1800-1836) (the first time series) are connected to the latent rates $\lambda_{1:37}$ via a Negative Binomial sampling distribution. This sampling model allows us to connect the smoothly varying rates to observed counts of new novels. Separating the latent rate from the observed counts in the model is particularly important before 1840 because there is considerable year-to-year variation in the observed counts of new novels which are due to the arbitrary assignment of novel publications into discrete years.[^10] In symbols, the sampling model is given for year $t = 1, \ldots, T,\, T = 37$ by $$\begin{aligned}
y_t \sim \text{NegativeBinomial}_2(\exp(\lambda_t), \phi_y)
\end{aligned}$$ where $\text{NegativeBinomial}_2$ is parameterized by a location parameter and a parameter controlling dispersion. (If $Y$ is distributed according to a $\text{NegativeBinomial}_2(\mu, \phi)$ distribution then $\operatorname{E}(Y) = \mu$ and $\operatorname{Var}(Y) = \mu +
\frac{\mu^2}{\phi}$.) We use a two-parameter Negative Binomial sampling model here rather than a simpler, single-parameter Poisson model. The former’s ability to model additional variation is important given the uncertainty about the latent process being modeled.
To incorporate the counts of *Publishers’ Circular* (PC) titles (the second time series), we introduce an additional Gaussian Process to model, for each year, the proportion of PC titles which are new novels. Background knowledge and @eliot1998patterns lead us to believe that the proportion will be certainly less than 50% and that it will increase modestly over the period. As we did for the rates of new novel appearance, we transform the proportions into units which are conveniently modeled using a linear trend. In this case, we express the proportions on the log odds scale, denoting the log odds as $\nu_t$ for year $t$. (The log odds is the logarithm of the odds, $\log(\frac{p}{1-p})$, where $p$ is a proportion between 0 and 1.) In contrast to our thinking about year-to-year variation in rates of new novel publication, we anticipate that the proportion of PC titles which are new novels will change comparatively slowly. Whereas an economic crisis or other kind of “shock” might affect the rate of new novel publication over a period of several years, it would likely not affect the proportion of books which are novels. In other words, we anticipate that factors influencing the economics of publishing novels as opposed to non-novels does not change as rapidly as factors influencing the rate of book publishing in general. To capture this belief, the characteristic length-scale for this second Gaussian Process is modeled with a prior distribution placing 90% probability on values between 8 and 36, expressing the belief that deviations from trend will tend to persist for between 8 and 36 years. In symbols, the proportions are modeled for year $t = 1, \ldots, T,\, T = 120$ on the log odds scale as follows:
$$\begin{aligned}
\nu_t &= \alpha_\nu + \beta_\nu t + \epsilon_\nu(t)\\
\epsilon_\nu(\vec t) &\sim \text{GP}(0, K_\nu) \\
k_\nu(t, t^\prime) &= \sigma_\nu^2 \exp\left(-\frac{|t - t^\prime|^2}{l_\nu^2} \right) \\
\end{aligned}$$
As with the yearly novel publication counts, observations of PC title counts (1843-1919) are connected to latent rates via a Negative Binomial sampling distribution. The latent rate of PC title appearance in year $t$, the mean of the sampling distribution, is $\exp(\lambda_t) /
\operatorname{logit}^{-1}(\nu_t)$, where $\operatorname{logit}^{-1}$, the inverse logistic function, is the inverse of the transformation of a proportion into log odds. For example, if the proportion of PC titles which are novels is 12% and the rate of new novel appearance is 300 then the observed PC title count will be modeled with a Negative Binomial distribution with mean 2,500.
The yearly *Nineteenth-Century Short Title Catalog* (LOCED) publication counts (the third time series) record, in essence, the same information as the PC title counts series. Because the two series are so similar—they both record total publications (novels and non-novels)—we model the LOCED rate in terms of the PC rate: we assume that the LOCED rate is a fixed multiple of the PC rate. The rate at which titles are recorded in LOCED is incorporated into the model by assuming that the rate is the same as the PC rate, multiplied by a constant factor, $\pi_\nu$. Because LOCED counts are always greater than PC counts, this factor will be greater than one. As before, a Negative Binomial sampling distribution connects this yearly rate to the observed LOCED counts (1801-1870). For reasons discussed earlier, LOCED counts from years which end in a ’0’ or ’5’ are ignored.
Counts of new novels reviewed in *The Athenaeum* (the fourth time series) are incorporated into the model using a similar strategy to the one just described for LOCED title counts. The rate at which novels are reviewed is assumed to be equal to the rate of new novel publication multiplied by a constant factor, $\pi_a$. The use of a constant factor reflects the assumption that the proportion of new novels reviewed in *The Athenaeum* was roughly the same during each of the nine years. As noted earlier, that *The Athenaeum*’s reviewing expands considerably during the period (from 137 in 1860 to 473 in 1900) lends this assumption superficial plausibility. As we know in advance that *The Athenaeum* does not review all new novels, an informative Gamma prior distribution placing 90% probability on a value between 30% and 70% is used. As with the other count-based time series, a Negative Binomial sampling model is used to model the relationship between latent rates and observed counts.
We relate the three distributions elicited from Bassett (the fifth data source) directly to new novel publication rates for the relevant years ($\lambda_{87}$, $\lambda_{92}$, and $\lambda_{94}$). This makes incorporating the distributions into the model straightforward: the three elicited distributions are used as prior distributions on the rate of new novel appearance during 1886, 1891, and 1894. Although a meticulous approach would associate the three distributions with the unobserved *counts* of new novel publications—this is, after all, what Bassett was asked about—such an approach would add considerably complexity to the model by requiring us to model latent discrete variables (the unobserved counts). Assuming that the Bassett estimates concern continuous latent rates rather than discrete counts has the consequence of modestly understating the variance of the elicited distributions. Given that the elicited distributions indicate a generous degree of uncertainty we think this is a reasonable price to pay for a simpler model.
#### Modeling author gender
The essential structure of the model has been introduced. The full model differs slightly from the version presented. In addition to estimating the number of new novels published each year, the full model also estimates the number of novels published by author gender. This is accomplished by adding, for each year, two parameters to the model. The first parameter, $\rho_t$, records the proportion of new novels associated with an author of unknown gender. The second parameter, $\sigma_t$, records the proportion of known-author-gender new novels associated with men authors (a proportion of a proportion). With these two parameters it is possible to calculate the proportion of new titles given each of the three author gender annotations. For example, new novels associated with women authors in year $t$ is given by $(1 - \rho_t) (1 -
\sigma_t)$. Each sequence, $\rho_{1:120}$ and $\sigma_{1:120}$, is modeled on the log odds scale using Gaussian Processes with a linear trend. Prior distributions for the characteristic length-scale parameters are the same as the prior distribution used for the length-scale parameter for the Gaussian Process model of $\nu_{1:120}$ (the proportion of PC titles which are new novels). Observed counts of new titles by author gender—available in *The Athenaeum* series and, for 1800 to 1829, in RFGS—are modeled with Negative Binomial sampling distributions.
#### New novels by author gender, 1789-1799
We estimate the number of new novels by author gender separately for the 11 years between 1789 and 1799. Because the number of new novels published during this period appears in RFGS, we need only estimate, for each year, the proportion of novels associated with men, women, and unknown gender authors. We accomplish this by collecting and manually annotating a random sample of 110 titles from RFGS (ten titles for each year). For each year we calculate a posterior distribution over proportions using a multinomial sampling model and an informative Dirichlet prior distribution loosely centered on observed proportions in 1800.
For the full model covering the period between 1800 and 1919, posterior inference is performed using Hamiltonian Monte Carlo as implemented in Stan [@carpenter2017stan]. All parameters whose prior distributions are not discussed are given reasonable, weakly informative priors. Source code and all datasets used accompany this paper.
New Novel Publications, 1789–1919 {#new-novel-publications-17891919 .unnumbered}
---------------------------------
The model provides estimates of the rate of novel publications for each year between 1789 and 1919 (Figure \[fig:new-novels-per-person\]). In aggregate between [40,000]{} and [63,000]{} new novels likely appeared between the years 1789 and 1919. (All intervals mentioned are 90% credible intervals.) A summary by decade appears in Table \[tbl:novels-by-decade\]. For comparison, the number of these titles which are still in print today is shown, by author gender and decade of publication, in Table \[tbl:novels-reprint-canon-by-decade\]. This “reprint canon” (borrowing the label from @bassett2017median) serves as an approximation of the body of works currently taught in universities. The reprint canon very likely represents less than one percent of novels published during the period. It is possible that it represents as little as one half of one percent of published titles.[^11]
One remarkable development which is visible by inspection is the rapid growth in per capita new novel publication between [1840]{} and [1855]{}. Figure \[fig:new-novels-per-person-log10\] shows a plot the number of new novels published per person on a logarithmic scale. Three regimes of growth in the 19th century are visible. Before [1840]{} there appears to be modest growth in the number of titles published each year. Between [1840]{} and [1855]{} there is rapid growth in the number of titles produced annually, from 3.1 to 6.4 novels per million persons. Average annual growth during this period is 5%. Between [1855]{} and 1900 there is likely steady, but markedly slower growth. The average annual growth during this period is likely 2%. The rapid growth during the second period ([1840]{}–[1855]{}) merits further investigation. How did it come about and how was it sustained? What consequences did it have for the network of actors involved in the literary market? The per capita rate of novel publication likely doubled in the space of a 15 year period, requiring a rapid expansion in a range of processes of interest to literary historians and historians of publishing. For example, this growth suggests a doubling of the labor of compositors, a doubling of paper used, and a doubling of the per capita rate at which manuscripts were developed for publication. How was this rate of growth sustained? Did one particular novel subgenre, group of intermediaries, or cohort of novelists benefit from this expansion? The rapid pace of growth seems likely to have left traces in a variety of places, not least in the lives of writers and in the morphology of literary texts.
![**New novels per million persons, 1820–1919.** Figure shows new novels published in the British Isles, per million persons, between 1820 and 1919. Population figures are from @maddison2009statistics. Population of the British Isles is calculated by adding UK and Ireland populations.\[fig:new-novels-per-person\]](build/figure_new_novels_per_person.pdf)
![**New novels per million persons, 1820–1919 (log scale).** Figure shows new novels published in the British Isles, per million persons, between 1820 and 1919 using a $log_{10}$ scale. Population figures are from @maddison2009statistics. Population of the British Isles is calculated by adding UK and Ireland populations.\[fig:new-novels-per-person-log10\]](build/figure_new_novels_per_person_log10.pdf)
Estimates of the yearly rate of new novel publication by men authors are shown in Figure \[fig:new-novels-gender-by-year-men-of-known\]. The estimates are consistent with the widely held belief that there was a demographic shift in the occupation of novel writing during the 19th century [@tuchman1989edging 5-11]. At the beginning of the 19th century a majority of novels with known author gender were associated with women novelists. By the end of the 19th century this percentage had likely declined to roughly 40%.[^12] Within the expected secular decline in the proportion of novels associated with women authors there is some evidence of a cyclical trend: the proportion of titles associated with men authors declines during the 1860s and 1870s before recovering again.[^13]
![**New novels by men authors, 1789–1919.** Percentage of new novels with known author gender which are novels by men authors. For years in 1800-1829 exhaustive gender annotations are known. For all other years model estimates are shown, with thick vertical bars indicating 50 percent credible intervals and thin vertical bars indicating 90 percent credible intervals.\[fig:new-novels-gender-by-year-men-of-known\]](build/figure_new_novels_gender_by_year_men_of_known.pdf)
The estimates also permit us to say that it is virtually certain that novels by men authors and novels first published in the 1860s are overrepresented in titles which are still in print today. That is, the proportion of novels associated with men authors in the reprint canon does not reflect the proportion of novels written by men during the period. It is very likely that between 40% and 58% of novels written between 1789 and 1919 were associated with men authors (Table \[tbl:novels-by-decade\]). In the reprint canon, however, 71% of novels from this period are associated with men authors (Table \[tbl:novels-reprint-canon-by-decade\]). The distribution of reprint canon titles by year of first publication is also not aligned with the distribution of titles published during the period. Titles published in the 1860s, in particular, appear to be overrepresented in the reprint canon. Titles published in the 1900s appear to be underrepresented. Although it is possible that the reprint canon does not reflect literary works used in research and taught in university classrooms, the reprint canon does represent the population of 19th century novels which continue to be sold and, presumably, read. Our estimates bring the question of how this important set of works is selected into sharp relief. The works are not representative of the population of published novels.
Next {#next .unnumbered}
----
The estimates presented here reduce uncertainty about the number of new novels published between 1789 and 1919. The reduction is significant enough that a variety of existing narratives of developments in the literary market and the text industry merit revisiting in light of the new estimates. The account offered by @tuchman1989edging of changes in the percentage of women pursuing careers as novelists is one example. The census data Tuchman uses to gauge changes between 1861 and 1919 are, by her own admission, unreliable [@tuchman1989edging 58]. Although the estimates presented here concern the annual number of titles published by author gender and not the number of working women novelists, our series presented here is more detailed and more relevant to the quantities of interest to Tuchman than any series available in the 1980s. Research on the social history of novel writing similarly merits revisiting in light of these new estimates.[^14] New research here would potentially complement any investigation into the period of particularly rapid per capita growth in new novel publication ([1840]{}–[1855]{}), as the factors driving this expansion may be illuminated by studying the differences between cohorts of writers before and after the expansion.
Although the estimates here give us greater confidence about the annual rates of new novel publication, much work remains to be done. The estimated intervals are wide, especially after 1850. Narrowing the intervals will require more precise information about novelistic production during the late 19th and early 20th centuries. One simple, effective strategy for gathering such information would involve conducting an exhaustive survey of novels published in a single year after 1850. Because knowing the rate of production in a given year provides information about plausible rates for neighboring years, accurate information about a single year would improve estimates of neighboring years. Although collecting an exhaustive list of novels published in, say, 1865 would be time-consuming, the work itself is straightforward: new novels need to be identified among all entries in *Publishers’ Circular* for the chosen year.
Another promising line of research these estimates support concerns the widely discussed project of studying prose style from below. Studying prose style from below requires, at the very least, a corpus which contains the text of a representative sample of published novels. Such a representative corpus is a prerequisite for research monitoring the flow of literary morphology across national and linguistic situations as well as for studies tracking the emergence and transmission of conventions and styles. (Existing corpora which claim to be representative are plainly not, as @bode2018world assiduously documents [@bode2018world Ch. 1].) The availability of estimates by year and author gender support the development of a representative corpus because they permit researchers to evaluate the plausibility of a claim that a given corpus is representative of published novels. Researchers may compare the characteristics of novels in the corpus with the characteristics of novels in the population.
The period of rapid growth between [1840]{} and [1855]{}, if evidence for its existence continues to accumulate, deserves further study. Did one or a small number of factors drive this growth? Was the growth attributable to, for example, lower per-unit costs arising out of technological changes (steam-powered presses and paper-making) and internal industrial developments which lowered firms’ cost of capital? Or, rather, was the growth attributable to an expansion in the number of novel readers or intensification of novel reading among the existing population of novel readers? The latter, at least, seems unlikely, because the gains of the industrial revolution—which might have enabled more people to purchase the luxury goods which novels and circulating library subscriptions unquestionably were—did not accrue meaningfully to the broader population until after [1840]{} [@allen2009engels].
Author Gender and Novel Subgenre Participation {#author-gender-and-novel-subgenre-participation .unnumbered}
==============================================
At the start of the 19th century, novels published in the British Isles tended to be written by women. On this point there seems to be agreement. Less clear is whether, by the end of the century, a majority of novels published each year tended to be written by men. That the question is unresolved can be attributed to the absence of comprehensive bibliographies of novels published after 1836. Those interested in answering the question of whether or not there was a decline in novels written by women include literary historians interested in the demography of novel writers and those working, like @tuchman1989edging, at the intersection of literary studies, publishing history, and labor studies.
In this section we turn to studies of trends in novel authorship by gender during the first decades of the century. Detailed information is available about novel authorship by gender for a subset of the years (1800-1829) covered by the exhaustive bibliography of RFGS. Data from these years indicate that the number of men authors grew in absolute and relative terms between 1820 and 1829 [@garside2000english_romantic_era 74]. Moretti attributes this growth to “a generation of military novels, nautical tales, and historical novels à la Scott attracting male writers” [@moretti2005graphs 27]. Moretti does not check this descriptive hypothesis against available bibliographic records of novels published in the 1820s. @tuchman1989edging (in which Moretti finds a competing hypothesis) argues that men writers were attracted to the profession of novel writing by the prospect of financial gain presented by an expanding industry [@tuchman1989edging 4-5]. In her account, Tuchman does not identify among new men novelists a tendency to write in specific subgenres.
To address this disagreement we examine the association between novel subgenre and author gender between 1800 and 1829. We find little change in the strength of the association after 1820, casting doubt on the descriptive hypothesis advanced by Moretti. We measure the association between subgenre and author gender by measuring how easily title words predict author gender. Title words are, we show, a reliable proxy for the subgenres of interest. Were Moretti’s descriptive hypothesis correct, we would expect it would become easier to predict author gender after 1820 since proportionally more men authors would be associated with novels in specific subgenres (to which men writers had been attracted, according to the hypothesis). We find, however, that men authors appear to be associated with military novels, nautical tales, and historical novels at roughly the same rate as they were before 1820.
#### Terminology
When discussing occupational gender segregation in the text industry it is important, in general, to distinguish between the advertised gender of a book’s author—the gender of the author associated with the book on the title page or elsewhere in the book—and the gender of the book’s writer(s) at the time the book was written. These two are often, but not always, the same. When the distinction is essential, the former might be labeled “author gender” and the latter “writer gender.” The former is far easier to work with since it is often mentioned in the printed edition and a digital facsimile of the novel is frequently available to facilitate development of intersubjective agreement about particular cases. The datasets we use in this paper record author gender directly or nearly equivalent annotations. The major source of disagreement in coding is in RFGS. In the RFGS annotations, titles authored anonymously whose writers were subsequently acknowledged are coded according to the new information about writer gender.
In the particular case of the [40,000]{}-[63,000]{} new novels published in the British Isles between 1789 and 1919, we assume that aggregate statistics about author gender are approximately equivalent to aggregate statistics about writer gender for all novels in which author gender is advertised. This assumption depends on the belief that cross-gender authorship was extremely rare, likely occurring in fewer than 1% of titles. What warrants this belief is the observation that cross-gender authorship is vanishingly rare in the exhaustive survey of novels published between 1800 and 1829. In a simple random sample of 40 novels from RFGS (1800-1829), cross-gender authorship never occurs. Were the practice common or even infrequent, it would likely have appeared at least once in such a sample. Given this evidence, we make the provisional assumption that, for this particular period, patterns observed in novels associated with a given author gender also hold for novels associated with the corresponding writer gender.
Background {#background-1 .unnumbered}
----------
The demographic characteristics of novel writers in the British Isles changed somewhat during the 19th century. Although literary historians are mostly ignorant about the nature and chronology of these changes, limited information is available, especially about changes in the first three decades of the 19th century. Thanks to the exhaustive bibliography available in RFGS, we know how many titles were associated with men and women authors between 1800 and 1829. Less useful information is available about the demographic characteristics of novel writers active after 1829. Working with the publisher Macmillan’s archives, @tuchman1989edging observes in the publisher’s internal records an increasing proportion of submitted fiction manuscripts associated with men writers and a decreasing proportion associated with women writers between 1870 and 1917. Tuchman’s sample is, however, sparse—not all years during the period are studied—and of limited relevance as Macmillan did not specialize in publishing fiction [@tuchman1989edging 57; @sutherland1989review 815]. @casey1996edging, drawing on all reviews of novels published in *The Athenaeum* during nine years, argues that Tuchman’s data overstates changes in the demography of novelists [@casey1996edging 157].
A related research question concerns changes in occupational gender segregation by novel subgenre. That titles associated with certain subgenres tend to be associated with specific author genders is familiar. The number of novels published in the late 18th century and first decade of the 19th century is small enough that this tendency can be checked by inspecting titles. Authors of titles mentioning military or nautical terms are, for example, very likely to be men. Authors of titles mentioning plot elements or settings associated with gothic novels tend to be women. What is far less certain is when and to what extent subgenre participation changed during the 19th century—if it changed at all. This question is of interest to those working in the sociology of literature as well as those studying the history of literary forms. Attempts to explore the question are blocked by the lack of comprehensive genre-specific bibliographies and, more generally, a lack of agreement about what subgenres are.[^15]
@moretti2005graphs offers a characterization of the association between writer gender and novel subgenre in the 19th century: “gender and genre are probably in synchrony with each other” [@moretti2005graphs 27]. The claim is specific in that it mentions periods during which gender and genre move together. For example, Moretti argues that there is one shift towards male writers “around 1820” when the success of men novelists such as Walter Scott attracted other men writers to the subgenres associated with the novelists. In this case, the subgenres mentioned are military novels, nautical tales, and historical novels. Moretti offers this claim while discussing the substantial increase in men novelists during the 1820s. This gives us a concrete claim to check: Do these novels, published after 1820 and associated with men authors, participate in the subgenres of military novels, nautical tales, and historical novels at a higher than expected rate?
Method {#method-1 .unnumbered}
------
We measure the association between subgenre and author gender using predictive accuracy. The more closely a variable tracks another, the easier it will be to predict the value of one given the value of the other. In this case the variable predicted is author gender and the variable used to predict author gender is title words. We use a standard logistic regression model for prediction and a standard transformation of title words into vectors of word frequencies. We exclude from this analysis the 354 novels listed in RFGS with a recorded author gender of “Unknown”, leaving the 1,922 novels published between 1800 and 1829 which have an author gender annotation.
A key assumption in this approach is that a novel’s title is a reliable—not necessarily perfect—indicator of subgenre membership. Occasionally the indication is explicit, with the subgenre named in the (sub)title. For example, the title page of Henry Duncan’s 1826 novel includes the following words “WILLIAM DOUGLAS; OR, THE SCOTTISH EXILES. A HISTORICAL NOVEL”. In most cases, the indication is less regular. Nautical tales, for example, will tend to feature words such as “naval”, “officer”, “freebooter”, “shipwreck”, and the like.
For the sake of continuity with the present debate, we accept the idea that a novel can be said to participate in the conventions of one or more discrete novel subgenres. The approach used here has, however, has no need of this assumption. The underlying claim investigated here is that novels with titles which contain the same words are likely to feature similar morphology in their texts. A more general version of the descriptive hypothesis would be the following: after 1820 men novelists increasingly wrote novels featuring morphology found in novels written by men authors before 1820.
One significant modification is made when gathering word frequencies from the title words. Because the titles of 19th century novels occasionally advertise the gender of the author in the novel’s title (e.g., “BY THE AUTHORESS OF …”, “…BY A GENTLEMAN“) we remove all words which directly signal author gender. These words are identified and removed in a preprocessing step. We locate words which directly indicate gender by counting words associated with titles of each of the two author genders and calculating a chi-squared test statistic for each word. The statistic measures the degree to which a word is associated with one group of texts relative to another group of texts. A word which appears frequently in one group and rarely in another will have a large chi-squared test statistic. All words with a statistic greater than 7 are removed. (In the nomenclature of classical statistics, this test statistic would be associated with a “p-value” of less than 0.01.) In addition to the desired words (e.g., “author”, “authoress”, “gentleman”, “lady”, “mr”, “miss”, “mrs”) the procedure identifies words associated with given names as well as several words associated with gothic novels (which tend to be associated with women authors). The number of other words caught up in this filter is small (165 words). Many words remain which are capable of signalling or hinting at novel subgenre.[^16] Words occurring in only one title are also ignored as they can provide no information about similarities among novels. The number of distinct title words whose frequencies are used to predict author gender is 2,003.
Measuring the association of two variables using predictive accuracy requires specifying a model which makes predictions. We use a logistic regression model to predict a novel’s author gender given the frequency of words in the novel’s title. As formal presentations of logistic regression is available elsewhere, we omit a description of the model.[^17] The particular model used incorporates an L2-regularization parameter of 1.0 due to the high dimensionality of the input variable (2,003) relative to the number of observations (1,922). Parameters of the model are found using maximum likelihood.
In order to see how the association changes over time, we need to measure the association between title words and author gender with respect to a specific year or time period. We accomplish this by calculating the leave-one-out predictive accuracy for each novel, noting the year of publication. Leave-one-out prediction, as the name suggests, involves giving a model records of the gender annotations and title words for all but one novel. Having built a predictive model using these records, the model is then asked to predict the gender annotation for the one held-out novel given the held-out novel’s title words. Whether or not the model makes the correct prediction is recorded. This process is repeated, holding out each novel in turn. The percentage of correct predictions (accuracy) is calculated for all novels published in the same year. Leave-one-out predictive accuracy will increase as the similarity between titles associated with the same author gender increases. For example, if in 1824 a greater percentage of men-authored titles feature words strongly associated with other men-authored titles—e.g., “naval”, “officer”, “military”—then leave-one-out predictive accuracy for titles published in 1824 will increase relative to other years.
The magnitude of change in the association between author gender and title needs to be weighed against the magnitude of change in the number of new titles associated with men authors. The change in the latter is considerable. Between 1815 and 1819, 90 novels are associated with men authors (38% of titles by men or women authors). Between 1820 and 1824 the number of titles associated by men authors increases to 198 (55% of titles by men or women authors). Numbers for 1825–1829 are similar to those for 1820–1824. If men writing after 1820 are being attracted to specific, pre-existing subgenres associated with men authors then leave-one-out predictive accuracy should increase.
There is, however, at least one way for the association between title words and author gender to remain constant even though men authors are writing in the subgenres of interest and doing so with greater regularity. This could occur if titles associated with women authors were increasingly being published in the same subgenres. In this case, we would expect the association between title words and subgenre to remain constant or even decline. This possibility, however, would also challenge the descriptive hypothesis, which implies that men authors are disproportionally attracted to the subgenres in which men authors (like Scott) had proven successful. If men authors are, in particular, increasingly associated with novels which use titles characteristic of military novels, nautical tales, and historical novels, we should still see an increase in the association between author gender and title words.
And there is at least one way for the association between title words and author gender to increase even while men-authored titles appear in the subgenres of interest at roughly the same rate as they did before 1820. The association would increase if titles authored by women became easier to predict on the basis of title words. That is, if titles by women used title words associated with women-authored titles with increasing regularity after 1820 then we would anticipate being able to predict the author gender of these titles with greater accuracy. We can, however, focus on the predictive utility of title words for predicting novels being associated with men authors by calculating the sensitivity of the classifier. Sensitivity records the proportion of men-authored titles correctly identified as such, ignoring correctly classified women-authored titles.
Subgenre and Occupational Gender Segregation, 1800-1829 {#subgenre-and-occupational-gender-segregation-1800-1829 .unnumbered}
-------------------------------------------------------
The association between author gender and title words does not increase noticeably after 1820 relative to earlier years, casting doubt on the descriptive hypothesis that men authors were attracted to writing novels in subgenres such as military novels, nautical tales, and historical novels after 1820. Measures of the association between author gender and title words between 1800 and 1829 are shown in Figure \[fig:gender-genre\]. Leave-one-out predictive accuracy increases modestly for novels published in 1820-1824 relative to 1815-1819 (from 75% to 78%) and it decreases for novels published in the following five year period to 70%. The sensitivity of the classifier exhibits essentially the same pattern. It increases very modestly in 1820–1824 relative to 1815-1819 (from 80% to 82%) before decreasing to 79% for 1825-1829. (Five-year “bins” are used in order smooth over the variability associated with individual years during which a small number of titles appeared.) These are the results we would expect if titles written by men in the 1820s participated in subgenres at roughly the same rate as titles written by men before 1820. Titles associated with men-author-linked subgenres appear to be little more frequent in the 1820s than in earlier decades. Even though the number of men-authored titles increases considerably in absolute and relative terms after 1820, we do not see a commensurate increase in the in the predictability of men-authored titles.
Subgenre-linked words contribute to the ease with which author gender can be predicted from title words, supporting the methodological assumption guiding this exercise. Words whose associated coefficient in the logistic regression model are positive—words whose presence increases the predictive probability of title being authored by a man—include words associated with nautical tales and related subgenres: “military”, “rank”, “subaltern”, “rebel”, and “outlaw”. Words with positive coefficients also include words likely associated with historical novels such as “seventeenth”, “royal”, “during”, and “chronicle”. Words linked to gothic novels (e.g., “horrors”) and to novels of manners (e.g., “lover”, “flirtation”, “infidelity”) are associated with negative coefficients in the model.
Accepting title words as an indicator of subgenre participation, we do not find the expected increase in the association between author gender and title words that the descriptive hypothesis leads us to expect. Men do not appear to be writing novels associated with military novels, nautical tales, and historical novels after 1820 at a markedly higher rate than they were before 1820.
Discussion {#discussion .unnumbered}
----------
The number of novel titles associated with men authors rises in absolute and relative terms after 1820. This rise merits an explanation. The account offered by Moretti loses some credibility in light of the analysis of the association between title words and author gender over time. The other account available, drawn from Tuchman, argues that men authors were attracted to the profession of novel writing by the prospect of financial gain. This account lacks the chronological precision of Moretti’s alternative hypothesis and was articulated before the detailed bibliography of RFGS became available. Further investigation seeking a more satisfactory account, likely as part of a broader investigation of the changing demographics of novel writers, remains an outstanding task for research. One particularly important task would be to to revisit and restate Tuchman’s general argument and chronology in *Edging Women Out* equipped with the resources presented in this paper and those available in RFGS and ATCL. A variety of the claims advanced in Tuchman’s book can be restated with greater chronological precision and more precise references to specific periods, works, and writers.
Further study of the gatekeeping functions of intermediaries such as reviewers and bookseller-publishers is also warranted and author gender seems likely to provide a useful vantage point. One line of inquiry that is unaddressed here would be a study of whether or not certain publishers displayed a marked preference for men-authored titles. Research on this question would be of broad interest because a precise understanding of the gatekeeping function of publishers, editors, and agents remains something of a mystery. Even to this day, there is a lack of reliable information that could address persistent allegations of bias against women authors and authors of color in the contemporary publishing industry [@sieghart2018why; @franklin2011literary; @vidawomeninliteraryarts20172016; @horning20182017; @milan2016speaking]. A better understanding of how durable biases among gatekeepers and intermediaries in the 19th century text industry emerged and were sustained will likely hold lessons for the present.
The impulse to study literary form alongside characteristics such as author gender is valuable and should be pursued in other settings. Literary morphology varies considerably across novels published within a given year. It also often varies across novels associated with the same bookseller-publisher. The results presented here are consistent with the idea that author gender and the use of specific conventions and literary forms are likely to be correlated. The existence of a reliable association between two variables suggests a strategy of studying the two together: a lack of information about one feature may be compensated by knowledge of the other. And studying elements of literary form—for example, the presence of specific morphology or conventions—is likely to become easier as the full text of complete and representative samples (e.g., a simple random samples) from the period become available. Analyses involving complete or representative samples would be able to address a broad range of questions. For example, Tuchman’s claim that “high-culture novels” were more likely to be associated with men authors in the late 19th century could be further investigated, supplemented by specific references to the population of novels published during the period.
Ordinary Literary History {#ordinary-literary-history .unnumbered}
=========================
This paper aims to demonstrate and to promote the study of the novel at multiple scales. Prerequisites for this genre of research include credible accounts of what novels were published as well as rudimentary information about novelists involved. For some, answering basic questions about novelistic production and conditions facing writers pursuing careers as novelists will simply count as a relevant task for literary historians. The history of the English novel is impoverished if it neglects the vast majority of novels which have been published and the vast majority of novelists who worked in the text industry. For others, having a rich description of these populations serves more specific goals including understanding the origins of practices in the contemporary culture industry, or giving valuable context to the study of specific literary forms or works.
Some research opportunities suggested here will have to wait for a time when information about material traces of literary works is better organized and more accessible. Chief among these opportunities is a task which this paper’s title references. Reassembling the novel involves, as suggested by @becker1995introduction, studying the network of actors whose interactions are necessary for the production of literary works and, moreover, recovering how literary works are shaped as the result of being made in such a network [@becker1995introduction xii]. One possible way of advancing this kind of research would focus on “repeopling” existing literary history, taking account of more of the actors involved in literary production, including intermediaries such as reviewers, booksellers, investors, and circulating libraries. Another avenue would supplement book history and sociologically-inclined studies of literary production with richer accounts of the literary morphology of individual texts [@svedjedal1996det 6; @becker1995introduction xii]. Either line of inquiry would likely overlap with research animated by—borrowing Moretti’s phrase—a “materialist conception of form” [@moretti2005graphs 92].
This paper contributes two analyses which demonstrate how a data-intensive, sociologically-inclined approach to literary history can address long-standing questions about the history of the novel. First, we estimate the yearly rate of new novel publications in the British Isles between 1789 and 1919, a period which includes the emergence of the modern text industry and mass literacy. These estimates facilitate current bibliographic work and support future work by researchers interested in studying a complete or a representative sample of literary works. Second, we explore changes in occupational gender segregation by novel subgenre between 1800 and 1829. By studying the association between title words and author gender, we collect evidence that casts doubt on a descriptive hypothesis we identify in @moretti2005graphs. Titles associated with men authors do not appear to concentrate in specific novel subgenres (military novels, nautical tales, historical novels) after 1820. The marked increase in appearance of men-authored titles after 1820 remains in need of an explanation.
The analyses performed here depend in large part on relatively recent developments: the availability of machine-readable bibliographic data and library digitization. These twin developments have created an increasingly hospitable environment for those interested in pursuing data-intensive bibliography and sociologically-inclined literary history.
Appendix {#appendix .unnumbered}
========
Source code and datasets {#source-code-and-datasets .unnumbered}
------------------------
Source code and data used accompany this paper.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank participants, in particular Fotis Jannidis and Karina van Dalen-Oskam, in the Symposium “Digitale Literaturwissenschaft” for helpful comments. Many thanks to Troy Bassett for his willingness to participate in the (lengthly) elicitation task and for his advice on collecting the reprint canon data. We are grateful to Laura Schneider for excellent research assistance.
[^1]: Indiana University, Bloomington
[^2]: Symplectomorphic, LLC. Research conducted while at Columbia University.
[^3]: There are many challenges associated with assembling an exhaustive list. A small number of books are published but never advertised in industry publications such as *Publishers’ Circular*. In other cases, novels may be advertised but never published, or published under a different title. Bibliographic work is complicated further by the fact that in a (very) small number of cases, no copies of a novel survive.
[^4]: Working with data from @eliot1994patterns, @weedon2003victorian combines the work of Eliot with other sources to offer a succinct description of publishing between 1836 and 1919 [@weedon2003victorian 46-51].
[^5]: To the best of our knowledge, @garside2006english includes corrections and additions to @raven2000english [@garside2006english] which have been published online from time to time (e.g., @garside2001english).
[^6]: “THE CASTLE OF PROBATION, OR, PRECEPTIVE ROMANCES; CHIEFLY TAKEN FROM LIFE. BY A CLERGYMAN. IN TWO VOLUMES.” (RFGS record no. 1802A002).
[^7]: Of course, ignoring counts in years ending with “0” or “5” means discarding potentially useful information about counts of dated publications. Separating counts of dated material from undated material in these years would be valuable.
[^8]: These years were chosen because a preliminary model made implausible predictions for these years. The predictions were implausible in that they were near or lower than a lower bound on the number of novels published in the relevant years. Lower bounds were available for these years because the ATCL database already contains records for many thousands of novels published in the 19th century.
[^9]: The distributions were elicited in a phone conversation between Allen Riddell and Troy Bassett on November 9th, 2015. The quartiles reported in the paper are discounted from the original quartiles (450, 550, 700). Discounting is required because ATCL uses a more inclusive definition of the novel than RFGS. (For example, RFGS exclude some religious and didactic fiction that ATCL includes.) Bassett reports that between 10% and 15% of the novels included in ATCL would not be counted as novels according to RFGS. For this reason we discount the reported quartiles by 12.5% (the midpoint between 10% and 15%). The matching of ideal distributions to the elicited distributions (implied by the quartiles) involves one additional step because we model the rate of new novel publication on the log scale. We use Gamma distributions which have quartiles as close as possible to the elicited distributions (now on the log scale). For example, the final representation of the distribution with quartiles 394, 482, and 613 is (on the log scale) a Gamma distribution with shape and rate parameters of 278 and 46.
[^10]: One way of appreciating the importance of modeling new novel publication with a continuous rate parameter is to imagine a situation where the aleatory variation in new novel counts is considerably greater. Imagine modeling new novel publication via weekly counts. In such a setting observing that zero new novels appeared in a given week would not be particularly meaningful. It would certainly not imply that there was zero activity associated with novel publishing during that week.
[^11]: The period between 1800 and 1899 is often the focus of discussion. Between [21,000]{} and [28,000]{} appeared between 1800 and 1899. Totals for other intervals may be calculated using annual publication rates shown in Table \[tbl:novels-by-year\]. Table \[tbl:novels-reprint-canon-by-year\] shows reprint canon titles by author gender and year.
[^12]: Our estimates concern the characteristics of the population of new novel titles, not novelists. If one assumes that novelist gender is uncorrelated with the number of novels they publish, then the share of novelists associated with each gender should be roughly the same as the share of novels associated with each gender. Estimating the demographic characteristics of the population of professional novelists should be addressed in subsequent research. This research may need to, for example, avoid double-counting novelists who used different—or even collective—pseudonyms.
[^13]: @moretti2005graphs suggests a connection between author gender and literary cycles during the 19th century. Moretti, however, does not appear to credit the possibility of a long-term secular decline in the proportion of novels written by women authors [@moretti2005graphs 27].
[^14]: A reference point for this kind of research, in addition to Tuchman, is @williams1965long. Past studies have explored—often with fragmentary or conspicuously partial or biased samples of writers—the social, educational, and geographic background of writers [@williams1965long 261-263; @tuchman1989edging 113-119].
[^15]: Assessing occupational gender segregation by subgenre (or other literary morphology) in the contemporary book publishing industry is made easier by the availability of publisher-assigned subject codes (e.g., Thema and BISAC subject headings). These codes are maintained by industry-sponsored international standards organizations such as the Book Industry Study Group. Although there is no guarantee that publisher-assigned codes align with reader judgments, publishers who assign codes without attention to works’ morphology or readers’ judgements of subgenres face considerable costs. For example, since codes are used by booksellers for organizing books on bookshelves, publishers who choose codes without regard to books’ contents run the risk of making it difficult for readers to discover books by browsing in the vicinity of books with similar morphology. In a competitive industry, such an act would likely reduce book sales.
[^16]: The following 165 words are removed using the chi-squared heuristic: “abbey”, “adventures”, “alicia”, “ann”, “anna”, “annals”, “anne”, “anthony”, “arthur”, “assassin”, “astonishment”, “augustus”, “author”, “authoress”, “baldivia”, “bandit”, “barons”, “bell”, “bluemantle”, “bride”, “bridget”, “cambrian”, “captain”, “caroline”, “castle”, “catherine”, “charles”, “charlotte”, “choice”, “clair”, “claremont”, “cliff”, “collected”, “college”, “confessions”, “conviction”, “cottage”, “daughter”, “de”, “decision”, “di”, “edgeworth”, “education”, “edward”, “elizabeth”, “emma”, “esq”, “eva”, “family”, “festival”, “five”, “fontaine”, “four”, “frances”, “francis”, “freebooter”, “french”, “friendship”, “from”, “general”, “genlis”, “gentleman”, “george”, “geraldine”, “german”, “glendowr”, “glenroy”, “guilty”, “gunning”, “happy”, “heart”, “heiress”, “helen”, “henry”, “her”, “hermit”, “himself”, “his”, “hofland”, “husband”, “isabella”, “italy”, “james”, “john”, “la”, “lady”, “lathom”, “lebrun”, “lefanu”, “legends”, “life”, “louisa”, “lussington”, “madame”, “manor”, “maria”, “marriage”, “married”, “mary”, “men”, “miriam”, “miss”, “monteith”, “mr”, “mrs”, “mystery”, “norwich”, “not”, “novel”, “nun”, “omer”, “opie”, “orphan”, “owen”, “parish”, “permission”, “picture”, “pigault”, “poems”, “porter”, “priory”, “rebecca”, “regina”, “richard”, “robert”, “roche”, “rosalia”, “rouviere”, “sarah”, “satirical”, “self”, “selina”, “series”, “she”, “sidney”, “sir”, “sisters”, “society”, “son”, “sophia”, “spy”, “st”, “stanhope”, “swansea”, “the”, “thomas”, “translated”, “true”, “uncle”, “unknown”, “vale”, “very”, “visit”, “ward”, “waverley”, “ways”, “widow”, “wieland”, “wife”, “william”, “with”, “woman”, “world”, “written”, “young”.
[^17]: Many standard statistics and machine learning texts cover logistic regression. For example, see Chapter 8 of @murphy2012machine or Chapter 4 of @bishop2007pattern.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'S. Krijt'
- 'M. Kama'
bibliography:
- 'refs.bib'
subtitle: 'An energy-constrained smallest fragment size'
title: A dearth of small particles in debris disks
---
[A prescription for the fragment size distribution resulting from dust grain collisions is essential when modelling a range of astrophysical systems, such as debris disks and planetary rings.]{} [While the slope of the fragment size distribution and the size of the largest fragment are well known, the behaviour of the distribution at the small size end is theoretically and experimentally poorly understood. This leads debris disk codes to generally assume a limit equal to, or below, the radiation blow-out size.]{} [We use energy conservation to analytically derive a lower boundary of the fragment size distribution for a range of collider mass ratios. Focussing on collisions between equal-sized bodies, we apply the method to debris disks.]{} [For a given collider mass, the size of the smallest fragments is found to depend on collision velocity, material parameters, and the size of the largest fragment. We provide a physically motivated recipe for the calculation of the smallest fragment, which can be easily implemented in codes for modelling collisional systems. For plausible parameters, our results are consistent with the observed predominance of grains much larger than the blow-out size in Fomalhaut’s main belt and in the *Herschel* cold debris disks.]{}
Introduction
============
Fragmenting collisions are important in a range of astrophysical systems. While the slope of the fragment size distribution and the size of the largest fragment are well characterized and can be used confidently in models, the smallest fragment size is less well understood and is usually assumed to be constant for all collisions. We provide a framework for self-consistently calculating the smallest fragment size as a function of material and collision parameters (Section \[sec:theory\]), and discuss its implications for modelling debris disks (Section \[sec:applications\]).
Numerous experimental studies have looked at the fragment size distribution of destructive collisions, focussing on the slope of the power law(s), and on the size of the largest fragment [@davis1990; @ryan1991; @nakamura1991; @ryan2000]. The smaller end of the size distribution has received considerably less attention; the smallest fragments are hard to count experimentally, and require a very high resolution to be captured in numerical simulations. Fragment distributions are therefore incomplete below sizes of $100$ $\mu$m, or masses below $10^{-3}$ gr [@fujiwara1977; @takagi1984]. Molecular dynamics [e.g. @dominiktielens1997] or smooth particle hydrodynamics [@geretshauser2010] simulations have limited resolution and tend to focus on the fragmentation threshold velocity rather than the smallest fragments.
Minimum fragment size in a single collision {#sec:theory}
===========================================
We consider collisions below the hypervelocity regime, i.e. the relative velocity of the colliders is much smaller than their internal sound speed, generally implying $v_{\rm rel}\lesssim1~$km/s. Based on experiments, we adopt the standard fragment size distribution
$$n(s) = C\cdot s^{-\alpha},$$
with $3 < \alpha < 4$, and $C$ a coefficient we express below. While the mass is dominated by the largest particles, the surface area and thus the surface energy is dominated by the smallest fragments. As the creation of infinitely small fragments would require an infinite amount of energy, while the amount of kinetic energy available in a collision is finite, the power law must stop or flatten at some small fragment size. To the best of our knowledge, however, the regime of fragment sizes relevant for the analysis below has not yet been probed by available experimental data nor described theoretically in an astrophysical context.
Assuming spherical fragments with sizes between $s_{\rm min}$ and $s_{\rm max} (\gg s_{\rm min})$, the total fragment mass and surface area are
$$\label{eq:M_frag}
M_{\rm frag} = \frac{4\pi \rho C}{3(4-\alpha)} s_{\rm max}^{4-\alpha}, ~~~~~~
A_{\rm frag} = \frac{4\pi C}{\alpha-3} s_{\rm min}^{3-\alpha}.$$
For a collision between two bodies of size $s_0$ and mass $M_0=(4\pi /3)\rho s_0^3$, mass conservation implies $M_{\rm frag}=2M_0$, and thus
$$\label{eq:C}
C = 2(4-\alpha)s_0^3 s_{\rm max}^{\alpha-4}.$$
The pre-collision kinetic energy is simply $U_{\rm K} = (1/2) \mu v_{\rm rel}^2$, where $\mu=M_0/2$ denotes the reduced mass. The difference in surface energy before and after the collision equals $\Delta U_{\rm S}=\gamma(A_{\rm frag} - 8\pi s_0^2)$, where $\gamma$ equals the surface energy per unit surface of the material. Assuming that only a fraction $\eta$ of the kinetic energy is used for creating new surface, we can combine Eqs. \[eq:M\_frag\] and \[eq:C\] to obtain a lower limit for the smallest fragment size. For the specific case of $\alpha=3.5$, this reduces to $$\label{eq:s_min3.5}
s_{\rm min} = \left(\frac{24 \gamma s_0}{\eta \rho s_0 v_{\rm rel}^2 + 24\gamma}\right)^2 s_{\rm max}^{-1},$$ and gives the size of the smallest fragments created in a collision at $v_{\rm rel}$, assuming $\alpha=3.5$ and $s_{\rm max} \gg s_{\rm min}$.
Instead of forming a fragment distribution, we imagine the limiting case in which the kinetic energy just suffices to split both colliders in half[^1], i.e. $\eta U_{\rm K} = 2\pi s_0^2 \gamma$. Solving for $s_0$, we obtain $$\label{eq:s_split}
s_0^{\rm split} = \frac{3 \gamma}{\eta \rho v_{\rm rel}^2},$$ which is the smallest particle that can be split in half. The smallest fragment is slightly smaller, but does not have a rigorously defined radius because we assume spherical particles. Equation \[eq:s\_split\] is similar to the result of @biermann1980 if $\eta=1$.
The same limit can be explored using Eq. \[eq:s\_min3.5\], by forcing $s_{\rm min} \sim s_{\rm max} \sim 2^{-1/3}s_0$. This results in $$\label{eq:biermann}
s_{\rm min} \simeq \frac{5 \gamma}{\eta \rho v_{\rm rel}^2},$$ which is very similar to Eq. \[eq:s\_split\]. To summarise, in an energetic collision in which many fragments are created, the size of the smallest fragment is given by Eq. \[eq:s\_min3.5\]. When the relative velocity is decreased, the fragment distribution becomes more and more discrete, until we reach the limit described by Eq. \[eq:biermann\], in which particles can only just be split into two.
Figure \[fig:fig1\] shows the minimum size from Eq. \[eq:s\_min3.5\] as a function of collider size, assuming $v_{\rm rel}=20$ m/s, $\eta=10^{-2}$, and a maximum fragment that carries half of the initial collider mass. Gravity is important for bodies larger than $100$ m (see below). Smaller bodies are weaker, and can produce fragments down to the $s_{\rm min}$ indicated by the solid curve. For example, SiO$_2$ fragments smaller than a micron can, at this velocity, *only* be formed by collisions of bodies larger than a few centimetres. The shaded region, top left, is forbidden, as there $M_{\rm frag} > 2M_0$ and mass is not conserved. Close to $s_{\rm min} \sim s_{0}$, the solid curves are non-linear as the pre- and post-collision surface areas become comparable.
Material $\rho$ () $\gamma$ ()
----------- --------------- -----------------------
Silicate 2.6 0.05
Ice 1.0 0.74
Aggregate $\sim10^{-1}$ $\sim10^{-4}-10^{-3}$
: Material properties for silicate and ice used in this work. The values for the typical aggregate are explained in Appendix \[sec:A\].[]{data-label="tab:materials"}
It is interesting to compare Eq. \[eq:biermann\] with the traditional form of the catastrophic fragmentation threshold velocity in equal-sized collisions: $v_f^2 = 8Q^*$. The critical energy, $Q^*$, has units of $\rm{erg/g}$, and varies with particle mass. For small bodies, the strength is dominated by cohesion, and for large ones by gravity [@benz1999]. For solid bodies, this transition occurs around 100 metres in size. Values of $Q^*\sim10^7 \rm{~erg/g}$ are often taken as typical for asteroids, and experimentally obtained values for small grains (mm to cm sizes) can be several orders of magnitude smaller [@blummunch1993; @beitz2011]. Figure \[fig:fig1\] shows the critical energy for splitting predicted by Eq. \[eq:biermann\] as a function of size for the materials in Table \[tab:materials\] assuming $\eta=10^{-3}$ and $\eta=1$. The solid lines indicate critical fragmentation energies for basalt and ice [@benz1999] and silicate aggregates [@beitz2011]. The critical fragmentation energies exceed the splitting energy, indicating that substantially more energy is required to destroy – rather than split – colliders. The values plotted for ice and basalt were obtained at collision velocities of 3 km/s, substantially higher than the velocities considered here, and $Q^*$ is known to depend on velocity [@Leinhardt2012]. While such a velocity dependence appears absent in the splitting energy, it might be implicitly included in $\eta$. In fact, $\eta$ is expected to vary with material and impact energy. We adopt a constant value of $\eta=10^{-2}$.
Appendix \[sec:B\] investigates similar limits for colliders with different mass ratios, and shows that collisions with a mass ratio close to unity are the most effective at creating small fragments.
Application to debris disks {#sec:applications}
===========================
Debris disks are leftovers of planet formation, and are usually described by a birth-ring of km-sized asteroid-like particles orbiting their parent star, together with a population of smaller bodies formed in a collisional cascade [for a recent review, see @matthews2014]. A steady-state and scale-independent population of bodies will follow a size distribution given by a power law with $\alpha=3.5$ [@dohnanyi1969]. Some variation in $\alpha$ has been found in different simulations. @PanSchlichting2012 find up to $\alpha=4$ for cohesion-dominated collisional particles, and up to $\alpha=3.26$ for gravity-dominated ones.
Models of debris disks most often assume a smallest fragment size equal to the blow-out size, $s_{\rm blow}$ [@WyattDent2002; @Wyattetal2010], or some constant, but arbitrary, $s_{\rm min} < s_{\rm blow}$ for all collisions [@Thebaultetal2003; @Krivovetal2008]. The blow-out size corresponds to particles with $\beta=1/2$, where $\beta~=~1.15 Q_{\rm pr} (L_{\star}/L_{\odot}) (M_{\star}/M_{\odot})^{-1} (\rho/\mathrm{g~cm^{-3}})^{-1}(s/\mathrm{\mu m})^{-1}$ is the ratio of the radiation and gravitational force. Particles with $\beta>1/2$ are removed from the system by radiation pressure. Alternatively, @Gasparetal2012a calculate a collision-dependent $s_{\rm min}$ from mass conservation, but do not study the surface energy.
If, however, for any relevant collision Eq. \[eq:s\_min3.5\] predicts $s_{\rm min}>s_{\rm blow}$, extrapolating the fragment size distribution down to these sizes is not justified. For example, starting from Eq. 2 of @Krivovetal2008, cm-sized bodies have $Q^*\simeq5\times10^6$ erg/g. In the Krivov et al. framework, a collision between two such bodies at 70 m/s will then result in fragmentation, as the kinetic energy ($\simeq 12$ J, assuming $\rho=2.35\rm{~g/cm^3}$) slightly exceeds the critical energy ($=2mQ^* \simeq 10$ J), and fragments will be created from a size comparable to the impactor [Eq. 21 of @Krivovetal2006] down to the blow-out size. For this particular collision, Eq. \[eq:s\_min3.5\] yields $s_{\rm min} \ll s_{\rm blow}$ for $\eta=0.1$ and $\gamma=0.1\rm{~J/m^2}$, but $s_{\rm min} \simeq 6\rm{~\mu m}$ for $\eta = 10^{-3}$. Thus, the difference between our results and the fragment sizes of Krivov et al. may be substantial, depending on the true value of $\eta$. We stress that our theory is valid for $3>\alpha>4$, and does not apply to models that use shallower power laws, for example Section 4.2 of @Krivovetal2013.
The importance of the limit given by Eq. \[eq:s\_min3.5\] depends on the parameters, and can vary per individual collision, depending on the collision velocity and choice for $s_{\rm max}$. In the rest of this section, we explore in which cases this limit is most relevant.
In a debris disk, a particle of size $s_0$ is most likely formed in a collision between only slightly larger particles. In addition, we focus on collisions between equal-sized particles, as these are most efficient at forming small fragments (Appendix \[sec:B\]). Therefore, we use Eq. \[eq:biermann\] as an indication for the lower limit of the particle size distribution. Quantitative comparisons require relative collision velocities, which for the largest bodies are often written in terms of the Keplerian orbital velocity at the corresponding distance from the central object. For bodies on orbits with identical semi-major axes, the relative velocity can then be written in terms of orbital eccentricity and inclination as $ f \equiv v_{\rm rel} / v_{\rm K} = (1.25 e^2 + i^2)^{1/2}$, with $v_{\rm K}=(GM_{\star} / a)^{1/2}$ the Keplerian orbital velocity [@WyattDent2002]. In a debris disk, a range of eccentricities and inclinations will be present. For a rough comparison, we use average quantities $\langle e \rangle$ and $\langle i \rangle$ to obtain typical collision velocities. In reality, $\langle e \rangle$ and $\langle i \rangle$ are poorly constrained. Estimates range from $\langle e\rangle \sim \langle i\rangle \sim10^{-3}-10^{-1}$, depending on the level of stirring [@matthews2014].
The ratio between the smallest grain size from Eq. \[eq:biermann\] and the blow-out size then becomes $$\label{eq:s_ratio}
\frac{s_{\rm min}}{s_{\rm blow}} = 2.4 \left( \frac{a}{5\mathrm{AU}} \right) \left( \frac{L_{\star}}{L_{\odot}} \right)^{-1} \left( \frac{f}{10^{-2}} \right)^{-2} \left( \frac{\eta}{10^{-2}} \right)^{-1} \left( \frac{\gamma}{0.1~\mathrm{J~m^{-2}}} \right),$$ where both the stellar mass and the material density drop out, and we assumed $Q_{\rm pr}=1$. Figure \[fig:arrows\] compares this ratio with observations of debris disks at large radii, where we predict the most pronounced effect. We have used a fixed $\gamma=0.1\mathrm{~J~m}^{-2}$, $f=10^{-2}$, and $\eta=10^{-2}$, and the arrows indicate the dependence of $s_{\rm min}/s_{\rm blow}$ on various parameters. For the main dust belt around , @Minetal2010 found the scattering properties to be consistent with predominantly $\sim100 \mu$m silicate grains [$s_{\rm blow}=13\mu$m @Ackeetal2012]. The relative velocities in Fomalhaut are typically taken a factor of 10 higher [@WyattDent2002]. For , @Donaldsonetal2012a derived $s_{\textrm{min}}=8.9 \mu$m ($s_{\rm blow}=0.5\mu$m) at orbital distances above $\sim50$ AU. Notably, very large grain sizes of $\sim100\mu$m ($s_{\rm blow}\lesssim1\mu$m) are inferred for the recently discovered “*Herschel* cold debris disks” [@Krivovetal2013], which are seen around F, G, and K type stars. @Krivovetal2013 were not able to model these systems with a collisional cascade reaching down to $s_{\rm min}=3\mu$m, and proposed that the large grains in these systems are primordial, unstirred material. Our calculations suggest that they can also be explained as the outcome of a collisional cascade. However, the model is highly degenerate, as material properties ($\eta$ and $\gamma$) and belt properties ($f$) are usually poorly known, and all have a large impact on $s_{\rm min}$.
In Figure \[fig:arrows\], we assume constant and equal relative velocities for all particles. In reality, radiation pressure will also increase the eccentricities of small particles with $\beta \lesssim 0.5$. The enhanced eccentricity can be written as $e_{\beta}=\beta / (1-\beta)$. The relative velocity of such a radiation-influenced particle scales with its size as $v_{\rm rel} \propto \beta \propto s^{-1}$, while Eq. \[eq:biermann\] predicts the fragmentation velocity scales as $v_{\rm rel}\propto s_0^{-1/2}$. Hence, the relative velocity between the smallest particles increases *faster* than the velocity needed for fragmentation. As a result, particles can reach arbitrarily small sizes in this regime. Particles smaller than $s_{\rm blow}$ are then removed on a short timescale. For a more detailed estimation, $\beta$ should be evaluated for each particular case, considering the optical properties of the material and the shape of the stellar spectrum.
A dearth of small grains in weakly stirred disks is also predicted by @Thebault2008, but the cause is not a limit on $s_{\rm min}$. In their scenario, $s_{\rm min}$ is fixed and the production rate of the smallest grains decreases with weaker stirring, while the destruction rate is determined by radiation forces and is unaffected by stirring. While the smallest grains present are always of blow-out size, their abundance is set by the balance between their creation and destruction (their Figure 7).
While the theory developed in this work predicts that the smallest particles that can fragment further can be quite large and sizes just below this will be depleted, some smaller particles will still be created as a result of erosive collisions, collisions between larger bodies, and collisions that occur above the average collision velocity. Detailed debris disk models implementing the surface energy constraint are needed to determine the resulting size distribution.
Discussion
==========
Of the fundamental parameters in our model, the largest uncertainty affects $\eta$, the fraction of kinetic energy used for the creation of new surface. While information may be available about the kinetic and surface energy of the largest fragments, it is hard to quantify whether the remainder of the available energy went into surface creation, heat generation, or kinetic energy of the smallest fragments. Experimentally, studying $\eta$ is challenging, since it requires sensitive and complete measurements down to very small sizes. Once the functional form of $\eta$ is quantified by laboratory and numerical experiments, observations of $s_{\textrm{min}}$ in a system of interest may constrain $f$ and thus the local relative velocities.
During the preparation of this manuscript, we discovered that a similarly defined $s_{\textrm{min}}$ to the one we present has been explored in a more abstract framework, and without elaborating on applications, by @Bashkirov1996. We note that the lack of data on the size distribution of small collision fragments, as well as the fraction of kinetic and internal energy in the fragments already noted by @Bashkirov1996, still prevails and we encourage further experiments to quantify these important parameters.
Thus far, we have focussed on equal-sized collisions. While collisions with a larger mass ratio might not lead to catastrophic fragmentation, cratering and erosion may still be important, and might be able to form small particles (Appendix \[sec:B\]). Assuming a fixed relative velocity, we focus on a particle of size $s_1$. We define a mass loss rate $\dot{m}(s)$ for the larger particle, dependent on collider size $s$. Assuming a collision with a particle of size $s<s_1$ erodes a mass $\propto s^3$, and noting the collision timescale is proportional to the particle density and collision cross-section, we obtain for the *total* mass loss rate $\dot{M} = \dot{m}(s)\intd s \propto s^{-3.5} s^3 (s+s_1)^2 \intd s$. If the collisional cross-section is dominated by $s_1$, we find $\dot{M} \propto s_1^2 s^{1/2} $, and thus the mass loss is dominated by the *larger* bodies. When $s \sim s_1$, we obtain $\dot{M}\propto s^{5/2}$.
We have adopted a constant value of 50% of the collider mass for the largest fragment. However, experiments show $s_{\rm max}$ can be substantially smaller as a function of material and impact velocity [e.g. @davis1990; @ryan1991]. Such results can easily be implemented in Eq. \[eq:s\_min3.5\] (and Eqs. \[eq:s\_min\_uneq\] and \[eq:s\_min\_eros\]) as necessary. Since $s_{\rm min} \propto s_{\rm max}^{-1}$, smaller sizes for the largest remnant will make the production of small particles even more difficult.
Other collisional systems where the proposed fragment size limit operates include planetary rings. Our calculations are consistent with the observed dominant grain sizes in the rings of Saturn, Jupiter, and Uranus. Because of additional relevant physics, such as tidal and electromagnetic effects, consistency does not directly imply the dominant grain size in all these rings is fragmentation-dominated.
The full implications of an energy-limited $s_{\textrm{min}}$ on systems such as debris disks and planetary rings can only be assessed with models tracking the full particle population with all relevant processes included. For example, if small particles cannot be destroyed in collisions, Poynting-Robertson (PR) drag will influence their orbits, and cause particles to drift towards the star on timescales of Gyrs in the outer parts of disks [@wyatt2005; @lieshout2014]. Such modelling is outside the scope of the present paper.
Conclusions
===========
We investigated the energetic constraints on the lower size limit in a distribution of collision fragments. A quantification of the lower limit of such size distributions is relevant for the modelling of debris disks and other astrophysical systems where collisional fragmentation is important.
1. [Based on surface energy constraints, we derive a parameterised recipe for the smallest fragment size in individual grain-grain collisions.]{}
2. [The smallest size in a distribution of fragments from a two-particle collision, constrained by the collision energy, is given by Eq. \[eq:s\_min3.5\], and illustrated in Figure \[fig:fig1\]. For example, at 20 m/s, submicron silicate particles can only be effectively produced by centimeter-sized colliders.]{}
3. [In the limit where the colliding bodies are split in half, the fragmentation threshold velocity is given by Eq. \[eq:biermann\].]{}
4. [While dedicated models are needed to reveal the full implications of the fragment size limit, Figure \[fig:arrows\] offers an indication of where the size distribution is expected to be influenced.]{}
5. [In systems where the collision velocities are low, our theory may offer a natural explanation for a paucity of small grains in debris at large orbital distances, such as observed in Fomalhaut and the *Herschel* cold debris disks (Figure \[fig:arrows\]).]{}
The authors wish to thank Carsten Dominik and Xander Tielens for comments and discussions. Dust studies at Leiden Observatory are supported through the Spinoza Premie of the Dutch science agency, NWO. Astrochemistry in Leiden is supported by NOVA, KNAW and EU A-ERC grant 291141 CHEMPLAN.
Applicability to aggregates {#sec:A}
===========================
For porous aggregates, the basic principles explored here are expected to hold, but some material properties have to be altered. First, aggregates have an internal filling factor $\phi = \rho_{\rm agg} / \rho$ that is $<1$, and might be as low as $10^{-4}$ in some extreme cases [@okuzumi2012; @kataoka2013b]. Second, the ’effective’ surface energy $\gamma_{\rm agg}$ will be smaller, since there is only limited contact between the aggregate’s constituents to begin with. Assuming the parent bodies are built up of spherical monomers, the effective surface energy can be estimated as $\gamma_{\rm agg} \sim (a/R)^2 \phi^{2/3} \gamma$, where $a$ and $R$ denote the radius of the contact area shared by monomers, and the radius of the monomers themselves. The fraction $(a/R)$ depends on the size of the monomers and the material properties, but for 0.1-micron-sized monomers, $(a/R) \sim 0.1$ is reasonable.
For aggregates, $N$-body simulations have been performed with particles containing up to $10^6$ monomers, and values of $\eta$ range from close to unity [@dominiktielens1997; @wada2009], to several orders of magnitude less [@ringl2012b], and depend on the employed contact model [@seizinger2013c].
Collisions with different mass ratios {#sec:B}
=====================================
Here we extend the theory to collisions between ’targets’ and ’projectiles’ of arbitrary sizes $s_{\rm t} > s_{\rm p}$. Assuming the fragment distribution can be described as before, we can still use Eq. \[eq:M\_frag\], while the pre-collision kinetic energy now equals $$U_{\rm K} = \frac{1}{2}\mu v_{\rm rel}^2 = \frac{1}{2} \frac{m_{\rm p} m_{\rm t}}{m_{\rm p}+m_{\rm t}} v_{\rm rel}^2.$$ For a given collision velocity, we might then think of two cases, complete fragmentation when $s_{\rm p} \sim s_{\rm t}$, and erosion/cratering when $s_{\rm t} \gg s_{\rm p}$.
Catastrophic fragmentation
--------------------------
Since both particles are destroyed completely we may set $M_{\rm frag} = m_{\rm p} + m_{\rm t}$. For simplicity we will assume in this section that the change in surface energy is dominated by the fragments $\Delta U_{\rm S}=\gamma A_{\rm frag}$. Using the same definition for $\eta$ as before and focussing on the $\alpha=3.5$ case, we obtain $$\label{eq:s_min_uneq}
s_{\rm min} = \left[ \frac{ 6 \gamma (s_{\rm p}^3+s_{\rm t}^3)^2}{\eta \rho v_{\rm rel}^2 (s_{\rm p}s_{\rm t})^3}\right]^2 s_{\rm max}^{-1}.$$
Erosion
-------
When the mass ratio becomes very large, it is no longer realistic to assume the target is completely disrupted. Rather, such collisions result in erosion, and the eroded mass is typically of the order of the mass of the projectile [@schrapler2011]. Thus, we write $M_{\rm frag} = \kappa m_{\rm p}$, with $\kappa$ of the order of unity. For large mass ratios $\mu \rightarrow m_{\rm p}$. Furthermore, assuming that the change in surface energy is dominated by the new fragments, $\Delta U_S = \gamma A_{\rm frag}$, we obtain $$\label{eq:s_min_eros}
s_{\rm min} = \left( \frac{ 6 \gamma \kappa}{\eta \rho v_{\rm rel}^2 }\right)^2 s_{\rm max}^{-1}.$$
Consider now a particle with a size $s_0$ close to the smaller end of the size distribution, colliding with particle of sizes $s_x$, ranging from slightly smaller to much larger than $s_0$. Figure \[fig:fig4\] shows the minimum size of the fragments produced as a function of collider size $s_x$. The $y$-axis is normalized to the value obtained in equal-sized collisions (i.e. between two $s_0$ particles). For mass ratios below unity, $s_0$ acts as the target, and the mass of the largest fragment is assumed to equal $m_0/2$. For large mass ratios, $s_0$ acts as a projectile instead, and the largest fragment mass is set to $m_x/2$. Collisions at mass ratios above $10^2$ are assumed to be erosive [@seizinger2013c], with the largest fragment equalling $m_0/2$. Since both the excavated mass, and the largest fragment mass, depend on the projectile, the curves in the erosive regime do not depend on the mass ratio directly. However, the size of the target does set an upper limit on $\kappa$.
[^1]: One could imagine splitting only one of the colliders, or indeed chipping off only small parts of one of the collider bodies. Since less surface area is created, this would still be allowed at very low velocities. However, in that case the largest fragment is of the same size as $s_0$. We refrain from identifying this as fragmentation, and use the size derived in Eq. \[eq:s\_split\] as the size below which fragmentation becomes inefficient.
| {
"pile_set_name": "ArXiv"
} |
---
address: |
Infineon Technologies, 2070 State Route 52, Hopewell Junction, NY, 12533\
MRAM Developement Alliance, IBM/Infineon Technologies, IBM Semiconductor Research and Developement Center, 2070 State Route 52, Hopewell Junction, NY, 12533
author:
- 'Daniel Braun[^1]'
title: Exact Activation Energy of Magnetic Single Domain Particles
---
Introduction
============
A lot of effort has been spent over the last few years to understand the magnetization reversal of small magnetic particles [@Lederman94; @Kent94; @Wernsdorfer95; @Koch98; @Bonet99; @Wernsdorfer00; @Schumacher03]. At sufficiently low temperatures, macroscopic quantum tunneling has been observed [@Coppinger95; @Friedmann96; @Wernsdorfer97; @Bokacheva00], while for higher temperatures thermally activated behavior may switch the magnetization of the particle [@Koch00]. For the anlaysis of the experiments one needs to know the activation energy. Moreover, with the advancing developement of integrated magnetoresistive memory devices (MRAM) the dependence of the energy barrier on the magnetic field has become of crucial technological importance as well. In a typical MRAM array, magnetic memory cells are written by a coincident field technique, where both a selected bitline (BL) and a selected wordline (WL) create magnetic fields, the sum of which are strong enough to switch the memory cell, whereas the fields from either BL or WL alone are not sufficient to switch the cell. Nevertheless, these fields do destabilize the non–selected cells to some extent, i.e. they reduce the energy barrier against thermally activated switching. Also, even for the selected cells the switching at finite temperatures happens before the actual zero temperature boundary of stability is reached, again due to thermal activation during the finite duration of a write pulse [@Koch00; @Braun93]. To estimate the life time of the information in the memory, one needs to know the dependence of the energy barrier on the applied fields as precisely as possible, as the energy barrier enters the switching rate exponentially.
In the study of the switching behavior of small size magnetic particles, the Stoner Wohlfarth model plays a central role [@Stoner48]. It describes a single domain particle with uni–axial anisotropy in an external magnetic field. The single domain approximation greatly simplifies the analysis, and becomes a good approximation if the size of the particle becomes comparable to or smaller than the exchange length, which in memory elements etched out of a thin magnetic film is typically of the order of 100nm. The activation energy in the Stoner Wohlfarth model can be calculated trivially if the external field is aligned either parallel to the preferred axis or perpendicular to it. However, so far no analytical solution has been known for the general case of the external field pointing in an arbitrary direction [@Wernsdorfer03]. Given the crucial importance of the field dependence of the activation energy, an exact analytical solution will be provided in the present paper.
The Stoner Wohlfarth Model
==========================
The energy of a uniformly magnetized particle with uniaxial symmetry, characterized by the anistropy energy density $K$ and saturation magnetization $M_s$ depends on its magnetization via the angle $\vartheta$ between the magnetization and the preferred axis, $$\label{Estowo}
E(\vartheta)=KV \sin^2\vartheta -V M_s H_x\cos\vartheta-V M_s H_y\sin\vartheta\,,$$ where $H_x$ and $H_y$ are the magnetic field components parallel and perpendicular to the preferred axis in a plane containing the magnetization and the preferred axis, respectively, and $V$ denotes the volume of the sample [@Stoner48]. In the following dimensionless variables will be used by writing the energy in terms of $2KV$, $e(\vartheta)=E(\vartheta)/(2KV)$, and the magnetic field components in terms of the switching field $H_c=2K/M_s$ as $h_x=H_x/H_c$, $h_y=H_y/H_c$, such that $$\label{estowo}
e(\vartheta)=\frac{1}{2}\sin^2\vartheta -h_x\cos\vartheta-h_y\sin\vartheta\,.$$ For vanishing magnetic fields, this model has a bistable ground state, whereas for very large fields ($h_x\gg 1$ or $h_y\gg 1$) the first term can be neglected, and there is only one minimum in the interval $-\pi\le\vartheta<\pi$, such that the magnetization tends to align parallel to the applied field. Thus, for increasing field, one of the original minima has to disappear, and the fields where this happens mark the boundary of bistability. These fields are easily obtained by setting both the first and second derivative of (\[estowo\]) to zero and eliminating $\vartheta$, whereupon the famous Stoner-Wohlfahrt astroid $h_x=\cos^3 \vartheta$, $h_y=\sin^3\vartheta$ is obtained [@Stoner48].
Activation Energy
=================
The activation energy can in principle be calculated in a straight forward manner by finding the two minima and the two maxima of the energy in the bistability range, determining the metastable of the two minima and the energy barrier to the smaller of the two maxima [@barrier]. In the case of $h_x=0$ or $h_y=0$ this is straight forward: For $h_y=0$, $$\label{d1e}
\frac{\partial e}{\partial
\vartheta}=\cos\vartheta\sin\vartheta+h_x\sin\vartheta-h_y\cos\vartheta=0$$ leads to $\vartheta=0$, $\vartheta=\pi$, or $\cos\vartheta=-h_x$. The solution $\vartheta=0$ is metastable for $-1\le h_x\le 0$, stable for $h_x>0$, and unstable for $h_x<-1$. Correspondingly, $\vartheta=\pi$ is metastable for $0\le h_x<1$, stable for $h_x<0$, and unstable for $h_x>1$. The solution $cos\vartheta=-h_x$ leads to two maxima which become complex for $|h_x|>1$. Thus, the energy barrier for switching from $\vartheta=\pi$ to $\vartheta=0$ is given by $E_A=e(\arccos(-h_x))-e(\pi)=(1-h_x)^2/2$. Similarly, for $h_x=0$ one finds the energy barrier $E_A=(1-h_y)^2/2$. Therefore, if one of the two field components vanishes, the activation energy depends quadratically on the distance from the stability boundary.
In the general case, where neither $h_x$ nor $h_y$ vanish, one may substitute $\sin\vartheta=u$, $\cos\vartheta=\pm\sqrt{1-u^2}$ into (\[d1e\]). Squaring the equation leads to $$\label{d1eu}
-u^4+2u^3h_y+u^2(1-h_x^2-h_y^2)-2uh_y+h_y^2=0\,.$$ In order to calculate the energy barrier, one needs to find the roots of this 4th order polynomial, which makes the analysis much more cumbersome than for $h_xh_y=0$. Still, the roots of a 4th order polynomial can be obtained analytically. The solutions $\vartheta_i$ are conveniently written in terms of the functions $$\begin{aligned}
f_1&=&h_x^2+h_y^2-1\,,\\
f_2&=&108 h_x^2h_y^2+2f_1^3\,,\\
f_3&=&f_2+\sqrt{f_2^2-4f_1^6}\,,\\
f_4&=&\sqrt{1+\frac{f_1}{3}+\frac{2^{1/3}}{3}\frac{f_1^2}{f_3^{1/3}}+\frac{1}{3}\frac{f_3^{1/3}}{2^{1/3}}-h_y^2}\,.\end{aligned}$$ Eq.(\[d1eu\]) has four solutions for $u$, but the ambiguity in the sign of $\cos\vartheta$ in terms of $u$ leads at this point to eight solutions for $\vartheta$. They differ by three signs $\mu,\nu,\sigma$ in various places, and are given by $$\begin{aligned}
\vartheta_i&=&\sigma \arccos\Bigg[-\frac{h_x}{2}+\frac{\mu}{2}f_4
+\frac{\nu}{2}\sqrt{1-\frac{f_1}{3}-\frac{2^{1/3}f_1^2}{3f_3^{1/3}}-\frac{1}{3}\frac{f_3^{1/3}}{2^{1/3}}+h_x^2-h_y^2+\frac{2\mu}{f_4}(2h_x+f_1h_x-h_x^3)}\Bigg]\,.\label{theti}\end{aligned}$$ The signs will be chosen according to the binary decomposition $(\mu,\nu,\sigma)=(-,-,-)$ for $\vartheta_1$, $(-,-,+)$ for $\vartheta_2$, $(-,+,-)$ for $\vartheta_3$, $\ldots$, $(+,+,+)$ for $\vartheta_8$. None of these solutions solves (\[d1e\]) for all $h_x,h_y$. Rather, each of them solves the equation only in two quadrants. However, it is possible to construct four uniform solutions out of the eight partially valid ones, which are continuous (after the identification of $\pi$ with $-\pi$) and solve (\[d1e\]) for all values of $h_x,h_y$. The four uniform solutions are $$\begin{aligned}
\tilde{\vartheta}_1&=&\theta(h_y)\vartheta_2+\theta(-h_y)\vartheta_1\,,\\
\tilde{\vartheta}_2&=&\theta(h_xh_y)\vartheta_5+\theta(-h_xhy)\vartheta_3\,,\\
\tilde{\vartheta}_3&=&\theta(h_xh_y)\vartheta_4+\theta(-h_xhy)\vartheta_6\,,\\
\tilde{\vartheta}_4&=&\theta(h_y)\vartheta_8+\theta(-h_y)\vartheta_7\,,\\\end{aligned}$$ where $\theta(x)=1$ for $x>0$ and zero elsewhere denotes the Heaviside function. Calculating the second derivative one finds that the first uniform solution is a minimum for $h_x<0$. It disappears as real solution for $h_x>0$ outside the astroid. Thus, this is the metastable state for $h_x>0$. On the other hand, $\tilde{\vartheta}_4$ remains a minimum even outside the astroid for $h_x>0$, but disappears as real solution outside the astroid for $h_x<0$. The uniform solutions $\tilde{\vartheta}_2$ and $\tilde{\vartheta}_3$ are both maxima inside the astroid, and become complex outside one half of the astroid (for $h_y<0$ and $h_y>0$, respectively [**??**]{}). One easily convinces oneself that for $h_y\ge 0$, $\tilde{\vartheta}_3$ is the relevant maximum ($e(\tilde{\vartheta}_3)\le e(\tilde{\vartheta}_2)$), whereas for $h_y<0$, escape over the maximum at $\tilde{\vartheta}_2$ is dominant ($e(\tilde{\vartheta}_3)>e(\tilde{\vartheta}_2)$). The activation energy out of the metastable state is therefore given by $$\begin{aligned}
E_A&=&e(\tilde{\vartheta}_3)-e(\tilde{\vartheta}_1)\mbox{ for }
h_y\ge 0\,,\label{eAhyg0}\\
E_A&=&e(\tilde{\vartheta}_2)-e(\tilde{\vartheta}_1)\mbox{ for }
h_y<0\,,\label{eAhyl0}\end{aligned}$$ Fig.\[fig.EA3D\] shows a plot of the activation energy as function of $h_x$ and $h_y$.
\
For $h_y=0$ both maxima lead to the same activation energy, and $E_A$ is an even function of $h_y$, as it should be. In the following the attention will therefore focus solely on the case $h_y\ge 0$, and to the case where the initial state $\vartheta=\pi$ is metastable, $h_x>0$, i.e. the first quadrant. In principle, the particle might be excited also out of the stable state and end up (for a finite time) in the metastable state, but this process is of much less importance, as the activation energy is much larger and it enters exponentially in the thermal switching rate.
Scaling
=======
The scaling of the activation energy as function of the distance $1-h$ from the astroid boundary is of particular interest. Here, $h$ is defined by $h_x=h
\cos^3\xi$, $h_y=h \sin^3\xi$ such that $h=1$ always corresponds to the astroid boundary. It has been shown before [@Wernsdorfer96] that the scaling must be of the form $$\begin{aligned}
E_A&=&c_{3/2}(\xi)(1-h)^{3/2}+c_{2}(\xi)(1-h)^{2}+c_{5/2}(\xi)(1-h)^{5/2}+\ldots\,.\label{E_As}\end{aligned}$$ All coefficients besides $c_{2}(\xi)$ vanish for $\xi=0$, and but it has been shown numerically [@Wernsdorfer96] and theoretically [@Victora89] that already for small values of $\xi$ the coefficient $c_{3/2}$ dominates the scaling. This is confirmed and made more precise by using the exact solution. Fig. \[fig.EAloglog\] shows $\ln E_A$ as function of $\ln(1-h)$. The plot reveals power law behavior to a good approximation even rather far away from the astroid boundary, $E_A\propto
(1-h)^a$ with an exponent $a=2$ for $\xi=0$, and an exponent close to $3/2$ for larger values of $\xi$.
\
According to the numerical evaluation of (\[eAhyg0\]) the exponent is symmetric with respect to $\xi=\pi/4$. The exact value of the exponent depends on the fitting range and on whether a quadratic term is included in the fit of $\ln E_A$ as function of $\ln(1-h)$. Fig. \[fig.alin\] shows the fitted exponent $a$ as a function of $x=\xi/\pi$ for three different fitting ranges, from $h=0.932$ to $h=0.99999853$, assuming a pure power law, $\ln E_A=a\ln(1-h)$. The observed dependence of the exponent on $x$ is very similar to what was previously calculated numerically [@Wernsdorfer96]. In particular, the exponent appears to become even slightly smaller than $3/2$ for values of $\xi$ close to $\pi/4$.
\
However, there is a substantial non–linear part in the scaling behavior, as becomes obvious when fitting to $\ln E_A=a \ln(1-h) +b \ln^2(1-h)$. The extracted linear part is plotted in Fig.\[fig.aquad\]. For small values of $\xi$, the exponent can now be substantially larger than $2$.
\
Summary
=======
I have derived the exact analytical expression of the activation energy for single domain switching of small magnetic particles in arbitrary magnetic fields (Stoner–Wohlfarth model). The activation energy scales approximately like a power law as a function of the distance of the switching boundary (astroid) up to distances of order unity, but also contains a substantial non–power law term.
[*Acknowledgement:*]{} I am grateful to Daniel Worledge for a useful discussion.
M. Lederman, S. Schultz, and M. Ozaki, Phys. Rev. Lett. [**73**]{}, 1986 (1994). A.D. Kent, S. von Molnar, S. Gider, and D.D. Awschalom, J. Appl. Phys. [**76**]{} 6656 (1994). W. Wernsdorfer [*et al.*]{}, J. Magn. Magn. Mater. [**151**]{}, 396 (1995). R.H. Koch, J.G. Dreak, D.W. Abraham, P.L. Trouilloud, R.A. Altman, Y. Lu, W.J. Gallagher, R.E. Scheuerlein, K.P. Roche, and S.S.P. Parkin, Phys. Rev. Lett., [**81**]{}, 4512 (1998). E. Bonet, W. Wernsdorfer, B. Barbara, A. Beno[î]{}t, D. Mailly, and A. Thiaville, Phys. Rev. Lett. [**83**]{}, 4188 (1999). W. Wernsdorfer, D. Mailly, A. Benoit, J. Appl. Phys., [**87**]{}, 5094 (2000). H.W. Schumacher, C. Chappert, R.C. Sousa, P.P. Freitas, and J. Miltat, Phys. Prev. Lett. [**90**]{}, 017204 (2003). F. Coppinger, J. Genoe, D.K. Maude, U. Gennser, J.C. Portal, K.E. Singer, P. Ruter,T. Taskin, A.R. Peaker, A.C. Wright, Phys. Rev. Lett, [**75**]{}, 3513 (1995). J.R. Friedmann, M.P. Sarachik, J. Tejada, R. Ziolo, Phys. Rev. Lett. [**76**]{}, 3830 (1996). W. Wernsdorfer, E. Bonet Orozco, K. Hasselbach, A. Benoit, D. Mailly, O. Kubo, H. Nakamo, and S. Barbara, Phys. Rev. Lett. [**79**]{}, 4014 (1997). L. Bokacheva, A.D. Kent, and M.A. Walters, Phys. Rev. Lett. [**85**]{}, 4803 (2000). R.H. Koch, G. Grinstein, Y. Lu, P.L. Trouilloud, W.J. Gallagher, and S.S.P. Parkin, Phys. Rev. Lett. [**84**]{}, 5415 (2000). H.-J. Braun, Phys. Rev. Lett. [**71**]{}, 3557 (1993). E.C. Stoner, E.P. Wolfarth, Phil. Trans. Roy. Soc. A [**240**]{}, 599 (1948). W. Wernsdorfer, private communication. Note that in principle there are two energy barriers, corresponding to the two escape paths out of the metastable minimum by clockwise or counterclockwise rotation of the magnetization. In the following the path with the higher activation energy will be neglected and the activation energy will always be defined as the smaller of the two energy barriers. W. Wernsdorfer, Ph.D. thesis, Université Joseph Fourier–Grenoble I (1996). R.H. Victora, Phys. Rev. Lett. [**63**]{}, 457 (1989)
[^1]: Corresponding author. Fax: +1 914 945 4421; email: v2braun@us.ibm.com
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Let $M$ be a noncommutative 2-torsion free semiprime $\Gamma$-ring satisfying a certain assumption and let $S$ and $T$ be left centralizers on $M$. We prove the following results:\
(i) If $[S(x),T(x)]_{\alpha }\beta S(x)+S(x)\beta [S(x),T(x)]_{\alpha }$=$0$ holds for all $x\in M$ and $\alpha ,\beta \in \Gamma $, then $[S(x),T(x)]_{\alpha }$=$0$.\
(ii) If $S\neq 0 (T\neq 0)$, then there exists $\lambda \in C$,(the extended centroid of $M$) such that $T$=$\lambda \alpha S(S=\lambda \alpha T)$ for all $\alpha \in \Gamma $.\
(iii) Suppose that $[[S(x),T(x)]_{\alpha },S(x)]_{\beta }$=$0$ holds for all $x\in M$ and $\alpha ,\beta \in\Gamma $. Then $[S(x),T(x)]_{\alpha }$=$0$ for all $x\in M$ and $\alpha \in\Gamma $.\
(iv) If $M$ is a prime $\Gamma $-ring satisfying a certain assumption and $S\neq 0(T\neq 0)$, then there exists $\lambda \in C$, the extended centroid, such that $T$=$\lambda \alpha S(S=\lambda \alpha T)$.
author:
- |
Md Fazlul Hoque\
School of Mathematics and Physics\
The University of Queensland, Brisbane, Australia\
E-mail: m.hoque@uq.edu.au\
\
A.C. Paul\
Department of Mathematics\
Rajshahi University, Bangladesh\
E-mail: acpaulrubd\_math@yahoo.com
title: Centralizers on Prime and Semiprime Gamma Rings
---
\[section\] \[section\] \[section\] \[section\]
[**2000 Mathematics Subject Classification**]{}, 16N60,16W25,16Y99.\
[**Keywords**]{}: prime $\Gamma $-ring, semiprime $\Gamma$-ring, central closure, extended centroid, left(right) centralizer.
Introduction
============
The notion of a $\Gamma $-ring was first introduced as an extensive generalization of the concept of a classical ring. From its first appearance, the extensions and generalizations of various important results in the theory of classical rings to the theory of $\Gamma $-rings have been attracted a wider attentions as an emerging area of research to the modern algebraists to enrich the world of algebra. All over the world, many prominent mathematicians have worked out on this interesting area of research to determine many basic properties of $\Gamma $-rings and have executed more productive and creative results of $\Gamma $-rings in the last few decades. We begin with the definition.
Let $M$ and $\Gamma $ be additive abelian groups. If there exists an additive mapping $(x,\alpha ,y)\rightarrow x\alpha y$ of $M\times\Gamma \times M\rightarrow M$, which satisfies the conditions $(x\alpha y)\beta z$=$x\alpha (y\beta z)$ for all $x,y,z\in M$ and $\alpha ,\beta \in \Gamma$, then $M$ is called a $\Gamma$-ring. Every ring $M$ is a $\Gamma $-ring with $M$=$\Gamma$. However a $\Gamma $-ring need not be a ring. Gamma rings, more general than rings\[8\]. Barnes\[1\] weakened slightly the conditions in the definition of $\Gamma $-ring in the sense of Nobusawa. Let $M$ be a $\Gamma $-ring. Then an additive subgroup $U$ of $M$ is called a left (right) ideal of $M$ if $M\Gamma U\subset U$($U\Gamma M \subset U$). If $U$ is both a left and a right ideal, then we say $U$ is an ideal of $M$. Suppose again that $M$ is a $\Gamma $-ring. Then $M$ is said to be a 2-torsion free if $2x$=$0$ implies $x$=$0$ for all $x\in M$. An ideal $P_{1}$ of a $\Gamma $-ring $M$ is said to be prime if for any ideals $A$ and $B$ of $M$, $A\Gamma B\subseteq P_{1}$ implies $A\subseteq P_{1}$ or $B\subseteq P_{1}$. An ideal $P_{2}$ of a $\Gamma $-ring $M$ is said to be semiprime if for any ideal $U$ of $M$, $U\Gamma U\subseteq P_{2}$ implies $U\subseteq P_{2}$. A $\Gamma$-ring $M$ is said to be prime if $a\Gamma M\Gamma b$=$(0)$ with $a,b\in M$, implies $a$=$0$ or $b$=$0$ and semiprime if $a\Gamma M\Gamma a$=$(0)$ with $a\in M$ implies $a$=$0$. Furthermore, $M$ is said to be commutative $\Gamma $-ring if $x\alpha y$=$y\alpha x$ for all $x, y\in M$ and $\alpha \in\Gamma $. Moreover, the set $Z(M)$ =$\{x\in M:x\alpha y=y\alpha x $ for all $ \alpha \in \Gamma, y\in M\}$ is called the centre of the $\Gamma $-ring $M$. For the definitions of the quotent $\Gamma $-ring, the extended centroid and the central closure, we refer to \[7\].\
If $M$ is a $\Gamma $-ring, then $[x,y]_{\alpha }$=$x\alpha y-y\alpha x$ is known as the commutator of $x$ and $y$ with respect to $\alpha $, where $x,y\in M$ and $\alpha \in\Gamma $. We make the basic commutator identities:\
$[x\alpha y,z]_{\beta }$=$[x,z]_{\beta }\alpha y+x[\alpha ,\beta ]_{z}y+x\alpha [y,z]_{\beta }$\
and $[x,y\alpha z]_{\beta }$=$[x,y]_{\beta }\alpha z+y[\alpha ,\beta ]_{x}z+y\alpha [x,z]_{\beta }$ for all $x,y.z\in M$ and $\alpha ,\beta \in\Gamma $. We consider the following assumption:\
$(A)$.................$x\alpha y\beta z$=$x\beta y\alpha z$ for all $x,y,z\in M$ and $\alpha ,\beta \in\Gamma $.\
According to the assumption $(A)$, the above two identites reduce to\
$[x\alpha y,z]_{\beta }$=$[x,z]_{\beta }\alpha y+x\alpha [y,z]_{\beta }$\
and $[x,y\alpha z]_{\beta }$=$[x,y]_{\beta }\alpha z+y\alpha [x,z]_{\beta }$, which we extensively used.\
An additive mapping $T : M\rightarrow M$ is a left(right) centralizer if $T(x\alpha y)$=$T(x)\alpha y$ $(T(x\alpha y)=x\alpha T(y))$ holds for all $x,y\in M$ and $\alpha\in \Gamma$. A centralizer is an additive mapping which is both a left and a right centralizer. For any fixed $a\in M$ and $\alpha \in\Gamma$, the mapping $T(x)=a\alpha x$ is a left centralizer and $T(x)=x\alpha a$ is a right centralizer. We shall restrict our attention on left centralizer since all results represented in this paper are true also for right centralizers because of left and right symmetry.\
Barnes \[1\], Lue \[6\] and Kyuno\[5\] studied the structure of $\Gamma$-rings and obtained various generalizations of corresponding parts in ring theory.\
Borut Zalar\[14\] worked on centralizers of semiprime rings and proved that Jordan centralizers and centralizers of these rings coincide. Joso Vukman\[11, 12, 13\] developed some remarkable results using centralizers on prime and semiprime rings.\
In \[2\], Hoque and Paul have proved that every Jordan centralizer of a 2-torsion free semiprime $\Gamma $-ring satisfying a certain assumption is a centralizer. Also, they proved in \[3\], if $T$ is an additive mapping on a 2-torsion free semiprime $\Gamma $-ring $M$ with a certain assumption such that $T(x\alpha y\beta x)=x\alpha T(y)\beta x$ for all $x, y\in M$ and $\alpha ,\beta \in\Gamma $, then $T$ is a centralizer and in \[4\], if $2T(x\alpha b\beta a)=T(x)\alpha y\beta x+x\alpha y\beta T(x)$ holds for all $x, y\in M$ and $\alpha ,\beta \in\Gamma $, then $T$ is also a centralizer.\
In this paper, we generalize some results of Joso Vukman\[11\] in Gamma rings.
Centralizers of Prime and Semiprime $\Gamma $-rings.
====================================================
To prove our main results, we need the following lammas: .2cm
Suppose that the elements $a_{i}$, $b_{i}$ in the central closure of a prime $\Gamma $-ring $M$ satisfy $\Sigma a_{i}\alpha _{i}x\beta _{i}b_{i}=0$ for all $x\in M$ and $\alpha _{i},\beta _{i}\in\Gamma $. If $b_{i}\neq 0$ for some $i$, then $a_{i}$’s are $C$-dependent, where $C$ is the extended centroid.
.2cm [**Proof.**]{} Let $M$ be a prime $\Gamma $-ring and let $C_{\Gamma }=C$ be the extended centroid of $M$. If $a_{i}$ and $b_{i}$ are non-zero elements of $M$ such that $\Sigma a_{i}\alpha _{i}x\beta _{i}b_{i}=0$ for all $x\in M$ and $\alpha _{i},\beta _{i}\in\Gamma $, then $a_{i}$’s (also $b_{i}$’s) are linearly dependent over $C$. Moreover, if $a\alpha x\beta b=b\alpha x\beta a$ for all $x\in M$ and $\alpha ,\beta \in\Gamma $, where $a(\neq 0)$, $b\in M$ are fixed, then there exists $\lambda \in C$ such that $a=\lambda \alpha b$ for $\alpha \in\Gamma $. Clearly, the lemma is proved. .2cm
Suppose that $M$ is a noncommutative prime $\Gamma$-ring satisfying the assumption $(A)$ and $T:M\rightarrow M$ is a left centralizer. If $T(x)\in Z(M)$ for all $x\in M$, then $T=0$.
.2cm [**Proof.**]{} Since $T$ is a left centralizer on $M$, we have $T(x\alpha y)=T(x)\alpha y$ holds for all $x,y\in M$ and $\alpha \in\Gamma $ and hence $[T(x),y]_{\alpha }=0$ for all $x,y\in M$ and $\alpha \in\Gamma $. Putting $x=x\beta z$ in the above relation, we have $$\begin{aligned}
&0&=[T(x\beta z),y]_{\alpha }\\&&=[T(x)\beta z,y]_{\alpha }\\&&=[T(x),y]_{\alpha }\beta z+T(x)\beta [z,y]_{\alpha }\\&&=T(x)\beta [z,y]_{\alpha }\end{aligned}$$ Hence $T(x)\beta [z,y]_{\alpha }=0$, which gives $T(x)\beta w \gamma [z,y]_{\alpha }=0$ for all $x,y.z,w\in M$ and $\alpha ,\beta ,\gamma \in\Gamma $, whence it follows that $T=0$, otherwise $M=0$. .2cm
Suppose that $M$ is a noncommutative prime $\Gamma$-ring satisfying the assumption $(A)$ and $S, T:M\rightarrow M$ are left centralizers. If $[S(x),T(x)]_{\alpha }=0$ holds for all $x\in M$ and $\alpha \in\Gamma $ and $T\neq 0$, then there exists $\lambda \in C$ such that $S=\lambda \alpha T$, where $C$ is the extended centroid.
.2cm [**Proof.**]{} First, we put $x=x+y$ in the relation $[S(x),T(x)]_{\alpha }=0$ and linearize, we have $$\begin{aligned}
&[S(x),T(y)]_{\alpha }+[S(y),T(x)]_{\alpha }&=0\end{aligned}$$. Replace $y$ by $y\beta z$ in (1), we have $$\begin{aligned}
&0&=[S(x),T(y)\beta z]_{\alpha }+[S(y)\beta z,T(x)]_{\alpha }\\&&=[S(x),T(y)]_{\alpha }\beta z+T(y)\beta [S(x),z]_{\alpha }+[S(y),T(x)]_{\alpha }\beta z+S(y)\beta [z,T(x)]_{\alpha }\\&&=T(y)\beta [S(x),z]_{\alpha }+S(y)\beta [z,T(x)]_{\alpha }\end{aligned}$$ Thus we have $$\begin{aligned}
&T(y)\beta [S(x),z]_{\alpha }+S(y)\beta [z,T(x)]_{\alpha }&=0\end{aligned}$$ Putting $y=y\gamma w$ in the above relation, we obtain, $$\begin{aligned}
&T(y)\gamma w\beta [S(x),z]_{\alpha }+S(y)\gamma w\beta [z,T(x)]_{\alpha }&=0\end{aligned}$$ Our assumption $T\neq 0$ follows from Lemma-2.2 that there exist $x,z\in M$ and $\alpha \in \Gamma $ such that $[T(x),z]_{\alpha }\neq 0$. Now, the relation (2) and Lemma-2.1 imply that $S(y)=\lambda (y)\alpha T(y)$, where $ \lambda (y)$ is from $C$. If we put $S(y)=\lambda (y)\alpha T(y)$ and $S(x)=\lambda (x)\alpha T(x)$ in the relation (2), we obtain $(\lambda (x)-\lambda (y))\alpha T(y)\gamma w\beta [T(x),z]_{\alpha }=0$ for all pairs $y,w\in M$, whence it follows $(\lambda (x)-\lambda (y))\alpha T(y)=0$, since $[T(x),z]_{\alpha }\neq 0$. Thus we have $\lambda (x)\alpha T(y)=\lambda (y)\alpha T(y)$ which completes the proof of the lemma. .2cm
Suppose that $M$ is a 2-torsion free noncommutative semiprime $\Gamma $-ring satisfying the assumption $(A)$ and $S$, $T$ are left centralizers on $M$. If $[S(x),T(x)]_{\alpha }\beta S(x)+S(x)\beta [S(x),T(x)]_{\alpha }=0$ holds for all $x\in M$ and $\alpha ,\beta \in\Gamma $. Then $[S(x),T(x)]_{\alpha }=0$ for all $x\in M$ and $\alpha \in\Gamma $. Also, if $M$ is prime $\Gamma $-ring satisfying the assumption $(A)$ and $S\neq 0(T\neq 0)$, then there esixts $\lambda \in C$,(the extended centroid of $M$) such that $T=\lambda \alpha S (S=\lambda \alpha T)$.
.2cm [**Proof.**]{} By the hypothesis, we have $$\begin{aligned}
&[S(x),T(x)]_{\alpha }\beta S(x)+S(x)\beta [S(x),T(x)]_{\alpha }&=0\end{aligned}$$ The lineariztion of the above relation, we have $$\begin{aligned}
0&=&[S(x),T(x)]_{\alpha }\beta S(y)+S(y)\beta [S(x),T(x)]_{\alpha }\nonumber\\&&+[S(x),T(y)]_{\alpha }\beta S(x)+S(x)\beta [S(x),T(y)]_{\alpha }\nonumber\\&&+[S(y),T(x)]_{\alpha }\beta S(x)+S(x)\beta [S(y),T(x)]_{\alpha }\nonumber\\&&+[S(y),T(y)]_{\alpha }\beta S(x)+S(x)\beta [S(y),T(y)]_{\alpha }\nonumber\\&&+[S(y),T(x)]_{\alpha }\beta S(y)+S(y)\beta [S(y),T(x)]_{\alpha }\nonumber\\&&+[S(x),T(y)]_{\alpha }\beta S(y)+S(y)\beta [S(x),T(y)]_{\alpha }\end{aligned}$$ Replacing $-x$ for $x$ in the above relation, we have $$\begin{aligned}
0&=&[S(x),T(x)]_{\alpha }\beta S(y)+S(y)\beta [S(x),T(x)]_{\alpha }\nonumber\\&&+[S(x),T(y)]_{\alpha }\beta S(x)+S(x)\beta [S(x),T(y)]_{\alpha }\nonumber\\&&+[S(y),T(x)]_{\alpha }\beta S(x)+S(x)\beta [S(y),T(x)]_{\alpha }\nonumber\\&&-[S(y),T(y)]_{\alpha }\beta S(x)-S(x)\beta [S(y),T(y)]_{\alpha }\nonumber\\&&-[S(y),T(x)]_{\alpha }\beta S(y)-S(y)\beta [S(y),T(x)]_{\alpha }\nonumber\\&&-[S(x),T(y)]_{\alpha }\beta S(y)-S(y)\beta [S(x),T(y)]_{\alpha }\end{aligned}$$ Adding (4) and (5), we have $$\begin{aligned}
0&=&2[S(x),T(x)]_{\alpha }\beta S(y)+2S(y)\beta [S(x),T(x)]_{\alpha }\nonumber\\&&+2[S(x),T(y)]_{\alpha }\beta S(x)+2S(x)\beta [S(x),T(y)]_{\alpha }\nonumber\\&&+2[S(y),T(x)]_{\alpha }\beta S(x)+2S(x)\beta [S(y),T(x)]_{\alpha }\end{aligned}$$ Hence by 2-torsion freeness of $M$, it follows that $$\begin{aligned}
0&=&[S(x),T(x)]_{\alpha }\beta S(y)+S(y)\beta [S(x),T(x)]_{\alpha }\nonumber\\&&+[S(x),T(y)]_{\alpha }\beta S(x)+S(x)\beta [S(x),T(y)]_{\alpha }\nonumber\\&&+[S(y),T(x)]_{\alpha }\beta S(x)+S(x)\beta [S(y),T(x)]_{\alpha }\end{aligned}$$ Replacing $y$ by $x\gamma y$ in the above relation , we have $$\begin{aligned}
0&=&[S(x),T(x)]_{\alpha }\beta S(x)\gamma y+S(x)\gamma y\beta [S(x),T(x)]_{\alpha }\nonumber\\&&+[S(x),T(x)\gamma y]_{\alpha }\beta S(x)+S(x)\beta [S(x),T(x)\gamma y]_{\alpha }\nonumber\\&&+[S(x)\gamma y,T(x)]_{\alpha }\beta S(x)+S(x)\beta [S(x)\gamma y,T(x)]_{\alpha }\nonumber\\&=&[S(x),T(x)]_{\alpha }\beta S(x)\gamma y+S(x)\gamma y\beta [S(x),T(x)]_{\alpha }\nonumber\\&&+[S(x),T(x)]_{\alpha }\gamma y\beta S(x)+T(x)\gamma [S(x),y]_{\alpha }\beta S(x)\\&&+S(x)\beta [S(x),T(x)]_{\alpha }\gamma y+S(x)\beta T(x)\gamma [S(x),y]_{\alpha }\nonumber\\&&+[S(x),T(x)]_{\alpha }\gamma y\beta S(x)+S(x)\gamma [y,T(x)]_{\alpha }\beta S(x)\\&&+S(x)\beta [S(x),T(x)]_{\alpha }\gamma y+S(x)\beta S(x)\gamma [y,T(x)]_{\alpha }\end{aligned}$$ According to (6), the above relation reduces to $$\begin{aligned}
0&=&S(x)\gamma y\beta [S(x),T(x)]_{\alpha }+2[S(x),T(x)]_{\alpha }\gamma y\beta S(x)\nonumber\\&&+T(x)\gamma [S(x),y]_{\alpha }\beta S(x)+S(x)\beta T(x)\gamma [S(x),y]_{\alpha }\nonumber\\&&+S(x)\gamma [y,T(x)]_{\alpha }\beta S(x)+S(x)\beta [S(x),T(x)]_{\alpha }\gamma y\nonumber\\&&+S(x)\beta S(x)\gamma [y,T(x)]_{\alpha }\end{aligned}$$ Putting $y=y\delta S(x)$ in (7), we obtain $$\begin{aligned}
0&=&S(x)\gamma y\delta S(x)\beta [S(x),T(x)]_{\alpha }+2[S(x),T(x)]_{\alpha }\gamma y\delta S(x)\beta S(x)\nonumber\\&&+T(x)\gamma [S(x),y]_{\alpha }\beta S(x)\delta S(x)+S(x)\beta T(x)\gamma [S(x),y]_{\alpha }\delta S(x)\nonumber\\&&+S(x)\gamma [y,T(x)]_{\alpha }\delta S(x)\beta S(x)+S(x)\gamma y\delta [S(x),T(x)]_{\alpha }\beta S(x)\nonumber\\&&+S(x)\beta [S(x),T(x)]_{\alpha }\gamma y\delta S(x)+S(x)\beta S(x)\gamma y\delta [S(x),T(x)]_{\alpha }\nonumber\\&&+S(x)\beta S(x)\gamma [y,T(x)]_{\alpha }\delta S(x)\end{aligned}$$ which gives according to (7) to $$\begin{aligned}
0&=&S(x)\gamma y\delta S(x)\beta [S(x),T(x)]_{\alpha }+S(x)\beta S(x)\gamma y\delta [S(x),T(x)]_{\alpha }\end{aligned}$$ Putting $y=T(x)\omega y$ in (8), we have $$\begin{aligned}
0&=&S(x)\gamma T(x)\omega y\delta S(x)\beta [S(x),T(x)]_{\alpha }\nonumber\\&&+S(x)\beta S(x)\gamma T(x)\omega y\delta [S(x),T(x)]_{\alpha }\end{aligned}$$ Also left multiplication of (8) by $T(x)\omega $ gives $$\begin{aligned}
0&=&T(x)\omega S(x)\gamma y\delta S(x)\beta [S(x),T(x)]_{\alpha }\nonumber\\&&+T(x)\omega S(x)\beta S(x)\gamma y\delta [S(x),T(x)]_{\alpha }\end{aligned}$$ From (9) and (10), we obtain, $$\begin{aligned}
0&=&[S(x),T(x)]_{\gamma }\omega y\delta S(x)\beta [S(x),T(x)]_{\alpha }+[S(x)\beta S(x),T(x)]_{\gamma }\omega y\delta [S(x),T(x)]_{\alpha }\nonumber\\&=&[S(x),T(x)]_{\gamma }\omega y\delta S(x)\beta [S(x),T(x)]_{\alpha }\nonumber\\&&+([S(x),T(x)]_{\gamma }\beta S(x)+S(x)\beta [S(x),T(x)]_{\gamma })\omega y\delta [S(x),T(x)]_{\alpha }\nonumber\\&=&[S(x),T(x)]_{\gamma }\omega y\delta S(x)\beta [S(x),T(x)]_{\alpha }\end{aligned}$$ Thus we have $$\begin{aligned}
0&=&[S(x),T(x)]_{\gamma }\omega y\delta S(x)\beta [S(x),T(x)]_{\alpha }\end{aligned}$$ Left multiplication of the above relation by $S(x)\beta $ gives $$\begin{aligned}
0&=&S(x)\beta [S(x),T(x)]_{\gamma }\omega y\delta S(x)\beta [S(x),T(x)]_{\alpha }\end{aligned}$$ for all $x,y\in M$ and $\alpha ,\beta ,\gamma ,\delta ,\omega \in\Gamma $. Hence from (11), it follows $$\begin{aligned}
S(x)\beta [S(x),T(x)]_{\alpha }&=&0\end{aligned}$$ From (3) and (12), we have also $$\begin{aligned}
_{\alpha }\beta S(x)&=&0\end{aligned}$$ From (12) one obtains the relation $$\begin{aligned}
0&=&S(y)\beta [S(x),T(x)]_{\alpha }+S(x)\beta [S(y),T(x)]_{\alpha }\nonumber\\&&+S(x)\beta [S(x),T(y)]_{\alpha }\end{aligned}$$(see the proof of (6)). Putting $y=x\gamma y$ in (14), we have $$\begin{aligned}
0&=&S(x)\gamma y\beta [S(x),T(x)]_{\alpha }+S(x)\beta [S(x)\gamma y,T(x)]_{\alpha }\nonumber\\&&+S(x)\beta [S(x),T(x)\gamma y]_{\alpha }\\&=&S(x)\gamma y\beta [S(x),T(x)]_{\alpha }+S(x)\beta [S(x),T(x)]_{\alpha }\gamma y\nonumber\\&&+S(x)\beta S(x)\gamma [y,T(x)]_{\alpha }+S(x)\beta [S(x),T(x)]_{\alpha }\gamma y+S(x)\beta T(x)\gamma [S(x),y]_{\alpha }\\&=&S(x)\gamma y\beta [S(x),T(x)]_{\alpha }+S(x)\beta S(x)\gamma [y,T(x)]_{\alpha }\\&&+S(x)\beta T(x)\gamma [S(x),y]_{\alpha }\end{aligned}$$ Thus we have the above relation $$\begin{aligned}
0&=&S(x)\gamma y\beta [S(x),T(x)]_{\alpha }+S(x)\beta S(x)\gamma [y,T(x)]_{\alpha }\\&&+S(x)\beta T(x)\gamma [S(x),y]_{\alpha }\end{aligned}$$ which can be written in the form $$\begin{aligned}
&0=&S(x)\gamma y\beta [S(x),T(x)]_{\alpha }+S(x)\beta S(x)\gamma y\alpha T(x)\\&&-S(x)\beta T(x)\gamma y\alpha S(x)+S(x)\beta [T(x),S(x)]_{\gamma }\alpha y\end{aligned}$$ whence it follows $$\begin{aligned}
&0=&S(x)\gamma y\beta [S(x),T(x)]_{\alpha }+S(x)\beta S(x)\gamma y\alpha T(x)\nonumber\\&&-S(x)\beta T(x)\gamma y\alpha S(x)\end{aligned}$$ according to (12). Taking $T(x)\delta $ of (15) on the left side, we have $$\begin{aligned}
&0=&T(x)\delta S(x)\gamma y\beta [S(x),T(x)]_{\alpha }+T(x)\delta S(x)\beta S(x)\gamma y\alpha T(x)\nonumber\\&&-T(x)\delta S(x)\beta T(x)\gamma y\alpha S(x)\end{aligned}$$ Putting $y=T(x)\delta y$ in (15) gives $$\begin{aligned}
&0=&S(x)\gamma T(x)\delta y\beta [S(x),T(x)]_{\alpha }+S(x)\beta S(x)\gamma T(x)\delta y\alpha T(x)\nonumber\\&&-S(x)\beta T(x)\gamma T(x)\delta y\alpha S(x)\end{aligned}$$ From (16) and (17), we have $$\begin{aligned}
&0=&[S(x),T(x)]_{\gamma }\delta y\beta [S(x),T(x)]_{\alpha }+[S(x)\beta S(x),T(x)]_{\gamma }\delta y\alpha T(x)\nonumber\\&&+[T(x),S(x)]_{\delta }\beta T(x)\gamma y\alpha S(x)\\&=&[S(x),T(x)]_{\gamma }\delta y\beta [S(x),T(x)]_{\alpha }+([S(x),T(x)]_{\gamma }\beta S(x)\\&&+S(x)\beta [S(x),T(x)]_{\gamma })\delta y\alpha T(x)+[T(x),S(x)]_{\delta }\beta T(x)\gamma y\alpha S(x)\end{aligned}$$ which reduces to $$\begin{aligned}
0&=&[S(x),T(x)]_{\gamma }\delta y\beta [S(x),T(x)]_{\alpha }+[T(x),S(x)]_{\delta }\beta T(x)\gamma y\alpha S(x)\end{aligned}$$ The substitution $y\omega S(x)\rho z$ for $y$ in (18) gives $$\begin{aligned}
0&=&[S(x),T(x)]_{\gamma }\delta y\omega S(x)\rho z\beta [S(x),T(x)]_{\alpha }\nonumber\\&&+[T(x),S(x)]_{\delta }\beta T(x)\gamma y\omega S(x)\rho z\alpha S(x)\end{aligned}$$ Again, right multiplication of (18) by $\omega z\rho S(x)$, we have $$\begin{aligned}
0&=&[S(x),T(x)]_{\gamma }\delta y\beta [S(x),T(x)]_{\alpha }\omega z\rho S(x)\nonumber\\&&+[T(x),S(x)]_{\delta }\beta T(x)\gamma y\alpha S(x)\omega z\rho S(x)\end{aligned}$$ From (19) and (20), we obtain $$\begin{aligned}
0&=&[S(x),T(x)]_{\gamma }\delta y\beta A(x,z)\end{aligned}$$ where $A(x,z)=[S(x),T(x)]_{\alpha }\omega z\rho S(x)-S(x)\rho z\omega [S(x).T(x)]_{\alpha }$. Replacing $y$ by $z\rho S(x)\omega y$ in (21) gives $$\begin{aligned}
0&=&[S(x),T(x)]_{\gamma }\delta z\rho S(x)\omega y\beta A(x,z)\end{aligned}$$ Left multiplication of (21) by $S(x)\rho z\omega $ gives $$\begin{aligned}
0&=&S(x)\rho z\omega [S(x),T(x)]_{\gamma }\delta y\beta A(x,z)\end{aligned}$$ Combining (22) and (23), we arrive at $$\begin{aligned}
0&=&A(x,z)\delta y\beta A(x,z)\end{aligned}$$ for all $x,y,z\in M$ and $\delta ,\beta \in \Gamma $. Hence by semiprimeness of $M$, it follows $A(x,z)=0$ and hence $$\begin{aligned}
_{\alpha }\omega z\rho S(x)&=&S(x)\rho z\omega [S(x),T(x)]_{\alpha }\end{aligned}$$ The substitution of $z$ by $T(x)\gamma y$ in (24) gives $$\begin{aligned}
_{\alpha }\omega T(x)\gamma y\rho S(x)&=&S(x)\rho T(x)\gamma y\omega [S(x),T(x)]_{\alpha }\end{aligned}$$ The relation (25) makes it possible to replace in (18), $[S(x),T(x)]_{\delta }\beta T(x)\gamma y\alpha S(x)$ by $S(x)\alpha T(x)\gamma y\beta [S(x),T(x)]_{\delta }$. Thus we have $$\begin{aligned}
0&=&[S(x),T(x)]_{\gamma }\delta y\beta [S(x),T(x)]_{\alpha }-S(x)\alpha T(x)\gamma y\beta [S(x),T(x)]_{\delta }\end{aligned}$$ which reduces to $$\begin{aligned}
0&=&T(x)\gamma S(x)\delta y\beta [S(x),T(x)]_{\alpha }\end{aligned}$$ Putting $y=T(x)\omega y$ in (26), we have $$\begin{aligned}
0&=&T(x)\gamma S(x)\delta T(x)\omega y\beta [S(x),T(x)]_{\alpha }\end{aligned}$$ Multiplying (26) from the left side by $T(x)\omega $, we have $$\begin{aligned}
0&=&T(x)\omega T(x)\gamma S(x)\delta y\beta [S(x),T(x)]_{\alpha }\end{aligned}$$ Subtrating (28) from (27), we have $$\begin{aligned}
0&=&T(x)\omega [S(x),T(x)]_{\gamma }\delta y\beta [S(x),T(x)]_{\alpha }\end{aligned}$$ which gives putting $y=y\omega T(x)$,$$\begin{aligned}
0&=&T(x)\omega [S(x),T(x)]_{\gamma }\delta y\omega T(x)\beta [S(x),T(x)]_{\alpha }\\&=&T(x)\omega [S(x),T(x)]_{\gamma }\delta y\beta T(x)\omega [S(x),T(x)]_{\alpha }\end{aligned}$$ whence it follows $$\begin{aligned}
0&=&T(x)\omega [S(x),T(x)]_{\alpha }\end{aligned}$$ The substitution $y=y\beta T(x)$ in (25) gives because of (29)$$\begin{aligned}
0&=&[S(x),T(x)]_{\alpha }\omega y\beta T(x)\rho S(x)\end{aligned}$$ From (13), we obtain the relation $$\begin{aligned}
0&=&[S(x),T(x)]_{\alpha }\beta S(y)+[S(x),T(y)]_{\alpha }\beta S(x)+[S(y),T(x)]_{\alpha }\beta S(x)\end{aligned}$$ (see the proof of (6)). Putting in the above relation $y=x\gamma y$, we have $$\begin{aligned}
0&=&[S(x),T(x)]_{\alpha }\beta S(x)\gamma y+[S(x),T(x)\gamma y]_{\alpha }\beta S(x)+[S(x)\gamma y,T(x)]_{\alpha }\beta S(x)\\&=&T(x)\gamma [S(x),y]_{\alpha }\beta S(x)+[S(x),T(x)]_{\alpha }\gamma y\beta S(x)\\&&+S(x)\gamma [y,T(x)]_{\alpha }\beta S(x)+[S(x),T(x)]_{\alpha }\gamma y\beta S(x)\end{aligned}$$ Thus we have $$\begin{aligned}
&0=&2[S(x),T(x)]_{\alpha }\gamma y\beta S(x)+T(x)\gamma [S(x),y]_{\alpha }\beta S(x)+S(x)\gamma [y,T(x)]_{\alpha }\beta S(x)\end{aligned}$$ which can be written after some calculation in the form $$\begin{aligned}
&0=&[S(x),T(x)]_{\alpha }\gamma y\beta S(x)+S(x)\gamma y\alpha T(x)\beta S(x)\nonumber\\&&-T(x)\gamma y\alpha S(x)\beta S(x)\end{aligned}$$ The relation (24) makes it possible to replace in (31), $[S(x),T(x)]_{\alpha }\gamma y\beta S(x)$ by $S(x)\beta y\gamma [S(x),T(x)]_{\alpha }$. Thus we have $$\begin{aligned}
&0=&S(x)\beta y\gamma [S(x),T(x)]_{\alpha }+S(x)\gamma y\alpha T(x)\beta S(x)\nonumber\\&&-T(x)\gamma y\alpha S(x)\beta S(x)\\&=&S(x)\beta y\gamma S(x)\alpha T(x)-T(x)\gamma y\alpha S(x)\beta S(x)\end{aligned}$$ Therefore, we have $$\begin{aligned}
S(x)\beta y\gamma S(x)\alpha T(x)&=&T(x)\gamma y\alpha S(x)\beta S(x)\end{aligned}$$ Putting in the above relation $y=T(x)\omega y$, we have $$\begin{aligned}
S(x)\beta T(x)\omega y\gamma S(x)\alpha T(x)&=&T(x)\gamma T(x)\omega y\alpha S(x)\beta S(x)\end{aligned}$$ Left multiplication of (32) by $T(x)\omega $ gives $$\begin{aligned}
T(x)\omega S(x)\beta y\gamma S(x)\alpha T(x)&=&T(x)\omega T(x)\gamma y\alpha S(x)\beta S(x)\end{aligned}$$ Combining (33) and (34), we have $$\begin{aligned}
0&=&[S(x),T(x)]_{\beta }\omega y\gamma S(x)\alpha T(x)\end{aligned}$$ which gives together with (30), $$\begin{aligned}
0&=&[S(x),T(x)]_{\alpha }\omega y\beta [S(x),T(x)]_{\alpha }\end{aligned}$$ Hence by semiprimeness of $M$, we have $$\begin{aligned}
_{\alpha }&=&0\end{aligned}$$
If $M$ is a prime $\Gamma $-ring, then the relation (35) and Lemma-2.3 complete the proof of the theorem.
.2cm
Suppose that $M$ is a 2-torsion free noncommutative semiprime $\Gamma $-ring satisfying the assumption $(A)$ and $S$, $T$ are left centralizers on $M$. If $[[S(x),T(x)]_{\alpha }, S(x)]_{\beta }=0$ holds for all $x\in M$ and $\alpha ,\beta \in\Gamma $. Then $[S(x),T(x)]_{\alpha }=0$ for all $x\in M$ and $\alpha \in\Gamma $. Moreover, if $M$ is prime $\Gamma $-ring satisfying the assumption $(A)$ and $S\neq 0(T\neq 0)$, then there esixts $\lambda \in C$,(the extended centroid of $M$) such that $T=\lambda \alpha S (S=\lambda \alpha T)$.
.2cm [**Proof.**]{} By the assumption $$\begin{aligned}
_{\beta }&=&0\end{aligned}$$ The linearization of (36) gives $$\begin{aligned}
0&=&[[S(x),T(x)]_{\alpha },S(y)]_{\beta }+[[S(x),T(y)]_{\alpha },S(x)]_{\beta }\nonumber\\&&+[[S(y),T(x)]_{\alpha },S(x)]_{\beta }\end{aligned}$$ Putting $y=x\gamma y$ in (37), we have $$\begin{aligned}
0&=&[[S(x),T(x)]_{\alpha },S(x)\gamma y]_{\beta }+[[S(x),T(x)\gamma y]_{\alpha },S(x)]_{\beta }\nonumber\\&&+[[S(x)\gamma y,T(x)]_{\alpha },S(x)]_{\beta }\\&=&[[S(x),T(x)]_{\alpha },S(x)]_{\beta }\gamma y+S(x)\gamma [[S(x),T(x)]_{\alpha },y]_{\beta }\nonumber\\&&+[[S(x),T(x)]_{\alpha }\gamma y+T(x)\gamma [S(x),y]_{\alpha },S(x)]_{\beta }\\&&+[[S(x),T(x)]_{\alpha }\gamma y+S(x)\gamma [y,T(x)]_{\alpha },S(x)]_{\beta }
\\&=&S(x)\gamma [[S(x),T(x)]_{\alpha },y]_{\beta }+[[S(x),T(x)]_{\alpha },S(x)]_{\beta }\gamma y\\&&+[S(x),T(x)]_{\alpha }\gamma [y,S(x)]_{\beta }+T(x)\gamma [[S(x),y]_{\alpha },S(x)]_{\beta }\\&&+[T(x),S(x)]_{\beta }\gamma [S(x),y]_{\alpha }+[[S(x),T(x)]_{\alpha },S(x)]_{\beta }\gamma y\\&&+[S(x),T(x)]_{\alpha }\gamma [y,S(x)]_{\beta }+S(x)\gamma [[y,T(x)]_{\alpha },S(x)]_{\beta }\end{aligned}$$ Therefore, we have $$\begin{aligned}
0&=&S(x)\gamma [[S(x),T(x)]_{\alpha },y]_{\beta }+3[S(x),T(x)]_{\alpha }\gamma [y,S(x)]_{\beta }\nonumber\\&&+T(x)\gamma [[S(x),y]_{\alpha },S(x)]_{\beta }+S(x)\gamma [[y,T(x)]_{\alpha },S(x)]_{\beta }\end{aligned}$$ Replacing $y$ by $y\delta S(x)$ in the above relation, we have $$\begin{aligned}
0&=&S(x)\gamma [[S(x),T(x)]_{\alpha },y\delta S(x)]_{\beta }+3[S(x),T(x)]_{\alpha }\gamma [y\delta S(x),S(x)]_{\beta }\nonumber\\&&+T(x)\gamma [[S(x),y\delta S(x)]_{\alpha },S(x)]_{\beta }+S(x)\gamma [[y\delta S(x),T(x)]_{\alpha },S(x)]_{\beta }\\&=&S(x)\gamma [[S(x),T(x)]_{\alpha },y]_{\beta }\delta S(x)+S(x)\gamma y\delta [[S(x),T(x)]_{\alpha },S(x)]_{\beta }\\&&+3[S(x),T(x)]_{\alpha }\gamma [y,S(x)]_{\beta }\delta S(x)+T(x)\gamma [[S(x),y]_{\alpha }\delta S(x),S(x)]_{\beta }\\&&+S(x)\gamma [[y,T(x)]_{\alpha }\delta S(x)+y\delta [S(x),T(x)]_{\alpha },S(x)]_{\beta }
\\&=&S(x)\gamma [[S(x),T(x)]_{\alpha },y]_{\beta }\delta S(x)+3[S(x),T(x)]_{\alpha }\gamma [y,S(x)]_{\beta }\delta S(x)\\&&+T(x)\gamma [[S(x),y]_{\alpha },S(x)]_{\beta }\delta S(x)+S(x)\gamma [[y,T(x)]_{\alpha },S(x)]_{\beta }\delta S(x)\\&&+S(x)\gamma [y,S(x)]_{\beta }\delta [S(x),T(x)]_{\alpha }+S(x)\gamma y\delta [[S(x),T(x)]_{\alpha },S(x)]_{\beta }\end{aligned}$$ Thus we have according to (36) and (38), $$\begin{aligned}
0&=&S(x)\gamma [y,S(x)]_{\beta }\delta [S(x),T(x)]_{\alpha }\end{aligned}$$ which can be written in the form $$\begin{aligned}
S(x)\gamma y\beta S(x)\delta [S(x),T(x)]_{\alpha }&=&S(x)\gamma S(x)\beta y\delta [S(x),T(x)]_{\alpha }\end{aligned}$$ Putting in the above relation $y=T(x)\omega y$, we have $$\begin{aligned}
S(x)\gamma T(x)\omega y\beta S(x)\delta [S(x),T(x)]_{\alpha }&=&S(x)\gamma S(x)\beta T(x)\omega y\delta [S(x),T(x)]_{\alpha }\end{aligned}$$ On the other hand, left multiplication of (39) by $T(x)\omega $, we have $$\begin{aligned}
\lefteqn{T(x)\omega S(x)\gamma y\beta S(x)\delta [S(x),T(x)]_{\alpha }=}\nonumber\\&&T(x)\omega S(x)\gamma S(x)\beta y\delta [S(x),T(x)]_{\alpha }\end{aligned}$$ Subtracting (41) from (40), we obtain $$\begin{aligned}
0&=&[S(x),T(x)]_{\gamma }\omega y\beta S(x)\delta [S(x),T(x)]_{\alpha }-[S(x)\gamma S(x),T(x)]_{\beta }\omega y\delta [S(x),T(x)]_{\alpha }\\&=&[S(x),T(x)]_{\gamma }\omega y\beta S(x)\delta [S(x),T(x)]_{\alpha }\\&&-([S(x),T(x)]_{\beta }\gamma S(x)+S(x)\gamma [S(x),T(x)]_{\beta }\omega y\delta [S(x),T(x)]_{\alpha }\end{aligned}$$ According to the requirement of the theorem one can replace in the above calculation $[S(x),T(x)]_{\beta }\gamma S(x)$ by $S(x)\gamma [S(x),T(x)]_{\beta }$ which gives $$\begin{aligned}
\lefteqn{[S(x),T(x)]_{\gamma }\omega y\beta S(x)\delta [S(x),T(x)]_{\alpha }}\\&&=2S(x)\gamma [S(x),T(x)]_{\beta }\omega y\delta [S(x),T(x)]_{\alpha }\end{aligned}$$ Left multiplication of the above relation by $S(x)\rho $ gives $$\begin{aligned}
\lefteqn{S(x)\rho [S(x),T(x)]_{\gamma }\omega y\beta S(x)\delta [S(x),T(x)]_{\alpha }}\nonumber\\&&=2S(x)\rho S(x)\gamma [S(x),T(x)]_{\beta }\omega y\delta [S(x),T(x)]_{\alpha }\end{aligned}$$ On the otherhand, putting $y=[S(x),T(x)]\rho y$ in (39), we have $$\begin{aligned}
\lefteqn{S(x)\gamma [S(x),T(x)]_{\omega }\rho y\beta S(x)\delta [S(x),T(x)]_{\alpha }}\nonumber\\&&=S(x)\gamma S(x)\beta [S(x),T(x)]_{\omega }\rho y\delta [S(x),T(x)]_{\alpha }\end{aligned}$$ Combining (42) with (43), we obtain $$\begin{aligned}
0&=&S(x)\rho [S(x),T(x)]_{\gamma }\omega y\beta S(x)\delta [S(x),T(x)]_{\alpha }\end{aligned}$$ Hence by semiprimeness of $M$, we obtain $$\begin{aligned}
S(x)\delta [S(x),T(x)]_{\alpha }&=&0\end{aligned}$$ From (44) and the assumption of the theorem, we have $$\begin{aligned}
_{\alpha }\delta S(x)&=&0\end{aligned}$$ The rest of the proof goes through in the same way as in the proof of the Theorem-2.1.
W.E.Barnes,On the $\Gamma$-rings of Nobusawa,Pacific J.Math.,18(1966),411-422. M.F.Hoque and A.C.Paul, On centralizers of semiprime gamma rings, International Mathematical Forum, 6(13)(2011), 627-638. M.F.Hoque and A.C.Paul, Centralizers of semiprime gamma rings, Italian J. Pure and Applied Mathematics, 30(2013), 289-302. M.F.Hoque and A.C.Paul, An equation related to centralizers in semiprime gamma rings, Annals of Pure and Applied Mathematics, 1(1)(2012), 84-90. S.Kyuno, On prime Gamma ring, Pacific J.Math.,75(1978),185-190. L.Luh, On the theory of simple Gamma rings, Michigan Math.J. 16(1969),65-75. W.S. Martindale, Prime rings satisfying ageneralized polynomial identity, Journal of Algebra 12(1969),576-584. N. Nobusawa, On the Generalization of the Ring Theory, Osaka J. Math.,1(1964),81-89. M.A. Ozturk and Y.B. Jun, On the centroid of the prime Gamma Rings-11, Turk, J.Math. 25(2001),367-377. Posner, Derivations in prime rings. Proc. Amer. Math. Soc. 8(1957),1093-1100. J.Vukman, Centralizers in prime and semiprime rings, Comment. Math. Univ. Carolinae 38(1997),231-240. J.Vukman, An identity related to centralizers in semiprime rings, Comment. Math. Univ. Carolinae 40,3(1999),447-456. J.Vukman, Centralizers on semiprime rings, Comment. Math. Univ. Carolinae 42,2(2001), 237-245. B.Zalar, On centralizers of semiprime rings, Comment.Math. Univ. Carolinae 32(1991),609-614.
| {
"pile_set_name": "ArXiv"
} |
---
address:
- 'LIMSI-CNRS, BP 133, 91403 Orsay, France.'
- |
Dpt. of Aeronautics and Astronautics, Massachusetts Institute of Technology,\
77 Massachusetts Av., Cambridge, MA 02139, USA.
author:
- Lionel Mathelin
bibliography:
- 'biblio.bib'
title: 'Quantification of uncertainty from high-dimensional experimental data'
---
Uncertainty Quantification ,Least Angle Regression ,High-Dimensional Model Reduction ,Total Least Squares ,Alternate Least Squares ,Polynomial Chaos.
Acknowledgement {#acknowledgement .unnumbered}
===============
The author gratefully acknowledges Tarek El Moselhy and Faidra Stavropoulou for stimulating discussions and useful comments. This work is part of the TYCHE project (ANR-2010-BLAN-0904) supported by the French Research National Agency (ANR).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We analyze the deformation and energetics of a thin elastic cylindrical tube compressed between two plates which are parallel to the tube axis. The deformation is studied theoretically using exact numerical simulations and the variational approach. These results are used to interpret the experimental data obtained by pressing plastic-foil tubes in an apparatus specially designed for this purpose.
[*A simplified variant of the physics that we present was used as a basis for one of the experimental problems posed on the 41st International Physics Olympiad held in Zagreb, Croatia in July 2010.*]{}
author:
- Antonio Šiber
- Hrvoje Buljan
title: 'Theoretical and experimental analysis of a thin elastic cylindrical tube acting as a non-Hookean spring'
---
Introduction
============
Springs are ubiquitous mechanical devices that can be used to absorb energy due to shock, vibration or exerted pressure (e.g. in vehicle suspension and clutch and brake systems, and some types of computer mouse devices and keyboards), to store and controlably release the energy (e.g. in winding clocks and children’s toys) and to measure forces (e.g. in scales, spring balances, dynamometers and similar). Springs store the energy by changing their shape, and they are useful because they can deform many times in practicaly the same way, acting always the same in response to the applied force, before the mechanical/elastical properties of the material that they are made of change and the springs become worn out. Before this point, measurements on springs in their deformed (forced) state can be used to precisely determine the applied force - this is the basic principle of a scale.
Deformation of the springs depends on the elastic properties of the material they are made of, but also on the construction i.e. geometry of the spring. There are many types of spring constructions, perhaps the best known being the helical spring. However, any piece of elastic material can be used in many different ways and settings so to deform in response to the applied force. The springs that are typically discussed in various courses of elementary physics are Hookean, i.e. their linear deformation is proportional to the force. For most students this becomes the hallmark of elasticity, but although the materials the springs are made of may be Hookean, the [*overall*]{} deformation of the spring can display all sorts of different functional dependences on the applied force, depending on the geometry of the spring. In particular, although a rod of a certain material may obey Hooke’s law, so that its extension is proportional to the force stretching it, this does not mean that the cylinder made of a thin sheet of such a material will deform in response to the force applied perpendicularly to its axis so that its displacements are proportional to the force.
In this paper, we introduce a particularly simple construction of a spring made by rolling a piece of thin sheet (used in strip and spiral book binding - binding cover) to form a cylindrical tube. We shall show how the deformation of such a spring can be predicted by applying an approximate, but completely adequate variant of the theory of elasticity to shells. Such an analysis enables one to construct a simple scale, but also to determine the elastic properties of the sheet material, as we shall demonstrate. The theoretical discussion of the deformation and elasticity of the spring is applied to the data obtained from the experimental setup that was made in order to gauge and test the spring properties.
Elasticity of bending {#sec:elements_theory_elasticity}
=====================
Two-dimensional moduli of elasticity and curvatures of deformed sheets
----------------------------------------------------------------------
The elementary theory of elasticity is usually exposed on the problem of extension of rods or the homogeneous compression of isotropic materials. In the two cases, the elasticity parameter that naturally appears is the Young’s modulus, that we shall denote by $E$ in the following.
In the case of deformation of thin sheets made of elastic material, an important simplification arises due to the fact that the thickness of the sheet is much smaller from the two other characteristic dimensions (mean length and width). In such a situation, it is of use to introduce “two-dimensional” elasticity parameters. These are known under different names, perhaps the most common ones being the bending (or flexural) rigidity ($\kappa$) and two-dimensional Young’s modulus ($Y$, sometimes also called the extensional stifness) [@Timoshenko_stability]. If the material the sheet is made of is isotropic, the two parameters are related to the bulk Young’s modulus and Poisson ratio ($\nu$) of the material as [@Timoshenko_stability] $$\begin{aligned}
Y &=& \frac{Ed}{1-\nu^2}\nonumber \\
\kappa &=& \frac{Ed^3}{12(1-\nu^2)},
\label{eq:2Dmoduli}\end{aligned}$$ where $d$ is the thickness of the sheet. The above equations are derived in the advanced textbooks on the theory of elasticity by examining the deformation of a small element of the sheet, which are either an in-plane (in-sheet) stretching and shear, or an out-of-plane bending. The two types of deformation are characterized by the two effective moduli ($Y$ for stretching and $\kappa$ for bending).
In general, the character of the deformation that will occur in a sheet depends on the magnitudes of the two elastic moduli, and the geometric constraints (which e.g. may or may not allow the bending type of deformation). It is typical for the sufficiently thin sheet to deform in a way that stores most of its energy in the bending type of deformations. In these cases, the elastic energy depends only on the magnitude of the bending rigidity ($\kappa$), and the in-plane stretching deformation that would depend on $Y$ is effectively “frozen”/forbidden (the deformation is [*inextensional*]{}). The energy of such deformations can be calculated from the curvatures of the sheet shape. As shown in the Appendix, for each point in a surface one can define [*two*]{} principal curvatures ($K_1$ and $K_2$) and from these, one can construct the mean ($K_M$) and gaussian ($K_G$) curvatures of the surface. However, in order that one surface can be transformed into some other by a pure bending (inextensional) type of deformation, it is necessary that their gaussian curvatures are the same. Thus, a planar sheet of material (with $K_G=0$) can be bent only in configurations that have a nonvanishing curvature only along [*one*]{} direction, while they need to remain flat in the perpendicular direction. Surfaces of cylinders and cones are examples of such configurations. In our case, the deformed spring will be a generalized cylinder, and the problem is thus effectively one-dimensional.
In such a case, the elastic energy of the sheet can be represented as a functional of the shape mean curvatures [@Timoshenko_stability] as $$E_{el} = \frac{\kappa}{2}\int_S K^2 dS,
\label{eq:Helfrich}$$ where $K=2 K_M$ is twice the mean curvature (see Appendix) that in general depends on the position of the point on the surface $S$. For the cylindrical surfaces, the above equation can also be written as $$E_{el} = \frac{\kappa h}{2} \int_{\cal C} K^2 dl,
\label{eq:Helf2}$$ where ${\cal C}$ is the curve outlining the shape of the cylinder base, $h$ is the cylinder height, and $dl$ is the inifinitesimal arc element of the curve ${\cal C}$ ($dS = h dl$). Our problem thus reduces to a planar problem, since the curve ${\cal C}$ with a given $h$ completely determines the shape of the deformed cylindrical spring.
It is implicit in equations (\[eq:Helfrich\]) and (\[eq:Helf2\]) that the principal radii of curvatures are much larger than the thickness of the sheet.
Experimental setup
==================
An essential part of our experiment is a thin sheet made of transparent polymer material. The sheets that we tested are typically used as transparencies in overhead projection systems or as transparent plastic covers for strip and spiral book binding. Their size is usually the same as those of standard paper sizes, A4 in our case; $W=210$ mm, $L=297$ mm. In order to make a spring, we roll the sheet in a cylinder, either along the width ($W$) or the length ($L$, $L>W$) of the sheet, and we use transparent adhesive tape to fix the cylindrical shape of the sheet. For accurate measurements, the width of the overlapping region of the sheet where the adhesive tape is applied should be as small as possible. For the sheets we tested, we have been able to reduce to width of overlapping region to $\sim 2$ mm. We have thus created the spring whose response we shall study by pressing the spring in a specially designed apparatus shown schematically in Fig. \[fig:fig\_apparatus\]. It is essentially a press driven by a wing nut with a known pitch which enables one to precisely determine the shift of the top press surface and thus the change in spring height, $2b$. As the spring is pressed, the scale measures the effective force that the spring exerts on it in terms of (effective) mass. The scale that we used was marketed as “electronic kitchen scale” (digital) with a precision of 1 g.
The process of measurement can be setup so to start from a point where the top press surface barely touches the spring - this can be checked by the mass reading on the scale which must remain zero. The height of the unladen spring ($2b_0$) should be measured at this point. As the wing nut is turned, the reading of the scale ($m_e$) should be recorded after each full turn of the nut. If needed, the readings can be recorded after quarter or a half turn or even after an arbitrary angle by mounting an angular scale above the nut which can be used to precisely determine the percentage of the turn. The smallest angular turns of the wing nut that we used were $\pi/3$. Since the pitch of the nut/thread is known, this measurement procedure (with determined $2b_0$) enables one to obtain many pairs of data, relating effective mass/force $m_{e}$ to the spring height $2b$.
Variational approach to determination of the shape of deformed cylindrical spring
=================================================================================
It is of use to think of this system by turning it upside-down in a sort of gedanken experiment. One can thus recognize that the effective mass measured by the scale can be thought of as a mass that presses the spring from “above” due to gravity (the mass of the spring being neglected here). This enables one to pose the physical problem of spring deformation as a minimization of energy. The two types of energies in the system are the gravitational energy of the effective mass and the elastic (bending) energy of the deformed spring. The shape that shall be adopted by the spring minimizes the total energy functional, $${\cal E} = E_p + E_{el} = 2 m_e g b + \frac{\kappa}{2} \int_S K^2 dS.
\label{eq:total_energy}$$
As already mentioned, the spring surface can be parametrized by the [*profile*]{} of its cross-section i.e. the closed curve ${\cal C}$ that defines it. In the language of variational calculus (see e.g. Ref. ), of all the possible curves that have the given length (so that the spring is inextensible) and height of $2b$, the curve that shall actually be adopted by the spring will be the one for which the functional in Eq. (\[eq:total\_energy\]) is minimal. This statement can also be written as $$\frac{\delta {\cal E}}{\delta {\cal C}} = 0,$$ where the combination $\delta / \delta {\cal C}$ symbolizes the variation of a functional ${\cal E}$ with respect to the curves ${\cal C}$. The problem is well posed in this way since the curve ${\cal C}$ can be written in a quite general fashion, so that the variation of the energy functional produces differential equation for the function parametrizing the curve (the Euler-Lagrange equations) [@variational_general]. However, such procedure typically results in equations that can be solved only numerically. Therefore, we resort to a simpler version of a variational approach that shall yield important analytical insights, however.
Instead of varying the functional ${\cal E}$ over [*all*]{} possible functions that represent ${\cal C}$, we can vary it over a particular [*subset*]{} of all functions. The subset that is chosen typically consists of functions that can be studied analytically and parametrized using small number of parameters whose variation spans the space of subset functions. A typical example of such procedure is a variational determination of the ground state energy of hydrogen atom that is studied in many textbooks (see e.g. Ref. ). In our case, we must choose the functions that are [*(i)*]{} closed, [*(ii)*]{} produce a given circumference of the spring profile, and [*(iii)*]{} are confined to a space between the two press surfaces, so that the profile height is $2b$. There are [*two*]{} substantially different categories of functions that may be tested as variational solutions to the cross-section of the deformed cylindrical sheet. The functions in the first category touch the upper and lower press cross-sections in single points. A typical representative of such functions is an ellipse. The functions in the second category touch the upper and lower press cross-sections along the lines of certain length. A typical representative of such functions is the function made of two semi-circles connected by lines whose length corresponds to the separation of the centers of the semi-circles. In the further, we shall term this profile as the [*stadium*]{}, since the same name is used in the literature on quantum and classical chaos for the so-called billiards of such a shape (sometimes also the Bunimovich stadium after the researcher who studied it [@chaos_billiards]).
That the two types of functions really exist as solutions to the real problem can be checked experimentally by using our setup. Profiles quite similar to the stadium are obtained in case of cylinder that is laden with sufficient mass. The profiles from the first category appear for quite small loads (effective masses), and it may be somewhat difficult to judge whether the rolled sheet touches the press surfaces only tangentially or along a region of finite area. In any case, one should remember that the profiles are variational functions and they thus only mimic the true solution. For example, the stadium profile may mimic a solution whose curvature is very low in the regions where the profile almost touches the press surfaces and it becomes large in the two regions where the profile separates from the press surface. Such a profile may in fact touch the press surfaces (that become lines in the plane of the cross-section) only in four points, yet the stadium approximation should still be an adequate variational try (or [*ansatz*]{} as it is sometimes called in the professional literature). One should keep in mind, however, that, since the subset of variational functions is a restricted one, the energy that is obtained is an upper limit for the exact energy of the system, as is always the case in a variational approach [@variational_general; @Schiff].
In the following two subsections we shall solve the variational problem for the two categories of spring cross-sections.
Stadium profile
---------------
The elastic energy of the stadium can be represented analytically by evaluating Eq. (\[eq:Helfrich\]) for the profile. Flat pieces of the profile contribute nothing to the energy and the energy of the curved parts is easily calculated since these are two halves of a cylinder of height $h$ and radius $b$ (note that the curvature of the profile shows a discontinuity along the lines where the flat pieces meet the cylindrical ones). The two principal curvatures along the cylindrical portions are constant and given as $K_1 = b^{-1}$, $K_2 = 0$ (since the radius of curvature along the cylinder height is infinite). The mean curvature is $K_M = b^{-1}/2$ ($K=b^{-1}$), so that the elastic energy is $$E_{el} = \frac{\kappa}{2} \frac{2b \pi h}{b^2} = \frac{\pi \kappa h}{b}.
\label{eq:elastic_stadium}$$ The total energy of the system for the chosen profile is now $${\cal E} = 2 m_e g b + \frac{\pi \kappa h}{b}.$$ Requiring that for given effective load $m_e$, the total energy be minimal leaves us with a simple variational condition on $b$, which is the only parameter of the profile: $$\frac{\partial {\cal E}}{\partial b} = 2 m_e g - \frac{\pi \kappa h}{b^2} \equiv 0.$$ This yields an equation for the profile characteristic radius (half of the height), $$b = \sqrt{\frac{\kappa \pi h}{2m_eg}}.
\label{eq:dep_b_m_stadium}$$ Obviously, the above solution can be expected to be correct only for sufficiently large loads $m_e$, as it diverges as $m_e^{-1/2}$ when $m_e \to 0$. A more careful analysis would require that the maximally allowed value for $b$ be bounded from above by $b_0$, i.e. the radius of the cylinder in its unladen state - this is a form of the nonextensibility requirement to the solution. This yields an inequality that defines a region of the applicability of the solution, $$\sqrt{\frac{\kappa \pi h}{2m_eg}} < b_0.
\label{eq:validity_stadium}$$ The inextensibility condition is easily satisfied whenever $b<b_0$ since the length of the flat parts of the profile can be adjusted as needed, without the change in the profile elastic energy. Note, however, that the region of validity of the solution is likely to be smaller, since the stadium shape may not be the best possible variational [*ansatz*]{} throughout the region defined by Eq. (\[eq:validity\_stadium\]).
Elliptic profile
----------------
The calculation of the elastic energy of an elliptic profile requires somewhat more mathematics than in the stadium case. We parametrize the ellipse as $x = a \cos t$, $y = b \sin t$, where $a$ and $b$ are the major and the minor axes, respectively. The curvature of such a profile can be calculated as explained in the Appendix, which yields $K_G=0$, as before, and $$K = \frac{ab}{(a^2 \cos^2 t + b^2 \sin^2 t)^{3/2}}.$$ Evaluation of the integral in Eq. (\[eq:Helfrich\]) with such a curvature yields $$E_{el} = \frac{2 \kappa h}{3 b} \left[ 2 \left( 1 + \frac{b^2}{a^2} \right) \mathpzc{E} \left( \sqrt{1 - \frac{a^2}{b^2}} \right)
- \mathpzc{K} \left( \sqrt{ 1 - \frac{a^2}{b^2} } \right) \right],
\label{eq:elipsa_energija}$$ where $\mathpzc{K}$ and $\mathpzc{E}$ are complete elliptic integrals of the first and second kinds, respectively [@notation_el_int]. The major and minor axes of the ellipse are connected via the condition of inextensibility of the sheet, so that the profile circumference always equals the circumference in the unladen state, when the radius of the cylinder is $b_0$. Equating the circumference of the ellipse with that of a circle (in the unladen state) yields $$\mathpzc{E} \left( \sqrt{ 1 - \frac{a^2}{b^2} } \right) = \frac{b_0 \pi}{2 b}.
\label{eq:elipsa_opseg}$$ The problem now reduces to varying Eq. (\[eq:elipsa\_energija\]), and at the same time requiring Eq. (\[eq:elipsa\_opseg\]) to hold. Instead of dealing with such a problem and all the inconveniences stemming from the appearance of special functions, we shall use the physical insight which tells us that elliptic profiles are likely to be observed for small deformations, i.e. when $a \approx b$. In that case, we can use the Taylor expansions of the elliptic integrals[@Wolfram] to relate the major and minor axes as $a^2 = 4b b_0 - 3b^2$, and obtain the elastic energy as $$\lim _{a \to b} E_{el} = \frac{\pi h \kappa}{b} \left[ \frac{5 (b_0/b) - 4 \left( b_0 / b \right)^2 - 2}{3 - 4 ( b_0 / b ) } \right].
\label{eq:atob}$$ The above expression differs from the elastic energy of the stadium profile \[Eq. (\[eq:elastic\_stadium\])\] by the multiplicative factor in square parentheses. Examination of this factor reveals that it is smaller than one in the interval $b \in [0.80 b_0, b_0]$, and it becomes larger than one when $b<0.80 b_0$. Thus, the elastic energy of the elliptic profile is smaller from the one corresponding to the stadium profile for initial deformation of the cylindrical spring, but for sufficiently large deformations ($b<0.80 b_0$), the stadium profile becomes more favorable energywise. Already at this point, we may suspect that the response of the spring will show two characteristic regimes separated at point where $b \approx 0.8 b_0$.
The characteristic profile parameter $b$ can be again obtained by varying the total energy which, with Eq. (\[eq:atob\]), yields a polynomial equation of fitfh degree for $b$. However, since we are interested in the solutions of this equation only for small deformations, we may again use Taylor expansion, but this time around the point where $b = b_0$, which then yields $$b = b_0 - \frac{m_e g b_0^3}{7 \pi h \kappa}.$$
Numerical solution to the problem: Universal energetics of the spring deformation {#sec:numerical}
=================================================================================
The problem of interest to us can be solved numerically. This can be done in many different ways. Perhaps the simplest one, and the one that we applied, is to discretize the profile of the deformed spring in $N$ points and to reformulate the energy functional so that it becomes a function of the coordinates of these points. Such a function can then be minimized using various numerical algorithms intended for such purpose. We shall use a particular variant of the conjugate gradient minimization that was successfully applied previously in Refs. . The constraints of the inextensibility of the sheet and the impenetrability of the top and bottom press surfaces can be implemented by suplementing the elastic energy with an energy penalty for all configurations that violate the constraints. Such a method is common in numerical optimization with constraints (see e.g. Ref. ). All these details are not really essential for our purposes, and we only need to know that the highly reliable numerical solution of our problem can be obtained.
The solution can be scaled so that it becomes universal i.e. applicable to appropriately scaled measurements of springs, irrespective of their equilibrium radii, $b_0$, heights $h$, and bending rigidities, $\kappa$. Such a scaled solution depends only on an adimensional parametrization of the shape, and the energy scale appears only as a multiplicative factor, scaling the universal solution to the concrete case with given spring dimensions and bending rigidity. The adimensional parameter that uniquely determines the spring shape is $b/b_0$, and an appropriate scale of elastic energy is $\pi \kappa h / b_0$ (one could also use $\kappa h / b$, but it is more convenient to have a fixed scale of energy that does not change during the spring deformation). The energy-shape dependence can thus be written as $$\frac{b_0}{\pi \kappa h} E_{el} = {\cal U} \left( \frac{b}{b_0} \right) \equiv \overline{E}_{el},$$ where ${\cal U} \left( b / b_0 \right)$ is the universal function characteristic for our problem \[note in particular that Eqs. (\[eq:elastic\_stadium\]) and (\[eq:elipsa\_energija\]) are of this form\]. The appropriately scaled energy (adimensional) is denoted by an overline ($\overline{E}_{el}$), as will be all the adimensional quantitites in the following.
In Fig. \[fig:fig\_energy\] we show the theoretical predictions for the spring energy. The circles show the numerical data. The dotted line is the prediction of the variational method based on the stadium profile, Eq. (\[eq:elastic\_stadium\]). The dashed line is the variational prediction for the elliptical profile in the limit of small eccentricities, Eq. (\[eq:atob\]). We see that the variational prediction based on the stadium profile quite nicely follows the trend of the numerical (exact) results in the range when $b/b_0 > 0.7$, as expected from the discussion in the previous section. The variational energy is, however, always [*above*]{} the exact results, as is always the case in variational approach. However, a simple scaling of the energies obtained from stadium variational results by a factor of 0.912 gives a full line that fits the numerical data to a precision better than 0.8 % in the range $0.15< b/b_0 < 0.7$. One can immediately note the power of variational approach - although it may overestimate energies (by about 9 % in our case), it gives strong clues regarding functional behavior of energies that are not always easy to interpret solely from the numerical results. The stadium variational prediction becomes worse as $b>0.7 b_0$, but the energies based on the elliptical profile are also quite unreliable in this interval, except quite close to the point where $b/b_0 = 1$. This was to be expected, having in mind the approximations that were used in deriving the variational prediction in Eq. (\[eq:atob\]), the assumption of small eccentricities in particular.
In Fig. \[fig:fig\_profiles\], we show the profiles obtained by the numerical method. Only quarters of the profiles are shown as the remaining parts can be obtained by appropriate reflections about $x$ and $y$ axes. An astute reader may note a slight depression, concavity of the profile curve centered around $(x,y)=(0,b)$ point for sufficiently compressed springs ($b/b_0 \lesssim 0.3$). This feature is also easily observed in experiments that we describe in the next section.
Experiment and theory: determining elastic properties of the sheet
==================================================================
In Fig. \[fig:figexp1\] we show four representative ”raw” experimental data, i.e. the half of the separation between the two press surfaces ($b$) as a function of the mass read on the scale ($m_e$). The measurements were performed on four different foils, all of them belonging to the same package. The thickness of the foils is nominally 200 $\mu$m, but our measurements of the average thickness of the foils in the package yielded 190 $\pm$ 7 $\mu$m. We shall denote these foils as belonging to the Set 1 in the following. We have rolled the foils along their longer side, so that $H=210$ mm. The separation between the two press surfaces just at the point where the foil barely touches the upper surface is $2 b_0 = 92$ $\pm$ 1 mm. Note that $2 b_0 \pi = 289$ mm which is 8 mm smaller from 297 mm (longer side of the A4 paper), and about half of this difference is due to small overlap of the foils that is necessary in order to apply the adhesive tape. The other half is due mostly to quite slight distortion of the foil under its own weight. We shall neglect this effect in the following, as we shall find no need for its inclusion in the data analysis.
The dashed line in the figure shows the slope expected for $b \propto m_e^{-1/2}$ dependence in the stadium regime, as predicted by Eq. (\[eq:dep\_b\_m\_stadium\]), but also by exact numerical solution shown in Fig. \[fig:fig\_energy\]. One can see that the predicted dependence is nicely obeyed by the data below the dotted line that shows $b=0.7 \langle b_0 \rangle$, where $\langle b_0 \rangle$ is the average half-height of the spring in the unladen state for the four sets of data. From the numerical analysis in the previous section, one finds that the easiest way to obtain the bending rigidity of the foils is to fit the experimental data to the $$b = \sqrt{\frac{0.912 \kappa \pi h}{2m_eg}}$$ dependence in the region $b < 0.7 b_0$. In addition to this, we shall perform a scaling analysis of the data, in accordance with the numerical results presented in Sec. \[sec:numerical\]. The scaling analysis provides a universal description of the spring response, for all magnitudes of deformation and regardless of the spring bending rigidity, its height, and its unladen radius. It is thus of interest to experimentally investigate the predicted universality by studying differently shaped springs, and springs of different bending rigidities. It is easiest to analyze the scaling with respect to $h$, as the same sheet can be rolled either along its length/longer side (so that $h=W$), or along its width (so that $h=L$). Concerning the scaling with $\kappa$, it is not necessary that the springs be of different materials, as $\kappa$ scales with the sheet thickness as in Eq. (\[eq:2Dmoduli\]), so that the sheets of the same material, but with different thicknesses are adequate in that respect. We have tested two additional sets of sheets, one of them sold again as binding covers (A4 format), but of smaller thickness (nominally 150 $\mu$m, but we measured 146 $\pm$ 8 $\mu$m). We denote the set of these foils by Set 2. Finally, the thickest set of sheets that we tested (A4 format, nominal thickness of 400 $\mu$m, but we measured 412 $\pm$ 4 $\mu$m) is marketed as ”flexible plastic film” and its intended purpose is to be used for cutting shapes out of it (we bought it in a hobby store). The set of these foils is denoted by Set 3.
For all the measurements we performed, we scaled the mass readings, so to produce the adimensional experimental force $\overline{F}_{exp}$ as $$\overline{F}_{exp} = \frac{2 m_e g b_0^2}{\pi \kappa h}.
\label{eq:scaled_exp_f}$$ The scale of force can be derived from the scale of energy, $\pi \kappa h / b_0$, simply by dividing it by the scale of length, $b_0$ in our case. The quantity in equation (\[eq:scaled\_exp\_f\]) can be directly compared to its counterpart obtained from the numerical analysis, $$\overline{F}_{num} = - \frac{d \overline{E}_{el}}{d ( b/b_0 )}.
\label{eq:scaled_theo_f}$$ Note that the factor of $2$ in Eq. (\[eq:scaled\_exp\_f\]) arises from the fact that compression of the spring where $b$ changes by $\Delta b$ requires applying the force of $m_e g$ on a distance of $2 \Delta b$ - the same factor of $2$ is present in Eq. (\[eq:total\_energy\]). The comparison of the scaled experimental readings with the numerical results is shown in Fig. \[fig:fig\_scaling\]. One can see that the scaling predicted by the numerical results is evident in the experimental data through an interval of almost four order of magnitude of the force ($y$-axis; in our experimental setup, this corresponds to effective masses from about a gram to several kilograms). Note also that experimental data confirm the numerical predictions through the whole interval of deformation, in the regime where the profile can be described as a stadium, but also in the regime of small deformations, where the stadium ansatz fails.
The summary of the bending rigidities obtained for the sheets shown in Fig. \[fig:fig\_scaling\] is shown in Table \[tab:summary\]. The fifth column of data contains the bulk Young modulus ($E$) of the sheets obtained from Eq. (\[eq:2Dmoduli\]), using a value of $\kappa$ determined in the experiments and Poisson ratio of $\nu = 0.3$ that is typical for most materials. The bulk Young moduli are indeed in the range expected for polymeric materials such as nylon, for example ($E \sim$ 2-4 GPa). One should note, though, that this analysis is approximate as we did not measure the Poisson ratio of the sheets.
Sheet $h$ \[cm\] $d$ \[$\mu$m\] $\kappa$ \[mJ\] $E$ \[GPa\]
------- ------------ ---------------- ----------------- -------------
Set 1 29.7 190 1.58 2.52
Set 1 21.0 190 1.59 2.53
Set 2 29.7 146 0.87 3.05
Set 2 21.0 146 0.71 2.48
Set 3 21.0 412 13.2 2.06
: \[tab:summary\]Summary of the properties of the sheets we used in experiments shown in Fig. \[fig:fig\_scaling\]. The fourth and fifth columns of data were calculated as explained in the text.
This completes our theoretical and experimental analysis of the response of a thin cylindrical tube used as a spring.
Energy, elasticity, and deformation of thin shells in nano- and bio-systems {#sec:applications}
===========================================================================
Sheet-like materials and shells made of them are not uncommon at the micro- and nano-scale. These structures are often theoretically studied using simplified variants of theory of elasticity [@elasticity_bio_nano], some of which are close to the one we presented. Here we mention several examples of recent research that can be understood in the context of physics that we presented in this work.
Graphene, fullerenes, carbon nanotubes, graphene cones, ...
-----------------------------------------------------------
An example of considerable recent interest is graphene - a single layer of carbon atoms in honeycomb arrangement [@graphene_rev]. It is interesting that other single-shell carbon structures, such as fullerenes and carbon nanotubes [@carbon_nanotube_rev], can also be thought of as nanoscopic pieces of graphene material, cut-out from an infinite graphene plane in certain way and rolled and “glued together” [@tersoff_1; @carbon_nanotube_rev]. With more elaborate cutting patterns, one can construct more complicated structures made of graphene, such as closed carbon cages, including fullerenes [@siber_nano_el] and carbon/graphene cones [@siber_cones]. Interestingly, it seems that one can calculate the energy of the structure, with respect to the energy of a planar piece of graphene, by accounting only for bending elastic energy of the shape in combination with the energy required to form [*pentagons*]{} in fullerenes, closed carbon nanotubes or carbon cones [@tersoff_1; @siber_nano_el; @siber_cones]. This is due to the fact that the shapes of these structures are locally pieces of cones and cylinders, so that the assumption of inextensibility holds ($K_G=0$). It is interesting that completely classical theory of elasticity can be succesfully applied to shapes in nano-domain (diameters of cylinders and shells $\sim$ 2 nm), but one should keep in mind that the bending rigidity parameter of graphene is determined by quantum physics - to calculate $\kappa$ for graphene, one should in principle account for change in energies of electrons in graphene that occurs when the planar piece of graphene is bent.
Not only the equilibrium shape and energy, but also deformation of nanotubes and fullerenes, can be studied using the theory of elasticity. For example in Ref. , the deformation of carbon nanotubes under hydrostatic pressure is studied using a variational approach similar to ours, but with more complicated and versatile profiles. This is a problem somewhat different from the one we presented, nevertheless, shapes of deformed cylinders/nanotubes quite similar to ours (Fig. \[fig:fig\_profiles\]) were obtained.
Viruses and microtubules
------------------------
Biological systems abound with different structures whose characteristic dimensions are $\sim$ 10 nm. These are typically the so-called protein quaternary structures in which many proteins arrange in precise ways to form a larger shape. The quaternary structures often look like cylinders and shells of other geometries. Some typical examples are microtubules and protein coatings of certain viruses, such as tobacco mosaic virus. These structures are hollow cylinders made of many identical proteins. A different type of symmetry is more common in case of viruses - protein coatings of large number of viruses look like more or less spherical shells. The symmetry of such structures is quite similar to those of the icosahedral fullerenes, and analogous to pentagons and hexagons in fullerenes, the protein coatings of viruses are made of clusters of five and six proteins.
Energetics and elastic response of viruses and microtubules has been studied in the literature using various simplifications of the theory of elasticity [@LMN; @Siber_vir1; @Siber_vir2; @microtubule1], some of which are similar to the one we presented. The experimental studies of such structures are typically performed using atomic force microscope (AFM) to press these structures against the substrate, which is an approach quite similar in spirit to the one we described. Such measurements yield information on the elastic response that can sometimes be difficult to interpret. In particular, when such experiments are used to measure the response of protein cylinders (tobacco mosaic virus or microtubule), the deformation depends also on the size of the AFM tip [@microtubule1; @virus_AFM]. In such case, the energy of stretching may become of importance in determining the overall deformation. In some cases, the shells of interest cannot be treated as thin.
Vesicles
--------
Vesicles are closed structures of different geometries formed by a sheet-like material that is usually a molecular bilayer. The typical diameters of vesicles are on the scale of micrometers [@Seifert97; @Helfrich_original], and due to this fact, some consider them to represent a model system for cells membranes. In some cases, these structures can also be investigated by the AFM technique, especially when they are coated with proteins. Such problems are often solved variationally, using methods similar to those we used in our problem [@clathrin_vesicles].
We thank Hrvoje Mesić for the ideas regarding the design of the experimental setup, that is, for suggesting to use a simple kitchen scale below the press for measuring the force. We thank Tomislav Vuletić for designing the setup that we actually used.
This work was supported in part by Ministry of Science, Education and Sports of Republic of Croatia (projects 035-0352828-2837 and 119-0000000-1015).
Curvatures of plane curves and surfaces
=======================================
Differential geometry is a part of mathematics that deals with curvatures of plane and space curves and surfaces. For our purposes, we shall introduce only the most elementary notions, that are sufficient to mathematically support the physics of our problem. Somewhat more general views on surface curvatures are also summarized in Ref. .
Curvature of surfaces
---------------------
Here we briefly introduce the basic concepts needed to define the curvature of surfaces. These are illustrated in Fig. \[fig:fig\_curvature\]. The normal vector at point $P$ of the surface $S$ is ${\bf n}$. The tangent plane $\tau$ at $P$ contains all the tangent vectors ${\bf t}$ perpendicular to ${\bf n}$. The intersection of a plane $\pi$ that contains ${\bf n}$ and a particular tangent vector ${\bf t}$ with a surface $S$ is a certain curve, $\Gamma$. This curve can be approximated by a circle around point $P$, so that [*(i)*]{} the circle passes through point $P$, [*(ii)*]{} the circle and the curve $\Gamma$ have a common tangent line at $P$, and [*(iii)*]{} the distance between the points the circle and on curve $\Gamma$ in the normal direction ${\bf n}$ decays as the cube of a higher power of the distance of these points to $P$ in the tangential direction ${\bf t}$. Such a circle is called [*the osculating circle*]{} of a curve $\Gamma$ (in direction ${\bf t}$). The minimum ($R_1$) and maximum ($R_2$) radii of all possible osculating circles at point $P$ (in all possible tangential directions) are called [*the principal radii of curvature*]{}. The two (principal) osculating circles belong to mutually perpendicular planes. The principal curvatures at point $P$ are given as $K_1 \equiv R_1^{-1}$ and $K_2 \equiv R_2^{-1}$.
The mean ($K_M$) and gaussian ($K_G$) curvatures at point $P$ are defined as $$\begin{aligned}
K_M &=& \frac{K_1 + K_2}{2} \nonumber \\
K_G &=& K_1 K_2\end{aligned}$$ We define $$K \equiv 2 K_M = K_1 + K_2,$$ i.e. twice the mean curvature.
Curvature of plane curves
-------------------------
The curvature of a plane curve given in a parametric form, $x=x(t)$, $y=y(t)$ can be obtained as $$K_1 = \frac{x'y'' - y'x''}{\left( x'^2 + y'^2 \right)^{3/2}},$$ where $x' \equiv dx / dt$, $y' \equiv dy / dt$.
[99]{}
Stephen P. Timoshenko and James P. Gere, *Theory of elastic stability* (McGraw-Hill International, 1963), 2nd. ed. - international student edition. Lee A. Segel, *Mathematics applied to continuum mechanics*, (Dover Publications, 1987). Leonard I. Schiff, *Quantum mechanics* (McGraw-Hill, Singapore, 1968), 3rd. ed. L.A. Bunimovich, “On the ergodic properties of nowhere dispersing billiards”, Comm. Math. Phys. [**65**]{}, 295–312 (1979). Wolfram Research, Inc., *Mathematica Edition: Version 7.0* (Wolfram Research, Inc., Champaign, Illinois, 2008). There are different notational conventions regarding the elliptic integrals. In our case, $\mathpzc{E}(k) \equiv \int_0^{1} (1-t^2)^{-1/2} (1 - k^2 t^2)^{1/2} dt$ and $\mathpzc{K}(k) \equiv \int_0^{1} [(1-t^2)(1 - k^2 t^2)]^{-1/2} dt$. A. Šiber, “Energies of sp$^2$ carbon shapes with pentagonal disclinations and elasticity theory”, Nanotechnology [**17**]{}, 3598–-3606 (2006). A. Šiber, “Continuum and all-atom description of the energetics of graphene nanocones”, Nanotechnology [**18**]{}, 375705-1–375705-6 (2007). A. Šiber, “Buckling transition in icosahedral shells subjected to volume conservation constraint and pressure: Relations to virus maturation”, Phys. Rev. E [**73**]{}, 061915-1–061915-10 (2006). A. Šiber and R. Podgornik, “Stability of elastic icosadeltahedral shells under uniform external pressure: Application to viruses under osmotic pressure”, Phys. Rev. E [**79**]{}, 011919-1-011919-5 (2009). Z. C. Tu and Z. C. Ou-Yang, “Elastic theory of low-dimensional continua and its applications in bio- and nano-structures”, J. Comput. Theor. Nanosci. [**5**]{}, 422–448 (2008). C. Lee, X. Wei, J. W. Kysar, and J. Hone, “Measurement of the elastic properties and intrinsic strength of monolayer graphene”, Science [**321**]{}, 385-388 (2008). M.S. Dresselhaus, G. Dresselhaus, and Ph. Avouris (eds.), *Carbon nanotubes: Synthesis, structure, properties, and applications* (Springer-Verlag, Berlin, Heidelberg, 2001). J. Tersoff, “Energies of fullerenes”, Phys. Rev. B [**46**]{}, 15546–15549 (1992). M. Hasegawa and K. Nishidate, “Radial deformation and stability of single-wall carbon nanotubes under hydrostatic pressure”, Phys. Rev. B [**74**]{}, 115401-1–115401-10 (2006). J. Lidmar, L. Mirny and D.R. Nelson, “Virus shapes and buckling transitions in spherical shells”, Phys. Rev. E [**68**]{}, 051910-1–051910-10 (2003). P. J. de Pablo, I. A.T. Schaap, F. C. MacKintosh, and C. F. Schmidt, “Deformation and collapse of microtubules on the nanometer scale”, Phys. Rev. Lett. [**91**]{}, 098101-1–098101-4 (2003). Y. Zhao, Z. Ge and J. Fang, “Elastic modulus of viral nanotubes”, Phys. Rev. E [**78**]{}, 031914-1–031914-5 (2008). U. Seifert, “Configurations of fluid membranes and vesicles”, Adv. Phys. [**46**]{}, 13–137 (1997). W. Helfrich, “Elastic properties of lipid bilayers: theory and possible experiments”, Z. Naturforsch. [**28**]{} c, 693–703 (1973). A. J. Jin,K. Prasad, P. D. Smith, E. M. Lafer, and R. Nossal, “Measuring the elasticity of clathrin-coated vesicles via atomic force microscopy”, Biophys. J. [**90**]{}, 3333–3344 (2006).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'My talk will consist of three parts: (a) what every atomic physicist needs to know about the physics of light nuclei \[and no more\]; (b) what nuclear physicists can do for atomic physics; (c) what atomic physicists can do for nuclear physics. A brief qualitative overview of the nuclear force and calculational techniques for light nuclei will be presented, with an emphasis on debunking myths and on recent progress in the field. Nuclear quantities that affect precise atomic measurements will be discussed, together with their current theoretical and experimental status. The final topic will be a discussion of those atomic measurements that would be useful to nuclear physics.'
address: 'Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545 USA. '
author:
- 'J. L. Friar'
title: The Structure of Light Nuclei and Its Effect on Precise Atomic Measurements
---
Introduction {#introduction .unnumbered}
============
> *....numerical precision is the very soul of science.....*
This quote[@darcy] from Sir D’Arcy Wentworth Thompson, considered by many to be the first biomathematician, could well serve as the motto of this conference, since precision is the [*raison d’être*]{} of this meeting and this field. I have always been in awe of the number of digits of accuracy achievable by atomic physics in the analysis of simple atomic systems[@2S1S]. Nuclear physics, which is my primary field and interest, must usually struggle to achieve three digits of numerical significance, a level that atomic physics would consider a poor initial effort, much less a decent final result.
The reason for the differing levels of accuracy is well known: the theory of atoms is QED, which allows one to calculate properties of few-electron systems to many significant figures[@review]. On the other hand, no aspect of nuclear physics is known to that precision. For example, a significant part of the “fundamental” nuclear force between two nucleons must be determined phenomenologically by utilizing experimental information from nucleon-nucleon scattering[@2gen], very little of which is known to better than 1%. In contrast to that level of precision, energy-level spacings in few-electron atoms can be measured so precisely that nuclear properties influence significant digits in those energies[@d-p]. Thus these experiments can be interpreted as either a measurement of those nuclear properties, or corrections must be applied to eliminate the nuclear effects so that the resulting measurement tests or measures non-nuclear properties. That is the purview of my talk.
The single most difficult aspect of a calculation for any theorist is assigning uncertainties to the results. This is not always necessary, but in calculating nuclear corrections to atomic properties it is essential to make an effort. That is just another way to answer the question,“What confidence do we have in our results?” Because it is important for you to be able to judge nuclear results to some degree, I have slanted my talk towards answers to two questions that should be asked by every atomic physicist. The first is: “What confidence should I have in the values of nuclear quantities that are required to analyze precise atomic experiments?” The second question is: “What confidence should I have that the nuclear output of my experiment will be put to good use by nuclear physicists?”
Myths of Nuclear Physics {#myths-of-nuclear-physics .unnumbered}
========================
Every field has a collection of myths, most of them being at least partially true at one time. Myths propagate in time and distort the reality of the present. I have collected a number of these, some of which I believed in the past. The resolution of these “beliefs” also serves as a counterpoint to the very substantial progress made in light-nuclear physics in the past 15 years, which continues unabated.
My myth collection includes:
$\bullet$ The strong interactions (and consequently the nuclear force) aren’t well understood, and nuclear calculations are therefore unreliable.
$\bullet$ Large strong-interaction coupling constants mean that perturbation theory doesn’t converge, implying that there are no controlled expansions in nuclear physics.
$\bullet$ The nuclear force has no fundamental basis, implying that calculations are not trustworthy.
$\bullet$ You cannot solve the Schrödinger equation accurately because of the complexity of the nuclear force.
$\bullet$ Nuclear physics requires a relativistic treatment, rendering a difficult problem nearly intractable.
All of these myths had some (even considerable) truth in the past, but today they are significant distortions of our current level of knowledge.
The Nuclear Force {#the-nuclear-force .unnumbered}
=================
Most of the recent progress in understanding the nuclear force is based on a symmetry of QCD, which is believed to be the underlying theory of the strong interactions (or an excellent approximation to it). It is generally the case that our understanding of any branch of physics is based on a framework of symmetry principles. QCD has “natural” degrees of freedom (quarks and gluons) in terms of which the theory has a simple representation. The (strong) chiral symmetry of QCD results when the quark masses vanish, and is a more complicated analogue of the chiral symmetry that results in QED when the electron mass vanishes. The latter symmetry explains, for example, why (massless or high-energy) electron scattering from a spherical (i.e., spinless) nucleus vanishes in the backward direction.
The problem with this attractive picture is that it does not involve the degrees of freedom most relevant to experiments in nuclear physics: nucleons and pions. Nevertheless, it is possible to “map” QCD (expressed in terms of quarks and gluons) into an “equivalent” or surrogate theory expressed in terms of nucleon and pion degrees of freedom. This surrogate works effectively only at low energy. The small-quark-mass symmetry limit becomes a small-pion-mass symmetry limit. In general this (slightly) broken-symmetry theory has $m_{\pi} c^2 <<
\Lambda$, where the pion mass is $m_{\pi} c^2 \cong$ 140 MeV and $\Lambda \sim$ 1 GeV is the mass scale of QCD bound states (heavy mesons, nucleon resonances, etc.). The seminal work on this surrogate theory, now called chiral perturbation theory (or $\chi$PT), was performed by Steve Weinberg[@QCD], and many applications to nuclear physics were pioneered by his student, Bira van Kolck[@nQCD]. From my perspective they demonstrated two things that made an immediate impact on my understanding of nuclear physics[@pc]: (1) There is an alternative to perturbation theory in coupling constants, called “power counting,” that converges geometrically like $(Q/\Lambda)^N$, where $Q \sim
m_{\pi}c^2$ is a relevant nuclear energy scale, and the exponent $N$ is constrained to have $N \geq 0$; (2) nuclear physics mechanisms are severely constrained by the chiral symmetry. These results provide nuclear physics with a well-founded rationale for calculation.
This scheme divides the nuclear-force regime in a natural way into a long-range part (which implies a low energy, $Q$, for the nucleons) and a short-range part (corresponding to high energy, $Q$, between nucleons). Since $\chi$PT only works at low energies, we expect that only the long-range part of the nuclear force can be treated by utilizing the pion degrees of freedom. We need to resort to phenomenology (i.e., nucleon-nucleon scattering data) to treat systematically the short-range part of the interaction.
The long-range nuclear force is calculated in much the same way that atomic physics calculates the interactions in an atom using QED. The dominant interaction is the exchange of a single pion, and is denoted $V_{\pi}$. Its atomic analogue is one-photon exchange (containing the dominant Coulomb force). Because it is such an important part of the nuclear potential, it is fair to call $V_{\pi}$ the “Coulomb force” of nuclear physics. Smaller contributions arise from two-pion exchange (the analogue of two-photon exchange). There is even an analogue of the atomic polarization force, where two electrons simultaneously polarize their nucleus using their electric fields. The nuclear analogue involves three nucleons simultaneously, and is called a three-nucleon force[@3NF]. Although relatively weak compared to $V_{\pi}$ (a few percent), it plays an important role in fine-tuning nuclear energy levels. The final ingredient is an important short-range interaction (which must be determined by phenomenology) that has no direct analogue in the physics of light atoms.
What are the consequences of exchanging a pion rather than a photon? The pseudoscalar nature of the pion mandates a spin-dependent coupling to a nucleon, and this leads to a dominant tensor force between two nucleons. Except for its radial dependence, the form of $V_{\pi}$ mimics the interaction between two magnetic dipoles, as seen in the Breit interaction, for example. Thus we have in nuclear physics a situation that is the converse of the atomic case: a dominant tensor force and a smaller central force. In order to grasp the difficulties that nuclear physicists face, imagine that you are an atomic physicist in a universe where magnetic (not electric) forces are dominant, and where QED can be solved only for long-range forces and you must resort to phenomenology to generate the short-range part of the force between electrons and nuclei.
Although this may sound hopeless, it is merely difficult. The key to handling complexities is adequate computing power, and that became routinely available only in the late 1980s or early 1990s. Since then there has been explosive development in our understanding of light nuclei. Underlying all of these developments is an improved understanding of the nuclear force. I like to divide the history of the nuclear force into three distinct time periods.
[**First-generation**]{} nuclear forces were developed prior to 1993. They all contained the one-pion exchange force, but everything else was relatively crude. The fits to the nucleon-nucleon scattering data (needed to parameterize the short-range part of the nuclear force) were indifferent.
[**Second-generation**]{} forces were developed in 1993 and in the years following[@2gen]. They were more sophisticated and generally very well fit to the scattering data. As an example of how well the fitting worked, the Nijmegen group (which pioneered this sophisticated procedure) allowed the pion mass to vary in the Yukawa function defining $V_{\pi}$, and then fit that mass. They also allowed different masses for the neutral and charged pions that were being exchanged and found[@pi-mass]
$$m_{\pi^\pm} = 139.4(10)\, {\rm MeV} \, ,$$
$$m_{\pi^{0}}\, = 135.6(13)\, {\rm MeV} \, ,$$
both results agreeing with free pion masses ($m_{\pi^{\pm}} = 139.57018(35)$ MeV and $m_{\pi^0} = 134.9766(6)$ MeV [@PDG]). It is both heartening and a bit amazing that the masses of the pions can be determined to better than 1% using data taken in reactions that have no free pions! This result is the best quantitative proof of the importance of pion degrees of freedom in nuclear physics.
[**Third-generation**]{} nuclear forces are currently under development. These forces are quite sophisticated and incorporate two-pion exchange, as well as $V_{\pi}$. All of the pion-exchange forces (including three-nucleon forces) are being generated in accordance with the rules of chiral perturbation theory. One also expects excellent fits to the scattering data. This is very much work in progress, but preliminary calculations and versions have already appeared[@3gen].
Calculations of Light Nuclei {#calculations-of-light-nuclei .unnumbered}
============================
Having a nuclear force is not very useful unless one can calculate nuclear properties with it. Such calculations are quite difficult. Until the middle 1980s only the two-nucleon problem had been solved with numerical errors smaller than 1%. At that time the three-nucleon systems $^3$H and $^3$He were accurately calculated using a variety of first-generation nuclear-force models[@3N]. Soon thereafter the $\alpha$-particle ($^4$He) was calculated by my colleague, Joe Carlson, who pioneered a technique that has revolutionized our understanding of light nuclei: Green’s Function Monte Carlo (GFMC)[@GFMC].
The difficulty in solving the Schrödinger equation for nuclei is easily understood, although it was not initially obvious to me. Nuclei are best described in terms of nucleon degrees of freedom. Nucleons come in two types, protons and neutrons, which can be considered as the up and down components of an “isospin” degree of freedom. If one also includes its spin, a single nucleon has four internal degrees of freedom. Two nucleons consequently have 16 internal degrees of freedom, which is roughly the number of components in the nucleon-nucleon force (coupling spin, isospin and orbital motion in a very complicated way). To handle this complexity one again requires fast computers, and that is a fairly recent development.
The GFMC technique has been used by Joe Carlson and his collaborators to solve for all of the bound (and some unbound) states of nuclei with up to 10 nucleons. One of Joe’s collaborators (Steve Pieper[@steve]) calculated that the ten-nucleon Schrödinger equation requires the solution of more than 200,000 coupled second-order partial-differential equations in 27 continuous variables, and this can be accomplished with numerical errors smaller than 1%! Their results are very impressive.
Although the nucleon-nucleon scattering data alone can predict the binding energy of the deuteron ($^2$H) to within about 1/2%, the experimental binding energy is used as input data in fitting the nucleon-nucleon potential. The nuclei $^3$H and $^3$He are slightly underbound without a three-nucleon force, and that force can be adjusted to remedy the underbinding. This highlights both the dominant nature of the nucleon-nucleon force and the relative smallness of three-nucleon forces, which is nevertheless appropriate in size to account for the small discrepancies that result from using only nucleon-nucleon forces in calculations.
The binding energy of $^4$He is then accurately predicted to within about 1%. The six-nucleon systems are also rather well predicted. There are small problems with neutron-rich nuclei, but only 3 adjustable parameters in the three-nucleon force allow several dozen energy levels to be quite well predicted. I recommend that everyone peruse the impressive results of the GFMC collaboration[@10A].
We note finally that power counting can be used to show that light nuclei are basically non-relativistic, and relativistic corrections are on the order of a few percent. Power counting is a powerful qualitative technique for determining the relative importance of various mechanisms in nuclear physics.
What Nuclear Physics Can Do for Atomic Physics {#what-nuclear-physics-can-do-for-atomic-physics .unnumbered}
==============================================
With our recently implemented computational skills we in nuclear physics can calculate many properties of light nuclei with fairly good accuracy. This is especially true for the deuteron, which is almost unbound and is computationally simple. Although nuclear experiments don’t have the intrinsic accuracy of atomic experiments, many nuclear quantities that are relevant to precise atomic experiments can also be measured using nuclear techniques, and usually with fairly good accuracy.
What quantities are we talking about? The nuclear length scale is $R \sim 1$ fm $= 10^{-5}$ Å. The much larger atomic length scale of $a_0 \sim $ 1 Åmeans that an expansion in powers of $R/a_0$ makes great sense, and a typical wavelength for an atomic electron is so large compared to the nuclear size that only moments of the nuclear observables come into play. An example is the nuclear charge form factor (the Fourier transform of the nuclear charge density, $\rho$), which is given by
$$F (\bq) = \int d^3 r \, \rho (\br) \, \exp ( i \bq \cdot \br )
\cong Z (1- \frac{\bq^2}{6}\langle r^2 \rangle_{\rm ch} + \cdots \, )
- \haf \, \bq^{\alpha} \bq^{\beta}\, Q^{\alpha \beta} + \cdots \, ,$$
where $\bq$ is the momentum transferred from an electron to the nucleus, $Q^{\alpha \beta}$ is the nuclear quadrupole-moment tensor, $Z$ is the total nuclear charge, and $\langle r^2 \rangle_{\rm ch}$ is the mean-square radius of the nuclear charge distribution. Since the effective $|\bq|$ in an atom will be set by the atomic scales and consequently will be very small, these moments should dominate the nuclear corrections to atomic energy levels. If one then uses $F$ to construct the electron-nucleus Coulomb interaction, one obtains
$$V_{\rm C} ( \br ) \cong - \frac{Z \alpha}{r} + \frac{2 \pi Z \alpha}{3}
\langle r^2 \rangle_{\rm ch} \, \delta^3 (\br) - \frac{Q \alpha}{2 r^3}
( 3 \, (\bS \cdot \hat{\br})^2 -\bS^2) + \cdots \, ,$$
where $\bS$ is the nuclear spin operator and $Q$ is the nuclear quadrupole moment (which vanishes unless the nucleus has spin $\geq 1$). The Fourier transform of the nuclear current density has a similar expansion
$$\bJ ( \bq ) = \int d^3 r \ \bJ (\br) \, \exp ( i \bq \cdot \br )
\cong -i \bq \times \bmu \, (1- \frac{\bq^2}{6} \langle r^2
\rangle_{\rm M}\, + \, \cdots \, ) \; + \, \cdots \, ,$$
where $\bmu$ is the nuclear magnetic-moment operator and $\langle r^2
\rangle_{\rm M}$ is the mean-square radius of the magnetization distribution. The first term generates the usual atomic hyperfine interaction.
Electron-nucleus scattering is the primary technique used to determine those nuclear moments of charge and current densities that are relevant to atomic physics[@ingo]. An exception is the measurement of the deuteron’s quadrupole moment \[$Q = 0.282(19)$ fm$^2$\] obtained by scattering polarized deuterons from a high-Z target at low energy[@Qnuc]. This result is consistent with the molecular measurement \[$Q = 0.2860(15)$ fm$^2$\][@Qhd1; @Qhd2], but its error is an order of magnitude larger. Although there is no reason to believe that the tensor polarizability of the deuteron[@tau] plays a significant role in the H-D (molecular) quadrupole-hyperfine splitting that was used to determine $Q$, that correction was not included in the analysis. It was included in the analysis of the nuclear measurement.
I highly recommend the recent review by Ingo Sick[@ingo], which contains values of the charge and magnetic radii of light nuclei. That review not only contains the best values of quantities of interest, but discusses reliability and many technical details for those who are interested. One qualitative result from that review is important for the discussion below. The errors of the tritium ($^3$H) radii are about an order of magnitude larger than those of deuterium. Of all the light nuclei tritium is the most poorly known experimentally, although the charge radius is relatively easy to calculate.
In addition to moments of the nuclear charge and current densities, various components and moments of the nuclear Compton amplitude can play a significant role. Examples are the (scalar) electric polarizability, $\alpha_E$, and the nuclear spin-dependent polarizability ($\sim \bS$). The latter term interacts with the electron spin to produce a contribution to the electron-nucleus hyperfine interaction. There exists a recent calculation of the latter for deuterium[@d-nu], and either calculations or measurements of $\alpha_E$ for $^2$H[@d-pol; @d-pol-g; @d-pol-x], $^3$H and $^3$He[@3pol], and $^4$He[@He4-x; @He4-t].
The Proton Size {#the-proton-size .unnumbered}
===============
One recurring problem in the hydrogen Lamb shift is the appropriate value of the mean-square radius of the proton, $\langle r^2 \rangle_{\rm p}$, to use in calculations. Some older analyses[@HMW] disagree strongly with more recent ones[@Simon]. As shown in Eqn. (3), the slope of the charge form factor (with respect to $\bq^2$) at $\bq^2$ = 0 determines that quantity. The form factor is measured by scattering electrons from the proton at various energies and scattering angles.
There are (at least) four problems associated with analyzing the charge form-factor data to obtain the proton size. The first is that the counting rates in such an experiment are proportional to the flux of electrons times the number of protons in the target seen by each electron. That product must be measured. In other words the measured form factor at low $\bq^2$ is ($a - b
\frac{\bq^2}{6} + \cdots$), where $b/a = \langle r^2 \rangle_{\rm p}$. The measured normalization $a$ (not equal to 1) clearly influences the value and error of $\langle r^2 \rangle_{\rm p}$. Most analyses unfortunately don’t take the normalization fully into account, and Ref.[@norm] estimates that a proper treatment of the normalization of available data could increase $\langle
r^2 \rangle_{\rm p}^{1/2}$ by about 0.015 fm and increase the error, as well. In an atom, of course, the normalization is precisely computable.
Another source of error is neglecting higher-order corrections in $\alpha$ (i.e., Coulomb corrections). Ref.[@Coulomb] demonstrates that this increases $\langle r^2 \rangle_{\rm p}^{1/2}$ by about 0.010 fm. A similar problem in analyzing deuterium data was resolved in Ref.[@ST]. Another difficulty that existed in the past was a lack of high-quality low-$\bq^2$ data. The final problem is that one must use a sufficiently flexible fitting function to represent $F ( \bq)$, or the errors in the radius will be unrealistically low. All of the older analyses had one or more of these flaws.
Most of the recent analyses[@Simon; @Coulomb; @fit] are compatible if the appropriate corrections are made. An analysis by Rosenfelder[@Coulomb] contains all of the appropriate ingredients, and he obtains $\langle r^2
\rangle_{\rm p}^{1/2}$ = 0.880(15) fm. There is a PSI experiment now underway to measure the Lamb shift in muonic hydrogen, which would produce the definitive result for $\langle r^2 \rangle_{\rm p}$ [@PSI; @savely]. I fully expect the results of that experiment to be compatible with Rosenfelder’s result. Extraction of the proton radius[@2loop] from the electronic Lamb shift is now somewhat uncertain because of controversy involving the two-loop diagrams. These diagrams are significantly less important in muonic hydrogen, where the relative roles of the vacuum polarization and radiative diagrams are reversed.
What Atomic Physics Can Do for Nuclear Physics {#what-atomic-physics-can-do-for-nuclear-physics .unnumbered}
==============================================
The single most valuable gift by atomic physics to the nuclear physics community would be the accurate determination of the proton mean-square radius: $\langle r^2 \rangle_{\rm p}$. This quantity is important to nuclear theorists who wish to compare their nuclear wave function calculations with measured mean-square radii. In order for an external source of electric field (such as a passing electron) to probe a nucleus, it is first necessary to “grab” the proton’s intrinsic charge distribution, which then maps out the mean-square radius of the proton probability distribution in the wave function: $\langle r^2
\rangle_{\rm wfn}$. Thus the measured mean-square radius of a nucleus, $\langle
r^2 \rangle$, has the following components:
$$\langle r^2 \rangle = \langle r^2 \rangle_{\rm wfn} + \langle r^2 \rangle_{\rm
p}+ \frac{N}{Z} \langle r^2 \rangle_{\rm n} + \frac{1}{Z} \langle r^2
\rangle_{\ldots} \, ,$$
where I have included the intrinsic contribution of the N neutrons as well as the Z protons, and $\langle r^2 \rangle_{\ldots}$ is the contribution of everything else, including the very interesting (to nuclear physicists) contributions from strong-interaction mechanisms and relativity in the nuclear charge density[@czech]. Because the neutron looks very much like a positively charged core surrounded by a negatively charged cloud, its mean-square radius has the opposite sign to that of the proton, whose core is surrounded by a positively charged cloud. It should be clear from Eqn. (6) that $\langle r^2 \rangle_{\rm p}$ (which is much larger than $\langle r^2
\rangle_{\rm n}$) is an important part of the overall mean-square radius. Its present uncertainty degrades our ability to test the wave functions of light nuclei.
The next most important measurements are isotope shifts in light atoms or ions. Since isotope shifts measure differences in frequencies for fixed nuclear charge Z, the effect of the protons’ intrinsic size cancels in the difference. This is particularly important given the current lack of a precise value for the proton’s radius. The neutrons’ effect is relatively small and can be rather easily eliminated, and thus one is directly comparing differences in wave functions, or of small contributions from $\langle r^2 \rangle_{\ldots}$. Isotope shifts are therefore especially “theorist-friendly” measurements, since they are closest to measuring what nuclear theorists actually calculate.
Precise isotope-shift measurements have been performed for $^4$He - $^3$He[@shiner] and for $^2$H - $^1$H (D-H)[@d-p]. A measurement of $^6$He-$^4$He is being undertaken[@ANL] at ANL. Gordon Drake has written about and strongly advocated such measurements in the Li isotopes[@Li-IS]. These are all highly desirable measurements. Because there are currently large errors in the $^3$H (tritium) charge radius, in my opinion the single most valuable measurement to be undertaken for nuclear physics purposes would be the tritium-hydrogen ($^3$H - $^1$H) isotope shift. An extensive series of calculations using first-generation nuclear forces found $\langle r^2
\rangle_{\rm wfn}^{1/2}$ for tritium to be 1.582(8) fm, where the “error” is a subjective estimate[@radius]. This number could likely be improved by using second-generation nuclear forces, although it will never be as accurate as the corresponding deuteron value, which we discuss next.
The D-H isotope shift in the 2S-1S transition reported by the Garching group[@d-p] was
$$\Delta \nu = 670 \ \, 994 \ \, 334.64 (15) \ {\rm kHz} \, .$$
$\uparrow$ $\uparrow$ $\uparrow$ $\uparrow$\
Most of this effect is due to the different masses of the two isotopes (and begins in the first significant figure, indicated by an arrow). Nevertheless, the precision is sufficiently high that the mean-square radius effect in the sixth significant figure (second arrow) is much larger than the error. The electric polarizability of the deuteron influences the eighth significant figure, while the deuteron’s magnetic susceptibility contributes to the tenth significant figure. It becomes difficult to trust the interpretation of the nuclear physics at about the 1 kHz level, so improving this measurement probably wouldn’t lead to an improved understanding of the nuclear physics.
Analyzing this isotope shift and interpreting the residue (after applying all QED corrections) in terms of the deuteron’s radius leads to the results[@iso] in Table 1. The very small binding energy of the deuteron produces a long wave function tail outside the nuclear potential (interpretable as a proton cloud around the nuclear center of mass), which in turn leads to an easy and very accurate calculation of the mean-square radius of the (square of the) wave function. Subtracting this theoretical radius from the experimental deuteron radius (corrected for the neutron’s size) determines the effect of $\langle r^2 \rangle_{\ldots}$ on the radius. Although this difference is quite small, it is nevertheless significant and half the size of the error in the corresponding electron-scattering measurement. This high-precision analysis in Table 1 of the content of the deuteron’s charge radius would have been impossible without the precision of the atomic D-H isotope-shift measurement. This measurement has given nuclear physics unique insight into small mechanisms that are at present poorly understood[@MEC].
[|cc|c||c|]{} $ \langle r^2 \rangle_{\rm wfn}^{1/2}\, ({\rm fm})$ &
------------------------------------------------------------------------
$ _{\rm exp}\langle r^2 \rangle_{\rm pt}^{1/2}\, ({\rm fm})$ & $ {\rm difference}\, ({\rm fm})$ & $ \Delta \langle r^2 \rangle_{\rm n}^{1/2}\, ({\rm fm})$\
1.9687(18)
------------------------------------------------------------------------
& 1.9753(10) & 0.0066(21) & -0.0291(7)\
I have said very little in my talk about how information from hyperfine splittings might provide insight into nuclear mechanisms. Partly this is due to a lack of background on my part, and partly because the necessary calculations haven’t been performed. Karshenboim and Ivanov[@sav-nu] have compared theoretical (QED only) and experimental results for various S-states in light atoms, results that are expected to be accurate to roughly 1 part in $10^{8}$. They find significant residual differences attributable to nuclear effects, which range from tens of ppm to several hundred ppm. It is likely that these differences contain interesting and useful nuclear information, in the form of Zemach moments[@zemach] and spin-dependent polarizabilities. The latter are related to the Drell-Hearn-Gerasimov sum rule[@DHG], a topic of considerable current interest in nuclear physics[@DHG-nuc]. Exploratory calculations are underway.
Summary and Conclusions {#summary-and-conclusions .unnumbered}
=======================
I hope that I have convinced you that nuclear forces and nuclear calculations in light nuclei are under control in a way never before attained. This progress has been possible because of the great increase in computing power in recent years. Many of the nuclear quantities that contribute to atomic measurements have been calculated or measured to a reasonable level of accuracy, a level that is improving with time. Isotope shifts are valuable contributions to nuclear physics knowledge, and are especially useful to theorists who are interested in testing the quality of their wave functions for light nuclei. In special cases such as the deuteron these measurements provide the only insight into the size of small contributions to the electromagnetic interaction that are generated by the underlying strong-interaction mechanisms. In my opinion the tritium-hydrogen isotope shift would be the most useful measurement of that type. I am especially hopeful for the success of the ongoing PSI experiment attempting to measure the proton size via the Lamb shift in muonic hydrogen. The absence of a stable, accurate proton radius has been particularly annoying.
Acknowledgements {#acknowledgements .unnumbered}
================
The work of J. L. Friar was performed under the auspices of the United States Department of Energy. The author would like to thank Savely Karshenboim for an invitation to this conference and for the opportunity to discuss the nuclear side to precise atomic measurements.
[999]{}
D’Arcy Wentworth Thompson. On Growth and Form. Cambridge \[Eng.\] Univ. Press. 1959, p. 2.
J. Reichert, M. Niering, R. Holzwarth, M. Wietz, Th. Udem, and T. W. Hänsch. Phys. Rev. Lett. [**84**]{}, 3232 (2000). The H 2S-1S frequency was measured to 14 significant figures in this work.
M. I. Eides, H. Grotch, and V. A. Shelyuto. Phys. Rep. [**342**]{}, 63 (2001). This is a recent and comprehensive review.
V. G. J. Stoks, R. A. M. Klomp, C. P. F. Terheggen, and J. J. de Swart. Phys. Rev. C [**49**]{}, 2950 (1994); R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla. Phys. Rev. C [**51**]{}, 38 (1995); J. L. Friar, G. L. Payne, V. G. J. Stoks, and J. J. de Swart. Phys. Lett. B[**311**]{}, 4 (1993).
A. Huber, Th. Udem, B. Gross, J. Reichert, M. Kourogi, K. Pachucki, M. Weitz, and T. W. Hänsch. Phys. Rev. Lett. [**80**]{}, 468 (1998). The deuteron’s size has a significant effect on the D-H isotope shift in the 2S-1S transition.
S. Weinberg. Physica A[**96**]{}, 327 (1979); [*in*]{} A Festschrift for I. I. Rabi. Transactions of the N. Y. Academy of Sciences [**38**]{}, 185 (1977); Nucl. Phys. B[**363**]{}, 3 (1991); Phys. Lett. B[**251**]{}, 288 (1990); Phys. Lett. B[**295**]{}, 114 (1992).
U. van Kolck. Thesis, University of Texas, (1993); C. Ordóñez and U. van Kolck. Phys. Lett. B[**291**]{}, 459 (1992); C. Ordóñez, L. Ray, and U. van Kolck. Phys. Rev. Lett. [**72**]{}, 1982 (1994); U. van Kolck. Phys. Rev. C [**49**]{}, 2932 (1994); C. Ordóñez, L. Ray, and U. van Kolck. Phys. Rev. C [**53**]{}, 2086 (1996).
J. L. Friar. Few-Body Systems [**22**]{}, 161 (1997).
S. A. Coon, M. D. Scadron, P. C. McNamee, B. R. Barrett, D. W. E. Blatt, and B. H. J. McKellar. Nucl. Phys. A[**317**]{}, 242 (1979).
R. A. M. Klomp, V. G. J. Stoks, and J. J. de Swart. Phys. Rev. C [**44**]{}, R1258 (1991); V. Stoks, R. Timmermans, and J. J. de Swart. Phys. Rev. C [**47**]{}, 512 (1993).
K. Hagiwara, [*et al.*]{} Phys. Rev. D [**66**]{}, 010001 (2002).
M. C. M. Rentmeester, R. G. E. Timmermans, J. L. Friar, J. J. de Swart. Phys. Rev. Lett. [**82**]{}, 4992 (1999); M. Walzl, U. -G. Meißner, and E. Epelbaum. Nucl. Phys. A[**693**]{}, 663 (2001); D. R. Entem and R. Machleidt. Phys. Lett. B[**524**]{}, 93 (2002).
C. R. Chen, G. L. Payne, J. L. Friar, and B. F. Gibson. Phys. Rev. C [**31**]{}, 2266 (1985); J. L. Friar, B. F. Gibson, and G. L. Payne. Phys. Rev. C [**35**]{}, 1502 (1987).
J. Carlson and R. Schiavilla. Rev. Mod. Phys. [**70**]{}, 743 (1998). This is a comprehensive review of light nuclei containing an elementary discussion of GFMC. What Joe Carlson pioneered in the late 1980s was the application of GFMC to solving the nuclear problem with realistic potentials, where difficulties of principle exist (the so-called “fermion problem”). The GFMC technique was pioneered for nuclear physics with very simplified potentials in [@GFMC-h]. Atomic and molecular applications began about 1967[@GFMC-at].
G. A. Baker, Jr., J. L. Gammel, B. J. Hill, and J. G. Wills. Phys. Rev. [**125**]{}, 1754 (1962); M. H. Kalos. Phys. Rev. [**128**]{}, 1791 (1962).
M. Kalos. J. Comp. Phys. [**1**]{}, 257 (1967).
S. C. Pieper. Private Communication.
S. C. Pieper, K. Varga, and R. B. Wiringa. Nuclear Theory Archive. nucl-th/0206061. This is only the latest in a series of calculations of light nuclei.
I. Sick. Prog. Part. Nucl. Phys. [**47**]{}, 245 (2001). This is an excellent review and is highly recommended.
J. E. Kammeraad and L. D. Knutson. Nucl. Phys. A[**435**]{}, 502 (1985).
R. V. Reid, Jr. and M. L. Vaida. Phys. Rev. Lett. [ **29**]{}, 494 (1972); (E) [**34**]{}, 1064 (1975).
D. M. Bishop and L. M. Cheung. Phys. Rev. A [**20**]{}, 381 (1979).
M. H. Lopes, J. A. Tostevan, and R. C. Johnson. Phys. Rev. C [**28**]{}, 1779 (1983).
A. I. Mil’shtein and I. B. Khriplovich. JETP [**82**]{}, 616 (1996) \[Zh. Eksp. Teor. Fiz. [**109**]{}, 1146 (1996)\].
J. L. Friar and G. L. Payne. Phys. Rev. C [**55**]{}, 2764 (1997). See the references in this work for earlier calculations.
J. L. Friar, S. Fallieros, E. L. Tomusiak, D. Skopik, and E. G. Fuller. Phys. Rev. C [**27**]{}, 1364 (1983).
N. L. Rodning, L. D. Knutson, W. G. Lynch, and M. B. Tsang. Phys. Rev. Lett. [**49**]{}, 909 (1982).
V. D. Efros, W. Leidemann, and G. Orlandini. Phys. Lett. B[**408**]{}, 1 (1997).
J. L. Friar. Phys. Rev. C [**16**]{}, 1540 (1977).
R. Rosenfelder. Nucl. Phys. A[**393**]{}, 301 (1983).
L. N. Hand, D. J. Miller, and R. Wilson. Rev. Mod. Phys., 335 (1963). The value of $\langle r^2 \rangle_{\rm p}^{1/2}$ = 0.805(11) fm is not reliable and should not be used.
G. G. Simon, C. Schmidt, F. Borkowski, V. H. Walter. Nucl. Phys. A[**333**]{}, 381 (1980).
C. W. Wong. Int. J. Mod. Phys. E [**3**]{}, 821 (1994).
R. Rosenfelder. Phys. Lett. B[**479**]{}, 381 (2000).
I. Sick and D. Trautmann. Phys. Lett. B[**375**]{}, 16 (1996).
P. Mergell, U. -G. Meißner, and D. Drechsel. Nucl. Phys. A[**596**]{}, 367 (1996).
D. Taqqu, [*et al.*]{} Hyperfine Int. [**119**]{}, 311 (1999).
S. G. Karshenboim. Can. J. Phys. [**77**]{}, 241 (1999).
K. Melnikov and T. van Ritbergen. Phys. Rev. Lett. [**84**]{} 1673 (2000). They find $\langle r^2 \rangle_{\rm p}^{1/2}$ = 0.883(14) fm.
J. L. Friar. Czech. J. Phys. [**43**]{}, 259 (1993); H.Arenhövel. Czech. J. Phys. [**43**]{}, 207 (1993). These are introductory articles treating meson-exchange currents \[strong-interaction contributions to electromagnetic currents\]. Neglecting such currents and other binding effects is commonly denoted the “impulse approximation”.
D. Shiner, R. Dixson, and V. Vedantham. Phys. Rev. Lett. [**74**]{}, 3553 (1995).
Z.-T. Lu. Private Communication.
Z. - C. Yan and G. W. F. Drake. Phys. Rev. A [**61**]{}, 022504 (2000).
J. L. Friar. Summary talk presented at “XIV$^{\underline{th}}$ International Conference on Few-Body Problems in Physics,” Williamsburg, VA, May 31, 1994, ed. by F. Gross. AIP Conference Proceedings [**334**]{}, 323 (1995). Theoretical values of $\langle r^2
\rangle_{\rm wfn}^{1/2}$ for $^3$He and $^3$H were obtained from the fits in Fig. (1). The $^3$He value of 1.769(5) fm was used in the talk.
J. L. Friar, J. Martorell, and D. W. L. Sprung. Phys. Rev.A [**56**]{}, 4579 (1997). A tiny center-of-mass contribution to the deuteron radius was overlooked in that work, but has been included in Table 1 above. I would like to thank Ingo Sick for pointing out the problem.
A. J. Buchmann, H. Henning, and P. U. Sauer, Few-Body Systems [**21**]{}, 149 (1996).
S. Kopecky, P. Riehs, J. A. Harvey, and N. W. Hill. Phys. Rev. Lett. [**74**]{}, 2427 (1995). The compiled value at the bottom of Table 1 was used.
S. G. Karshenboim and V. G. Ivanov. Phys. Lett. B[ **524**]{}, 259 (2002).
C. Zemach. Phys. Rev. [**104**]{}, 1771 (1956).
S. D. Drell and A. C. Hearn. Phys. Rev. Lett. [**16**]{}, 908 (1966); S. B. Gerasimov. Sov. J. Nucl. Phys. [**2**]{}, 430 (1966).
J. L. Friar. Phys. Rev. C [**16**]{}, 1540 (1977).
| {
"pile_set_name": "ArXiv"
} |
---
address: |
A.M.Prokhorov General Physics Institute of the Russian Academy of Sciences,\
38 Vavilov Street, Moscow, 119991, Russia
author:
- 'L.V.Arapkina, V.A.Yuryev[^1]'
title: 'Atomic structure of Ge quantum dots on the Si(001) surface'
---
Dense arrays of the self-assembled Ge quantum dots (QD) deposited on the Si(001) surface at moderate temperatures and compatible with the Si technology (Fig. \[fig:arrays\]) stay an attractive subject of investigations for a number of years (see e.g. Refs. [@Mo; @Chem_Rev; @Kastner; @Fujikawa; @Pyramid_to_dome]) due to their promising potential applications in microelectronics and primarily in microphotoelectronics [@Wang-Cha; @Wang-properties]. Significant technological achievements of the recent years enabled the controllable formation of the QD arrays with the desired cluster densities [@Smagina; @classification]. However, the uniformity of cluster types in the arrays (or array defectiveness) and the dispersion of cluster sizes are still issues of the day.
It was shown recently that the {105} faceted clusters usually referred to as hut clusters are subdivided into two main species—pyramids and wedges. Their densities are equal at the initial stage of the array formation. Then, as the Ge coverage is increased, the later become dominating in the arrays whereas the former exponentially rapidly disappear [@classification]. (The authors of Ref. [@Kastner] observed the analogous process—the growth of Ge coverage resulted in the increase of density of elongated hut clusters.) An additional peculiarity is the difference of heights of the pyramidal and wedge-shaped clusters. The formed pyramids are always higher than the wedges and the later are limited in their heights [@classification]. The limit height depends on the growth temperature. The dispersion of the cluster sizes in the arrays was found to be governed by the lengths of the wedge-like clusters which appeared to be distributed rather uniformly over a wide interval of values of the cluster base aspect ratio [@classification]. In order to control the parameters of the arrays a deep knowledge of the cluster morphology is important.
(a) (b)
In this article, we investigate the structure of both main species of the hut clusters on the atomic level. It can be shown based on the bulk Ge lattice that the {105} facet is composed by monoatomic steps and (001) terraces [@Kastner]. The step height is $\sim 1.4$ , the terrace width is $\sim 7$ . The first model of the hut cluster faces was proposed in Ref. [@Mo]. Later another model appeared in Ref. [@Fujikawa]. Both models are founded on the assumption that a reconstruction formed by the Ge ad-dimers is present on the free surfaces of the (001) terraces. The main difference between the models is in the determination of this reconstruction. Now a new model of the hut cluster faces is brought forward on the basis of our [*in situ*]{} investigations of the {105} facet carried out by the high resolution scanning tunnelling microscopy (STM). In addition, crystallographic models of pyramidal and wedge-shaped clusters are proposed based on the facet structure and a fine structure of the vertices of the pyramids and the ridges of the wedges.
(a) (b)
The experiments were carried out using an ultra high vacuum instrument consisting of the molecular beam epitaxy (MBE) chamber coupled with STM which enables the sample study at any stage of processing serially investigating the structure and giving additional treatments to the specimen. The surface of the silicon substrates was completely purified of the oxide film as a result of short annealing at the temperature of $\sim 925^\circ$C [@our_Si(001)_en]. Germanium was deposited directly on the purified silicon (001) surface from the source with the electron beam evaporation. The rate of Ge deposition was $\sim 0.15$ /s, the Ge coverage ($h_{\rm Ge}$) (or more accurately the thickness of the Ge film measured by the graduated in advance film thickness monitor with the quartz sensor installed in the MBE chamber) was varied from 6 to 14 for different samples. The substrate temperature $T_{\rm gr}$ was 360 or $530^\circ$C during the process. The rate of the sample cooling down to the room temperature was $\sim 0.4^\circ$C/s after the deposition. Images were scanned at room temperature in the constant tunnelling current ($I_{\rm t}$) mode. The STM tip was zero-biased while a sample was positively or negatively biased ($U_{\rm s}$). The details of the sample preparation as well as the experimental techniques can be found in Refs. [@classification] and [@our_Si(001)_en]. The WSxM software [@WSxM] was used for processing of the STM images.
(a) (b)
(a) (b)
(a) (b) (c) (d) (e) (f)
(a) (b)
The obtained experimental results are as follows.
Fig. \[fig:arrays\] demonstrates STM images of the Ge QD arrays formed at 360 and 530$^\circ$C and the same value of $h_{\rm Ge} = 10$ . Ge clusters grown at these temperatures and the mentioned value of $h_{\rm Ge}$ are seen to have different structure of faces—the facets of the clusters grown at 360$^\circ$C are completed whereas the faces of the clusters formed at 530$^\circ$C appeared to be uncompleted and contain additional steps of the solidifying interfaces (Fig. \[fig:arrays\](b)). This means that Ge clusters reached some limit height determined by the growth temperature begin to non-uniformly ad the migrating Ge atoms to the faces. As a result new incomplete planes are formed on the facets. No small clusters or nuclei were found on the wetting layer (WL) in the arrays grown at 530$^\circ$C. Simultaneous growth and nucleation of several phases of Ge clusters is typical for the arrays growing at lower temperatures (e.g. 360$^\circ$C) [@classification]. So at 530$^\circ$C the process of the QD growth dominates over the process of nucleation of new clusters after the act of the initial cluster nucleation whereas at 360$^\circ$C these processes compete during the array growth. The details of classification of Ge hut clusters and the properties of QD arrays grown at moderate temperatures can be found in Ref. [@classification].
Fig. \[fig:profile\] demonstrates an STM image of a small Ge QD, the wetting layer in the nearest vicinity of its base (a) and a profile of its facet (b). The WL consists of $(M\times N)$ blocks with the $p(2\times 2)$ and $c(4\times 2)$ reconstruction (Fig. \[fig:profile\](a) and Ref. [@Chem_Rev]). The QD is seen to be formed by the monoatomic steps (inclined regions of the curve) and the (001) terraces (horizontal regions of the curve). Heights of the steps are $\sim 1.4$ and total widths of the terraces are $\sim 7$ that confirms the assumption made in Ref. [@Kastner] regarding the sizes of the steps and terraces. The line connecting the maximum and the minimum of the profile has a slope of $\sim$ 11.3$^\circ$. The slopes of the inclined regions of the profile are much greater.
The STM investigations of faces of the clusters grown at 360 and 530$^\circ$C evidenced that their fine structure depends on neither the growth temperature nor the hut cluster species. A typical STM image of the QD facet is presented in Fig. \[fig:facet\]. Characteristic distances on the facets are as follows: $\sim 10.5$ in the <100> directions (along the corresponding side of the base) and $\sim 14$ in the normal (<051>) directions. The facets are composed by structural units which are outlined by ellipses in Fig. \[fig:facet\](a) and can be arranged along either \[110\] or \[1${\overline 1}$0\] direction on the (001) plane. The spatial arrangement of the units is changed to perpendicular on the adjacent terraces and remains unchanged within the terrace. Following the authors of Refs. [@Mo] and [@Fujikawa], we suppose this structural unit to be a pair of dimers composed by Ge atoms on the terrace surface. The QD itself may be considered as a set of successive monoatomic steps and (001) terraces which form the {105} faces [@Mo; @Chem_Rev; @Kastner; @Fujikawa]. We observe pairs of dimers which are the “reminders” of dimer rows covering the free “outcrops” of the (001) planes. As the dimer rows change their direction to the perpendicular on the adjacent terraces, the observed structural units turn at 90$^\circ$ as well.
A model of facets of the Ge hut clusters may be built on the basis of the above experimental observations and the previous data of Refs. [@Mo] and [@Fujikawa]. The model is based on the following assumptions: (i) the hut clusters are faceted by the {105} planes, (ii) the hut cluster faces consist of the monoatomic steps and (001) terraces, and (iii) a structure is formed on the free “outcrops” of the (001) terraces which consists of pairs of ad-dimers and is similar to the structures arising on the pure Si(001) or Ge(001) surfaces [@Chadi]. A schematic of this model is shown in Fig. \[fig:face\_structure\](a). The sides of the cluster base are alined with the <100> directions. The terraces are bounded by the S$_{\rm A} $ and nonrebonded S$_{\rm B} $ monoatomic steps (the terminology by Chadi [@Chadi]). Fig. \[fig:face\_structure\](b) demonstrates the drawing of the structure superposed with its STM image.
As mentioned above there are two main species of Ge hut clusters—pyramidal and wedge-shaped ones [@classification]. Images of the clusters of both species are shown in Figs. \[fig:top\_view\](a,b). It is seen that the structures of the top vertex of the pyramid and the top edge of the wedge (the ridge) are different. To build models of atomic structure of Ge hut clusters the knowledge of architecture of tops of each species is required. Figs. \[fig:top\_view\](c,d) demonstrate high resolution topographs of the vertex and a nucleus (1 ML over WL) of the pyramid-like cluster and Figs. \[fig:top\_view\](e,f) show the topographs of the ridge and a very small (only 2 ML over WL) wedge-like QD. These images allowed us to find out the required structures of tops which are shown as the highest atomic layers in the structural schematics of each species of the clusters demonstrated in Fig. \[fig:schematic\]. We used the following procedure to build the models of the clusters: (i) we drew each of four facets in accordance with the schematic shown in Fig. \[fig:face\_structure\](a) (ii) then we combined the faces according to the results of the STM study of tops. For the simplicity and clarity of the drawings (Fig. \[fig:schematic\]) the structure of edges is not shown and only the highest layer of atoms is shown on the vertex and the ridge.
It should be remarked in the conclusion that according to our STM data pyramides and wedges have different structures of tops. It is seen from Figs. \[fig:top\_view\] and \[fig:schematic\] that one cannot transform the structure of clusters of one species to the structure of the other. The structure of the pyramid cannot be obtained from the structure of the wedge simply combining four equal faces as the resultant model would have a wrong structure of the vertex. Hence, the conditions and processes of nucleation and growth should be expected to be different for pyramidal and wedge-like Ge hut clusters.
In summary, we investigated the structure of the {105} faceted Ge quantum dots (hut clusters) grown on the Si(001) surface at the temperatures of 360 and 530$^\circ$C. Despite the difference of the formation processes at these temperatures both species of hut clusters (pyramidal and wedges-like ones) were found to always have the same structure of the {105} facets which was visualized. Structures of the vertexes of the pyramidal clusters and the ridges of the wedge-shaped clusters were revealed as well. This allowed us to bring forward a crystallographic model of the {105} facets as well as the models of the atomic structure of both species of Ge hut clusters.
[99]{}
Y.-W. Mo, D. E. Savage, B. S. Swartzentruber, M. G. Lagally. **65**, 1020 (1990).
F. Liu, F. Wu, M. G. Lagally. **97**, 1045 (1997).
M. K[ä]{}stner, B. Voigtl[ä]{}nder. **82**, 2745 (1999).
Y. Fujikawa, *et al.* **88**, 176101 (2002).
F. Montalenti, *et al.* **93**, 216102 (2004).
K. L. Wang, D. Cha, J. Liu, C. Chen. **95**, 1866, (2007).
K. L. Wang, S. Tong, H. J. Kim. **8**, 389 (2005).
J. V. Smagina, *et al.* **106**, 517 (2008).
L. V. Arapkina, V. A. Yuryev. ; .
L. V. Arapkina, V. M. Shevlyuga, V. A. Yuryev. **87**, 215 (2008).
I. Horcas, *et al.* **78**, 013705 (2007).
D. J. Chadi. **59**, 1691 (1987).
Fig.1. STM images of arrays of Ge hut clusters grown on the Si(001) surface, $h_{\rm Ge}=10$ , $T_{\rm gr}$ is (a) $360^{\circ}$C ($U_{\rm s}=+2.5$ V, $I_{\rm t}=80$ pA) and (b) $530^{\circ}$C ($U_{\rm s}=+2.1$ V, $I_{\rm t}=80$ pA).
Fig.2. STM image (a) of Ge hut clusters ($h_{\rm Ge}=6$ , $T_{\rm gr}=360^{\circ}$C, $U_{\rm s}=+1.8$ V, $I_{\rm t}=80$ pA), blocks of the Ge wetting layer $(M\times N)$ structure with the $p(2\times 2)$ and $c(4\times 2)$ reconstructions [@Chem_Rev] are seen in the lower right quarter of the field. A profile of the cluster facet (c) taken along the white line shown in the image (a), the monoatomic steps and (001) terraces are clearly seen.
Fig.3. 2-D (a) and 3-D (b) STM images of the same area on Ge hut cluster facet ($h_{\rm Ge}=10$ , $T_{\rm gr}=360^{\circ}$C, $U_{\rm s}=+2.1$ V, $I_{\rm t}=80$ pA). The sides of the cluster base lie along the \[100\] direction, structural units arising on the free surfaces of the (001) terraces are marked out.
Fig.4. A structural model of the $\{105\}$ facet of hut clusters (a), S$_{\rm A}$ and S$_{\rm B}$ are commonly adopted designations of the monoatomic steps [@Chadi], atoms situated on higher terraces are shown by larger circles. (b) The schematic of the facet superimposed on its STM image ($4.3\times 4.4$ nm, $U_{\rm s}=+3.0$ V, $I_{\rm t}=100$ pA), the \[100\] direction coincides with that of the cluster base side.
Fig.5. STM topographs of (a) pyramidal ($T_{\rm gr}$ = 530$^\circ$C, $h_{\rm Ge} = 11$ Å) and (b) wedge-shaped ($T_{\rm gr}$ = 360$^\circ$C, $h_{\rm Ge} = 10$ Å) clusters, (c) the vertex, face and (d) a nucleus \[the left of two features, 1ML over WL\] of a pyramid ($T_{\rm gr} = 360^\circ$C, $h_{\rm Ge} = 6$ ), (e) the ridges and long facets of two closely neighbouring wedges ($T_{\rm gr} = 360^\circ$C, $h_{\rm Ge} = 8$ ) and (f) a small wedge \[in the center of the field of view, 2ML over WL\] ($T_{\rm gr} = 360^\circ$C, $h_{\rm Ge} = 6$ ).
Fig.6. Schematic drawings of atomic structures of Ge (a) pyramidal and (b) wedge-shaped hut clusters composed of 6 monoatomic steps on the wetting layer.
[^1]: http://www.gpi.ru/eng/staff\_s.php?eng=1&id=125, e-mail: vyuryev@kapella.gpi.ru
| {
"pile_set_name": "ArXiv"
} |
---
address: |
Arnold Sommerfeld Center for Theoretical Physics,\
Ludwig-Maximilians-Universität München, Theresienstra[ß]{}e 37, 80333 München, Germany.
author:
- 'Emanuel Malek[^1]'
title: Dualising consistent truncations
---
Introduction
============
The consistent Kaluza-Klein truncation of higher-dimensional supergravity to lower-dimensional theories is an old and generically difficult problem due to the highly non-linear gravitational field equations [@Duff:1984hn]. In the case of truncations on group manifolds via a Scherk-Schwarz Ansatz, consistency follows from the group structure. However, consistent reductions can also be obtained without compactifying on a group manifold, as is the case for the sphere reductions of 11-dimensional SUGRA [@deWit:1986iy; @Nastase:1999kf; @Cvetic:2000dm]. Recent progress in understanding these consistent truncations has come via generalised Scherk-Schwarz reductions [@Aldazabal:2011nj; @Geissbuhler:2011mx; @Berman:2012uy; @Hohm:2014qga; @Lee:2014mla; @Lee:2015xga] in double field theory (DFT) [@Hull:2009mi] and exceptional field theory (EFT) [@Berman:2010is], and their associated generalised geometries [@Coimbra:2011ky].
In [@Malek:2015hma] we use this framework to establish a duality relating consistent IIA and IIB truncations for certain gaugings of maximal 7-dimensional supergravity. We then employ this duality to derive new consistent truncations of type IIB theory on the three sphere $S^3$, as well as on hyperboloids $H^{p,q}$, which lead to compact $SO(4)$, non-compact $SO(p,q)$ and non-semisimple $CSO(p,q,r)$ gaugings.
Seven-dimensional maximal gauged SUGRA
======================================
Maximal seven-dimensional gauged SUGRA can be conveniently formulated using the embedding tensor formalism [@Samtleben:2005bp], which describes the embedding of the gauge group into the global symmetry group of the ungauged theory, in this case ${\mathrm{SL}(5)}$. Let us denote fundamental ${\mathrm{SL}(5)}$ indices by ${\bar{a}}, {\bar{b}}= 1, \ldots 5$ and observe that the gauge group can at most be 10-dimensional because there are only 10 vector fields in seven-dimensional maximal SUGRA. If we denote by $X_{{\bar{a}}{\bar{b}}}$ the 10 generators of the gauge group and $t_\alpha$ the generators of ${\mathrm{SL}(5)}$, we have $$X_{{\bar{a}}{\bar{b}}} = \theta_{{\bar{a}}{\bar{b}}}{}^\alpha t_\alpha \,,$$ where $\alpha$ labels the adjoint of ${\mathrm{SL}(5)}$ and $\theta_{{\bar{a}}{\bar{b}}}{}^\alpha$ is the embedding tensor.
Consistency of the seven-dimensional theory requires the embedding tensor to satisfy two constraints. The linear constraint restricts the representations in which the embedding tensor must lie: $$\theta_{{\bar{a}}{\bar{b}}}{}^\alpha \in \mathbf{15} \oplus \mathbf{40}' \oplus \mathbf{10} \subset \mathbf{10} \otimes \mathbf{24} \,,$$ and ensures supersymmetry. We will denote these irreps by $S_{{\bar{a}}{\bar{b}}}$, $Z^{{\bar{a}}{\bar{b}},{\bar{c}}}$ and $\tau_{{\bar{a}}{\bar{b}}}$ for the $\mathbf{15}$, $\mathbf{40}'$ and $\mathbf{10}$, respectively. Let us note that gaugings in the $\mathbf{10}$, known as “trombone gaugings”, lead to gauged SUGRAs which can only be defined at the level of the equations of motion since they do not admit an action principle [@LeDiffon:2008sh]. The quadratic constraint ensures closure of the gauge group $$\left[ X_{{\bar{a}}{\bar{b}}}, X_{{\bar{c}}{\bar{d}}} \right] = - \left(X_{{\bar{a}}{\bar{b}},{\bar{c}}{\bar{d}}}\right)^{{\bar{e}}{\bar{f}}} X_{{\bar{e}}{\bar{f}}} \,, \label{eq:QC}$$ where $$\left(X_{{\bar{a}}{\bar{b}}}\right)_{{\bar{c}}}{}^{{\bar{d}}} = \frac{1}{2} \epsilon_{{\bar{a}}{\bar{b}}{\bar{c}}{\bar{e}}{\bar{f}}} Z^{{\bar{e}}{\bar{f}},{\bar{d}}} + 2 \delta^{{\bar{d}}}_{[{\bar{a}}} S_{{\bar{b}}]{\bar{c}}} + \frac{1}{3} \delta_{{\bar{c}}}^{{\bar{d}}} \tau_{{\bar{a}}{\bar{b}}} + \frac{2}{3} \delta^{{\bar{d}}}_{[{\bar{a}}} \tau_{{\bar{b}}]{\bar{c}}} \,,$$ and $\epsilon_{{\bar{a}}{\bar{b}}{\bar{c}}{\bar{d}}{\bar{e}}} = \epsilon^{{\bar{a}}{\bar{b}}{\bar{c}}{\bar{d}}{\bar{e}}} = \pm 1$ is the 5-d alternating symbol. Here we will primarily be interested in the scalar sector of the gauged SUGRA, which parameterises the coset space ${\mathrm{SL}(5)}/ \mathrm{SO}(5)$ and can thus be represented by a unit-determinant $5\times 5$ matrix $M_{{\bar{a}}{\bar{b}}}$. In addition, the theory also contains 10 vectors $A_\mu{}^{{\bar{a}}{\bar{b}}}$, five two-forms $B_{\mu\nu\,{\bar{a}}}$, five three-forms $C_{\mu\nu\rho}{}^{{\bar{a}}}$ and the seven-dimensional metric $G_{\mu\nu}$.
Interestingly, the quadratic constraint allows for two inequivalent types of $\textrm{CSO}(p,q,r)$ gaugings [@Samtleben:2005bp]. The first is triggered by gaugings in the $\mathbf{15}$ and is obtained by truncating M-theory on the hyperboloid $H^{p,q} \times T^r$, while the second is triggered by certain gaugings in the $\mathbf{40}'$ and is believed to be related to truncations of IIB supergravity on $H^{p,q} \times T^r$. We will now use EFT [@Berman:2010is] to show that these IIA / IIB truncations can be dualised into each other.
Generalised Scherk-Schwarz truncations
======================================
Our key tool for studying seven-dimensional consistent truncations is the ${\mathrm{SL}(5)}$ EFT [@Berman:2010is]. This is a reformulation of 11- and 10-dimensional supergravity which renders manifest the ${\mathrm{SL}(5)}$ symmetry known to appear under dimensional reduction to seven dimensions. In order to do this, extra coordinates are introduced so that there are 10 “internal coordinates” $Y^{ab}$ transforming in the antisymmetric representation of ${\mathrm{SL}(5)}$. In the case of a toroidal compactification of 11-dimensional supergravity, the extra six coordinates can be understood as duals to the wrapping modes of membranes.
With respect to the seven-dimensional “external” spacetime, the degrees of freedom of 11- and 10-dimensional supergravity coincide with those of the maximal gauged SUGRA and we denote them by the same symbols in calligraphic font: ${\cal M}_{ab}$, ${\cal A}_\mu{}^{ab}$, ${\cal B}_{\mu\nu\,a}$ and ${\cal C}_{\mu\nu\rho}{}^a$, ${\cal G}_{\mu\nu}$. These fields can depend on all of the external coordinates, $x^\mu$, but their dependence on the internal coordinates, $Y^{ab}$, is restricted by the so-called “section condition” [@Berman:2011cg] $$\partial_{[ab} f(x,Y)\, \partial_{cd]} g(x,Y) = \partial_{[ab} \partial_{cd]} f(x,Y) = 0 \,,$$ for all physical fields which we symbolically denote here by $f(x,Y)$ and $g(x,Y)$. The section condition can be solved by only allowing dependence on four coordinates $Y^{i5}$ where $i = 1, \ldots 4$ upon which the theory reduces to the full 11-dimensional supergravity.
Here we wish to consider the type II theories and so it is useful to decompose ${\mathrm{SL}(5)}\rightarrow {\mathrm{SL}(4)}\simeq \mathrm{Spin}(3,3)$, the relevant “T-duality” group. The coordinate representation then splits as $\mathbf{10} \rightarrow \mathbf{6} \oplus \mathbf{4}$ where the $\mathbf{6}$ consists of the 3 coordinates of IIA plus the 3 coordinates of IIB. Thus to consider only the type II theories, we restrict the coordinate dependence to the $\mathbf{6}$ by demanding all fields to be independent of $Y^{\alpha 5}$, where $\alpha = 1, \ldots 4$. Now the section condition reduces to $$\partial_{\alpha\beta} f(x,Y) \partial^{\alpha\beta} g(x,Y) = \partial_{\alpha\beta} \partial^{\alpha\beta} f(x,Y) = 0 \,,$$ where we define $\partial^{\alpha\beta} = \frac{1}{2} \epsilon^{\alpha\beta\gamma\delta} \partial_{\gamma\delta}$. There are two inequivalent solutions to this constraint, where all fields only depend on either $$y^m \equiv Y^{m4} ~~ (\mathrm{IIA}) \,, \qquad \mathrm{or } \qquad \tilde{y}_m \equiv \frac{1}{2} \epsilon_{mnp} Y^{np} ~~ (\mathrm{IIB}) \,,$$ with $m = 1, \ldots 3.$ Upon restricting the field dependence as above, the EFT reduces to either IIA or IIB SUGRA, as has been shown for the scalar sector in [@Blair:2013gqa], and for the full EFT in other dimensions, for example in [@Hohm:2013uia] for the $E_{7(7)}$ EFT. An important observation for what follows is that these two solutions of the section condition are related by the $\mathbb{Z}_2$ outer automorphism of ${\mathrm{SL}(4)}$ which takes\
$\partial_{\alpha\beta} \longleftrightarrow \partial^{\alpha\beta}$.
Given this framework, a natural way to generalise the Scherk-Schwarz Ansatz is to allow for a ${\mathrm{SL}(5)}$-valued twist $U_a{}^{{\bar{a}}}(Y)$ of the EFT fields, as follows [@Hohm:2014qga] $$\begin{split}
\mathcal{M}_{ab}\left(x,Y\right) &= U_{a}{}^{{\bar{a}}}(Y)\,U_{b}{}^{{\bar{b}}}(Y)\,{M}_{{\bar{a}}{\bar{b}}}(x)\,, \\
G_{\mu\nu}(x,Y) &= \rho^{-2}(Y)\,G_{\mu\nu}(x)\,,\\
{\cal A}_{\mu}{}^{ab}(x,Y) &= \rho^{-1}(Y)\, A_{\mu}{}^{\bar{a}\bar{b}}(x)U_{\bar{a}\bar{b}}{}^{ab}(Y) \,, \\
{\cal B}_{\mu\nu\,a}(x,Y) &= \rho^{-2}(Y)\, B_{\mu\nu\,\bar{a}}(x)\,U_{a}{}^{\bar{a}}(Y) \,, \\
{\cal C}_{\mu\nu\rho}{}^a(x,Y) &= \rho^{-3}(Y)\, C_{\mu\nu\rho}{}^{\bar{a}}(x)\,U_{\bar{a}}{}^{a}(Y) \,,
\label{eq:SSAnsatz}
\end{split}$$ where $\rho(Y)$ is some scalar function and $U_{{\bar{a}}{\bar{b}}}{}^{ab} = U_{{\bar{a}}}{}^{[a} U_{{\bar{b}}}{}^{b]}$. The metric and p-form fields in the internal directions can be read off from ${\cal M}_{ab}$, with the explicit formulae given in [@Malek:2015hma], while the remaining degrees of freedom of the 10- or 11-dimensional supergravity are encoded in ${\cal A}_{\mu}{}^{ab}$, ${\cal B}_{\mu\nu\,a}$ and ${\cal C}_{\mu\nu\rho}{}^a$.
This Ansatz leads to a maximal gauged SUGRA with the embedding tensor as a function of the twists [@Berman:2012uy], which must be nowhere vanishing. In particular, the embedding tensor is realised as the torsion of the flat connection of the extended space [@Coimbra:2011ky; @Berman:2013uda; @Blair:2014zba] $$\begin{split}
S_{\bar{a}\bar{b}} &= \frac{1}{\rho} \partial_{ab} U_{(\bar{a}}{}^a U_{\bar{b})}{}^{b} \,, \\
Z^{\bar{a}\bar{b},\bar{c}} &= \frac{1}{2\rho} \epsilon^{abcef} \left( U_{ef}{}^{\bar{a}\bar{b}} \, \partial_{ab} U_c{}^{\bar{c}} - U_{ef}{}^{[\bar{a}\bar{b}}\,\partial_{ab} U_c{}^{\bar{c}]} \right) \,, \\
\tau_{\bar{a}\bar{b}} &= - \frac{1}{2\rho}\left( \partial_{cd} U_{\bar{a}\bar{b}}{}^{cd} - 6\,\rho^{-1} \, U_{\bar{a}\bar{b}}{}^{cd} \,\partial_{cd} \rho \right) \,.
\end{split}
\label{eq:consistent}$$ One can show that the reduction is consistent if the embedding tensor is constant [@Aldazabal:2011nj; @Geissbuhler:2011mx; @Berman:2012uy], thus providing a set of differential equations which have to be solved for the twists $U_a{}^{{\bar{a}}}(Y)$ and $\rho(Y)$. A sufficient requirement for the embedding tensor to satisfy the quadratic constraint is that the twists obey the section condition.
Dualising consistent truncations
================================
Having reviewed the set-up for generalised Scherk-Schwarz truncations, let us now turn to the question of whether and when truncations of IIA have dual truncations of IIB and vice versa. To do so, let us decompose the embedding tensor under ${\mathrm{SL}(5)}\rightarrow {\mathrm{SL}(4)}\simeq \mathrm{Spin}(3,3)$ to find $$\begin{split}
\mathbf{15} &\rightarrow \mathbf{10} \oplus \mathbf{4} \oplus \mathbf{1} \,, \qquad \mathbf{10}' \rightarrow \mathbf{6} \oplus \mathbf{4} \,, \\
\mathbf{40}' &\rightarrow \mathbf{20}' \oplus \mathbf{10}' \oplus \mathbf{6} \oplus \mathbf{4}' \,.
\end{split}$$
Let us for now make a $\mathrm{GL}(4)$ Ansatz for the twist matrix $$U_a{}^{{\bar{a}}} = \begin{pmatrix}
\omega^{-1/2} V_\alpha{}^{{\bar{\alpha}}} & 0 \\ 0 & \omega^2
\end{pmatrix} \,, \qquad |V| = 1 \,. \label{eq:DiagAnsatz}$$ From one finds that this only generates non-zero gaugings in the $\mathbf{10}$’s and $\mathbf{6}$’s: $$\begin{split}
S_{{\bar{\alpha}}{\bar{\beta}}} &\equiv M_{{\bar{\alpha}}{\bar{\beta}}} = \rho^{-1} \omega V_{({\bar{\alpha}}}{}^\alpha \partial_{|\alpha\beta|} V_{{\bar{\beta}})}{}^\beta \,, \\
Z^{{\bar{5}}({\bar{\alpha}},{\bar{\beta}})} &\equiv \tilde{M}^{{\bar{\alpha}}{\bar{\beta}}} = \rho^{-1} \omega V_\alpha{}^{({\bar{\alpha}}} \partial^{|\alpha\beta|} V_\beta{}^{{\bar{\beta}})} \,, \\
2 \tau_{{\bar{\alpha}}{\bar{\beta}}} &= - \rho^{-1} \omega \left( \partial_{\alpha\beta} V_{{\bar{\alpha}}{\bar{\beta}}}{}^{\alpha\beta} -5 V_{{\bar{\alpha}}{\bar{\beta}}}{}^{\alpha\beta} \partial_{\alpha\beta} \ln \omega \right. \\
& \left. \quad + \,6 V_{{\bar{\alpha}}{\bar{\beta}}}{}^{\alpha\beta} \partial_{\alpha\beta} \ln \left(\rho^{-1}\omega\right) \right)\,, \\
6Z^{{\bar{5}}[{\bar{\alpha}},{\bar{\beta}}]} &\equiv 2\xi^{{\bar{\alpha}}{\bar{\beta}}} = \rho^{-1} \omega \left( \partial^{\alpha\beta} V_{\alpha\beta}{}^{{\bar{\alpha}}{\bar{\beta}}} - 5 V_{\alpha\beta}{}^{{\bar{\alpha}}{\bar{\beta}}} \partial^{\alpha\beta} \ln \omega \right) \,, \label{eq:CC10+6}
\end{split}$$ We now see that the outer automorphism of ${\mathrm{SL}(4)}$ $$V_\alpha{}^{{\bar{\alpha}}} \longleftrightarrow \left(V^{-T}\right)_{{\bar{\alpha}}}{}^{\alpha} \,, \qquad \partial_{\alpha\beta} \longleftrightarrow \partial^{\alpha\beta} \,, \label{eq:Duality}$$ induces a duality on the embedding tensor taking $$M_{{\bar{\alpha}}{\bar{\beta}}} \longleftrightarrow \tilde{M}^{{\bar{\alpha}}{\bar{\beta}}} \,, \qquad \tau_{{\bar{\alpha}}{\bar{\beta}}} \longleftrightarrow \tau^{{\bar{\alpha}}{\bar{\beta}}} \,, \qquad \xi^{{\bar{\alpha}}{\bar{\beta}}} \longleftrightarrow \xi_{{\bar{\alpha}}{\bar{\beta}}} \,, \label{eq:DualGaugings}$$ where we define $\tau^{{\bar{\alpha}}{\bar{\beta}}} = \frac12 \epsilon^{{\bar{\alpha}}{\bar{\beta}}{\bar{\gamma}}{\bar{\delta}}} \tau_{{\bar{\gamma}}{\bar{\delta}}}$ and $\xi_{{\bar{\alpha}}{\bar{\beta}}} = \frac{1}{2} \epsilon_{{\bar{\alpha}}{\bar{\beta}}{\bar{\gamma}}{\bar{\delta}}} \xi^{{\bar{\gamma}}{\bar{\delta}}}$. Thus, given a IIA / IIB truncation gauging only the $\mathbf{10}$’s and $\mathbf{6}$’s there is a consistent truncation of IIB / IIA with the dual gaugings as in . Furthermore, from the Ansatz , one can check that the reduction formulae for the NS-NS sector remains invariant under this duality. Note that the duality does not mix theories with trombone gaugings and those without.
This structure can also be seen in the half-maximal theory, where the $\mathbf{10} \oplus \mathbf{10}' \oplus \mathbf{6}$ correspond to the usual gaugings while the second $\mathbf{6}$ is the trombone of the half-maximal theory. As first observed in [@Dibitetto:2012rk], the quadratic constraint of the half-maximal theory is invariant under the duality , showing that in the half-maximal theory these gaugings come in pairs of equivalent theories. Here we realise this as a duality between IIA and IIB truncations, which in the maximal theory have inequivalent gaugings.
Let us now discuss more general embedding tensors, where the gaugings are not restricted to lie in the $\mathbf{10}$’s and $\mathbf{6}$’s. Such gaugings can be generated by relaxing the twist Ansatz . In general, such gaugings cannot be dualised as discussed in more detail in [@Malek:2015hma]. In particular, gaugings which involve the $\mathbf{4}$’s cannot be dualised. The differential equations governing the gaugings of the $\mathbf{4} \subset \mathbf{15}$ and $\mathbf{4} \subset \mathbf{10}$ are more restrictive than those governing the $\mathbf{4}' \subset \mathbf{40}'$. Thus, while reductions that lead to gaugings in the $\mathbf{4}$ can be dualised to give gaugings in the $\mathbf{4}'$, the reverse is no possible in general. Furthermore, one can check that unlike exchanging the $\mathbf{10}$’s and $\mathbf{6}$’s as in , exchanging $\mathbf{4} \longleftrightarrow \mathbf{4}'$ is not a symmetry of the quadratic constraint .
Example: IIA / IIB on $S^3$ and $H^{p,q}$
=========================================
Let us revisit the twist matrices of [@Hohm:2014qga] for the consistent truncation of IIA on $S^3$ and $H^{p,q}$. In the notation used here, they correspond exactly to a twist matrix of the form and provide gaugings in the $\mathbf{10} \subset \mathbf{15}$. Thus, the duality discussed above can be used to dualise these truncations to obtain the consistent truncation of IIB on the same internal spaces. The resulting seven-dimensional maximal gauged SUGRAs have an embedding tensor in the $\mathbf{10}' \subset \mathbf{40}'$ leading to the gauge group $\mathrm{CSO}(p,q,r) \times \left(U(1)\right)^{4-p-q}$.
A no-go theorem
===============
So far we have seen that the $\mathbf{10} \subset \mathbf{15}$ and $\mathbf{10}' \subset \mathbf{40}'$ gaugings correspond to $H^{p,q}$ truncations of IIA and IIB, respectively. These are related by the duality . A natural question is whether it is possible to obtain a gauging in the $\mathbf{10} \subset \mathbf{15}$ by a truncation of IIB or equivalently a gauging in the $\mathbf{10}' \subset \mathbf{40}'$ by a truncation of IIA.
To answer this question, let us study the consistency equations again. Assume that we have a IIA gauging, i.e. only $\partial_{m4} \neq 0$. Then we find that in order for a gauging to be uplifted to IIA it must satisfy $$W^{{\bar{a}}{\bar{b}},{\bar{c}}} E_{{\bar{a}}{\bar{b}}}{}^{54} = W^{{\bar{a}}{\bar{b}},{\bar{c}}} E_{{\bar{a}}{\bar{b}}}{}^{4m} = 0 \,, \label{eq:NoGoIIA}$$ where $$W^{{\bar{a}}{\bar{b}},{\bar{c}}} = Z^{{\bar{a}}{\bar{b}},{\bar{c}}} - 3 \epsilon^{{\bar{a}}{\bar{b}}{\bar{c}}{\bar{d}}{\bar{e}}} \tau_{{\bar{d}}{\bar{e}}} \,, \quad E_{{\bar{a}}{\bar{b}}}{}^{ab} = \rho^{-1} U_{{\bar{a}}{\bar{b}}}{}^{ab} \,.$$ This restricts the possible twist matrices given a certain gauging. In particular let us take $\tilde{M}^{{\bar{\alpha}}{\bar{\beta}}} = \eta^{{\bar{\alpha}}{\bar{\beta}}}$ as required for the $\mathbf{10}' \subset \mathbf{40}'$. If $\eta^{{\bar{\alpha}}{\bar{\beta}}}$ is non-degenerate we find requires the twist matrix to vanish. Thus, there is no IIA truncation yielding a non-degenerate gauging in the $\mathbf{10}' \subset \mathbf{40}'$ and by the duality there is no IIB truncation with gauging in the $\mathbf{10} \subset \mathbf{15}$. For the degenerate case, the truncation depends on less than three of the internal coordinates so that IIA / IIB coincide.
Let us conclude this section by giving the equivalent requirement for a gauging to be obtained by a IIB truncation. This is given by $$W^{{\bar{a}}{\bar{b}},{\bar{c}}} E_{{\bar{a}}{\bar{b}}}{}^{mn} = 0 \,.$$
Conclusions
===========
We showed how to dualise consistent truncations of IIA and IIB supergravity to seven-dimensional maximal gauged SUGRA. The duality is generated by an outer automorphism of ${\mathrm{SL}(4)}\simeq \mathrm{Spin}(3,3)$. Using the results of [@Hohm:2014qga] this allows us to prove the consistent truncation of IIB on $H^{p,q}$ and deduce its non-linear truncation Ansätze.
The duality leaves the NS-NS sector invariant and can be used to dualise IIA / IIB truncations with vanishing embedding tensor in the $\mathbf{4}'$ of ${\mathrm{SL}(4)}$. We also gave a necessary requirement for a seven-dimensional gauged SUGRA to be uplifted to IIA / IIB and used this to show that there is no IIA truncation gauging that $\mathbf{10}' \subset \mathbf{40}'$ and by duality no IIB truncation gauging the $\mathbf{10} \subset \mathbf{15}$. This is related to the fact that the half-maximal gaugings mixing these two irreps must violate the section condition [@Lee:2015xga].
It would be interesting to study this duality in other dimensions. For $d \le 7$ the embedding tensor has only one irreducible component in addition to the trombone. It is then not immediately clear whether the IIA / IIB truncations would define inequivalent lower-dimensional gauged SUGRAs. An interesting starting point would be truncations to three dimensions where there are known to be two inequivalent $\mathrm{SO}(8)$ gaugings believed to be related to $S^7$ truncations of IIA / IIB.
[10]{} <span style="font-variant:small-caps;">M. Duff</span>, <span style="font-variant:small-caps;">B. Nilsson</span>, <span style="font-variant:small-caps;">C. Pope</span>, and <span style="font-variant:small-caps;">N. Warner</span>, **149**, 90 (1984).
<span style="font-variant:small-caps;">B. de Wit</span> and <span style="font-variant:small-caps;">H. Nicolai,</span> **281**, 211 (1987).
<span style="font-variant:small-caps;">H. Nastase</span>, <span style="font-variant:small-caps;">D. Vaman</span>, and <span style="font-variant:small-caps;">P. van Nieuwenhuizen</span>, **581**, 179 (2000).
<span style="font-variant:small-caps;">M. Cvetic</span>, <span style="font-variant:small-caps;">H. Lu</span>, and <span style="font-variant:small-caps;">C. N. Pope</span>, **62** 060428 (2000).
<span style="font-variant:small-caps;">G. Aldazabal</span>, <span style="font-variant:small-caps;">W. Baron</span>, <span style="font-variant:small-caps;">D. Marqués</span>, and <span style="font-variant:small-caps;">C. Núñez</span>, **1111**, 052 (2011).
<span style="font-variant:small-caps;">D. Geissbühler</span>, **111**, 116 (2011).
<span style="font-variant:small-caps;">D. S. Berman</span>, <span style="font-variant:small-caps;">E. T. Musaev</span>, and <span style="font-variant:small-caps;">D. C. Thompson</span>, **1210**, 174 (2012).
<span style="font-variant:small-caps;">K. Lee</span>, <span style="font-variant:small-caps;">C. Strickland-Constable</span>, and <span style="font-variant:small-caps;">D. Waldram</span>, arXiv:1401.3360 \[hep-th\].
<span style="font-variant:small-caps;">K. Lee</span>, <span style="font-variant:small-caps;">C. Strickland-Constable</span> and <span style="font-variant:small-caps;">D. Waldram</span>, arXiv:1506.03457 \[hep-th\].
<span style="font-variant:small-caps;">O. Hohm</span> and <span style="font-variant:small-caps;">H. Samtleben</span>, **1501**, 131 (2015).
<span style="font-variant:small-caps;">C. Hull</span> and <span style="font-variant:small-caps;">B. Zwiebach</span>, **0909**, 099 (2009).
<span style="font-variant:small-caps;">D. S. Berman</span> and <span style="font-variant:small-caps;">M. J. Perry</span>, **1106**, 074 (2011).
<span style="font-variant:small-caps;">A. Coimbra</span>, <span style="font-variant:small-caps;">C. Strickland-Constable</span> and <span style="font-variant:small-caps;">D. Waldram</span>, **1402**, 054 (2014).
<span style="font-variant:small-caps;">E. Malek</span> and <span style="font-variant:small-caps;">H. Samtleben</span>, arXiv:1510.03433 \[hep-th\] (to appear in JHEP).
<span style="font-variant:small-caps;">H. Samtleben</span> and <span style="font-variant:small-caps;">M. Weidner</span>, **725**, 383 (2005).
<span style="font-variant:small-caps;">D. S. Berman</span>, <span style="font-variant:small-caps;">C. D. A. Blair</span>, <span style="font-variant:small-caps;">E. Malek</span>, and <span style="font-variant:small-caps;">M. J. Perry</span>, **29**, 1450080 (2014).
<span style="font-variant:small-caps;">C. D. A. Blair</span> and <span style="font-variant:small-caps;">E. Malek</span>, **1503**, 144 (2015).
<span style="font-variant:small-caps;">D. S. Berman</span>, <span style="font-variant:small-caps;">H. Godazgar</span>, <span style="font-variant:small-caps;">M. Godazgar</span> and <span style="font-variant:small-caps;">M. J. Perry</span>, **1201**, 012 (2012).
<span style="font-variant:small-caps;">A. Le Diffon</span> and <span style="font-variant:small-caps;">H. Samtleben</span>, **811**, 1 (2009).
<span style="font-variant:small-caps;">C. D. Blair</span>, <span style="font-variant:small-caps;">E. Malek</span>, and <span style="font-variant:small-caps;">J.-H. Park</span>, **1401**, 172 (2014).
<span style="font-variant:small-caps;">O. Hohm</span> and <span style="font-variant:small-caps;">H. Samtleben</span>, **89**, 066017 (2014).
<span style="font-variant:small-caps;">G. Dibitetto</span>, <span style="font-variant:small-caps;">J. Fernández-Melgarejo</span>, <span style="font-variant:small-caps;">D. Marqués</span> and D. Roest, **60** 1123 (2012).
[^1]: E-mail:
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The electronic structures of [$\mathrm{U}X_3$ ($X=\mathrm{Al}$, $\mathrm{Ga}$, and $\mathrm{In}$)]{}were studied by photoelectron spectroscopy to understand the the relationship between their electronic structures and magnetic properties. The band structures and Fermi surfaces of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}were revealed experimentally by angle-resolved photoelectron spectroscopy (ARPES), and they were compared with the result of band-structure calculations. The topologies of the Fermi surfaces and the band structures of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}were explained reasonably well by the calculation, although bands near the Fermi level ([$E_\mathrm{F}$]{}) were renormalized owing to the finite electron correlation effect. The topologies of the Fermi surfaces of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}are very similar to each other, except for some minor differences. Such minor differences in their Fermi surface or electron correlation effect might take an essential role in their different magnetic properties. No significant changes were observed between the ARPES spectra of [$\mathrm{UGa}_3$]{}in the paramagnetic and antiferromagnetic phases, suggesting that [$\mathrm{UGa}_3$]{}is an itinerant weak antiferromagnet. The effect of chemical pressure on the electronic structures of $\mathrm{U}X_3$ compounds was also studied by utilizing the smaller lattice constants of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}than that of [$\mathrm{UIn}_3$]{}. The valence band spectrum of [$\mathrm{UIn}_3$]{}is accompanied by a satellite-like structure on the high-binding-energy side. The core-level spectrum of [$\mathrm{UIn}_3$]{}is also qualitatively different from those of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}. These findings suggest that the [$\mathrm{U}~5f$]{}states in [$\mathrm{UIn}_3$]{}are more localized than those in [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}.'
author:
- 'Shin-ichi Fujimori'
- Masaaki Kobata
- Yukiharu Takeda
- Tetsuo Okane
- Yuji Saitoh
- Atsushi Fujimori
- Hiroshi Yamagami
- Yoshinori Haga
- Etsuji Yamamoto
- Yoshichika Ōnuki
bibliography:
- 'UGa3\_UAl3.bib'
title: 'Electronic structures of $\mathrm{U}X_3$ ($X=\mathrm{Al}$, $\mathrm{Ga}$, and $\mathrm{In}$) studied by photoelectron spectroscopy'
---
INTRODUCTION
============
Magnetism in $f$-electron materials is a test stand of the modern concepts of magnetism. Hybridization between $f$ electrons and ligand states results in a competition between itinerant and localized natures of $f$ electrons, which manifests as the complex magnetic behaviors of these compounds. To unveil the microscopic origins of these magnetic properties, systematic control of $f$-ligand hybridization in $f$-electron materials is desirable. Binary uranium compounds with $\mathrm{U}X_3$ ($X$ is a group 13 or 14 element) stoichiometry and $\mathrm{AuCu_3}$-type crystal structure comprise an ideal model system to understand the relationship between hybridization and the origin of a rich variety of magnetic properties in uranium-based compounds. Figure \[UX3\_crystal\] summarizes the physical parameters and properties of this series of compounds [@UX3_SSC; @UX3; @UX3_dHvA; @Onuki_review_JPSJ; @UIn3_NMR]. They exhibit various physical properties depending on $X$, and their specific heat coefficients range from $\gamma = 14~\mathrm{mJ/mol K^2}$ in $\mathrm{USi_3}$ to $\gamma =170~\mathrm{mJ/mol K^2}$ in $\mathrm{USn_3}$. Their lattice constants are considerably larger than the Hill limit ($\sim3.4~\mathrm{\AA}$), suggesting that the hybridization between $f$-state and ligand $X$ states is a key parameter for these compounds. Generally, as $X$ becomes heavier, the lattice constant increases, and the compound tends to be magnetic. Among the considered series of compounds, [$\mathrm{U}X_3$ ($X=\mathrm{Al}$, $\mathrm{Ga}$, and $\mathrm{In}$)]{}has very different physical properties depending on $X$, namely, enhanced Pauli paramagnetism in [$\mathrm{UAl}_3$]{}[@UAl3_dHvA], itinerant antiferromagnetism in [$\mathrm{UGa}_3$]{}[@UGa3_Kaczorowski], and localized antiferromagnetism in [$\mathrm{UIn}_3$]{}[@UIn3_NMR]. Therefore, they comprise an excellent model system to study the origin of magnetism in $f$-electron systems.
![(Online color) Crystal structure and summary of physical properties of $\mathrm{U}X_3$ compounds [@UX3_SSC; @UX3; @UX3_dHvA; @Onuki_review_JPSJ; @UIn3_NMR]. []{data-label="UX3_crystal"}](UX3_crystal.pdf)
In the present work, we studied the electronic structures of [$\mathrm{U}X_3$ ($X=\mathrm{Al}$, $\mathrm{Ga}$, and $\mathrm{In}$)]{}to reveal the origin of their different magnetic properties by using valence-band and core-level photoelectron spectroscopies. Furthermore, angle-resolved photoelectron spectroscopy (ARPES) with photon energies of [$h\nu =575-650~\mathrm{eV}$]{} were performed for [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}. Their detailed band structures and Fermi surfaces were revealed experimentally, and they were compared with the result of the band-structure calculation treating [$\mathrm{U}~5f$]{}electrons as being itinerant.
[$\mathrm{UAl}_3$]{}is a paramagnetic compound with a relatively large specific heat coefficient of [$\gamma = 43~\mathrm{mJ/mol K^2}$]{} [@UX3_dHvA]. In a dHvA study of [$\mathrm{UAl}_3$]{}, several branches were observed, and a few of them were explained by band-structure calculations based on itinerant [$\mathrm{U}~5f$]{}states [@UX3_dHvA; @UAl3_dHvA]. In recent years, fully relativistic band-structure calculations could explain the origin of most of other branches, suggesting that [$\mathrm{U}~5f$]{}electrons have very itinerant characteristics in this compound [@UAl3_Maehira].
[$\mathrm{UGa}_3$]{}is an antiferromagnet with [$T_\mathrm{N}=67~\mathrm{K}$]{} [@UGa3_AF]. Its specific heat coefficient is slightly higher than that of [$\mathrm{UAl}_3$]{}. Many experimental and theoretical studies on [$\mathrm{UGa}_3$]{}have suggested that [$\mathrm{UGa}_3$]{}is an itinerant antiferromagnet [@UGaSn3; @UX3_dHvA; @UGa3_optical; @UGa3_Hiess; @UGa3_Usuda; @UGa3_positron]. Its magnetic ordering is of type-II with the magnetic propagating vector $\vec{Q}=\lfloor \frac{1}{2} \frac{1}{2} \frac{1}{2} \rfloor$ [@UGa3_neutron]. There exists another phase transition at $T=40~\mathrm{K}$, and it has been ascribed to the reorientation of magnetic moments, although the direction of magnetic moments has been controversial [@UGa3_NMR; @UGa3_Mossbauer]. Several experimental methods have been applied for studying electronic structure of [$\mathrm{UGa}_3$]{}such as resonant photoemission [@UGa3_RPES], dHvA measurement [@UX3_dHvA; @UGa3_Aoki], positron annihilation [@UGa3_positron], and magnetic X-ray scattering [@UGa3_MXS]. Despite these extensive studies, the details of its electronic structure have not been well understood. An interesting question is the difference in the magnetic properties of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}. They have very similar lattice constants, but [$\mathrm{UAl}_3$]{}is a paramagnet, while [$\mathrm{UGa}_3$]{}is an antiferromagnet. Therefore, the magnetic ordering of [$\mathrm{UGa}_3$]{}originates from the tiny differences in their electronic structures.
[$\mathrm{UIn}_3$]{}is an antiferromagnet with [$T_\mathrm{N}=88~\mathrm{K}$]{} [@UIn3_NMR]. The lattice constants of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}are almost identical, while that of [$\mathrm{UIn}_3$]{}is about 8 % larger than theirs. This leads to the weaker hybridized nature of [$\mathrm{U}~5f$]{}in [$\mathrm{UIn}_3$]{}compared to those in [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}. $\mathrm{^{115}In}$-NMR and NQR studies revealed that [$\mathrm{U}~5f$]{}electrons have localized natures well above the Néel temperature [@UIn3_NMR]. They further suggested that a plausible ordering vector is along the $\lfloor 110 \rfloor$ direction. Meanwhile, there are only a few studies on the electronic structure of [$\mathrm{UIn}_3$]{}. Sarma [*et al.*]{}conducted a resonant photoemission study of $\mathrm{U(Sn,In)_3}$ and found that the spectral profiles of the [$\mathrm{U}~5f$]{}contributions do not show any significant changes within the series [@UIn3_RPES]. In a dHvA study of [$\mathrm{UIn}_3$]{}, several branches originating from closed Fermi surfaces and multiply-connected Fermi surfaces were observed [@UIn3_dHvA]. The estimated electron masses of these branches were $10-33 m_0$, suggesting the existence of heavy quasi-particle bands. By contrast, comparison with band-structure calculation has not been performed yet, and the overall topology of the Fermi surface is not well understood. Therefore, knowledge about its electronic structure is very limited at present.
An interesting standpoint is that [$\mathrm{UIn}_3$]{}is considered as [$\mathrm{UGa}_3$]{}or [$\mathrm{UAl}_3$]{}under negative pressures. Pressure is a clean tuning parameter for controlling the physical properties of strongly correlated materials. In particular, it has been used to tune the electronic structure of $f$-electron materials to explore their quantum-criticality and unconventional superconductivities. Meanwhile, the effect of pressure on their electronic structures has been not well studied because spectroscopic studies are very difficult to perform in high-pressure cells. Therefore, [$\mathrm{U}X_3$ ($X=\mathrm{Al}$, $\mathrm{Ga}$, and $\mathrm{In}$)]{}can be used as a model system for studying the pressure effect by using photoelectron spectroscopy.
EXPERIMENTAL PROCEDURES
=======================
Photoemission experiments were performed at the soft X-ray beamline BL23SU in SPring-8 [@BL23SU; @BL23SU2]. High-quality single crystals were grown by the self-flux method, as described in Refs. [@UAl3_dHvA; @UGa3_Aoki; @UIn3_dHvA]. Clean sample surfaces were obtained by cleaving the samples [*in situ*]{} with the surface under an ultra-high vacuum (UHV) condition. Among the series of compounds, we could not obtain a flat cleaving surface in the case of [$\mathrm{UIn}_3$]{}. The ARPES spectra of [$\mathrm{UIn}_3$]{}were dominated by non-dispersive features that might have originated from elastically scattered photoelectrons from irregular surface. Therefore, only angle-integrated photoemission spectra are shown in the present paper. The overall energy resolution in the angle-integrated photoemission experiments at $h\nu = 800$ and $850~\mathrm{eV}$ was about $120~\mathrm{meV}$, while that in the ARPES experiments at [$h\nu =575-650~\mathrm{eV}$]{} was about $100~\mathrm{meV}$. The position of [$E_\mathrm{F}$]{}was determined carefully by measuring of the vapor-deposited gold film. During the measurements, the vacuum was typically $<1 \times 10^{-8}~\mathrm{Pa}$, and the sample surfaces were stable for the duration of measurements ($1-2$ days) because no significant changes were observed in the ARPES spectra during the measurement period. The positions of ARPES cuts were determined by assuming a free-electron final state with an inner potential of $V_{0}=12~\mathrm{eV}$. Background contributions in ARPES spectra originated from elastically scattered photoelectrons due to surface disorder or phonons were subtracted by assuming momentum-independent spectra. The details of the procedure are described in Ref. [@UGe2_UCoGe_ARPES].
RESULTS
=======
![(Online color) Angle-integrated photoemission spectra of [$\mathrm{U}X_3$ ($X=\mathrm{Al}$, $\mathrm{Ga}$, and $\mathrm{In}$)]{}. (a) Valence band spectrum of [$\mathrm{U}X_3$ ($X=\mathrm{Al}$, $\mathrm{Ga}$, and $\mathrm{In}$)]{}measured at [$h\nu =800~\mathrm{eV}$]{}. The spectrum of [$\mathrm{UAl}_3$]{}is superimposed onto the spectra of [$\mathrm{UGa}_3$]{}and [$\mathrm{UAl}_3$]{}. (b) [$\mathrm{ U } ~ 4f $]{} core-level spectra of [$\mathrm{U}X_3$ ($X=\mathrm{Al}$, $\mathrm{Ga}$, and $\mathrm{In}$)]{}. The spectra of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}were measured at [$h\nu =800~\mathrm{eV}$]{} while that of [$\mathrm{UIn}_3$]{}was measured at [$h\nu =850~\mathrm{eV}$]{} to avoid the contribution of $\mathrm{In}$-originated Auger signals, which overlap with [$\mathrm{ U } ~ 4f $]{} core-level. The spectrum of [$\mathrm{UAl}_3$]{}was superimposed onto the spectra of [$\mathrm{UGa}_3$]{}and [$\mathrm{UIn}_3$]{}as shown by the dotted line. []{data-label="AIPES"}](UX3_AIPES.pdf)
Angle-integrated photoemission spectra of [$\mathrm{U}X_3$ ($X=\mathrm{Al}$, $\mathrm{Ga}$, and $\mathrm{In}$)]{}
-----------------------------------------------------------------------------------------------------------------
Figure \[AIPES\] summarizes the angle-integrated photoemission spectra of [$\mathrm{U}X_3$ ($X=\mathrm{Al}$, $\mathrm{Ga}$, and $\mathrm{In}$)]{}. Figure \[AIPES\](a) shows the valence band spectra of [$\mathrm{U}X_3$ ($X=\mathrm{Al}$, $\mathrm{Ga}$, and $\mathrm{In}$)]{}. The spectra were measured at $20~\mathrm{K}$, and [$\mathrm{UAl}_3$]{}was in the paramagnetic phase while [$\mathrm{UGa}_3$]{}and [$\mathrm{UIn}_3$]{}were in the antiferromagnetic phase. According to the calculated cross-sections of atomic orbitals [@Atomic], the cross-section of [$\mathrm{U}~5f$]{}orbitals is more than one-order larger than those of [$\mathrm{ Al } ~ 3s, p $]{}, [$\mathrm{ Ga } ~ 4s, p $]{}, and [$\mathrm{ In } ~ 5s, p $]{} orbitals. Therefore, the signals from [$\mathrm{U}~5f$]{}states are dominant in these spectra. These spectra exhibit an asymmetric shape with a sharp peak just below the Fermi energy. Their spectral profiles are very similar to those of itinerant uranium compounds, such as $\mathrm{UB_2}$ [@UB2_ARPES] and $\mathrm{UN}$ [@UN_ARPES], suggesting that [$\mathrm{U}~5f$]{}states have an itinerant character in these compounds. By the contrast, the spectral profile of [$\mathrm{UIn}_3$]{}is slightly different from those of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}. To understand the differences in their spectral profiles, we superimposed the spectrum of [$\mathrm{UAl}_3$]{}on the spectra of [$\mathrm{UGa}_3$]{}and [$\mathrm{UIn}_3$]{}. The spectrum of [$\mathrm{UGa}_3$]{}is almost identical to that of [$\mathrm{UAl}_3$]{}, while that of [$\mathrm{UIn}_3$]{}has a shoulder structure at [$E_{\mathrm{B}} \sim 0.5~\mathrm{eV}$]{} whose tail extends [$E_{\mathrm{B}} \sim 2~\mathrm{eV}$]{}. Note that the structure cannot be due to the antiferromagnetic transition because a similar structure does not exist in the valence band spectrum of [$\mathrm{UGa}_3$]{}, which was also measured in the antiferromagnetic phase. Therefore, this shoulder structure has originated from the incoherent part of the correlated [$\mathrm{U}~5f$]{}states. A similar satellite structure was observed in the valence band spectrum of $\mathrm{UBe_{13}}$ [@Laub_UBe13] which is also considered as the contribution of correlated [$\mathrm{U}~5f$]{}states.
To further understand the nature of [$\mathrm{U}~5f$]{}states in these compounds, we measured the $\mathrm{U}~4f_{7/2}$ core-level spectra of [$\mathrm{U}X_3$ ($X=\mathrm{Al}$, $\mathrm{Ga}$, and $\mathrm{In}$)]{}, which are shown in Fig. \[AIPES\](b). These spectra were replotted from Ref. [@SF_review_JPSJ]. The core-level spectrum is a sensitive probe of the local electronic structure [@Ucore; @SF_review_JPSJ]. The spectra of these compounds have an asymmetric line shape with a tail toward the higher binding energies. In addition to the main line located at [$E_{\mathrm{B}} = 377.0-377.3~\mathrm{eV}$]{}, a weak satellite structure can be observed in the high-binding-energy side of the main line at about $7~\mathrm{eV}$ ([$E_{\mathrm{B}} \sim 384~\mathrm{eV}$]{}). This is called as the $7~\mathrm{eV}$ satellite [@Laub_U4f], and it has been observed in many uranium based compounds [@Ucore; @SF_review_JPSJ]. The core-level spectrum of [$\mathrm{UAl}_3$]{}was superimposed on the spectra of [$\mathrm{UGa}_3$]{}and [$\mathrm{UIn}_3$]{}, as in the case of their valence band spectra. The spectra exhibit considerable differences. The tail of the main line of [$\mathrm{UGa}_3$]{}is more enhanced than that of [$\mathrm{UAl}_3$]{}, suggesting that the asymmetry of the main line is larger than the case of [$\mathrm{UAl}_3$]{}. In general, the asymmetry of the main line is associated with the density of states (DOS) at [$E_\mathrm{F}$]{}in itinerant uranium compounds [@Ucore]. Compounds with higher DOSs at [$E_\mathrm{F}$]{}have main lines with greater asymmetry. Therefore, the larger asymmetry in the main line of [$\mathrm{UGa}_3$]{}than that of [$\mathrm{UAl}_3$]{}suggests that [$\mathrm{UGa}_3$]{}has higher DOS at [$E_\mathrm{F}$]{}than that does [$\mathrm{UAl}_3$]{}. By contrast, the spectrum of [$\mathrm{UIn}_3$]{}is broadened and is located on the higher-binding-energy side. In addition, its spectral shape becomes more symmetric. A detailed analysis of the main line of [$\mathrm{UIn}_3$]{}suggests that it consists of two components, and the one on the high-binding-energy side becomes dominant [@SF_review_JPSJ]. This is a characteristic feature of the main lines of localized compounds, suggesting that [$\mathrm{U}~5f$]{}in [$\mathrm{UIn}_3$]{}is more localized than those in [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}. The intensities of the satellite structures of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}are similar to those of other itinerant compounds [@Ucore; @SF_review_JPSJ], but that of [$\mathrm{UIn}_3$]{}is more enhanced than those of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}. In general, the intensity of the satellite structure is more enhanced in localized compounds [@Ucore], and this slightly enhanced satellite intensity also indicates that [$\mathrm{U}~5f$]{}electrons are more localized in [$\mathrm{UIn}_3$]{}than in [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}. It should be noted that there are some theoretical attempts to reproduce these structures by the single impurity Anderson model [@Okada_core; @Zwicknagl_core]. A systematic trend observed in the valence band and core-level spectra of these compounds would be helpful to further understand the microscopic origin of these satellite structures.
Accordingly, both valence band and core-level spectra indicate that the [$\mathrm{U}~5f$]{}states of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}have an essentially itinerant character. [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}have similar degrees of delocalization of the [$\mathrm{U}~5f$]{}states, although the correlation effect is somewhat enhanced in [$\mathrm{UGa}_3$]{}. By contrast, the nature of the electronic structure of [$\mathrm{UIn}_3$]{}is clearly different. The valence band spectrum of [$\mathrm{UIn}_3$]{}is accompanied with the satellite structure, and the main lines and satellite structures in the core-level spectra are also different from those of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}. This should be due to the reduced hybridization in [$\mathrm{UIn}_3$]{}originated from its larger lattice constant than those of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}.
Band-structure calculation
--------------------------
Before showing the experimental ARPES spectra, we overview the result of our band-structure calculations of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}. Figure \[3DFS\] shows the calculated Fermi surfaces of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}in the paramagnetic phase. In the present study, band-structure calculation in the paramagnetic phase is used for comparison between ARPES spectra and the calculation of [$\mathrm{UGa}_3$]{}because the changes in the spectral profiles due to antiferromagnetic transition are very small, as discussed in Sec. \[UGa3\_AF\].
Figure \[3DFS\] (a) shows the calculated Fermi surfaces of [$\mathrm{UAl}_3$]{}. The calculated Fermi surface of [$\mathrm{UAl}_3$]{}consists of multiply-connected hole-type Fermi surfaces formed by band 11 and small electron pocket formed by band 12. The Fermi surface formed by band 11 consists of two large Fermi surfaces centered at the [$\mathrm{\Gamma}$]{}and the [$\mathrm{R}$]{} points, and they are connected along the $\lfloor 111 \rfloor$ direction. Note that the topology of the calculated Fermi surface is basically consistent with that obtained from previous band-structure calculations [@UX3_dHvA; @UAl3_dHvA; @UAl3_Maehira], although there exist minor differences. For example, the Fermi surfaces centered at [$\mathrm{\Gamma}$]{}and [$\mathrm{R}$]{} points are not connected in Ref. [@UAl3_dHvA]. Furthermore, the electron pocket centered at the [$\mathrm{M}$]{} point does not exist in these previous calculations. These differences might have originated from tiny differences in the band structure near [$E_\mathrm{F}$]{}. There exist very narrow bands with energy dispersions of less than 50 meV in these calculations, and tiny changes in the structures of these bands due to different computational factors can alter the shapes of Fermi surface very easily. Experimentally, several branches originating from this Fermi surface were observed by dHvA measurement [@UAl3_dHvA], and they were interpreted reasonably as signals from the two large Fermi surfaces centered at the [$\mathrm{\Gamma}$]{}and the [$\mathrm{R}$]{} points.
Figure \[3DFS\] (b) shows the calculated Fermi surfaces of [$\mathrm{UGa}_3$]{}. The topologies of the calculated Fermi surfaces of [$\mathrm{UGa}_3$]{}are very similar to those of [$\mathrm{UAl}_3$]{}, but those of [$\mathrm{UGa}_3$]{}have considerably complicated structures. Band 25 forms a thin cubic frame-like Fermi surface centered at the [$\mathrm{R}$]{} point which has no equivalent in the Fermi surface of [$\mathrm{UAl}_3$]{}. Band 26 forms a cubic Fermi surface centered at the [$\mathrm{R}$]{} point, which is very similar to the Fermi surface formed by band 11 in the case of [$\mathrm{UAl}_3$]{}, although its shape is closer to cubic in the case of [$\mathrm{UGa}_3$]{}. It also forms a small hole pocket at the [$\mathrm{\Gamma}$]{}point, but the size of which is considerably smaller than the corresponding hole pocket in [$\mathrm{UAl}_3$]{}. As a result, the large Fermi surface centered at the [$\mathrm{R}$]{} and the hole pocket centered at the [$\mathrm{\Gamma}$]{}point are disconnected in [$\mathrm{UGa}_3$]{}. Band 27 forms a hollow spherical Fermi surface around the [$\mathrm{\Gamma}$]{}point, which has no equivalent in the Fermi surface in [$\mathrm{UAl}_3$]{}. The topology of these calculated Fermi surfaces is essentially identical to the topology obtained in previous calculations [@UGa3_positron], although there are some minor differences. dHvA measurement of [$\mathrm{UGa}_3$]{}was performed, and the branches observed were compared with the results of band-structure calculation in the antiferromagnetic phase, but any satisfactory agreement was not obtained between them [@UGa3_Aoki]. A positron annihilation study suggested overall agreement between the experimental data and the band-structure calculation in the paramagnetic phase, but the details of the Fermi surface were not understood [@UGa3_positron]. Therefore, information about the Fermi surface of [$\mathrm{UGa}_3$]{}is very limited.
Band structure and Fermi surface of [$\mathrm{UAl}_3$]{}
--------------------------------------------------------
We begin with the overall band structure of [$\mathrm{UAl}_3$]{}. In Fig. \[UAl3\_band\], we summarize the experimental ARPES spectra and the result of the band-structure calculation of [$\mathrm{UAl}_3$]{}along several high-symmetry lines. Figure \[UAl3\_band\] (a) shows the ARPES spectra of [$\mathrm{UAl}_3$]{}measured along the $\mathrm{X} - \mathrm{\Gamma}-\mathrm{M} - \mathrm{X}$ high-symmetry line at the photon energy of [$h\nu =645~\mathrm{eV}$]{}. Note that the locations of the [$\mathrm{X}$]{} point in the leftmost and the rightmost sides have different measurement configurations, and the spectra show different profiles owing to matrix element effects. Clear energy dispersions can be observed. The strongly dispersive features corresponding to the higher-binding energy of [$E_{\mathrm{B}} >1~\mathrm{eV}$]{} are mainly the contributions of the $\mathrm{Al}~3s, 3p$ states. By contrast, the less dispersive features near [$E_\mathrm{F}$]{}are quasi-particle bands with dominant contribution of the [$\mathrm{U}~5f$]{}states. They have finite energy dispersions, and form the Fermi surface of [$\mathrm{UAl}_3$]{}. Figure \[UAl3\_band\] (b) shows the band dispersions and the simulated ARPES spectra based on the band-structure calculation along the same high-symmetry lines. The color coding of the bands is the projection of the contributions of the [$\mathrm{U}~5f$]{}states. The overall experimental band structure is well explained by the band-structure calculation. The dispersive features located in the higher binding energies correspond well with bands 5–8 in the calculation. The feature near [$E_\mathrm{F}$]{}seems to correspond to bands 9–12. The overall shapes of these bands agree reasonably well between experimental and calculation results. Figure \[UAl3\_band\] (c) shows the experimental ARPES spectra of [$\mathrm{UAl}_3$]{}measured along the $\mathrm{M}-\mathrm{X}-\mathrm{R}-\mathrm{M}$ high-symmetry line at the photon energy of [$h\nu =575~\mathrm{eV}$]{}. There exist similar types of energy dispersions to the spectra shown in Fig. \[UAl3\_band\] (a), and their overall structure can be explained by the band-structure calculation and the simulated ARPES spectra shown in Fig. \[UAl3\_band\](d).
To further understand the electronic structure near [$E_\mathrm{F}$]{}, blow-up of the ARPES spectra and the experimental and calculated Fermi surfaces of [$\mathrm{UAl}_3$]{}are shown in Fig. \[UAl3\_band\_FS\]. Figure \[UAl3\_band\_FS\] (a) shows a comparison of the ARPES spectra and the calculated energy dispersions together with the simulated ARPES spectra along the $\mathrm{X} - \mathrm{\Gamma} - \mathrm{M} - \mathrm{X}$ high-symmetry line. These spectra are divided by the Fermi – Dirac function broadened by the instrumental energy resolution to avoid the influences of Fermi cut-off. There exist one-to-one correspondences between the experimentally observed bands and the calculated bands. Band 11 forms a hole pocket centered at the [$\mathrm{\Gamma}$]{}point in the calculation, and there exist very similar structures in the experimental spectra. In addition, a hole pocket centered at the [$\mathrm{X}$]{} point is recognized in the profiles at the leftmost [$\mathrm{X}$]{} point of the experimental spectra, but there is no corresponding Fermi surface in the calculated results. Although the experimental spectra around the [$\mathrm{M}$]{} point are more featureless than the calculated result, there exists a similar high-intensity region near [$E_\mathrm{F}$]{}in the calculation. Figure \[UAl3\_band\_FS\] (b) shows a comparison between the experimental Fermi surface map (upper panel) and the simulated ARPES spectra (lower panel). Both the experimental and the calculated Fermi surface maps have very similar features. Especially, very similar squared features centered at the [$\mathrm{\Gamma}$]{}point were observed in both the experimental and the calculated Fermi surface map. By contrast, the experimental and calculated features around the [$\mathrm{M}$]{} point are somewhat different. The experimental Fermi surface map has a large circular region with enhanced intensity, while the calculation predicts a more complicated structure consisting of a small circular region with enhanced intensity at the [$\mathrm{M}$]{} point surrounded by arcs. This difference originates from plainer feature of bands near [$E_\mathrm{F}$]{}around the [$\mathrm{M}$]{} point in the experiment than in the calculation as shown in Fig. \[UAl3\_band\_FS\] (a). This might be due to the renormalization of experimental band corresponding to band 11, which leads to the featureless structure in the experimental Fermi surface map.
Figure \[UAl3\_band\_FS\] (c) shows the same comparison of Fig. \[UAl3\_band\_FS\] (a), but along the $\mathrm{M} - \mathrm{X} - \mathrm{R} - \mathrm{M}$ high-symmetry line. Very similar correspondence between the experimental and the calculated results can be seen in these spectra. There is overall agreement between the experimental and the calculated results. Especially, the feature around the [$\mathrm{M}$]{} point shows good agreement among them. In the calculation, two bands form Fermi surfaces around the [$\mathrm{M}$]{} point along the $\mathrm{R} - \mathrm{M}$ high-symmetry line, but they cannot be resolved in the experimental spectra. Meanwhile, as in the case of the $\mathrm{X} - \mathrm{\Gamma} - \mathrm{M} - \mathrm{X}$ high-symmetry line shown in Fig. \[UAl3\_band\_FS\] (a), bands near [$E_\mathrm{F}$]{}are pushed toward the lower-binding-energy side, and their profiles become more featureless. Figure \[UAl3\_band\_FS\] (d) shows a comparison between the experimental Fermi surface map and the simulated ARPES spectra. There is a similar relationship between the experimental Fermi surface map and the result of the band-structure calculation within the $\mathrm{\Gamma} - \mathrm{M} - \mathrm{X}$ high-symmetry plane shown in Fig. \[UAl3\_band\_FS\] (b). Although the overall shape of the features is very similar between the experiment and the calculation, the calculated map has a more complicated structure. This more featureless nature of the experimental Fermi surface map might be also due to the renormalized nature of bands near [$E_\mathrm{F}$]{}. Nevertheless, the feature corresponding to the cubic Fermi surface centered at the [$\mathrm{R}$]{} point is observed experimentally, which corresponds well to the Fermi surface formed by band 11 in the calculation.
Accordingly, the band structure and the Fermi surface of [$\mathrm{UAl}_3$]{}were explained well by the band-structure calculation although the bands near [$E_\mathrm{F}$]{}were renormalized considerably. Especially, the topology of the Fermi surface is mostly identical to the result of the calculation, although there are a few minor differences.
Band structure and Fermi surface of [$\mathrm{UGa}_3$]{}
--------------------------------------------------------
We summarize the the experimental ARPES spectra of [$\mathrm{UGa}_3$]{}measured along several high-symmetry lines in Fig. \[UGa3\_band\]. Figure \[UGa3\_band\] (a) shows the experimental ARPES spectra along the $\mathrm{X} - \mathrm{\Gamma} - \mathrm{M} - \mathrm{X}$ high-symmetry line measured at [$h\nu =650~\mathrm{eV}$]{}. The spectra exhibit clear energy dispersions, and their overall structure is very similar to the structure in the spectra of [$\mathrm{UAl}_3$]{}shown in Fig. \[UAl3\_band\]. There are dispersive bands with weak intensity on the high binding energy side, and they are ascribed to the [$\mathrm{ Ga } ~ 4s, p $]{} states. Less dispersive bands with enhanced intensity located near [$E_\mathrm{F}$]{}are ascribed to [$\mathrm{U}~5f$]{}states, which form narrow quasi-particle bands. Figure \[UGa3\_band\] (b) shows the band structure and the simulated ARPES spectra based on the band-structure calculation. The experimental ARPES spectra are explained quantitatively by the calculated results. Bands 20–24 consist mainly of the [$\mathrm{ Ga } ~ 4s,p $]{} states, and they have one-to-one correspondence with the experimentally observed band dispersions. Figure \[UGa3\_band\] (c) shows the experimental ARPES spectra of [$\mathrm{UGa}_3$]{}along the $\mathrm{M}-\mathrm{X}-\mathrm{R}-\mathrm{M}$ high-symmetry line measured at [$h\nu =580~\mathrm{eV}$]{}. The nature of energy dispersion is very similar to that in the case of the $\mathrm{X}-\mathrm{\Gamma}-\mathrm{M}$ high-symmetry line shown in Fig. \[UGa3\_band\] (a). Figure \[UGa3\_band\] (d) shows the calculated band structure and the simulated ARPES spectra based on the band-structure calculation. There is overall agreement between the experimental and the calculated results as in the case of the $\mathrm{X} - \mathrm{\Gamma} - \mathrm{M} - \mathrm{X}$ high-symmetry shown in Figs. \[UGa3\_band\] (a) and (b). In particular, bands 20–24 correspond well with the experimental ARPES spectra shown in Fig. \[UGa3\_band\] (c). Note that overall band structures of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}are very similar to each other. Especially, the dispersive bands of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}at high binding energies ([$E_{\mathrm{B}} \gtrsim 1~\mathrm{eV}$]{}) show apparent one-to-one correspondences.
We further focus on the electronic structure near [$E_\mathrm{F}$]{}. Figure \[UGa3\_band\_FS\] summarizes the band structure near [$E_\mathrm{F}$]{}and the Fermi surface of [$\mathrm{UGa}_3$]{}. Figure \[UGa3\_band\_FS\] (a) shows a comparison between the experimental ARPES spectra and the calculated energy dispersions together with simulated ARPES spectra along the $\mathrm{X} - \mathrm{\Gamma} - \mathrm{M} - \mathrm{X}$ high-symmetry line. The behavior of the quasi-particle bands near [$E_\mathrm{F}$]{}can be recognized well from this comparison. A parabolic dispersion forming the Fermi surface is observed clearly in the middle of the [$\mathrm{X} - \mathrm{\Gamma} $]{} high-symmetry line, and this band corresponds well to the calculated band 26. On the other hand, along the [$\mathrm{\Gamma} - \mathrm{M} $]{} high-symmetry line, there is hole-type energy dispersion around the [$\mathrm{\Gamma}$]{}point. The band-structure calculation shows hole-type dispersions formed by band 26 and small electron pockets formed by band 27 around the area. The experimentally observed dispersion corresponds well to the calculated band 26. Because band 27 forms very small electron pocket, it is not clear whether the corresponding band exists in the experimental ARPES spectra. The structure around the [$\mathrm{M}$]{} point is very similar between experimental and calculated results although its detail was not well resolved in the experimental ARPES spectra. Note that the intensity at [$E_\mathrm{F}$]{}at the middle of the [$\mathrm{\Gamma} - \mathrm{M} $]{} high-symmetry line is featureless while the calculation predicts an energy dispersion of about $\sim 0.1~\mathrm{eV}$. This difference can be understood that the experimental feature corresponds to the calculated band 26, but it is pushed toward [$E_\mathrm{F}$]{}. Therefore, the band forms Fermi surface similar to the calculated band 26, but it is strongly renormalized owing to the electron correlation effect. To summarize, the experimental Fermi surface map is reasonably well explained by the band-structure calculation although the bands near [$E_\mathrm{F}$]{}are renormalized.
Figure \[UGa3\_band\_FS\] (b) shows a comparison between the experimental Fermi surface maps obtained by integrating $100~\mathrm{meV}$ over [$E_\mathrm{F}$]{}of ARPES spectra, and the result of the band-structure calculation within the $\mathrm{\Gamma}-\mathrm{M}-\mathrm{X}$ plane. There is reasonable agreement between experimental map and the simulation results. Meanwhile, the size of the hole pocket formed by band 26 around the [$\mathrm{X}$]{} point is smaller than the one deduced from the experimental Fermi surface map. The size of the hole pocket around the [$\mathrm{\Gamma}$]{}point is also smaller in the calculation than that in the experiment. The existence of the small Fermi surface formed by band 27, which is located at the middle of $\mathrm{\Gamma-X}$ high-symmetry line, is not clear in the experimental Fermi surface map, but there exists a similar high intensity region along the [$\mathrm{\Gamma} - \mathrm{M} $]{} high-symmetry line, suggesting there might be a similar Fermi surface in the experiment.
Figure \[UGa3\_band\_FS\] (c) is identical to Fig. \[UGa3\_band\_FS\] (a) but along the $\mathrm{M}-\mathrm{X}-\mathrm{R}-\mathrm{M}$ high-symmetry line. The experimental ARPES spectra are very similar to those of [$\mathrm{UAl}_3$]{}shown in Fig. \[UAl3\_band\_FS\] (c). There exists a hole-like dispersion along the [$\mathrm{X} - \mathrm{R} $]{} high-symmetry line as indicated by dotted curves. Furthermore, the intensity at [$E_\mathrm{F}$]{}is particularly enhanced around the [$\mathrm{M}$]{} point, implying that there exist Fermi surfaces. In addition, the intensity at [$E_\mathrm{F}$]{}is enhanced around the [$\mathrm{M}$]{} point, suggesting that there exists some Fermi surfaces around the [$\mathrm{M}$]{} point as in the case of [$\mathrm{UAl}_3$]{}. In the calculation, bands 25 and 26 form Fermi surfaces. Band 25 forms a hole pocket in the middle of the [$\mathrm{X} - \mathrm{R} $]{} high-symmetry line, while band 26 forms a tiny hole pocket around the [$\mathrm{X}$]{} point and large cubic Fermi surface around the [$\mathrm{R}$]{} point as shown in the three-dimensional Fermi surface. The hole pocket formed by band 25 corresponds well to the experimentally observed hole pocket along the [$\mathrm{X} - \mathrm{R} $]{} high-symmetry line. In addition, the feature formed by band 26 also has a good correspondence to the experimentally observed feature in the vicinity of [$E_\mathrm{F}$]{}although their details were not well resolved experimentally.
In Fig. \[UGa3\_band\_FS\] (d), we present a comparison of the Fermi surface map and the result of the band-structure calculation, which is the same as that in Fig. \[UGa3\_band\_FS\] (b) but within the [$\mathrm{X} - \mathrm{R} - \mathrm{M} $]{} high-symmetry plane. There is a reasonable agreement between the experimental and the calculated results. The calculated Fermi surface within this plane consists mainly of band 26, which forms a large square-shaped Fermi surface around the [$\mathrm{R}$]{} point. This structure agrees with the experimental Fermi surface map. The hole pocket formed by band 26 around the [$\mathrm{\Gamma}$]{}point also agrees with the experimental Fermi surface map. Band 25 forms a tiny hole pocket, and it appears as spots with somewhat enhanced intensity midway along the [$\mathrm{X} - \mathrm{R} $]{} high-symmetry line. There is a similar feature in the corresponding location of the experimental Fermi surface map, and there exists a similar Fermi surface in the experimental spectra. To summarize, the topology of the Fermi surface of [$\mathrm{UGa}_3$]{}is also essentially explained by the band-structure calculation although the experimental band structure in the vicinity of [$E_\mathrm{F}$]{}is renormalized due to the finite electron correlation effect.
Antiferromagnetic transition in [$\mathrm{UGa}_3$]{} {#UGa3_AF}
----------------------------------------------------
To further understand the nature of the antiferromagnetic transition in [$\mathrm{UGa}_3$]{}, we present the comparison of ARPES spectra of [$\mathrm{UGa}_3$]{}measured in the paramagnetic and the antiferromagnetic phases. Figure \[UGa3\_tmpdep\] summarizes the ARPES spectra of [$\mathrm{UGa}_3$]{}measured in the paramagnetic phase ($75~\mathrm{K}$) and the antiferromagnetic phase ($20~\mathrm{K}$). Figure \[UGa3\_tmpdep\] (a) shows the Brillouin zones of [$\mathrm{UGa}_3$]{}in the paramagnetic and the antiferromagnetic phases. The symmetry of the Brillouin zone is changed from simple cubic in the paramagnetic phase to face-centered cubic in the antiferromagnetic phase. Note that the [$\mathrm{M}$]{} and the [$\mathrm{R}$]{} points in the paramagnetic Brillouin zone become equivalent to the [$\mathrm{X}$]{} and the [$\mathrm{\Gamma}$]{}points in the antiferromagnetic Brillouin zone, respectively. Correspondingly, the [$\mathrm{R} - \mathrm{M} - \mathrm{R} $]{} and the [$\mathrm{M} - \mathrm{X} - \mathrm{M} $]{} high-symmetry lines in the paramagnetic phase become equivalent to the [$\mathrm{\Gamma} - \mathrm{X} - \mathrm{\Gamma} $]{} and the $\mathrm{X} - \mathrm{W}- \mathrm{X} - \mathrm{W}- \mathrm{X}$ high-symmetry lines in the antiferromagnetic phase, respectively. Figures \[UGa3\_tmpdep\](b)–(d) show comparisons of the ARPES spectra measured in the paramagnetic and the antiferromagnetic phases along three high-symmetry lines. The spectra are divided by the Fermi–Dirac function broadened by the instrumental energy resolution to avoid the influence of Fermi cut-off. There are no recognizable changes in these spectra. Especially, the [$\mathrm{M}$]{} point in the paramagnetic phase becomes equivalent to the [$\mathrm{X}$]{} point in the antiferromagnetic phase, but the enhanced intensity at the [$\mathrm{M}$]{} point in the paramagnetic phase does not appears at the [$\mathrm{X}$]{} point in the antiferromagnetic phase as shown in Fig. \[UGa3\_tmpdep\] (d). Generally, antiferromagnetic transition is observed as the emergence of back-folded replica bands owing to changes in the Brillouin zone, and formation of a hybridization gap at the crossing point [@Cr_ARPES1; @Cr_ARPES2]. By contrast, changes in spectral profiles due to an antiferromagnetic transition in a system with low $T_\mathrm{N}$ and small ordered moments are very small [@UN_ARPES]. The absence of clear changes in the spectral profiles suggests that [$\mathrm{UGa}_3$]{}is a weak itinerant magnet, as in the case of $\mathrm{UN}$ [@UN_ARPES].
Discussion
----------
The [$\mathrm{U}~5f$]{}states in [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}have a very itinerant nature in the ground state, suggesting that the antiferromagnetism of [$\mathrm{UGa}_3$]{}originates from itinerant [$\mathrm{U}~5f$]{}electrons. This is consistent with macroscopic properties of [$\mathrm{UGa}_3$]{}, such as its small ordered moment in the antiferromagnetic phase. The experimentally obtained Fermi surfaces of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}were explained reasonably well by the band-structure calculation. By contrast, the band in the vicinity of [$E_\mathrm{F}$]{}exhibits noticeable deviations from the results of the band-structure calculations, that is, bands are renormalized near [$E_\mathrm{F}$]{}owing to the weak but finite electron correlation effect. The core-level spectra of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}also suggest the existence of finite electron correlation effects. Therefore, the band-structure calculation is a reasonable starting point for describing the electronic structures of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}, but the electron correlation effect must be considered for describing their electronic structures. Furthermore, no significant temperature dependencies across $T_\mathrm{N}$ were observed in the ARPES spectra of [$\mathrm{UGa}_3$]{}; thus, [$\mathrm{UGa}_3$]{}can be considered as weak itinerant antiferromagnet.
An important question is the relationship between their electronic structures and magnetic properties in these compounds. We have shown experimentally that the Fermi surfaces of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}are very similar to each other, although there are a few minor differences in their details. For example, the cubic hole-type Fermi surface centered at the [$\mathrm{R}$]{} point was observed in both [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}, but the hole-type Fermi surface formed by band 25 and the electron-type Fermi surface formed by band 27 in [$\mathrm{UGa}_3$]{}were not observed in the case of [$\mathrm{UAl}_3$]{}. If the magnetism in [$\mathrm{UGa}_3$]{}originates of such minor differences in their Fermi surfaces, these two Fermi surfaces might play an important role in the emergence of antiferromagnetism in [$\mathrm{UGa}_3$]{}. By contrast, the electron correlation effect might play an essential role in the magnetic properties of this series of compounds. As seen in the core-level spectra of these compounds in Fig. \[AIPES\] (b), the electron correlation effect is enhanced in the order of [$\mathrm{UAl}_3$]{}to [$\mathrm{UIn}_3$]{}. The Néel temperature is also enhanced in the same order if one assumes that the Néel temperature of [$\mathrm{UAl}_3$]{}is $T_\mathrm{N}<0~\mathrm{K}$. The slightly larger specific coefficient of [$\mathrm{UGa}_3$]{}than that of [$\mathrm{UAl}_3$]{}indicates that the density of states at [$E_\mathrm{F}$]{}$N(E_\mathrm{F})$ are larger in [$\mathrm{UGa}_3$]{}than in [$\mathrm{UAl}_3$]{}, which might satisfy the Stoner criterion $IN(E_\mathrm{F}) > 1$ in [$\mathrm{UGa}_3$]{}where $I$ is the energy reduction due to the electron correlation. Nevertheless, the very similar electronic structures of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}suggest that they are located at the boundary between magnetic and non-magnetic states. This is consistent with the weak itinerant magnetic nature of [$\mathrm{UGa}_3$]{}observed in the present study and the spin-fluctuation nature inferred by the resistivity and magnetic susceptibility measurements [@UAl3_dHvA]. Notably, the rare-earth based compounds of the same crystal structure, which have very localized $4f$ states, also exhibit antiferromagnetic transitions. Their Fermi surfaces have some similarities with these of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}, although they have considerably more complex structures [@Onuki_review]. Therefore, the mechanism of antiferromagnetic transition in actinide and rare-earth compounds might have different origins, even though both types of compounds exhibit antiferromagnetic transitions.
CONCLUSION
==========
The electronic structures of [$\mathrm{U}X_3$ ($X=\mathrm{Al}$, $\mathrm{Ga}$, and $\mathrm{In}$)]{}were studied using photoelectron spectroscopy. The valence band and the core-level spectra showed that the electron correlation effect increases in the order of [$\mathrm{UAl}_3$]{}to [$\mathrm{UIn}_3$]{}. Especially, the core-level spectrum of [$\mathrm{UIn}_3$]{}is qualitatively different from those of [$\mathrm{UAl}_3$]{}and [$\mathrm{UIn}_3$]{}, suggesting that the electron correlation effect is strongly enhanced in [$\mathrm{UIn}_3$]{}. The detailed band structures and the Fermi surfaces of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}were clarified by ARPES, and their essential structures were explained by the band-structure calculations. The topologies of the Fermi surfaces of [$\mathrm{UAl}_3$]{}and [$\mathrm{UGa}_3$]{}are very similar, but there exist a few differences. These differences or the electron correlation effect might play an essential role in their different magnetic properties. No noticeable changes were observed in the ARPES spectra of [$\mathrm{UGa}_3$]{}across the antiferromagnetic transition, suggesting that the magnetism of [$\mathrm{UGa}_3$]{}is of the weak-itinerant type.
The authors thank D. Kaczorowski for stimulating discussion. The experiment was performed under Proposal Numbers 2013A3820, 2014A3820, 2014B3820, 2015A3820, 2015B3820, and 2016A3810 at SPring-8 BL23SU. The present work was financially supported by JSPS KAKENHI Grant Numbers 26400374 and 16H01084 (J-Physics).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study the Hall response of quasi-two-dimensional lattice systems of charged fermions under a weak transverse magnetic field, in the ballistic coherent limit. We identify a setup in which this response vanishes over a wide range of parameters: the paradigmatic “Landauer-Büttiker” setup commonly studied for coherent quantum transport, consisting of a strip contacted to biased ideal reservoirs of charges. We show that the effect does not rely on particle-hole symmetry, and is robust to a variety of perturbations including variations of the transverse magnetic field, chemical potential, and temperature. We trace this robustness back to a topological property of the Fermi surface: the number of Fermi points (central charge) of the system. We argue that the mechanism responsible for the vanishing Hall response can operate both in noninteracting and interacting systems, which we verify in concrete examples using density-matrix renormalization group (DMRG) simulations.'
author:
- Michele Filippone
- 'Charles-Edouard Bardyn'
- Sebastian Greschner
- Thierry Giamarchi
bibliography:
- 'bibliography.bib'
title: Vanishing Hall Response of Charged Fermions in a Transverse Magnetic Field
---
Transport properties induced by electromagnetic fields are an active area of study in condensed matter physics. The Hall response, $\sigma_{\rm H}$, is of particular interest: It represents the off-diagonal response of a current density $\mathbf J$ to an electric field $\mathbf E$, $\sigma_{\rm H} = \varepsilon_{ij} \sigma_{ij}$, where $J_i = \sigma_{ij} E_j$, and $\varepsilon_{ij}$ is the Levi-Civita symbol. The Hall response probes important geometric and topological properties of quantum systems: the Fermi-surface curvature of metals under weak magnetic fields [@ong91_geometric_hall; @tsuji58_hall_effect_cubic; @haldane05_geometrical_hall_effect], the Berry curvature of anomalous Hall systems [@haldane2004__berry_anomalous_hall_effect], and related topological invariants of band insulators (e.g., the band-integrated Berry curvature) [@thouless1982_quantized_hall_conductance; @niu1985_quantized_hall_topological]. Studies of $\sigma_{\rm H}$ are ubiquitous in fields focused on topological quantum matter [@xiao2010_berry_phase_review] and synthetic realizations thereof [@bloch2008_many_body_cold_atoms_review; @*jotzu2014_haldane_model_cold_fermions; @*mancini2015_chiral_edges_neutral_fermions; @*tai2017_microscopy_HHmodel; @*miyake2013_Harper_model_optical_lattices; @*genkina2018_imaging; @*jaksch2003_synthetic_gauge_field; @haldane2008_photonic_crystals; @*wang2009_topological_photonic_crystal; @*hafezi2011_optical_delay_topological; @*ningyuan2015_time_site_resolved_topo_circuit].
Scattering is an essential ingredient in conventional studies of the Hall response: In the two-dimensional (2D) Hall effect [@hall_1879_hall_effect], e.g., Boltzmann-type approaches [@ziman_book] allow us to reproduce the observed Hall constant, $R_{\rm H}$, in the limit of weak transverse magnetic fields $B$: $R_{\rm H} \equiv -\sigma_{\rm H}/(\sigma_{xx} \sigma_{yy} B) \sim -1/(ne)$, where $n$ is the density of carriers with charge $e$, and $x$ $(y)$ denotes the longitudinal (transverse) direction. Scattering is also key to explaining the plateaus of quantized Hall conductance ($\sigma_{\rm H} = \nu e^2/h$ for filling factor $\nu$) appearing at stronger magnetic fields, in the quantum Hall regime [@klitzing1980_quantum_hall; @tsui1982_fractional_quantum_hall; @bernevig2013_book].
As ballistic quantum systems become more and more accessible in experiments [@ella2018_simultaneous_voltage_current; @bachmann2019_super_gemoetric_focusing; @genkina2018_imaging], new challenges are emerging for theory beyond Boltzmann-type approaches, despite significant efforts for mesoscopic systems [@roukes1987_quench_hall_effect; @*ford1988_quench_hall; @baranger1991_classicla_quantum_ballistic; @*beenakker1988_quench_hall; @*kirczenow88_ballistic_hall_effect]. For example, $\sigma_{ii}$ can be infinite in clean interacting systems, even at finite temperature [@prosen2011_openxxz_ballistic; @ljubotina2017_spin_diffusion_integrakble]. The connection between Hall response and carrier density is not even clear in the presence of interactions [@hagen1990_anomalous_hall_superconductors; @badoux2016_change_carrier_superconductor; @smith1994_sign_reversal_hall], remaining an important theoretical issue [@kapitulnik2017_anomalous_metals_supra]. Recent progress was made with the calculation of $R_{\rm H}$ in dissipative metallic systems [@auerbach2018_hall_number], where $\sigma_{ii}$ is finite at zero frequency (in contrast to gapped systems where $\sigma_{ii} = 0$, and $\sigma_{\rm H}$ can be calculated in torus geometry [@thouless1982_quantized_hall_conductance; @niu1985_quantized_hall_topological; @hatsugai1993_chern_number; @avron1985_hall]). Nevertheless, the Hall response of coherent ballistic systems remains a largely unexplored field.
In this Letter, we identify a ballistic coherent setup in which charged fermions under a weak transverse magnetic field exhibit a strictly vanishing Hall response. We demonstrate this effect in noninteracting quasi-2D lattice systems at zero temperature, in a transport setup where the system is connected to weakly-biased ideal reservoirs of charges. We show that the Hall response remains suppressed under a wide variety of perturbations: variations of the magnetic field, chemical potential, temperature, and particle-hole symmetry breaking. We relate this remarkable robustness to the topological nature of a key property underpinning the effect: the number of Fermi points (central charge) of the system. We extend our results to interacting systems, demonstrating similar effects using DMRG.
[*Hall response of ballistic systems —*]{} We consider lattice systems in a quasi-2D geometry, i.e., with edges as well as a finite number of fermionic degrees of freedom (lattice sites, in particular) in the $y$ direction. Edges imply that the transverse current $J_y$ vanishes in the low-frequency limit $\omega \rightarrow 0$ of the longitudinal electric field $E_x$. In that case, the Hall response is described by the transverse polarization difference $\Delta P_y(x, t) = \int_{t_0}^t dt' J_y(x, t')$. We set the initial polarization $P_y(x, t_0)$ (at time $t_0$ right before applying $E_x$) to zero, which corresponds to a gauge choice [@resta1992_theory_polarization; @*resta1994_review_polarization; @king_smith1993_theory_polarization; @watanabe2018_inequivalent_berry_phases], and denote $\Delta P_y(x, t) \equiv P_y(x, t)$.
The relation between $P_y$ and $\sigma_{\rm H}$ can be derived within standard linear response theory [@kubo1957_kubo_formula; @*greenwood1958_conduction_metals]: Expressing the electric field as $E_x = -\partial_t A_x$ (with $e = \hbar = c = 1$), without loss of generality, one can write $$\label{eq:Hrelation}
P_y(k, \omega) = -\sigma_{\rm H}(k, \omega) A_x(k, \omega)\,,$$ where $k$ is the crystal momentum along $x$. As we detail in the Supplemental Material (SM) [@SM], this can be seen as a Kubo formula for the transverse polarization induced by a time-dependent vector potential [^1], $P_y(x, t) = i\sum_{x'} \int dt' \theta(t-t') \langle [P_y(x, t), J_x(x', t')] \rangle A_x(x', t')$.
Equation enables very [*different*]{} Hall responses $\sigma_{\rm H}$ depending on the nature of $E_x$, for the [*same*]{} longitudinal current $J_x$. Here we consider a paradigmatic setup for coherent quantum transport: a system with two ends in the $x$ direction, where $E_x$ (or $J_x$) is generated via ideal contacts to two external reservoir of charges (left and right) with chemical potentials $\mu_{\rm L}$ and $\mu_{\rm R}$ \[Fig. \[fig:setups\](a)\]. In this “Landauer-Büttiker” (LB) setup [^2], $J_x$ is related to the potential difference $eV \equiv \mu_{\rm L} - \mu_{\rm R}$ via the conductance $G$ of the system, i.e., $J_x = GV$. Without interactions, the polarization $P_y^{\rm LB}$ can be calculated using conventional scattering theory [@lesovik11_scattering_review], with conductance $G$ derived from the Landauer formula. Kubo’s formalism \[Eq. \] provides an instructive equivalent approach [@stone1988_kubo_landauer; @*baranger1089_kubo_landauer_magnetic]. As we detail in the SM [@SM], the LB setup can be described by $A_x(x,t) = -V e^{-i\omega t} \delta(x)$, corresponding to a potential drop of amplitude $V$ at the position $x = 0$ of contact between the system and the left reservoir. Since $A_x(x,t)$ is local, the stationary transverse polarization $P_y^{\rm LB}$ takes the form of an [*integral*]{} of the Hall response [*over all momenta*]{} [@SM]: $$\label{eq:PyLB}
\frac{P_y^{\rm LB}}{J_x} = -G^{-1} \lim_{\omega\rightarrow 0} \frac{1}{2\pi} \int dk \, e^{ikx}\frac{\sigma_{\rm H}(k, \omega)}{\omega + i0^+}\,,$$ where $i0^+$ is a small positive imaginary part.
To illustrate how different the Hall responses of ballistic coherent systems can be, we investigate an additional “Aharonov-Bohm” (AB) setup: a contactless system forming a ring in the $x$ direction, where $J_x$ is induced by a time-dependent magnetic flux \[Fig. \[fig:setups\](b)\]. In that case, $A_x(x,t)$ corresponds to the magnetic vector potential describing the flux, i.e., $A_x(x,t) = e^{i\omega t} \Phi/N_x$, where $N_x$ is the number of lattice sites in the $x$ direction. The flux induces a persistent current [@buttiker1983_persistent_current; @levy1990_persistent_currents; @*kulik2010_review_persistent_currents; @*saminadayar2004_equilibrium_mesoscopic; @*bleszynski2009_persistent_currents], $J_x = D \Phi/N_x$, where $D$ is the Drude weight [@kohn64_drude; @*shastry90_drude_weight; @*millis1990_drude], which in turn generates a [*reactive*]{} Hall response [@prelovsek99_hall_correlated; @*zotos00_reactive_hall; @greschner2018_universal_hall_response] \[see Fig. \[fig:bandStructureAndResponses\](b)\]. In contrast to Eq. , and in agreement with known transport results [@luttinger1964_theory_transport_coefficients], the stationary transverse polarization $P_y^{\rm AB}$ is then related to the [*zero-momentum*]{} component of $\sigma_{\rm H}$ alone [@SM]: $$\label{eq:PyAB}
\frac{P_y^{\rm AB}}{J_x} = -D^{-1} \lim_{\omega \rightarrow 0} \sigma_{\rm H}(0,\omega)\,.$$
[*Hall response in the LB setup —*]{} We now detail the LB setup and derive an explicit formula for the response $P_y^{\rm LB}$ at zero temperature, in the low-bias limit $\mu_{\rm L} \rightarrow \mu_{\rm R} \equiv \mu$. We focus on weak magnetic fields $B \lesssim 1/N_y$, where $N_y$ is the number of lattice sites in the $y$ direction. This ensures that energy bands hybridize in a single Brillouin zone despite momentum shifts induced by the minimal coupling of system charges to $B$ [^3]. The resulting spectrum generically consists of $M_y$ bands, where $M_y$ is the number of fermionic degree of freedom (d.o.f) along $y$. We assume that $M_y = N_y$ (one d.o.f per lattice site), for simplicity, and consider systems whose energy spectrum is symmetric under momentum reversal $k \rightarrow -k$.
Our results apply to generic lattice models in this framework. For concreteness, however, we consider the Harper-Hofstadter (HH) model [@harper55_harper_model; @*hofstadter76_hofstadter_model] with Hamiltonian $H_{\rm HH} = -\sum_{x,y} [t_x e^{iBy} c^\dagger_{x,y} c_{x+1,y} + t_y c^\dagger_{x,y} c_{x,y+1}]/2 + \mbox{H.c}$ in the Landau gauge, where $t_x, t_y$ are hopping amplitudes and $c^\dagger_{x,y}$ creates a fermion on site $(x, y)$. The spectrum of $H_{\rm HH}$ is symmetric under $k \rightarrow -k$ by a combination of time reversal (TR) and spatial inversion in the $y$ direction. The corresponding effective TR symmetry is described by the operator $\Theta = I_y \mathcal K$, where $I_y$ permutes positions $y$ around the center of the system, and $\mathcal K$ describes complex conjugation. As $[H_{\rm HH}, \Theta] = 0$, the action of $\Theta$ on an eigenstate $|\psi_k(E)\rangle$ of $H_{\rm HH}$ with momentum $k$ and energy $E$ gives a (non-necessarily distinct [^4]) eigenstate $\Theta |\psi_{k'}(E)\rangle$ with $k' = -k$ and the same energy.
The Hall response can be derived using scattering theory [^5]: In the low-bias zero-temperature limit, the conductance is $G = G_0 \sum_j T_j$, where $G_0 = e^2/h = 1/(2\pi)$ is the conductance quantum, and $T_j$ is the transmission probability, at the chemical-potential energy $\mu$, of scattering modes $\psi_j(x, y)$ incoming from the left reservoir. We assume that the reservoirs are large (infinite) regions described by $H_{\rm HH}$ (with chemical potentials $\mu_L$ and $\mu_R$, respectively), so that scattering modes take a similar form as the system’s eigenmodes. In that case, $T_j = 1$ for all modes $\psi_j(x, y)$ available at energy $\mu$. Relevant modes have the asymptotic form $\psi_j(x \rightarrow -\infty, y) = e^{ik_{F,j}x} w_j(y)/v_{F,j}$, where $k_{F,j}$ ($v_{F,j}$) denote the system’s Fermi momenta (velocities), and $w_j(y)$ its transverse wavefunctions. The conductance is $G = c(\mu) G_0$, where $c(\mu)$ is the number of Fermi points at $k > 0$ \[see Fig. \[fig:bandStructureAndResponses\](a)\]. As we detail in the SM [@SM], Eq. becomes $$\label{eq:pollandauer}
\frac{P_y^{\rm LB}(\mu)}{J_x} = \frac{1}{c(\mu) G_0} \sum_{j = 1}^{c(\mu)} \sum_{y} y \frac{w_j(y)^2}{v_{F,j}} \,.$$ More explicit expressions can be found in the SM for (i) finite $B$, $N_y = 2$, and (ii) $B \rightarrow 0$, $N_y \geq 2$.
We used the quantum-transport simulation package “Kwant” [@groth14_kwant] to verify the above analytical results, to compute $P_y^{\rm LB}$ for arbitrary $B$ and $N_y$, and to compare $P_y^{\rm LB}$ to $P_y^{\rm AB}$. Our results, illustrated in Fig. \[fig:bandStructureAndResponses\](b), demonstrate two key points: First and foremost, $P_y^{\rm LB}$ [*vanishes identically*]{} whenever $c(\mu) = N_y$, i.e., whenever the chemical potential $\mu$ crosses $N_y$ times the system’s energy bands at $k > 0$. This is in stark contrast to what may be expected of particles with charges of the same sign in a finite magnetic field. Provided that $c(\mu) = N_y$, the Hall response remains zero irrespective of particle-hole symmetry (generically absent here), and of the specific values of $\mu$ and $B$ \[see the region around $\mu/t_x = \pm 0.5$, e.g., in Fig. \[fig:bandStructureAndResponses\](b)\]. Second, the responses $P_y^{\rm LB}$ to $P_y^{\rm AB}$ are strikingly different, which reflects the clear differences between Eqs. and . Intuitively, this comes from the fact that the stationary states found in the LB and AB setups are different \[Fig. \[fig:bandStructureAndResponses\](a, b)\], with distinct polarizations $P_y$, though they carry the same current $J_x$. The two responses only coincide when $c(\mu) = 1$, which can be understood from simple analytical considerations [@SM].
[*Topological origin of the vanishing Hall response —*]{} We now demonstrate that $P_y^{\rm LB} = 0$ arises from two elements: (i) the topological nature of $c(\mu)$, and (ii) the traceless nature of the operator $\hat P_y$ describing the polarization. The number $c(\mu)$ of Fermi points with $k > 0$ is topological in the sense that it coincides with the [*central charge*]{} of the system—the number of gapless modes at $k > 0$, if we interpret the system as a Luttinger liquid (without interactions, for now). The polarization operator is $\hat P_y = e Y$, where $Y = \sum_{x, y} y c_{x, y}^\dagger c_{x, y}$ describes the “center-of-mass” position along $y$. To ensure that $\langle \psi_i^{\rm LB}(\mu) | \hat P_y | \psi_i^{\rm LB}(\mu) \rangle = 0$ in the initial state $| \psi_i^{\rm LB}(\mu) \rangle \equiv | \psi_i \rangle$ with zero bias ($V, J_x = 0$), corresponding to our gauge choice for the polarization, we set $y = 0$ at the center of the system. The operator $\hat P_y$ then satisfies $I_y^T \hat P_y I_y = -\hat P_y$. It is traceless, and $P_y^{\rm LB} = 0$ at $V = 0$ is enforced by the symmetry between $k$ and $-k$: Indeed, $| \psi_i \rangle$ is the many-body ground state of $H_{\rm HH}$ with single-particle states occupied symmetrically around $k = 0$ up to the chemical potential $\mu$. It is symmetric under $\Theta$ (i.e., $\Theta | \psi_i \rangle = \pm | \psi_i \rangle$), such that $\langle \psi_i | \hat P_y | \psi_i \rangle = \langle \psi_i | \Theta^\dagger \hat P_y \Theta | \psi_i \rangle = \langle \psi_i | I_y^T \hat P_y I_y | \psi_i \rangle = - \langle \psi_i | \hat P_y | \psi_i \rangle$.
Upon applying a finite bias $V \neq 0$ to generate a stationary current $J_x$ in the “final” state $| \psi_f^{\rm LB}(\mu) \rangle \equiv | \psi_f \rangle$, the symmetry $\Theta$ breaks: The state $| \psi_f \rangle$ is a many-body stationary state consisting of single-particle states occupied symmetrically around $k = 0$ [*except*]{} at the chemical potential $\mu$ where single-particle states are occupied at $k > 0$ only. By symmetry, noncancelling contributions to the polarization must come from these $c(\mu)$ Fermi points. We index the latter with $j = 1, 2, \ldots, c(\mu)$, and denote by $|j\rangle$ the corresponding single-particle states ($|j\rangle \equiv |k_{F,j},s_j\rangle$ here, where $k_{F,j}$ and $s_j$ are the Fermi momentum and band index of the Fermi point $j$). The polarization takes the generic form $$\label{eq:PyLBFermiPoints}
P_y^{\rm LB}(\mu) = \sum_{j=1}^{c(\mu)} n_j \langle j | \hat P_y | j \rangle,$$ where $n_j = 1$ is the stationary occupation of $|j\rangle$.
We are now in position to show that $P_y^{\rm LB}(\mu)$ vanishes in a robust way whenever $c(\mu) = N_y$: The states $|j\rangle$ in Eq. belong to the eigenspace of $H_{\rm HH}$ with energy $\mu$, and are characterized by distinct momenta. Since they are not related by any symmetry [^6], they form a basis for a Hilbert (sub)space of dimension $c(\mu)$. Consequently, when $c(\mu) = N_y$, Eq. reduces to $$\label{eq:PyLBFermiPoints2}
P_y^{\rm LB}(\mu)|_{c(\mu) = N_y} = \sum_{j=1}^{N_y} \langle j | \hat P_y | j \rangle = \mathrm{Tr} \hat P_y = 0 \,.$$ This key result represents a conservation law for the Hall response of the LB setup, which differs from the response of the AB setup, e.g., where [*all*]{} occupied single-particles states contribute to $J_x$ and $P_y$ (see Fig. \[fig:bandStructureAndResponses\](b) and SM [@SM]). Note that other potentially observable conservation laws can be derived from the tracelessness of $\hat P_y$ in bases of dimension $N_y$: In particular, replacing the set $\{ |j\rangle \}$ by a basis of Bloch eigenstates $\{ | k, s \rangle \}$ (with momentum $k$ and band index $s = 1, 2, \ldots, N_y$), one finds $$\label{eq:conservationLawFixedk}
P_y(k) \equiv \sum_{s=1}^{N_y} \langle k, s | \hat P_y | k, s \rangle = 0 \,,$$ implying that the transverse polarization (Hall response) of a system with $N_y$ bands vanishes in any momentum sector $k$ where all bands are equally occupied. This conservation law is directly related to the known zero-sum rule for the Berry curvature of all eigenstates of a Hamiltonian [^7]. The conservation law in Eq. can be seen as an analog with fixed energy (and $k > 0$), instead of $k$.
[*Robustness to perturbations —*]{} The vanishing of $P_y^{\rm LB}$ for $c(\mu) = N_y$ is protected against temperature by an energy gap: the energy $\Delta \mu$ corresponding to the smallest chemical-potential variation required for $c(\mu) \neq N_y$. The Hall response is suppressed as $e^{-\beta |\Delta \mu|}$ at finite temperature $T = 1/\beta > 0$ (setting $k_B = 1$), accordingly, as we illustrate in the SM in an example with $N_y = 2$ [@SM].
We remark that deviations from a strictly vanishing Hall response are also expected in the presence of disorder, as the effect relies on the ballistic coherent nature of the system [^8]. Disorder in quasi-1D systems generally leads to Anderson localization [@abrahams79_anderson_localization; @*abrahams10_anderson_localization; @*lagendijk09_anderson_localization]. Nevertheless, provided that the scattering region connecting the two reservoirs is shorter than the corresponding localization length (scaling as $N_y t_x^2/W^2$ with disorder strength $W$ [@kappus1981_anomaly_anderson]; see SM [@SM]), disorder can be regarded as a weak perturbation. In this regime, deviations of the disorder-averaged polarization $\langle P_y \rangle$ from zero scale as $W^2/t_x^2$ [@SM], with large fluctuations around the average (as for typical conductance fluctuations in disordered systems [@altshuler1985_conductance_fluctuations; @*lee1985_universal_conductance_fluctuations]).
[*Generalization to interacting systems —*]{} Equation applies whenever $c = N_y$ independent and equally occupied fermionic d.o.f (the Fermi points at $k > 0$ discussed so far) are responsible for the current $J_x \neq 0$. The noninteracting nature of the underlying many-body state is irrelevant. To demonstrate that our results extend to interacting systems with $c = N_y$, we consider the above HH model with $N_y = 2$ (two-leg ladder) and additional intra- and inter-leg interactions described by Hamiltonian terms $U_\parallel \sum_{x, y = \pm 1} n_{x, y} n_{x+1, y} + U_\perp \sum_x n_{x, 1} n_{x, -1}$, where $n_{x, y}$ is the density on site $(x, y)$.
To simulate stationary transport conditions in the LB setup, we compute the time evolution of the full system with reservoirs described by a quenched steplike potential $-\epsilon \sum_{x < L_{\rm res}, y} n_{x, y} + \epsilon \sum_{x > L_{\rm sys} + L_{\rm res}, y} n_{x, y}$, where $L_{\rm sys/res}$ denotes the length of the system/reservoirs. We first set $\epsilon = 0$ and prepare the full system in its ground state using DMRG [@White1992; @Schollwoeck2011]. We then set $\epsilon > 0$, at time $\tau = 0$, and compute its evolution using tDMRG [@Schollwoeck2011] and the ITensor library [@ITensor]. We choose $L_{\rm sys} = 2$, for simplicity, and compute the Hall response $P_y^{\rm LB}/J_x$ in the middle of the system at times $1 \lesssim \tau/t_x \lesssim L_{res}$ [@Einhellinger2012], averaging over a time window where $J_x$ is approximately stationary. Figure \[fig:numerics\] illustrates typical results for $U_\parallel = U_\perp = t_x/2$ and different magnetic fields. For comparison, we simulate transport in the AB setup by quenching, instead, a small linear potential $-(\epsilon/N_x) \sum_{x, y} x n_{x, y}$. While $J_x$ increases linearly in time in that case \[Fig. \[fig:numerics\](b)\], the ratio $P_y^{\rm AB}/J_x$ oscillates around a constant value corresponding to the stationary Hall response [@greschner2018_universal_hall_response].
![Numerical estimates of the Hall response of interacting fermions in the LB and AB setups (tDMRG simulations of the interacting HH model presented in the text, with $t_x = t_y = 1$, $U_\parallel = U_\perp = 1/2$, and $\epsilon = 0.01$, for $10$ particles in a full system of length $L_x = 60$). [**(a)**]{} Time evolution of $P_y^{\rm LB}/J_x$ (filled symbols) and $P_y^{\rm AB}/J_x$ (empty symbols) for a total magnetic flux $\chi \equiv B/N_x = 0.2\pi$ (Luttinger-liquid phase with central charge $c = 1$), and $0.7\pi$ and $0.8\pi$ ($c = 2$). Lines interpolate a finer set of data points than shown symbols. [**(b)**]{} Time evolution of $J_x$ alone for the same parameters as in (a). [**(c)**]{} Average of $P_y^{\rm LB}/J_x$ ($\times$) and $P_y^{\rm AB}/J_x$ ($\Box$) over times $10 < \tau < 30$. The dashed line indicates the estimated transition between $c = 1$ and $c = 2$ phases. Averages coincide for $\chi = 0.4\pi$, while no stationary regime was reached for $0.5\pi$.[]{data-label="fig:numerics"}](fig3.pdf){width="1\linewidth"}
The results presented in Fig. \[fig:numerics\] are consistent with our theoretical expectations: First, the Hall responses found in the LB and AB setups are identical (for time averages, within errorbars) when the ground state prepared at $\tau = 0$ is a phase with central charge $c = 1$ [@SM; @Holzhey1994; @*Korepin2004; @*Calabrese2004]. Second, the two responses differ completely when the system enters a $c = 2 = N_y$ phase, at some larger magnetic field \[Fig. \[fig:numerics\](c)\]: While $P_y^{\rm LB}/J_x$ shows large oscillations around an average value consistent with $P_y^{\rm LB}/J_x = 0$, the response $P_y^{\rm AB}/J_x$ is finite. Along with additional data presented in the SM [@SM], our results are fully consistent with our theoretical arguments that the Hall response of the LB setup vanishes when $c = 2 = N_y$.
[*Discussion —*]{} Our results exemplify the rich and sometimes counterintuitive quantum transport phenomena that can occur in the ballistic coherent regime. The effect discovered here could be observed in solid-state or synthetic-matter experiments [@ella2018_simultaneous_voltage_current; @bachmann2019_super_gemoetric_focusing; @genkina2018_imaging]. In fact, an experimental platform for the realization of the LB setup has recently been proposed [@salerno2018_quantized_hall]. We emphasize that our results equally apply to bosons: In photonic systems [@carusotto_2013; @*hafezi2011robust; @*kruk2017; @*bellec2013; @*poddubny2014; @*downing2017], for example, the conservation law found in this work could be observed by selectively populating the $c=N_y$ states responsible for the effect [@Bardyn_2014].
Our results could lead to additional clues towards a better understanding of the Hall response of strongly-correlated (non-Fermi-liquid) systems, for which low-energy-quasiparticle descriptions of quantum transport inexorably fail. More presently, they raise important questions regarding the behavior of the transverse polarization $P_y$ of interacting systems at finite temperatures: Although a transition to dissipative/metallic regimes is expected in such systems, explicit calculations of $P_y$ remain challenging [@auerbach2018_hall_number]. Recent studies have shown the persistence of ballistic and superdiffusive regimes in specific cases [@prosen2011_openxxz_ballistic; @ljubotina2017_spin_diffusion_integrakble]. It will be interesting to investigate whether analogs exist in quasi-1D lattice systems.
[*Acknowledgments —*]{} We thank Jean-Philippe Brantut, Nigel Cooper, and Nathan Goldman for fruitful discussions, and acknowledge support by the Swiss National Science Foundation (FNS/SNF) under Division II. M.F. also acknowledges support from the FNS/SNF Ambizione Grant PZ00P2\_174038.
[^1]: We refer to the function $A_x(x, t)$ in $E_x = -\partial_t A_x$ as a “vector potential”, though it need not coincide with the magnetic vector potential.
[^2]: Name chosen in connection with the transport formalism of the same name [@landauer1970_scatt_theory; @lesovik11_scattering_review].
[^3]: The system contains at most one magnetic unit cell in the $y$ direction.
[^4]: Since $\Theta^2 = +1$ (due to the spinless nature of the fermions that we consider), Kramers’ theorem does not hold.
[^5]: Or, equivalently, by using Eq. (see SM [@SM]).
[^6]: The symmetry $\Theta$ relates states with [*opposite*]{} momenta.
[^7]: The quantity can be seen as a Berry connection in the continuum limit where the position operator $Y = \hat P_y$ is represented by $Y = -i\partial_{k_y}$; see also Ref. [@xiao2010_berry_phase_review].
[^8]: Generic disorder also breaks the symmetry $\Theta$ connecting momentum sectors $k$ and -$k$.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- Moumita Maiti
- Michael Schmiedeberg
title: The thermal jamming transition of soft harmonic disks in two dimensions
---
[leer.eps]{} gsave 72 31 moveto 72 342 lineto 601 342 lineto 601 31 lineto 72 31 lineto showpage grestore
This is the final author’s version of the article.\
The final version of the article is published as\
Eur. Phys. J. E [**42**]{} (2019), 38\
and is available at Springer via\
http://dx.doi.org/10.1140/epje/i2019-11802-3.
Introduction {#intro}
============
The dynamics of particulate systems can dramatically slow down if the density is increased or the temperature is decreased (for reviews see, e.g., [@BerthierandBiroli; @hunter2012]). The properties of this phenomena that is also known as glass transition are the subject of a lot of research and discussions [@BerthierandBiroli; @hunter2012]. A related phenomena is the transition of an athermal system from a state where all overlaps between particles can be removed by minimizing the energy to a disordered state where overlaps cannot be avoided and that is termed a jammed state [@ohern2002; @ohern2003]. Interestingly, while the packing fraction of this athermal jamming transition might depend on the starting conditions [@Chaudhury2010], the critical behavior close to the transition is universal [@ohern2002; @ohern2003; @Chaudhury2010].
Recently, the method employed for athermal jamming in [@ohern2002; @ohern2003] has been modified in order to explore thermal jamming in three dimensions [@maiti2018; @corwin2017] for a system with finite-ranged repulsive interactions. While in [@corwin2017] the particles during the minimization process are kept sticking together, in [@maiti2018] we allowed the rare crossing of energy barriers that is impossible in case of an athermal system. Therefore, while our approach in [@maiti2018] neglects thermal fluctuations within the valleys of the energy landscape, rare thermal rearrangement events in principle are possible. Both methods [@maiti2018; @corwin2017] revealed that in case of spheres with finite-ranged harmonic repulsive interactions there is a spatial percolation transition at a packing fraction of $\phi_G=0.55\pm 0.01$ if random initial configurations are used. Therefore, the corresponding transition takes place at much smaller packing fractions than athermal jamming, which occurs at a packing fraction of $\phi_J=0.639$ in case of similar starting conditions and interactions [@ohern2002; @ohern2003]. Furthermore, we have shown that due to this spatial percolation the system cannot explore the full energy landscape and therefore effectively is non-ergodic at packing fractions above $\phi_G$ in the limit of small but non-zero probabilities for barrier crossing events [@maiti2018]. The thermal jamming transition as transition from a fluid state to a non-ergodic glass state is in the universality class of directed percolation [@maiti2018] and is similar to a modified random organization transition [@milz2013]. The predictions of our energy landscape exploration method have been shown to be in agreement with simulation results [@maiti2018b].
Here we use the method introduced in [@maiti2018] in order to study the thermal jamming transition in two-dimensional bidisperse soft disk systems with finite-ranged harmonic repulsive interactions. We find that the transition packing fraction of thermal jamming $\phi_G\approx 0.74$ is much smaller than the one of athermal jamming that in two dimensions occurs at $\phi_J=0.842$ in case of the same starting conditions and interaction potentials [@ohern2002]. Furthermore, as in three dimensions the thermal jamming transition in two dimensions also is in the universality class of directed percolation.
At a first glance the dynamics of glassy systems in two dimensions look fundamentally different from the dynamics of comparable systems in three dimensions [@flenner2015]. However, it has been revealed that the differences are due to Mermin-Wagner-like fluctuations [@mermin1966], i.e., long-ranged fluctuations that in two dimensions occur in addition to the glassy dynamics [@shiba2016; @vivek2017; @keim2017]. Since we neglect the fluctuation within energy valleys in this article, we cannot observe any long-ranged density fluctuations and therefore our results are directly related to the pure glassy behavior.
Our article is structured as follows, In sec. \[sec:methods\] we describe the model system and the employed method in detail. The results are presented in sec. \[sec:Results\] where we first determine the state diagram and then study the critical behavior. Finally, we conclude in sec. \[sec:conc\].
Model system and methods {#sec:methods}
========================
Bidisperse harmonic soft disks
------------------------------
We consider a bidisperse system of soft disks. Motivated by the mixture that is often employed, e.g., in [@ohern2003], half of the disks have the diameter $\sigma$ and the other half the diameter $1.4\sigma$. Crystallization is suppressed due to this bidispersity. The discs do not interact if they do not overlap. Overlapping particles repel each other according to the harmonic pair potential $V_{ij} = \epsilon(1-\frac{r_{ij}}{\sigma_{ij}})^2$, where $\epsilon$ sets the energy scale, $r_{ij}$ denotes the distance between the two particle and $\sigma_{ij}=(\sigma_i+\sigma_j)/2$ their average diameter. We employ systems with periodic boundary conditions and system sizes ranging from $10^5$ to $6 \times 10^5$ particles.
![\[fig:fig1\]State diagram as a function of packing fraction $\phi$ and probability $p$ for steps where energy barriers might be crossed. The thermal jamming transition between a fluid state with no remaining overlaps at small packing fractions and a state where overlaps cannot be removed (corresponding to a glass state for small $p$) at large densities is shown by red error bars that indicate the largest observed packing fraction of a fluid and the lowest observed packing fraction of a jammed state not affected by finite size effects in our simulations. The blue arrow indicates the packing fraction $\phi_J=0.842$ [@ohern2002] of the athermal jamming transition. The magenta triangles indicate the transition packing fraction obtained by fitting a critical power laws to the fraction of overlaps $f_{ov,\infty}$ at long times. The green circles are obtained from power law fits to the relaxation times $\tau$. The analysis of the critical behavior is explained in detail in sec. \[sec:critical\].](fig1.eps){width="0.9\linewidth"}
{width="\linewidth"}
Exploration of the energy landscape
-----------------------------------
We use the same approach that we have employed for monodisperse systems in three dimensions [@maiti2018]. Starting with an random starting configuration we usually employ energy minimization steps that are also used in athermal systems [@ohern2002; @ohern2003]. However, in each step each particles with an overlap can be selected with a small given probability $p$. A selected particle does not perform a minimization step but is displaced in a randomly chosen direction until it crosses the nearest energy minimum or maximum in that direction. Therefore, such a displacement can lead to the crossing of an energy barrier. The minimization steps or random steps are repeated until all overlaps have been removed or until the fraction of overlapping particles reaches a plateau value and does not further decrease. As for athermal jamming systems that can reach a configuration without overlaps are called unjammed while systems with remaining overlaps are termed jammed.
As discussed in [@maiti2018] in the limit of small but non-zero $p$ the observed transition corresponds to a weak ergodicity-breaking transition because in the jammed state the ground state no longer is accessible within a reasonable time. Such an effective ergodicity breaking transition usually is referred to as dynamical glass transition (see. e.g., [@BerthierandBiroli; @hunter2012]). We want to point out that we consider a dynamical glass transition and not any ideal structural glass transition or the Kauzmann temperature [@Kauzmann]. Note that for larger $p$ there are significant rearrangements due to the randomly displaced particles that in principle correspond to ageing but for even large $p$ lead to a thermal fluidization even of the state with remaining overlaps. Therefore, in order to make predictions about the glass transition as a function of the packing fraction, we usually consider the limit of small $p$. For three dimensional systems we have demonstrated that in case of small $p$ this barrier crossing probability can be related to a real temperature $T$ and predictions concerning the temperature-dependence of the glass transition packing fraction are obtained that are in agreement with simulation results [@maiti2018b].
The minimization steps in our protocol are done by using the conjugate gradient algorithm implemented in LAMMPS [@lammps]. The minimization is stopped when the energy per particle is equal to $10^{-16}\epsilon$ or smaller and otherwise two particles are considered to overlap (instead of just being in contact) if $\sigma_{ij} - r_{ij}> 10^{-7} \sigma_{ij}$. Note that for three dimensional systems we have tested various modifications of the protocol, e.g., employing steepest descent minimization or using other ways of barrier crossing. However, all modifications led to the same thermal jamming transition [@maiti2018]. Furthermore, we have shown that the transition packing fraction can be larger if other starting conditions are employed. Note that this behavior is the well-known history-dependence of the glass transition [@BerthierandBiroli]. However the critical behavior does not depend on the initial conditions [@maiti2018].
Results {#sec:Results}
=======
For a probability $p=0$, i.e., without barrier crossings the athermal jamming transition is obtained that in case of of random initial conditions occurs at a packing fraction $\phi_J=0.842$ [@ohern2002]. In the following we study the thermal jamming transition that occurs for small but non-zero $p$.
Thermal jamming transition
--------------------------
In fig. \[fig:fig1\] the state diagram for the thermal jamming transition is shown. The transition between states where all overlaps can be removed at small packing fractions and states with remaining overlaps at large packing fractions is determine by different methods as described in the figure caption. For all methods of analysis, a small transition packing fraction is observed for large probabilities $p$ for barrier crossings and the transition packing fraction increases for decreasing $p$. For small $p$ it stays close to a value of $\phi_G=0.74\pm 0.01$ which is far below the transition packing fraction of athermal jamming that $\phi_J=0.842$ [@ohern2002] (indicated by a blue arrow).
In figs. \[fig:fig2n\] the relaxation curves for the fraction of overlapping particles $f_{ov}$ as a function of the number of steps $t$ are shown for various packing fractions and probabilities. The employed system sizes are given in the figure caption. Note that we have checked that similar curves are obtained for larger or slightly smaller system sizes. If one wanted to study relaxation curves even closer to the transition, much larger systems would be necessary.
The behavior of the thermal jamming transition is similar to the one that we have found for the three-dimensional systems in our previous work [@maiti2018]. For $p$ approaching zero $\phi_{G}$ is the packing fraction where the ergodicity is broken because above this transition the ground state effectively no longer is accessible. Usually this transition is also referred to as dynamical glass transition. Our result $\phi_G$ is lower than the glass transition density obtained from simulation, where, e.g., a packing fraction of $0.8$ has been reported in [@donev2006] for a bidisperse mixture. However, it is larger than the one obtained from mode-coupling theroy, where $0.697$ has been determined in [@bayer2007] for a monodisperse packing. Note that we expect that for other starting configurations our approach probably leads to another, usually larger transition packing fraction similar to the behavior that we have observed in three dimensions [@maiti2018].
Critical behavior {#sec:critical}
-----------------
![\[fig:fig2\]Analysis of the critical behavior close to the thermal jamming transition. (a,c) Fraction $f_{ov,\infty}$ of particles with overlaps at long times as a function of the difference of the packing fraction $\phi$ from the transition packing fraction $\phi_c$ in (a) linear-linear and (c) log-log-representation. (b,d) Relaxation times $\tau$ as function of $\phi-\phi_c$ in the same representations. Triangles pointing upwards or downwards are used to indicate data above or below the transition, respectively. (e,f) Exponents found by the fitting procedure. Colors in all panels indicate the probability $p$ as can be read of for the points in (e,f). The red and blue lines indicate the power laws or exponents expected for a directed percolation transition [@hinrichsen2000]. Note that for (a-d) $f_{ov,\infty}$ and $\tau$ are determined by fits to the relaxation curves as explained in the text. The transition packing fraction $\phi_c$ as well as the rescaling factors $1/A_p$ and $1/B_{p,\pm}$ are obtained by fitting power laws to the data in (a,b). The obtained values of $\phi_c$ are also shown in fig. \[fig:fig1\]. ](fig3.eps){width="\linewidth"}
In order to analyze the critical behavior close to the transition, we employ the same analysis as in [@reichhardt2009; @milz2013]. To be specific, we fit the function $$\label{eq:fovt}
f_{ov}(t)=C\frac{e^{-t/\tau}}{t^{\gamma}}+f_{ov,\infty}$$ with fit parameter $C$, $\tau$, $\gamma$, and $f_{ov,\infty}$ to the relaxation curves that are exemplary shown in figs. \[fig:fig2n\], i.e., to the fraction of overlapping particles $f_{ov}$ as a function of the numbers of steps $t$. The fitting constant $\tau$ indicates the time until the system reaches a steady state, $f_{ov,\infty}$ is the plateau value of the relaxation curve giving the fraction of remaining overlapping particles in jammed systems. In case the system is not jammed, $f_{ov,\infty}=0$. Close to the transition the relaxation curves can be described by a power law with exponent $\gamma$.
For $f_{ov,\infty}$ and $\tau$ as functions of the packing fractions, we now fit power laws $$\label{eq:fovu}
f_{ov,\infty} = A_p (\phi - \phi_{c})^{\beta}$$ and $$\tau = B_{p,\pm} \left|\phi - \phi_{c}\right|^{-\nu}$$ for each value of the probability $p$, where $A_p$ and $B_{p,\pm}$ are factors that can depend on $p$ and in case of $B_{p,\pm}$ also on whether one is below or above the transition. The data and a power law curve with the exponents $\beta=0.583$ and $\nu=1.295$ as expected for a directed percolation transition in 2+1 dimensions [@hinrichsen2000] are shown in figs. \[fig:fig2\](a-d). The transition packing fractions $\phi_{c}$ determined by fitting are shown in fig. \[fig:fig1\]. Note that since we obtain $\phi_{c}$ as a fitting parameter, error bars might be quite large. However, we believe that this is less prejudiced analysis than if we just assumed some values for $\phi_{c}$. The exponents extracted by fits are displayed in figs. \[fig:fig2\](e,f), where the horizontal lines indicate the mentioned values expected for a directed percolation transition. The error bars indicate the statistical errors obtained from the fitting procedure. However, the true error can be different because small variations of the fitting procedure like excluding the data points that have the largest distance to the transition might lead to similar or even slightly larger changes of the fit value.
Overall, our results are in good agreement with the critical behavior expected for a directed percolation transition and that has been observed for the corresponding three-dimensional system (where the directed percolation transition is in 3+1 dimensions) [@maiti2018]. Concerning the two dimensional system considered here, we find that the exponent $\beta$ for $10^{-4}\leq p<10^{-3}$ seem to be slightly larger than the literature value. Though the deviations are outside the statistical error, they are still within the variations that might arrise from different ways of fitting. Note that it is known that for $p=1$ a random organization-like transition is obtained that is in the universality class of directed percolation [@milz2013]. Here we do not see any systematic changes of $\beta$ as $p$ is decreased.
Concerning the analysis of the critical behavior related to the relaxation times, we first want to point out that for relaxation curves below the transition, we only have enough data that is sufficiently close to the transition in order to fit a power law for $p=10^{-2}$, $p=10^{-3}$, and $p=10^{-4}$. Above the transition, we have sufficient data for all cases. However, for all data we observe a large scattering. While for small $\left|\phi - \phi_{c}\right|$ the data is in good agreement to the expected power law behavior, there are significant deviations for larger $\left|\phi - \phi_{c}\right|$ as can be seen in the right part of fig. \[fig:fig2\](d). These deviations probably are due to problems that occur when fitting the function of eq. (\[eq:fovt\]) to relaxation curves that are never close to the power law-like behavior that is assumed by the $t^{\gamma}$-term in eq. (\[eq:fovt\]). The reason that the power laws does not show up (especially for many cases below the transition) is that the relaxation times might be to small, i.e., that we are still not close enough to the transition. Getting closer to the transition would require much larger system sizes that we cannot simulate at the moment within a reasonable time. Note that for the three-dimensional system we have observed the occurrence of an additional power law decay with exponent $-1.5$ that seems to be unrelated to the transition but made it impossible to analyze relaxation times below the transition for all cases [@maiti2018]. Here, in the two-dimensional case, we do not find any any additional power law decays and therefore we can analyze the relaxation times below the transition for the previously mentioned cases. The scattering of the data results in large error bars in fig. \[fig:fig2\](f) at least in the cases where a larger number of data points further away from the transition is still part of the analysis. Nevertheless, though we cannot determine $\nu$ with an accuracy that would be sufficient to rule out other universality classes if we only considered $\nu$ in our analysis, our results show that the critical behavior concerning the relaxation times is in agreement with the critical behavior of a directed percolation transition.
Finally, we look on the power laws with exponents $\gamma$ that are expected close to the transition. Averaging the values of $\gamma$ as determined by fits according to eq. (\[eq:fovt\]) over all results for curves with $|\phi-\phi_c|<0.001$, where $\phi_{c}$ is taken from the fit in eq. (\[eq:fovu\]), we find $\gamma = 0.44\pm 0.02$, which is close to the literature value $\gamma=\nu/\beta=0.45$ [@hinrichsen2000].
In summary, all exponents $\beta$, $\nu$, and $\gamma$ agree to the critical behavior of a directed percolation transition and there is now systematic change when the probability is changed.
Conclusions {#sec:conc}
===========
We have employed our previously introduced approach of exploring the energy landscape in order to study the thermal jamming transition. In this approach we usually minimize the energy but with a small, non-zero probability introduce a step where energy barriers can be crossed. If all overlaps between particles can be removed by this protocol, the system is called unjammed. If the system is stuck in a state with remaining overlaps, this state is termed jammed. In the latter case our protocol was not able to access the ground state of the system within our simulation time and as a consequence all simulation methods that consist of - usually less efficient - energy minimization and thermal fluctuations cannot reach the ground state as well. Therefore, the system effectively is non-ergodic and the thermal jamming transition in the limit of small barrier crossing probabilities corresponds to the dynamical glass transition (cf. discussion in [@maiti2018]).
As in our previous work [@maiti2018] where we considered a three-dimensional system, the critical behavior of the thermal jamming transition corresponds to the one known for directed percolation transitions. Furthermore, the thermal jamming transition occurs at a packing fraction that is much smaller than the one of the athermal transition. However, we expect that for different initial conditions the differences can be smaller as we have demonstrated in three dimensions [@maiti2018].
Note that it is well known that the glass transition can occur at packing fractions below the transition packing fraction of athermal jamming (see, e.g., [@ikeda; @Wang]). The properties of glasses that occur at packing fractions above the glass transition but below athermal jamming have been analyzed in [@Wang]. For example, in the hard sphere limit just above the glass transition the glasses can only carry longitudinal but no transverse phonons [@Wang].
An extension of our approach to systems with more complex pair interactions probably would be interesting, e.g., in order to study the temperature-dependence of soft disks [@BerthierandWitten; @BerthierandWitten1; @haxton2011; @schmiedeberg2011; @maiti2018b] or maybe even the reentrant glass transitions that occur at very large packing fractions [@berthier10; @schmiedeberg13; @miyazaki16]. Recently, a minimization protocol has been used in order to study the jamming transition of attractive systems revealing a second order transition except for weakly attractive systems where a first order transition has been reported that might only occur due to finite size effects [@Koeze18]. It would be interesting, to try to study complex gel networks in a similar way [@gel; @kohl]. Finally, exploring the energy landscape can also be used to study systems that are driven out of equilibrium, e.g., by shearing the system [@heuer1; @heuer2] or by using self-propelling particles [@maiti2018aktiv].
The project was supported by the Deutsche Forschungsgemeinschaft (Grant No. Schm 2657/3-1). We thank Sebastian Ruß for helpful discussions and gratefully acknowledge the computer resources and support provided by the Erlangen Regional Computing Center (RRZE).
Authors contributions {#authors-contributions .unnumbered}
=====================
M.M. carried out the simulations and M.S. designed the research and the model system. Both authors analyzed the results and wrote the article.
L. Berthier and G. Biroli, Rev. Mod. Phys. [**83**]{}, (2011) 587.
G. L. Hunter and E. R. Weeks, Rep Prog Phys [**75**]{}, (2012) 066501.
C. S. O’Hern, S. A. Langer, A. J. Liu, and S. R. Nagel, Phys. Rev. Lett. [**88**]{}, (2002) 075507.
C. S. O’Hern, S. A. Silbert, A. J. Liu, and S. R. Nagel, Phys. Rev. E [**68**]{}, (2003) 011306.
P. Chaudhuri, L. Berthier, and S. Sastry, Phys. Rev. Lett. 104, (2010) 165701.
M. Maiti and M. Schmiedeberg, Scientific Reports [**8**]{}, (2018) 1837.
P. K. Morse and E. I. Corwin, Phys. Rev. Lett. [**119**]{}, (2017) 118003.
L. Milz and M. Schmiedeberg, Phys. Rev. E [**88**]{}, (2013) 062308.
M. Maiti and M. Schmiedeberg, J. Phys.: Condens. Matter [**31**]{}, (2019) 165101.
E. Flenner and G. Szamel, Nat. Commun. [**6**]{}, (2015) 7392.
N. D. Mermin and H. Wagner, Phys. Rev. Lett. [**17**]{}, (1966) 1133.
H. Shiba, Y. Yamada, T. Kawasaki, and K. Kim, Phys. Rev. Lett. [**117**]{}, (2016) 245701.
S. Vivek, C. P. Kelleher, P. M. Chaikin, and E. R. Weeks, Proc. Natl. Acad. Sci. USA [**114**]{}, (2017) 1850.
B. Illing, S. Fritschi, H. Kaiser, C. L. Klix, G. Maret, and P. Keim, Proc. Natl. Acad. Sci. USA [**114**]{}, (2017) 1856.
W. Kauzmann, Chem. Rev. [**43**]{}, (1948) 219.
S. Plimpton, J. Comp. Phys. [**117**]{}, (1995) 1.
H. Hinrichsen, Adv. Phys. [**49**]{}, (2000) 815.
C. Reichhardt, and C. J. O. Reichhardt, Phys. Rev. Lett. [**103**]{}, (2009) 168301.
A. Donev, F. H. Stillinger, and S. Torquato, Phys. Rev. Lett. [**96**]{}, (2006) 225502.
M. Bayer, J. M. Brader, F. Ebert, M. Fuchs, E. Lange, G. Maret, R. Schilling, M. Sperl, and J. P. Wittmer, Phys. Rev. E [**76**]{}, (2007) 011508.
A. Ikeda, L. Berthier, and P. Sollich, Phys. Rev. Lett. [**109**]{}, (2012) 018301.
X. Wang, W. Zheng, L. Wang, and N. Xu, Phys. Rev. Lett. [**114**]{}, (2015) 035502.
L. Berthier and T. A. Witten, Europhys. Lett. [**86**]{}, (2009) 10001.
L. Berthier and T. A. Witten, Phys. Rev. E [**80**]{}, (2009) 021502.
T. K. Haxton, M. Schmiedeberg, and A. J. Liu, Phys. Rev. E [**83**]{}, (2011) 031503.
M. Schmiedeberg, T. K. Haxton, S. R. Nagel, and A. J. Liu, Europhys. Lett. [**96**]{}, (2011) 36010.
L. Berthier, A. J. Moreno, and G. Szamel, Phys. Rev. E [**82**]{}, (2010) 060501(R).
M. Schmiedeberg, Phys. Rev. E [**87**]{}, (2013) 052310.
R. Miyazaki, T. Kawasaki, and K. Miyazaki, Phys. Rev. Lett. [**117**]{}, (2016) 165701.
D. J. Koeze and B. P. Tighe, Phys. Rev. Lett. [**121**]{}, (2018) 188002.
J. M. van Doorn, J. Bronkhorst, R. Higler, T. van de Laar, and J. Sprakel, Phys. Rev. Lett. [**118**]{}, (2017) 188001.
M. Kohl, R. F. Capellmann, M. Laurati, S. U. Egelhaaf, and M. Schmiedeberg, Nat. Commun. [**7**]{}, (2016) 11817.
A. Heuer, C. F. E. Schroer, D. Diddens, C. Rehwald, and M. Blank-Burian, Eur. Phys. J. Special Topics [**226**]{}, (2017) 3061.
M. Blank-Burian and A. Heuer, Phys. Rev. E [**98**]{}, (2018) 033002.
M. Maiti and M. Schmiedeberg, Energy landscape description of the clustering transition for active soft spheres, ArXiv:1812.06839.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We investigate the spreading of information in a one-dimensional Bose-Hubbard system after a sudden parameter change. In particular, we study the time-evolution of correlations and entanglement following a quench. The investigated quantities show a light-cone like evolution, i.e. the spreading with a finite velocity. We discuss the relation of this veloctiy to other characteristic velocities of the system, like the sound velocity. The entanglement is investigated using two different measures, the von-Neuman entropy and mutual information. Whereas the von-Neumann entropy grows rapidly with time the mutual information between two small sub-systems can as well decrease after an initial increase. Additionally we show that the static von Neuman entropy characterises the location of the quantum phase transition.'
address:
- 'Institut Romand de Recherche Numérique en Physique des Matériaux (IRRMA), CH-1015 Lausanne, Switzerland'
- ' Centre de Physique Théorique, Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, France '
author:
- 'Andreas M. Läuchli'
- Corinna Kollath
bibliography:
- 'references130308.bib'
title: 'Spreading of correlations and entanglement after a quench in the one-dimensional Bose-Hubbard model'
---
Introduction
============
Entanglement and correlations are important properties of quantum systems. The interest in these quantities is manifold. Correlations have been used since a long time to characterize the low energy properties of quantum many body states. In particular in the study of quantum phase transitions, correlations played a leading role. Entanglement has been the resource for quantum computing and quantum cryptography. Only recently entanglement has also become a major tool to understand and characterize ground states of interacting quantum many body systems. Driven by these applications the study of entanglement has become an active field of research and many different measures have been proposed in quantum many body systems [@AmicoVedral2008].
Recently the interest in the fundamental questions of quantum dynamics was reinforced by the experimental realization of strongly interacting gases [@BlochZwerger2007]. In these systems the precise and rapid tunability of the system parameters and the very good decoupling from the environment open the possibility to study the quantum evolution of a system far from equilibrium. In this context the following questions arise: How does a quantum system react to a parameter change? How fast can information propagate through the system? How do correlations and entanglement between different parts of the system build up?
These and similar questions have been asked in different situations. Lieb and Robinson found for one-dimensional spin systems that the quantum dynamics generated by local operators obeys a light-cone like evolution, i.e. the information spreads with a finite velocity [@LiebRobinson1972]. They proved that only exponentially small corrections exist outside this light-cone in the considered systems. Many generalizations of this Lieb Robinson theorem have been developed over the last years (see Ref. [@CramerEisert2008] and references therein). Further the finding of the spreading of correlations with a finite velocity after a global quench has been verified in specific integrable models [@IgloiRieger2000; @RigolOlshanii2006a; @Cazalilla2006; @CubittCirac2007; @CramerOsborne2008], as well as for critical systems which can be described by conformal field theory [@CalabreseCardy2006; @CalabreseCardy2007].
Motivated by the experimental investigation of a sudden parameter change across the superfluid to Mott-insulator transition [@GreinerBloch2002] we investigate here the time-evolution of a one-dimensional non-integrable Bose-Hubbard system after a sudden quench of the interaction strength. For this model the recent generalization of the Lieb-Robinson theorem of Cramer et al. [@CramerEisert2008] applies deep in the superfluid regime. However, no generalization exists in the presence of sizable interaction strength between the bosons.
We start our investigation with the static properties of the von Neumann entanglement entropy in the Bose-Hubbard model. In particular we show that previous predictions from conformal field theory [@HolzeyWilczek1994; @CalabreseCardy2004] are in very good agreement with our numerical results in the critical superfluid phase assuming a central charge $c=1$. We further point out how the knowledge on the dependence of the entanglement entropy on the system and block length in the critical phase can be used to locate the quantum phase transition.
We then turn to the discussion of the time-evolution of the quantum system after a sudden parameter change. Hereby we mainly consider a quench from the superfluid to the Mott-insulating phase. However, we also show some examples of quenches inside the superfluid or the Mott-insulating regime. In a previous work [@KollathAltman2007] we focussed on the relaxation of the system to a quasi-stationary state after a quench. Here in contrast we discuss the short time behaviour in detail. We study the spreading of information by the time-evolution of correlations, the von Neumann entropy, and the mutual information. We find that for onsite quantities the time-scale of relaxation is set by the hopping time of the bosons. In contrast for longer range correlations, like the one-particle density matrix or the density-density correlations, a front spreads with almost constant velocity which causes a ‘light-cone’ like evolution. The front travels at a finite speed, such that in an infinite system equilibrium can never be reached. We discuss the time evolution of the von Neumann entropy of different contiguous blocks of length $l$. We find that the von Neumann entropy of a certain block shows a sharp linear rise for short time and saturates for longer times. To a good approximation the rate of the block entropy increase is shown to follow a boundary law, while the entropy value at saturation depends only on the block volume. Our results nicely corroborate recent analytical results [@CalabreseCardy2005; @EisertOsborne2006; @BravyiVerstraete2006]. In contrast to the von Neumann entanglement, the mutual information between two contiguous blocks of different length after a quench to a Mott-insulating regime drops substantially in time compared to the values in the initial superfluid state. So for the considered quench the growth of the entropy and the loss of correlations go hand in hand.
Model and methods
=================
Ultracold bosons subjected to an optical lattice can be described by the Bose-Hubbard model [@JakschZoller1998; @Zwerger2003; @FisherFisher1989], here in its one-dimensional form $$\label{eq:bh}
H(J,U)= -J \sum_{j} \left(b_j^\dagger b^{\phantom{\dagger}}_{j+1}+h.c.\right) + \frac{U}{2} \sum _{j} n_j (n_j-1),$$ where $b^\dagger_j$ and $b_j$ are the boson creation and annihilation operators, and ${n}_j= b^\dagger_j
b^{\phantom{\dagger}}_j$ the number operators on site $j$. The first term in models the kinetic energy of the atoms and the parameter $J$ is called the hopping parameter. The second term stems from the short range interaction between the atoms and the parameter $U$ characterizes its strength. In the experimental setup of ultracold bosons in an optical lattice the parameters $U$ and $J$ can be directly related to the experimental parameters. In particular, by varying the lattice height in the experiment the parameter $U/J$ can be changed by several orders of magnitude and by anisotropic lattices the dimensionality of the system can be varied. In equilibrium at integer filling the Bose-Hubbard model shows a quantum phase transition at a critical value of $u_c:=(U/J)_c$ between a superfluid state ($U/J<u_c$), in which the atoms are delocalized, to a Mott insulating state ($U/J>u_c$) in which the atoms are localized [@FisherFisher1989]. We use $\hbar =1$ and $a=1$, where $a$ denotes the lattice spacing, throughout the work. The theoretical tools we use to treat the ground state and the dynamic properties of the Bose-Hubbard model are analytical approximations and numerical methods. Static results for the entanglement entropy are obtained using the density matrix renormalization group method (DMRG) [@White1992; @Hallberg2006; @Schollwoeck2005] with up to 1000 states. The results for time-evolution are calculated using the exact diagonalization (ED) based on Krylov methods [@park:5870; @manmana-2005-789] and the adaptive time-dependent DMRG (t-DMRG) [@Vidal2004; @WhiteFeiguin2004; @DaleyVidal2004]. The ED is used to study one-dimensional chains with periodic boundary conditions up to 16 sites, while the adaptive t-DMRG is used for one dimensional systems with open boundaries and with up to 64 sites keeping several hundred DMRG states.
Static von Neumann Entanglement Entropy {#sec:static}
=======================================
\[ht\]
![ Left panel: The static von Neumann block entropy $S_L(l)$ for an open system of length $L=1024$ and a range of different interaction values $U/J$, located in the superfluid and the Mott insulating phases. The average filling is $n=1$. Right panel: the same data as a function of the logarithmic conformal distance $\lambda:=\log \left[(2L/\pi)\sin(\pi l/L)\right]$. The linear behavior of the curves of $U/J=$ 1, 2 and 3 reveals the $c=1$ critical theory. The rapid saturation of the entropy for $U/J=$ 4 and 5 is a consequence of the short correlation length in the Mott insulating phase. []{data-label="fig:entropystat"}](BoseHubbardEntropy_L_1024_v2.pdf){width="0.8\linewidth"}
Before we discuss the behaviour of the system after a sudden change of the parameters we would like to discuss the static properties of entanglement in the Bose-Hubbard model. In recent years many different measures have been proposed for the entanglement in a system [@AmicoVedral2008]. One of the proposed measures is the von Neumann block entropy. Let A be a block of length $l$ and B be the remaining system. Then the von Neumann entropy of a block A is defined by $S_A=-\Tr_A\rho_A \log \rho_A$, where $\rho_A=\Tr_B \rho$ is the reduced density matrix of the block A and $\rho={|{\Psi}\rangle}{\langle{\Psi}|}$ the pure-state density matrix of the whole system. $\Tr_X$ denotes the trace over X.
In the superfluid phase the low energy physics of the one-dimensional Bose-Hubbard model can be described by a Luttinger liquid, which is a conformal field theory with central charge $c=1$. For such a 1+1 critical system the von Neumann entropy has been derived for different geometries of the subsystems A and B [@HolzeyWilczek1994; @CalabreseCardy2004]. If A is a single block of length $l$ in a periodic system of length $L$ the von Neumann entropy is given by $$S_A=\frac{c}{3} \log \left[\frac{L}{\pi }\sin\left(\frac{\pi l}{L}\right)\right] +s_1.
\label{eq:S_pb}$$ Here $s_1$ is a non-universal constant. In a system with open boundary conditions which is divided at some interior point into an interval A of length $l$ and its complement the von Neumann entropy is $$S_L(l)=\frac{c}{6} \log \left[\frac{2L}{\pi } \sin\left(\frac{\pi l}{L}\right)\right] + \log g +s_1/2 .
\label{eq:S_ob}$$ Here $\log g$ is the boundary entropy of Affleck and Ludwig [@AffleckLudwig1991]. Note that an oscillating correction term beyond the conformal field theory prediction has been found for open boundary conditions in the critical phase of the $S=1/2$ XXZ spin model [@LaflorencieAffleck2006]. This is a specific example of Friedel-like oscillations decaying away from the boundaries. These, in turn, lead to an oscillating, algebraically decaying correction term to the block entropy. In the case of the Bose Hubbard model considered here, which is not particle-hole symmetric, the open boundaries can induce a slightly non-uniform particle density, but the amplitude of the difference with respect to the nominal density decays rapidly away from the boundaries. The effect is most pronounced for small $U/J$ and becomes negligible for large $U/J$. This inhomogeneous particle density distribution is therefore expected to affect the entanglement entropy on small open systems.
For stronger interactions the quantum phase transition to the Mott-insulating phase takes places. The Mott-insulating state is characterized by an energy gap above the ground state and has a finite correlation length $\xi$. For gapped systems with a finite correlation length $\xi$, the block entropy is expected to saturate at a finite value $S_L(l) \sim \log (\xi)$ for $l\gg\xi$ [@VidalKitaev2003; @CalabreseCardy2004].
We show in Fig. \[fig:entropystat\] the von Neumann entropy of the static Bose-Hubbard model with open boundary conditions for different interaction strengths $U/J$ obtained using DMRG calculations on a system of length $L=1024$. In the left panel we show the entropy as a function of the block length $l$, while in the right panel we rescale the $x$-axis according to the logarithmic conformal distance $\lambda:=\log \left[(2L/\pi )\sin(\pi l/L)\right]$. Eq. \[eq:S\_ob\] then simplifies as $S_L(\lambda)=c/6\ \lambda + \log g + s_1/2$.
For small interaction strength $U/J=1,2,3$ our results agree very well with the prediction of the conformal field theory for large block length $l$. For $U/J=1$ deviations at small $l$ can be seen. They are due to the slightly inhomogeneous density in the open system. Linear fits to the curves in the right panel for the larger values of $\lambda$ yield slopes which are very close to the expected value of $1/6$. The extracted value of $c$ for the three critical $U/J$ values is shown close to the corresponding curve in the left panel. The values of $c$ are very accurately 1. The constant contribution $\log g + s_1/2$ becomes smaller for increasing $U/J$ due to the reduced onsite number fluctuations. For $U/J=$ 4 and 5 a very clear saturation of the entropy as a function of the block length $l$ can be seen, revealing the finite correlation length in the Mott insulator. The entanglement entropy vanishes completely in the limit of $U/J \rightarrow \infty$ when the groundstate becomes a trivial product wave function of singly occupied sites.
\[ht\]
![ Difference $\Delta S(L)=S_L(L/2)-S_{L/2}(L/4)$ versus $U/J$ for different system lengths $L$ at average filling $n=1$. For $U/J<u_c$ the expected scaling behaviour $\Delta S=c/6 \log 2$ with $c=1$ is seen. In the $L \rightarrow \infty$ limit the data converges to a step function located at $u_c$. The vertical line shows the previously obtained critical values $u_c\approx 3.37(12)$ [@KuehnerMonien2000] (black line with grey uncertainty) and $u_c\approx 3.380(4)$ [@ZakrzewskiDelande2008] (red line) for comparison. The deviation of small system length for $U\le 2J$ stems from inhomogeneous density distribution due to the open boundaries and is not captured in the CFT approach. Inset: $\Delta S$ as a function of $1/L$ for selected values of $U/J$ close to the phase transition. []{data-label="fig:entropyscal"}](BoseHubbardMidSystemEntropyVsU_diff_v2.pdf){width="0.8\linewidth"}
In the following we investigate whether it is possible to locate the critical value $u_c$ purely based on the properties of the von Neumann entropy. Earlier high precision work with DMRG mostly used a priori knowledge about the Kosterlitz-Thouless transition to accurately locate the critical value [@KuehnerMonien2000; @ZakrzewskiDelande2008]. A recent investigation based on quantum information related quantities such as the fidelity and single site entropies and their derivatives arrived at the conclusion that the single site entropy would not allow to locate the critical value $u_c$ precisely [@BuonsanteVezzani2007].
In order to locate the critical point we devise the quantity $$\Delta S(L):=S_L(L/2)-S_{L/2}(L/4)$$ the increase of the entropy at the mid-system interface upon doubling the system size. In a region of parameter space described by a conformal field theory with central charge $c$ one expects simply $\Delta S=c/6\ \log 2$ based on . On the other hand in a gapped region with a finite correlation length one obtains $\Delta S=0$ for $L\gg \xi$, because the entropy saturates for block lengths $l$ larger than the correlation length $\xi$. As the system size increases one therefore expects $\Delta S$ for the one-dimensional Bose-Hubbard model to scale to a step function as a function of $U/J$.
In we display $\Delta S$ as a function of $U/J$ for different system sizes $L$ from 32 up to 1024 sites. In the Luttinger liquid regime at small $U/J$ the curves for different system sizes nicely converge towards the expected value of $1/6 \log 2$. For larger $U/J$ values the $\Delta S$ curves scale to zero for increasing $L$, indicative of the gapped phase. Based on our available data we can safely infer that the point $U/J=3.4$ is already in the Mott insulating phase, while $U/J=3.3$ is still critical, see inset of . This result is in full agreement with the previously obtained critical values $u_c\approx 3.37(12)$ [@KuehnerMonien2000] and $u_c\approx 3.380(4)$ [@ZakrzewskiDelande2008], based on fits of the decaying bosonic Green’s function. In future studies using a fine grid of $U/J$ values one could perform a finite size scaling of the inflection point to extract an even more accurate value of $u_c$. So we conclude that an appropriate scaling plot of the von Neumann entanglement entropy provides competitive results on the location of the quantum phase transition in the one-dimensional Bose Hubbard model without relying on a priori knowledge of the Kosterlitz-Thouless nature of the transition.
Description of the parameter quench
===================================
We implement the quench by an abrupt change of the interaction strength from an initial value $U_i$ to a final value $U_f$ at fixed hopping parameter $J$ at time $t=0$. In most cases we start from a superfluid phase ($U_i/J<u_c$) and change to the Mott-insulating regime ($U_f/J>u_c$). The initial wave function ${|{\psi_0}\rangle}$ for $t\le 0$ is the ground state of the Hamiltonian $H_i=H(J,U_i)$. We investigate its time evolution for times $t>0$ subject to the Hamiltonian $H_f=H(J,U_f)$. The time evolution of the wave function is governed by the Schrödinger equation, i.e. $${|{\Psi(t) }\rangle}= \exp\left(-\textrm{i}H_f t\right) {|{\Psi_0}\rangle}.$$ The time evolution of expectation values of relevant operators can be expressed as $$\langle \hat{O}(t)\rangle = \sum_{m,m'} c^*_m c^{\phantom{*}}_{m'}
\exp\left[{-\textrm{i}(E_m-E_{m'}) t}\right] {\langle{m}|} \hat{O} {|{m'}\rangle}.$$ Hereby ${|{m}\rangle}$ are the eigenstates of the final Hamiltonian $H_f$ and $E_m$ the corresponding energy eigenvalues. The expansion of the initial state ${|{\psi_0}\rangle}$ into the eigenstates of the final Hamiltonian, i.e. ${|{\psi_0}\rangle}= \sum_{m}c_m {|{m }\rangle}$ with the weights $c_m$, is used. In a realistic case the tunneling between lattice sites and the interaction strength are non-zero and to determine the time-evolution of the wave function is not an easy task. We calculated the time-evolution numerically using ED and DMRG methods and analytical approximation in certain limits.
Light cone effect in correlations {#sec:light_cone}
=================================
\[ht\]
![Time-evolution of correlation functions after a quench from $U_i=2J$ to $U_f=40J$. The upper panel shows the single particle correlation functions ${\ensuremath{\big<b^\dagger_0 b_{r} \big>}}$ for different distances $r$. The correlations show partial revivals up to a time $t_r$ when they start to reach a quasi-steady state. This time $t_r$ grows approximately linearly with the distance $r$ as marked by the vertical lines. The central panel shows the same correlations functions after filtering out the high frequencies, see text for details. The lowest panel shows the density density correlations function ${\ensuremath{\big<n_0 n_{r} \big>}}$ after shifting and rescaling their amplitude for better visibility. The common vertical dashed lines denote the arrival of the minima as determined from the density-density correlations. The data shown is ED for a $L=14$ and DMRG data for $L=32$ and filling $n=1$. []{data-label="fig:outcoupling"}](PropagationPhenomena_Ui_2_Uf_40_v2.pdf){width="0.8\linewidth"}
![Fourier transform of the equal-time single-particle and density-density correlation functions for different distances $r$. The same parameters as in are used. In the single-particle correlation function clear frequency bands located at multiples of the interaction strength $U_f=40J$ can be seen. The density-density correlation is dominated by the contributions at low frequencies from zero up to $\sim J$. []{data-label="fig:shorttime"}](BdagB-time-FT_L_13_2_40.pdf){width="0.8\linewidth"}
In this section we investigate how correlations over a distance $r$ react to the sudden parameter change. We consider two different types of correlations, the single particle correlations ${\ensuremath{\big<b^\dagger_j b^{\phantom{\dagger}}_{j+r} \big>}}$ and the density-density correlations ${\ensuremath{\big<n_jn_{j+r} \big>}}$ at equal time. In Fig. \[fig:outcoupling\] we show the time-evolution of the different correlations after a quench from the superfluid, $U_i=2$, to the Mott-insulating, $U_f=40$, parameter regime.
#### Single-particle correlations
The upper panel shows the correlations ${\ensuremath{\big<b^\dagger_0
b^{\phantom{\dagger}}_{r} \big>}}$ for different distances $r$[^1]. For short times the single particle correlations oscillate with the period $2\pi/U_f$. The origin of these oscillations lies in the integer spectrum of the operator $\hat{n}_j(\hat{n}_j-1)/2$. Consider the limit of very strong interactions, where the time-evolution is totally dominated by the interactions. The time evolution of the single particle correlations is given by $${\ensuremath{\big<b^\dagger_i b^{\phantom{\dagger}}_{j} \big>}}(t) =
\sum_{\{m\},\{m'\}} \delta_{m_i,m'_i+1} \times \delta_{m_j,m'_j-1} \times
\e^{\textrm{i} U_f (m'_j-m'_i-1)t} c^*_m c_{m'} {\langle{\{m \}}|} b^\dagger_i b_j
{|{\{m' \} }\rangle}.$$ Here we use the notation $\{m\}$ for the Fock state with $m_i$ particles on site $i$. The time-evolution of the correlation function is determined by the non-vanishing cross terms ${\langle{\{m \}}|} b^\dagger_i b_j {|{\{m' \}}\rangle}$ of Fock states whose occupations vary by removing one particle from site $j$ and adding it at site $i$. The frequencies occuring in the time-evolution will be given by $U_f(m'_j-m'_i-1)$. This results in oscillations of period $T\approx 2\pi/U_f$ for equally occupied lattice sites $j$ and $i$ of the state ${|{\{m'\}}\rangle}$. For unequal onsite occupations higher multiples of $U_f$ can occur in the frequency spectrum.
Thus the distribution of the different frequencies in the Fourier transformation of the single particle correlation functions (left panel in Fig. \[fig:shorttime\]) and the occupation difference in the initial state are intimately connected. For the shown quench ($U_i=2J, U_f=40J$) sizable contributions of the lowest three frequency bands can be seen, whereas the occupation of higher frequency bands becomes very small. The distribution among the frequency bands changes by varying the initial value of the interaction strength $U_i$. If the initial state is deep in the superfluid regime, the peaks at higher frequencies show more weight due to the presence of strong particle number fluctuations. In contrast if the initial state is close to the Mott-insulating transition the particle fluctuations are suppressed and the weight of the peaks at higher frequencies decreases accordingly. The width of the frequency bands is due to the finite value of $J$ and is responsible for the decay of the oscillations in time [@KollathAltman2007].
Let us now come back to the real-time evolution of the single particle correlations for different distances. After a time $t\approx 0.5/J$ the correlation corresponding to the smallest distance shown, $r=2$, reaches a quasi-steady value (marked by the leftmost vertical line). The correlations of distance $r=3$ deviate from the correlation with $r>3$ for a time $t\approx 0.7/J$ (second left vertical line). The same can be observed for correlations with increasing distances at longer times. In the central part of the the same single particle correlations are shown, but the high frequency oscillations at frequencies $\omega_n \sim n \times U_f $ with $n\gtrsim 1$ are filtered out, and only the $n=0$ contributions are kept. In these low-pass filtered correlations the saturation to a quasi-steady state value can be seen much more clearly.
\[ht\]
![ Time-evolution of the rescaled density-density correlations ${\ensuremath{\big<n_0 n_{r} \big>}}(t)$ after a quench with the same parameters as in . The front of the evolution evolves approximately with a constant velocity. []{data-label="fig:densitydensity"}](nn_manual_colorbar_aml.pdf){width="0.8\linewidth"}
#### Density-density correlations
The lowest panel of shows the density-density correlations ${\ensuremath{\big<n_jn_{j+r} \big>}}-{\ensuremath{\big<n_j \big>}}{\ensuremath{\big<n_{j+r} \big>}}$ at equal time. The amplitudes of the correlations are rescaled and shifted for better readability [^2].
The density-density correlations do not show strong oscillations, but remain almost constant in time up to the moment, where a pronounced signal arrives. In their Fourier spectrum (right panel of ) mostly low frequencies $\sim J$ occur. This is due to the fact that the interaction term of the Hamiltonian commutes with the correlation function. Thus, in the strong coupling limit the interaction term does not give rise to oscillations. Approximately at the same time as the single particle correlations saturate, the spreading of a signal (here the reaching of a minimum, loci of the common vertical dashed lines in ) can be found in the density-density correlation ${\ensuremath{\big<n_0 n_{r} \big>}}$.
In we show a contour plot of the rescaled density correlations. In this representation a clear light cone evolution can be seen, i.e. a front travels through the system at almost constant speed. Let us note, that the same light cone effect occurs in the evolution of the correlations for different quench parameters (cf. ) and also in incommensurate systems, e.g. at filling $n=1/2$.
\[ht\]
![Characteristic time $t_r$ for arrival of the signal in the density-density correlations depending on the distance $r$ are summarized for different quench parameter $U_i$, $U_f$, and average density $n$. A clear linear relation between time and distance is observed for all shown quenches. To give an order of the uncertainties, typical error bars are plotted for chosen points. They take into account both the difficulty to identify a sharp signal and deviations between different system length ($L=14$ up to $L=64$). []{data-label="fig:velocity"}](SpeedPlot_140308.pdf){width="0.8\linewidth"}
\[ht\]
![Dependence of the velocity $v_s/2$ on the final interaction strength after the quench. Results are shown for different values of the density and compared to an analytical approximation described in the text. The initial value is $U_i=2J$.[]{data-label="fig:vel"}](vel_comp_v5.pdf){width="0.8\linewidth"}
#### Propagation velocity $v_s$
To extract the velocity $v_s$ of the signal propagation in the density-density correlations, we plot in the time $t_r$, when a signal, e.g. the minimum, occurs in the density-density correlations versus the distance $r$. The curves for different values of the initial and final interaction strength $U_i$ and $U_f$ are presented. A clear linear behaviour of the time $t_r$ versus the distance is seen. The shift between the curves, e.g. for $U_f=20,40$ and $U_f=4,6$, stems from the different signatures that have been tracked in the density-density correlations. The inverse of the slope is the signal propagation velocity $v_s$ of the signature. We extract the velocity $v_s$ by a linear fit $t_r=r/v_s+b$, where $v_s$ and $b$ are the fitting parameters. The results for $v_s$ are shown in .
[2]{}
$n$ $U_i/J$ $U_f/J$ $v_s [J] $
----- --------- --------- ---------------
1 2 2.1 4.4
1 2 3 5
1 2 4 $5.6\pm0.4$
1 2 5 $6.2$
1 2 6 6.8
1 2 10 5.2
1 2 40 $5.2 \pm 0.6$
1 1 4 5.9
1 3 4 5.6
1 6 40 4.8
1 1 3 5.2
: Velocity extracted from a linear fit.
$n$ $U_i/J$ $U_f/J$ $v_s [J] $
----- --------- --------- ------------
0.5 2 2.1 3.6
0.5 2 3 $3.8$
0.5 2 4 4
0.5 2 6 4.4
0.5 2 10 3.8
0.5 2 20 $3.8$
0.5 2 40 3.6
: Velocity extracted from a linear fit.
\[tab:vel\]
The velocity of the spreading of correlations after a quench has been identified by Calabrese and Cardy [@CalabreseCardy2006] within conformal field theory and for different integrable models to be twice the maximal mode velocity. The simple picture given is that the modes depart from both considered sites and the signal arrives, if both modes interfere. In the description by a conformal field theory this velocity agrees with the sound velocity of the system, since the dispersion relation is linear. In a chain of harmonic oscillators or in a spin chain lattice effects occur, which cause a curved dispersion relation. Thus the maximal velocity of a mode can be distinct from the sound velocity in the system. For the transverse Ising model a velocity equal to one has been found [@IgloiRieger2000].
In we show the results for the rescaled signal velocity $v_s/2$ for one initial interaction strength $U_i=2J$. We present results for different densities. At the commensurate density $n=1$ a phase transition to the Mott-insulator takes place at a critical value. In contrast for the incommensurate density $n=1/2$ [^3] the system stays in the superfluid phase for all interaction strengths. The signal velocity shows a strong increase for low interaction strength. After reaching a maximum around $U/J \approx 6$ it saturates for strong interactions to an almost constant value.
We further compare our results to different characteristic velocities of the Bose-Hubbard model: (i) the sound velocity in the superfluid regime, (ii) the maximal mode velocity of a simplified model in the Mott-insulator, (iii) the maximal mode velocity of a fermionic model applicable at low filling.
\(i) For an infinitesimal quench inside the superfluid phase, the rescaled signal velocity $v_s/2$ for the spreading of the correlations can be described by the sound velocity of the system. However, for finite quenches we expect the rescaled signal velocity to be larger than the sound velocity, since modes with higher velocities can be excited. In we approximate the sound velocity in the superfluid regime by the sound velocity of the corresponding continuous Lieb-Liniger model [@Lieb1963]. For small values of $\gamma$, it is given by $v_{LL}=2 n \sqrt{\gamma}\sqrt{1-\frac{\sqrt{\gamma}}{2\pi}}$, where $\gamma=U/(2Jn)$. This expression approximates the sound velocity for the Bose-Hubbard model up to an interaction strength $\gamma \lesssim 4$ [@KollathZwerger2004].
\(ii) An idea of the expected signal velocity in the Bose-Hubbard model in the Mott-insulating state can be obtained by mapping the system onto a simpler model using only three local states, e.g. occupation by $n_0-1$, $n_0$, and $n_0+1$ bosons per site, where $n_0$ is an integer [@AltmanAuerbach2002; @HuberBlatter2007]. In the Mott-insulating phase the dispersion relation for a particle hole excitations in this effective model can be determined as $\epsilon(k)=\sqrt{U^2-U \epsilon_0(k) (4n_0+2)+\epsilon_0(k)^2}/2$ [@HuberBlatter2007]. Here $\epsilon_0(k)=2J \cos(ka)$ is the band dispersion for the non-interacting case. The group velocity in this case is given by $ v=\frac{\partial \epsilon(k)}{\partial k} $. The maximum velocity $v_g$ of a mode given by this model is shown in . In the strong coupling limit this velocity agrees with the velocity extracted from a perturbative calculation where the maximal group velocity is given by $aJ/\hbar (2n_0+1)$. For the case of $n=1$ this results in $3aJ/\hbar$. In particular we see that towards the critical value in this model the velocity slightly increases. The momentum at which the maximum velocity is reached changes compared to the strong coupling limit.
\(iii) For strong interaction and low filling the Bose-Hubbard system can be mapped onto a fermionic system [@Cazalilla2004]. In this fermionic system the velocity is given by $v_f=2\sin(\pi n) (1-8J n\cos(\pi n)/U)$.
In we compare our numerical results to the different velocities (i)-(iii). For small interaction strength we see that the velocity $v_s/2$ is always larger than the sound velocity showing a similar rise with $U/J$. Comparing further the velocity of different quenches in the superfluid regime, e.g. $U_i=1,2$ and $U_f=4$ in , the velocity $v_s/2$ seems to approach the sound velocity if the parameter changes becomes smaller.
In the regime of strong interaction the qualitative features of the signal velocity are well reproduced by the given approximations. In particular, the velocity is almost constant for high interaction values and increases if $U/J$ is lowered towards $U/J\approx 6$. Further the order of magnitude of the values is in good agreement.
However, the approximation has to be taken with care. It does not take all higher energy excitations into account, which will be neccessary to quantitatively describe the situation under consideration. Further it does not take into account the initial state, i.e. which of the modes are actually excited by the quench. This is in contrast to the numerical results in which suggest that the value of the observed signal velocity might depend as well on the initial state. In particular if the higher particle fluctuations are present in the initial state the observed signal velocity seems to be larger.
Dynamics of entanglement and mutual information
===============================================
We turn in the following to the time-evolution of the entanglement and the amount of correlations between different subsystems after a quench. As a measure for the entanglement we use the von Neumann block entropy (see section \[sec:static\]) and the mutual information. Whereas the von Neumann entropy describes the entanglement of a region of the system with the remaining part, the mutual information gives a measure about the amount of correlations between different subsystems [@GroismanWinter2005]. In the following we first analyze the time-evolution of the von Neumann entropy before we turn to the evolution of the mutual information.
#### von Neumann entropy
\[ht\]
![ Time-evolution of the block entropy for different block lengths for a quench from $U_i=2J$ to $U_f=20 J$. The initial linear rise is the same for different block lengths, but depends on the boundary condition of the block (PBC: two interface links per block, faster rise of the entropy; OBC: one interface link per block, slower rise of the entropy). For longer times a saturation of the entropy depending linearly on the block length is observed. The left panel contains the original time evolution, while in the right panel, the obc data has been plotted for times $\tilde{t}=2t$, so as to illustrate the slower increase of the entropy due to the smaller boundary. The PBC results are from ED on $L=14$ (circles) and $L=16$ (squares) systems, while the OBC data has been obtained using $t-$DMRG up to $L=64$. []{data-label="fig:entropy"}](DMRG_ED_Comparison_Entropies.pdf){width="0.8\linewidth"}
In Fig. \[fig:entropy\] we show the time-evolution of the von Neumann entropy for Bose-Hubbard systems with periodic (ED, $L=14,16$ [^4]) and open boundary conditions (t-DMRG, 5 bosons per site are allowed). In both cases we focus on contiguous blocks of length $l$. For open boundary conditions the blocks are aligned with one of the boundaries. We plot the difference $S_L(l,t)-S_L(l,0)$ so that the curves for all block sizes start at zero at $t=0$.
In the left panel we show data for ED ($l=1,2,3,4,5$) and DMRG ($l=4,5$) results for a specific quench from $U_i=2 J$ to $U_f=20 J$. Let us first discuss the ED results: Immediately after the quench for $t\lesssim 0.4 J^{-1}$ a small dip shows up in all block entropies. However, from time $t\approx 0.4 \hbar /J$ up to $t^*(l)$ a linear rise of the block entropies can be observed. Interestingly all block sizes for $l>1$ have the same slope, until they bend over to an almost flat behaviour at successively later times. The dip in the entropy at larger times is a finite size effect, as can be seen by comparing the data for different system sizes (circles for $L=14$ and squares for $L=16$). The saturation value of the different block entropies depends linearly on the block size, defining an entropy propagation velocity $v_\mathrm{e}$, which is roughly equal to $v_\mathrm{s}$ determined based on the density-density correlation functions in the preceding . In a next step it is instructive to compare the entanglement dynamics between different block geometries (two interface links in ED, one interface link in $t-$DMRG). The $t-$DMRG results are also shown in the left panel of . The entropy of these open blocks increases more slowly, but converges to about the same value at late times as the periodic blocks of the same length. To illustrate this convincingly we show in the right panel the same data, but where the entropy of the open blocks is shown on a time scale which is twice as fast. Indeed the results of the two block geometries agree reasonably well. This nice result lends direct support to the picture developed by Calabrese and Cardy [@CalabreseCardy2005], where they predicted that the saturation of the entropy occurs at $t^*_\mathrm{PBC}(l) =l/2v$ for periodic boundary conditions, while $t^*_\mathrm{OBC}=l/v$ for blocks aligned with an open boundary.
These characteristics of the entanglement evolution are similar to the results obtained for different models in Refs. [@CalabreseCardy2005; @AmicoMassimapalma2004; @ChiaraFazio2006; @EisertOsborne2006; @AmicoVedral2008; @BarmettlerGritsev2008]. A linear growth of the entropy has been seen up to times $t=l/2v$, where $v$ is the maximal velocity of the excitations. Afterwards a saturation of the entropy is seen for $t\to \infty$, and the rate of the approach to the saturation value is related to the dispersion relation of the underlying model. At a fixed time the entanglement saturates for increasing block length, i.e. fullfils a boundary law with the boundary increasing with time. This shows that the boundary law for the dynamics of entanglement which has been proven mostly for 1D spin-systems [@EisertOsborne2006; @BravyiVerstraete2006] seems to be valid in more general systems such as the Bose-Hubbard system considered here.
#### Mutual information
\[ht\]
![(Color online) Time-evolution of the mutual information between two blocks of equal length $l=1$ (left panel) and $l=2$ (right panel) for different spatial separations $r$. The quench parameters are $U_i=2J$, $U_f=20J$, and $n=1$. The results have been obtained by ED for $L=14$ (bold, dashed lines) and $L=16$ (thin, straight lines). []{data-label="fig:mutual"}](MutualInfo_1-1_2-2.pdf){width="0.8\linewidth"}
While the von Neumann entropy is rapidly increasing in time and finally leads to an extensive entropy scaling with the block volume at large times, it is interesting to ask whether the vast entanglement entropy also leads to increased correlations between subparts of the system. A very useful quantity to address this question is the mutual information. The mutual information $I(A:B)$ between two subsystems A and B of the system is defined by $$I(A:B)= S_A+S_B-S_{A \cup B}.
\label{eqn:mutualinfo}$$ Here $S_X$ is the von Neumann entropy for the subsystem $X$. The mutual information $I(A:B)$ measures the total amount of information of system $A$ about system $B$ [@GroismanWinter2005]. Note that $A \cup B$ is not required to be equal to the total system. Interestingly the mutual information can fulfill an area law at finite temperatures, while the entropy is expected to follow a volume law [@WolfCirac2008] [^5].
In we show the time evolution of the mutual information between two blocks of $l=1$ (left panel) or $l=2$ (right panel) sites each, shifted by a distance $r$. At $t=0$ we expect the mutual information to decay slowly with the block separation $r$, since the starting state is basically a scale invariant critical state and the mutual information is just a complicated function of the basic critical correlation functions, such as $\langle b^\dagger_0 b_{r}\rangle$ and $\langle n_0 n_{r}\rangle$, which all decay algebraically. As a function of time $t$, the mutual information is decreasing rapidly with some slow oscillations on top. For later times it seems as though the mutual information levels off to a finite value at a time which depends again linearly on the distance $r$. The mutual information at late times decays much faster as a function of distance than in the initial state. So even though the entanglement entropy is vastly growing in time, this does not lead to enhanced entanglement between different subsystems.
Conclusion
==========
In our work we show that correlations and entanglement are very useful quantities to characterize the equilibrium and dynamic properties of a quantum many body system.
In the first part of our work we showed that the static von Neumann entanglement signals the quantum phase transition between the superfluid and Mott-insulating state without previous knowledge on the type of the phase transition. In the superfluid, the von Neuman entropy is in very good agreement with previous predictions by conformal field theory [@HolzeyWilczek1994; @CalabreseCardy2004] with a central charge $c=1$. Deviation are only found close to the boundaries of the system. These deviations are induced by the inhomogeneous density distribution caused by the open boundary conditions. A saturation of the entropy with the block length is found in the gapped Mott-insulating phase as predicted [@VidalKitaev2003; @CalabreseCardy2004].
In the second part of our work, the time-evolution after a sudden parameter change is analyzed with a focus on the spreading of information. Hereby different parameter changes are discussed ranging from the change between the superfluid to Mott-insulating phase over a quench inside the superfluid to a quench inside the Mott-insulating regime. Our study proposes that the Lieb Robinson theorem is valid as well in the considered situation of the Bose-Hubbard model. This relies on our findings that a light-cone like evolution takes place in different correlation functions. The velocity of the front evolving in the correlation functions is discussed and compared to different characteristic velocities of the system. The validity of the Lieb-Robinson theorem is further supported by the von Neumann entropy which shows for a certain block length a linear growth for short times. To a good approximation the rate of the block entropy increase is shown to follow a boundary law, while the entropy value at saturation seems only to depend on the block volume. Our results nicely corroborate recent analytical results [@CalabreseCardy2005; @EisertOsborne2006; @BravyiVerstraete2006]. However, in contrast to the von Neumann entropy after a quench from the superfluid to the Mott-insulating regime, the mutual information between two spatially separated small blocks relaxes to a lower value than in the starting state. So even though the entanglement entropy is vastly growing in time, this does not necessarily lead to enhanced entanglement between different regions of the system.
Acknowledgement
===============
We would like to thank E. Altman, J. Eisert, S. Huber, S. Manmana, A. Muramatsu, R. Noack, A. Rosch, and S. Wessel for fruitful discussions. This work was partly supported by the Swiss National Science Foundation. We wish to thank the Institute Henri Poincare-Centre Emile Borel for hospitality and support. Further CK acknowledges support by the RTRA network ’Triangle de la Physique’ and the DARPA OLE program.
The ED simulations have been performed on the machines of the CSCS (Manno).
References {#references .unnumbered}
==========
[^1]: To extract these correlations from the DMRG data with open boundary conditions the average over central sites is taken. Note that for periodic boundary conditions this quantitiy is real due to symmetry, whereas for open boundary conditions an imaginary part can develop. However for the shown functions and times the imaginary part is negligible.
[^2]: We will denote the rescaled function ${\ensuremath{\big<n_jn_{j+r} \big>}}-{\ensuremath{\big<n_j \big>}}{\ensuremath{\big<n_{j+r} \big>}}$ averaged over different sites $j$ in the center of the chain by ${\ensuremath{\big<n_0n_{r} \big>}}$.
[^3]: In the open $L=32$ system used in the t-DMRG, the density in the middle of the system is $n\approx 0.53$.
[^4]: Due to the computational challenge of calculating density matrices in ED for large blocks we chose to limit the local boson occupancy to 3 at most.
[^5]: In this reference the authors study the case of the total system equal to $A\cup B$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We find the first non-octahedral balanced 2-neighborly 3-sphere and the balanced 2-neighborly triangulation of the lens space $L(3,1)$. Each construction has 16 vertices. We also show that the rank-selected subcomplexes of a balanced simplicial sphere do not necessarily have an ear decomposition.'
author:
- |
Hailun Zheng\
Department of Mathematics\
University of Washington\
Seattle, WA 98195-4350, USA\
`hailunz@math.washington.edu`
title: 'Ear Decomposition and Balanced 2-neighborly Simplicial Manifolds'
---
Introduction
============
A simplicial complex is called $k$-neighborly if every subset of vertices of size at most $k$ is the set of vertices of one of its faces. Neighborly complexes, especially neighborly polytopes and spheres, are interesting objects to study. In the seminal work of McMullen [@M] and Stanley [@St], it was shown that in the class of polytopes and simplicial spheres of a fixed dimension and with a fixed number of vertices, the cyclic polytope simultaneously maximizes all the face numbers. The $d$-dimensional cyclic polytope is $\lfloor \frac{d}{2}\rfloor$-neighborly. Since then, many other classes of neighborly polytopes have been discovered. We refer to [@G], [@Sh] and [@P] for examples and constructions of neighborly polytopes. Meanwhile, the notion of neighborliness was extended to other classes of objects: for instance, neighborly cubical polytopes were defined and studied in [@JZ], [@JR], and [@SZ], and neighborly centrally symmetric polytopes and spheres were studied in [@B], [@J], [@LN], and [@DT].
In this paper we discuss a similar notion for balanced simplicial complexes. Balanced complexes were defined by Stanley in [@St2], where they were called completely balanced. A $(d-1)$-dimensional simplicial complex is called balanced if its graph is $d$-colorable. For instance, the barycentric subdivision of regular CW complexes and order complexes are balanced. We say that a balanced simplicial complex is *balanced $k$-neighborly* if every set of $k$ or fewer vertices with *distinct* colors forms a face. The joins of balanced neighborly spheres give balanced neighborly spheres. However, apart from the cross-polytopes, it is not known whether “join-indecomposable" balanced $k$-neighborly polytopes or spheres exist. To the best of our knowledge, no examples of such objects appear in the current literature, even for $k=2$. As for balanced 2-neighborly manifolds, one such construction that triangulates the sphere bundle is given in [@KN]; it is also a minimal triangulation of the underlying topological space.
This more or less explains why so far there is even no plausible sharp upper bound conjectures for balanced spheres or manifolds. The goal of this paper is to partially remedy this situation by searching for balanced neighborly spheres and manifolds of lower dimensions. It turns out that even the lower dimensional cases are rather involved: we show that the octahedral spheres are the only balanced $k$-neighborly $(2k-1)$-spheres with less or equal to $6k$ vertices for $k=2, 3$. However, we find two constructions of balanced 2-neighborly 3-manifolds with 16 vertices; one triangulates the sphere, and the other triangulates the lens space $L(3,1)$.
In a different direction, it is also interesting to ask whether every rank-selected subcomplex of a balanced simplicial polytope or sphere has a convex ear decomposition. This statement, if true, would imply that rank-selected subcomplexes of balanced simplicial polytopes possess certain weak Lefschetz properties, see Theorem 3.9 in [@Sw]. As a consequence, it would also provide an alternative proof of the balanced Generalized Lower Bound Theorem, see Theorem 3.3 and Remark 3.4 in [@MS]. We present an example giving a negative answer to this question for 3-dimensional spheres.
The structure of this manuscript is as follows. In Section 2, after reviewing basic definitions, we establish basic properties of balanced neighborly spheres; in particular, we prove that for some values of $f_0$, such spheres cannot exist. In Section 3, we construct a balanced 2-neighborly 3-sphere with 16 vertices. In Section 4, we present the balanced 2-neighborly triangulation of $L(3,1)$ with 16 vertices. In Section 5 we provide a way to construct balanced spheres whose rank-selected subcomplex does not have an ear decomposition.
Basic properties of balanced neighborly spheres
===============================================
A *simplicial complex* $\Delta$ with vertex set $V$ is a collection of subsets $\sigma\subseteq V$, called *faces*, that is closed under inclusion, and such that for every $v \in V$, $\{v\} \in \Delta$. For $\sigma\in \Delta$, let $\dim\sigma:=|\sigma|-1$ and define the *dimension* of $\Delta$, $\dim \Delta$, as the maximum dimension of the faces of $\Delta$. A *facet* is a maximal face under inclusion. We say that a simplicial complex $\Delta$ is *pure* if all of its facets have the same dimension.
If $\Delta$ is a simplicial complex and $\sigma$ is a face of $\Delta$, the *star* of $\sigma$ in $\Delta$ is $\operatorname{\mathrm{st}}_\Delta \sigma:= \{\tau \in\Delta: \sigma\cup\tau\in\Delta \}$. We also define the *link* of $\sigma$ in $\Delta$ as $\operatorname{\mathrm{lk}}_\Delta \sigma:=\{\tau-\sigma\in \Delta: \sigma\subseteq \tau\in \Delta\}$, and the *deletion* of a subset of vertices $W$ from $\Delta$ as $\Delta\backslash W:=\{\sigma\in\Delta:\sigma\cap W=\emptyset\}$. If $\Delta_1$ and $\Delta_2$ are simplicial complexes on disjoint vertex sets, then the join of $\Delta_1$ and $\Delta_2$, denoted $\Delta_1*\Delta_2$, is the simplicial complex with vertex set $V(\Delta_1)\cup V(\Delta_2)$ whose faces are $\{\sigma_1\cup\sigma_2:\sigma_1\in \Delta_1,\sigma_2\in\Delta_2\}$.
If $\Delta$ is a pure $(d-1)$-dimensional complex such that every $(d-2)$-dimensional face of $\Delta$ is contained in at most 2 facets, then the *boundary complex* of $\Delta$ consists of all $(d-2)$-dimensional faces that are contained in exactly one facet, as well as their subsets. A simplicial complex $\Delta$ is a *simplicial sphere* (resp. *simplicial ball*) if the geometric realization of $\Delta$ is homeomorphic to a sphere (resp. ball). The boundary complex of a simplicial $d$-ball is a simplicial $(d-1)$-sphere. A simplicial sphere is called *polytopal* if it is the boundary complex of a convex polytope. For instance, the boundary complex of an octahedron is a polytopal sphere; we will refer to it as an octahedral sphere.
For a fixed field ${{\bf k}}$, we say that $\Delta$ is a $(d-1)$-dimensional *${{\bf k}}$-homology sphere* if $\tilde{H}_i(\operatorname{\mathrm{lk}}_\Delta \sigma;{{\bf k}})\cong \tilde{H}_i(\mathbb{S}^{d-1-|\sigma|};{{\bf k}})$ for every face $\sigma\in\Delta$ (including the empty face) and $i\geq -1$. A *homology $d$-ball* (over a field ${{\bf k}}$) is a $d$-dimensional simplicial complex $\Delta$ such that (i) $\Delta$ has the same homology as the $d$-dimensional ball, (ii) for every face $F$, the link of $F$ has the same homology as the $(d-|F|)$-dimensional ball or sphere, and (iii) the boundary complex, $\partial\Delta:=\{F\in \Delta\mid \tilde{H}_i(\operatorname{\mathrm{lk}}_\Delta F)=0, \forall i\}$, is a homology $(d-1)$-sphere. The classes of simplicial $(d-1)$-spheres and homology $(d-1)$-spheres coincide when $d\leq 3$. From now on we fix ${{\bf k}}$ and omit it from our notation.
Next we define a special structure that exists in some pure simplicial complexes.
An *ear decomposition* of a pure $(d-1)$-dimensional simplicial complex $\Delta$ is an ordered sequence $\Delta_1,\Delta_2,\cdots, \Delta_m$ of pure $(d-1)$-dimensional subcomplexes of $\Delta$ such that:
1. $\Delta_1$ is a simplicial $(d-1)$-sphere, and for each $j=2,3,\cdots,m$, $\Delta_j$ is a simplicial $(d-1)$-ball.
2. For $2\leq j\leq m$, $\Delta_j\cap (\cup_{i=1}^{j-1}\Delta_i)=\partial\Delta_j$.
3. $\cup_{i=1}^m \Delta_i=\Delta.$
We call $\Delta_1$ the *initial complex*, and each $\Delta_j$, $j\geq 2$, an *ear of this decompostion*. Notice that this definition is more general than Chari’s original definition of a *convex ear decomposition*, see [@C Section 3.2], where the $\Delta_i$’s are required to be subcomplexes of the boundary complexes of polytopes. In particular, if a complex has no ear decomposition, then it has no convex ear decomposition. However, by the Steinitz theorem, all simplicial 2-spheres are polytopal, and hence also all simplicial 2-balls can be realized as subcomplexes of the boundary complexes of 3-dimensional polytopes. So for 2-dimensional simplicial complexes, the notion of an ear decomposition coincides with that of a convex ear decomposition.
A $(d-1)$-dimensional simplicial complex $\Delta$ is called *balanced* if the graph of $\Delta$ is $d$-colorable, or equivalently, there is a coloring map $\kappa: V\to [d]$ such that $\kappa(x)\neq \kappa(y)$ for any edge $\{x,y\}\in\Delta$. Here $[d]=\{1,2,\cdots,d\}$ is the set of colors. We denote by $V_i$ the set of vertices of color $i$. A balanced simplicial complex is called *balanced $k$-neighborly* if every set of $k$ or fewer vertices with distinct colors forms a face. For $S\subseteq [d]$, the subcomplex $\Delta_S:=\{F\in \Delta: \kappa(F)\subseteq S\}$ is called the *rank-selected subcomplex* of $\Delta$. We also define the *flag $f$-vector* $(f_S(\Delta):S\subseteq [d])$ and the *flag $h$-vector* $(h_S(\Delta):S\subseteq [d])$ of $\Delta$, respectively, by letting $f_S(\Delta):=\#\{F\in \Delta: \kappa(F)=S\}$, where $f_{\emptyset}(\Delta)=1$, and $h_S(\Delta):=\sum_{T\subseteq S}(-1)^{\#S-\#T}f_S(\Delta)$. The usual $f$-numbers and $h$-numbers can be recovered from the relations $f_{i-1}(\Delta)=\sum_{\#S=i}f_S(\Delta)$ and $h_{i}(\Delta)=\sum_{\#S=i}h_S(\Delta)$.
In the reminder of this section, we establish some restrictions on the possible size of color sets of balanced neighborly spheres.
\[prop: d=2k k-neighborly\] Let $\Delta$ be a balanced $k$-neighborly homology $(2k-1)$-sphere. Then $\Delta$ has the same number of vertices of each color. In particular, $f_0(\Delta)=2kl$ for some $l\geq 2$.
Let $W\subseteq [2k]$ be an arbitrary subset of the set of the colors with $|W|=k$. Since $\Delta$ is balanced $k$-neighborly, $\Delta_W$ is also balanced $k$-neighborly, and hence $\Delta_W$ is the join of $k$ color sets of colors in $W$, each considered as a 0-dimensional complex. By the fact that $h_k(\Delta_1*\Delta_2)=\sum_{j=0}^k h_j(\Delta_1)h_{k-j}(\Delta_2)$, we obtain that $$\prod_{i\in W}(|V_i|-1)=\prod_{i\in W}h_1(\Delta_{\{i\}})=h_{|W|}(\Delta)=h_{2k-|W|}(\Delta)=\prod_{i\in[2k]\backslash W}h_1(\Delta_{\{i\}})=\prod_{i\in[2k]\backslash W}(|V_{i}|-1).$$ Since $W\subseteq [2k]$ can be chosen arbitrarily, it follows that each color set in $\Delta$ must have the same size.
The above lemma is not sufficient to tell whether a balanced $k$-neighborly homology $(2k-1)$-sphere with $2kl$ vertices can exist for given $k, l\geq 2$. In the first non-trivial case $l=3$ we propose the following conjecture.
For an arbitrary $k\geq 2$, there does not exist a balanced $k$-neighborly homology $(2k-1)$-sphere with 6k vertices.
In the remaining of this section, we prove this conjecture for $k\leq 3$.
\[lm: intersection of links is sphere\] Let $d\geq 4$. If $\Delta$ is a balanced homology $(d-1)$-sphere and $V_d=\{v_1,v_2,v_3\}$ is the set of vertices of color $d$, then $\operatorname{\mathrm{lk}}_\Delta v_i\cap \operatorname{\mathrm{lk}}_\Delta v_j$ is a homology $(d-2)$-ball for any $1\leq i<j\leq 3$, and $\cap_{k=1}^{3}\operatorname{\mathrm{lk}}_\Delta v_k$ is a homology $(d-3)$-sphere.
Let $\Sigma=\operatorname{\mathrm{lk}}_\Delta v_i\cap \operatorname{\mathrm{lk}}_\Delta v_j$ and $\Gamma=\cap_{k=1}^{3}\operatorname{\mathrm{lk}}_\Delta v_k$. Every facet $\sigma$ of $\Gamma$ is a $(d-3)$-face whose link in $\Delta$ is a 6-cycle that contains the vertices $v_1,v_2,v_3$. Hence $\sigma$ is contained in exactly one facet $\sigma\cup\{w\}$ of $\Sigma$, where $w$ is the unique vertex adjacent to both $v_i, v_j$ in $\operatorname{\mathrm{lk}}_\Delta \sigma$. We conclude that $\Gamma$ is the boundary complex of $\Sigma$ and it is pure.
We first prove that $\Sigma$ and $\Gamma$ have the same homology as a $(d-2)$-ball and $(d-3)$-sphere respectively. Since each $(d-2)$-face of $\Delta$ is contained in exactly 2 facets, it follows that $\operatorname{\mathrm{lk}}_\Delta v_i\cup \operatorname{\mathrm{lk}}_\Delta v_j=\Delta_{[d-1]}$. By the Mayer-Vietoris sequence, for any $n\geq 0$, $$\label{eq:1}
\cdots\to H_{n+1}(\Delta_{[d-1]})\to H_n(\Sigma)\to H_n(\operatorname{\mathrm{lk}}_\Delta v_i)\oplus H_n(\operatorname{\mathrm{lk}}_\Delta v_j)\to H_{n}(\Delta_{[d-1]})\to\cdots.$$ Note that $\Delta_{[d-1]}$ is a deformation retract of $\Delta$ minus three points, hence $\beta_{d-2}(\Delta_{[d-1]})=2$ and $\beta_k(\Delta_{[d-1]})=0$ for $0\leq k\leq d-3$. We conclude from (\[eq:1\]) that $\beta_k(\Sigma)=0$ for all $k\geq 0$. Again by the Mayer-Vietoris sequence and the fact that $\operatorname{\mathrm{lk}}_\Delta v_{[3]-\{i,j\}}\cup \Sigma=\Delta_{[d-1]}$, we obtain $$\cdots\to H_{n+1}(\Delta_{[d-1]})\to H_n(\Gamma)\to H_n(\operatorname{\mathrm{lk}}_\Delta v_{[3]-\{i,j\}})\oplus H_n(\Sigma)\to H_{n}(\Delta_{[d-1]}) \to\cdots.$$ Hence $\beta_{d-3}(\Gamma)=1$ and $\beta_{k}(\Gamma)=0$ for $0\leq k\leq d-4$.
Next, for any $\tau\in \Gamma$, we have $\operatorname{\mathrm{lk}}_\Sigma \tau=\operatorname{\mathrm{lk}}_{\operatorname{\mathrm{lk}}_\Delta \tau} v_i \cap \operatorname{\mathrm{lk}}_{\operatorname{\mathrm{lk}}_\Delta \tau} v_j$ and $\operatorname{\mathrm{lk}}_\Gamma \tau=\cap_{i=1}^{3}\operatorname{\mathrm{lk}}_{\operatorname{\mathrm{lk}}_\Delta \tau} v_i$. Since $\operatorname{\mathrm{lk}}_\Delta \tau$ is a balanced homology $(d-1-|\tau|)$-sphere, using the same argument as above, we may show that $\operatorname{\mathrm{lk}}_\Sigma \tau$ and $\operatorname{\mathrm{lk}}_\Gamma\tau$ has the same homology as a $(d-2-|\tau|)$-ball and $(d-3-|\tau|)$-sphere respectively. Therefore $\Gamma$ is a homology $(d-3)$-sphere. Finally, for any interior face $\sigma$ of $\Sigma$, $\operatorname{\mathrm{lk}}_\Sigma \sigma=\operatorname{\mathrm{lk}}_{\operatorname{\mathrm{lk}}_\Delta v_i} \sigma=\operatorname{\mathrm{lk}}_{\operatorname{\mathrm{lk}}_\Delta v_j} \sigma$, and hence $\operatorname{\mathrm{lk}}_\Sigma \sigma$ is a homology sphere. By definition we conclude that $\Sigma$ is a homology $(d-2)$-ball.
\[rm: 1\] The complex $\Gamma$ in Lemma \[lm: intersection of links is sphere\] is not balanced, since $\Gamma$ is $(d-1)$-colorable instead of being $(d-2)$-colorable.
\[prop: 2-nbly 3-sphere 12 ver\] No balanced 2-neighborly homology 3-spheres with 12 vertices exist.
![Left: triangulation of the vertex link $\operatorname{\mathrm{lk}}_\Delta v_i$ for $v_i\in V_4$, where $\{u_1,u_2,u_3\}$, $\{w_1,w_2,w_3\}$ and $\{z_1,z_2,z_3\}$ are the three other color sets. Right: the missing edges between vertices of different color of $\operatorname{\mathrm{lk}}_\Delta v_i$.[]{data-label="Figure: triangulation of the vertex link"}](f_0=12case "fig:") ![Left: triangulation of the vertex link $\operatorname{\mathrm{lk}}_\Delta v_i$ for $v_i\in V_4$, where $\{u_1,u_2,u_3\}$, $\{w_1,w_2,w_3\}$ and $\{z_1,z_2,z_3\}$ are the three other color sets. Right: the missing edges between vertices of different color of $\operatorname{\mathrm{lk}}_\Delta v_i$.[]{data-label="Figure: triangulation of the vertex link"}](missingedgesf_0=12 "fig:")
Assume that $\Delta$ is such a sphere. By Lemma \[prop: d=2k k-neighborly\], each color set of $\Delta$ has three vertices. We let $V_4=\{v_1,v_2,v_3\}$ be the set of vertices of color 4. Since $\Delta$ is balanced 2-neighborly, each $\operatorname{\mathrm{lk}}_\Delta v_i$ is a 2-sphere with 9 vertices, its $f$-vector is (1,9,21,14). Furthermore, the balancedness of $\Delta$ implies that every vertex $v\in \operatorname{\mathrm{lk}}_\Delta v_i$ has $\deg_{\operatorname{\mathrm{lk}}_\Delta v_i} v=4$ or 6. If $x$ is the number of vertices of degree 6 in $\operatorname{\mathrm{lk}}_\Delta v_i$, then $$4(9-x)+6x=\sum_{u\in \operatorname{\mathrm{lk}}_\Delta v_i}\deg(\operatorname{\mathrm{lk}}_{\operatorname{\mathrm{lk}}_\Delta v_i}u)=2 f_1(\operatorname{\mathrm{lk}}_\Delta v_i)=42,$$ and hence $x=3$. A balanced 2-sphere with 9 vertices, 3 of which have degree 6, is unique up to isomorphism, as shown in Figure \[Figure: triangulation of the vertex link\]. It is immediate that the missing edges between vertices of different colors in this sphere form a 6-cycle.
On the other hand, $\Sigma:=\operatorname{\mathrm{lk}}_\Delta v_1\cap \operatorname{\mathrm{lk}}_\Delta v_2$ is a triangulated 2-ball by Lemma \[lm: intersection of links is sphere\]. If we delete all of the boundary edges from $\Sigma$, the resulting complex $\Sigma'$ is still contractible. However, $\Sigma$ does not have interior vertices. (An interior vertex of $\Sigma$ would not be in $V(\operatorname{\mathrm{lk}}_\Delta v_3)$, which would contradict the 2-neighborliness of $\Delta$.) Hence the missing edges of $\operatorname{\mathrm{lk}}_\Delta v_3$ that form a 6-cycle form the only interior edges of $\Sigma$, i.e., $\Sigma'$ is a 6-cycle. This contradicts that $\Sigma'$ is contractible, so no such sphere exists.
In fact, a stronger result holds.
\[lm: 3 triangulations of balanced 3-sphere\] Up to an isomorphism, there are three triangulations of balanced 3-spheres with each color set of size 3 and no more than 50 edges.
Let $\Delta$ be such a sphere and let $V_4=\{v_1,v_2,v_3\}$. Each vertex link of $\Delta$ is a balanced 2-sphere with at most 9 vertices, hence it is either the octahedral sphere, the suspension of a 6-cycle, or the connected sum of two octahedral spheres. We denote these three 2-spheres as $\Sigma_1$, $\Sigma_2$ and $\Sigma_3$ respectively. By Lemma \[lm: intersection of links is sphere\], $\Delta_{[3]}$ is the union of three triangulated 2-balls $B_i=\operatorname{\mathrm{lk}}_\Delta v_j\cap \operatorname{\mathrm{lk}}_\Delta v_k$, where $\{i,j,k\}=[3]$, glued along their common boundary complex $c$. Assume that $f_0(\operatorname{\mathrm{lk}}_\Delta v_i)\leq f_0(\operatorname{\mathrm{lk}}_\Delta v_j)$ when $i\leq j$. An easy counting leads to $$f_0(\Delta_{[3]})=f_0(c)+\sum_{i=1}^{3} f_0(B_i\backslash c)=9, \quad f_0(\operatorname{\mathrm{lk}}_\Delta z_i)=f_0(c)+f_0(B_j\backslash c)+f_0(B_k\backslash c)\in \{6,8,9\},$$ where $f_0(B_i\backslash c)$ counts the number of interior vertices of $B_i$. We enumerate all possible values of the triple $(f_0(\operatorname{\mathrm{lk}}_\Delta v_1), f_0(\operatorname{\mathrm{lk}}_\Delta v_2), f_0(\operatorname{\mathrm{lk}}_\Delta v_3))$ under the condition $f_1(\Delta)=2\sum_{i=1}^{3}f_0(\operatorname{\mathrm{lk}}_\Delta v_i)\leq 48$:
1. $(f_0(\operatorname{\mathrm{lk}}_\Delta v_1), f_0(\operatorname{\mathrm{lk}}_\Delta v_2), f_0(\operatorname{\mathrm{lk}}_\Delta v_3))=(6,6,9)$ or $(6,8,9)$. Since $\operatorname{\mathrm{lk}}_\Delta v_1$ is combinatorially equivalent to the octahedral sphere, it follows that $\operatorname{\mathrm{lk}}_\Delta v_3$ is obtained from $\operatorname{\mathrm{lk}}_\Delta v_2$ by a cross flip (see [@IKN] for a reference). So in the former case $\operatorname{\mathrm{lk}}_\Delta v_2\cong \Sigma_1$, $\operatorname{\mathrm{lk}}_\Delta v_3\cong \Sigma_3$, and the cross flip replaces a 2-face of $\operatorname{\mathrm{lk}}_\Delta v_2$ with its complement in the octahedral sphere. In the latter case $\operatorname{\mathrm{lk}}_\Delta v_2\cong \Sigma_2$, $\operatorname{\mathrm{lk}}_\Delta v_3\cong \Sigma_3$, and the cross flip replaces the union of three 2-faces of $\operatorname{\mathrm{lk}}_\Delta v_2$ with its complement in the octahedral sphere.
2. $(f_0(\operatorname{\mathrm{lk}}_\Delta v_1), f_0(\operatorname{\mathrm{lk}}_\Delta v_2), f_0(\operatorname{\mathrm{lk}}_\Delta v_3))=(8,8,8)$. Then $c$ is a 6-cycle and $\Delta_{[3]}\backslash c$ consists of three disjoint vertices. It is easy to see that at least one of these vertices has degree 6. Then since $\operatorname{\mathrm{lk}}_\Delta v_1\cong \operatorname{\mathrm{lk}}_\Delta v_2\cong \Sigma_2$, the other two vertices must be of degree 6 as well, and hence $\Delta_{[3]}$ is the join of $c$ and three disjoint vertices.
3. $(f_0(\operatorname{\mathrm{lk}}_\Delta v_1), f_0(\operatorname{\mathrm{lk}}_\Delta v_2), f_0(\operatorname{\mathrm{lk}}_\Delta v_3))=(8,8,9)$. Since the vertices of degree 6 in $\operatorname{\mathrm{lk}}_\Delta z_3$ form a 3-cycle, the two disjoint vertices in $\Delta_{[3]}\backslash c$ cannot both have degree 6 or 4. However, if one vertex of $\Delta_{[3]}\backslash c$ is of degree 6, then since $\operatorname{\mathrm{lk}}_\Delta z_1$ and $\operatorname{\mathrm{lk}}_\Delta z_2$ are combinatorially equivalent to $\Sigma_2$ and $c$ is a 7-cycle, $B_3$ must be the join of one vertex $u$ and a path of length 6. Then $u$ is not connected to any vertex of $\Delta_{[3]}-c$, a contradiction.
In sum, we obtain three balanced 3-spheres with 12 vertices: $S_1$, the connected sum of two octahedral 3-spheres; $S_2$, the join of two 6-cycles, and $S_3$, with $\operatorname{\mathrm{lk}}_\Delta v_i\cong \Sigma_i$ for $1\leq i\leq 3$.
The above lemma implies that all balanced 3-spheres with each color set of set 3 can only have $f_1=42, 46, 48, 52$. (The first three numbers are attained by $S_i$.)
\[prop: 2-nbly 4-sphere 15 ver\] No balanced 2-neighborly homology $4$-spheres with each color set of size 3 exist.
Let $\Delta$ be such a sphere and let its color set $V_5=\{v_1,v_2,v_3\}$. By Alexander Duality, $\tilde{H_i}(\Delta_{\{4,5\}})\cong \tilde{H}_{3-i}(\Delta_{[3]})$. In particular, since $\Delta_{\{4,5\}}$ is balanced 2-neighborly, $\beta_2(\Delta_{[3]})=\beta_1(\Delta_{\{4,5\}})=4$ and $\beta_1(\Delta_{[3]})=0$. Hence $$f_2(\Delta_{[3]})=(f_1-f_0+\chi)(\Delta_{[3]})=\frac{9\cdot 6}{2}-9+5=23.$$ By double counting, $\sum_{i=1}^{3} f_1(\operatorname{\mathrm{lk}}_\Delta v_i)=\sum_{W=\{i,j,5\}\subseteq [5]} f_2(\Delta_{W})=\binom{4}{2}f_2(\Delta_{[3]})=138$. But $f_0(\operatorname{\mathrm{lk}}_\Delta v_i)\in \{42, 46, 48, 50,52,54\}$, it follows that either $138=42+48*3$, that is, $\operatorname{\mathrm{lk}}_\Delta v_1\cong S_1$ and $\operatorname{\mathrm{lk}}_\Delta v_2, \operatorname{\mathrm{lk}}_\Delta v_3\cong S_2$; or $138=46*3$ and $\operatorname{\mathrm{lk}}_\Delta v_i\cong S_3$ for all $i$.
Consider the first case above. It can be checked that for any $W=\{i,j\}$, $f_1((\operatorname{\mathrm{lk}}_\Delta v_1)_W)=7$ and $f_1((\operatorname{\mathrm{lk}}_\Delta v_2)_W)=6$ or 9, depending on whether $(\operatorname{\mathrm{lk}}_\Delta v_2)_W$ is a 6-cycle or not. Hence $f_2(\Delta_{W\cup\{5\}})=\sum_{i=1}^{3}f_1((\operatorname{\mathrm{lk}}_\Delta v_i)_W)\neq 23$, a contradiction. As for the second case, since $\operatorname{\mathrm{lk}}_\Delta v_1\cap \operatorname{\mathrm{lk}}_\Delta v_2$ is a homology 3-ball with 12 vertices on the boundary, by Lemma \[lm: 3 triangulations of balanced 3-sphere\] there is a unique balanced 3-sphere combinatorially equivalent to $S_3$ that contains $\operatorname{\mathrm{lk}}_\Delta v_1\cap \operatorname{\mathrm{lk}}_\Delta v_2$ as a subcomplex. It follows that $\operatorname{\mathrm{lk}}_\Delta v_1=\operatorname{\mathrm{lk}}_\Delta v_2$, a contradiction. Hence no balanced 2-neighborly homology 4-spheres with 15 vertices exist.
\[cor: 3-nbly 5-sphere 18 ver\] The only balanced 3-neighborly homology 5-sphere with $\leq 18$ vertices is the octahedral 5-sphere.
Let $\Delta$ be such a sphere. The vertex links of $\Delta$ are balanced 2-neighborly 4-spheres with $\leq 16$ vertices. By Proposition \[prop: 2-nbly 4-sphere 15 ver\], each link must be the suspension of a balanced 2-neighborly 3-sphere with $\leq 14$ vertices. Then the result follows from Lemma \[prop: d=2k k-neighborly\] and Proposition \[prop: 2-nbly 3-sphere 12 ver\].
First Construction
==================
In this section we provide a balanced 2-neighborly triangulation of the 3-sphere.
\[Second Example\]
[[R]{}]{} [[R]{}]{}=1.9cm (0:[[R]{}]{}) – cycle (360:[[R]{}]{}) node\[right\] [$w_3$]{} – cycle (30:[[R]{}]{}) node\[above right\] [$u_3$]{} – cycle (60:[[R]{}]{}) node\[above right\] [$v_3$]{} – cycle (90:[[R]{}]{}) node\[above\] [$w_1$]{} – cycle (120:[[R]{}]{}) node\[above left\] [$u_2$]{} – cycle (150:[[R]{}]{}) node\[above left\] [$v_1$]{} – cycle (180:[[R]{}]{}) node\[left\] [$u_1$]{} – cycle (210:[[R]{}]{}) node\[below left\] [$w_2$]{} – cycle (240:[[R]{}]{}) node\[below left\] [$v_4$]{} – cycle (270:[[R]{}]{}) node\[below\] [$u_4$]{} – cycle (300:[[R]{}]{}) node\[below right\] [$w_4$]{} – cycle (330:[[R]{}]{}) node\[below right\] [$v_2$]{}; (330:[[R]{}]{}) – (30:[[R]{}]{}) – (90:[[R]{}]{})–cycle; (330:[[R]{}]{}) – (120:[[R]{}]{}) – (300:[[R]{}]{})–(150:[[R]{}]{}); (150:[[R]{}]{}) – (270:[[R]{}]{}) – (210:[[R]{}]{})–cycle; (0:[[R]{}]{})–(30:[[R]{}]{}) –(60:[[R]{}]{})– (90:[[R]{}]{})–(120:[[R]{}]{})–(150:[[R]{}]{})–(180:[[R]{}]{})–(210:[[R]{}]{}) –(240:[[R]{}]{})–(270:[[R]{}]{})–(300:[[R]{}]{})–(330:[[R]{}]{})–(0:[[R]{}]{});
[[R]{}]{} [[R]{}]{}=1.9cm (0:[[R]{}]{}) – cycle (360:[[R]{}]{}) node\[right\] [$w_3$]{} – cycle (30:[[R]{}]{}) node\[above right\] [$u_3$]{} – cycle (60:[[R]{}]{}) node\[above right\] [$v_3$]{} – cycle (90:[[R]{}]{}) node\[above\] [$w_1$]{} – cycle (120:[[R]{}]{}) node\[above left\] [$u_2$]{} – cycle (150:[[R]{}]{}) node\[above left\] [$v_1$]{} – cycle (180:[[R]{}]{}) node\[left\] [$u_1$]{} – cycle (210:[[R]{}]{}) node\[below left\] [$w_2$]{} – cycle (240:[[R]{}]{}) node\[below left\] [$v_4$]{} – cycle (270:[[R]{}]{}) node\[below\] [$u_4$]{} – cycle (300:[[R]{}]{}) node\[below right\] [$w_4$]{} – cycle (330:[[R]{}]{}) node\[below right\] [$v_2$]{}; (0:[[R]{}]{}) – (60:[[R]{}]{}) – (120:[[R]{}]{})–cycle; (150:[[R]{}]{}) – (0:[[R]{}]{}) – (180:[[R]{}]{})–(330:[[R]{}]{}); (180:[[R]{}]{}) – (300:[[R]{}]{}) – (240:[[R]{}]{})–cycle; (0:[[R]{}]{})–(30:[[R]{}]{}) –(60:[[R]{}]{})– (90:[[R]{}]{})–(120:[[R]{}]{})–(150:[[R]{}]{})–(180:[[R]{}]{})–(210:[[R]{}]{}) –(240:[[R]{}]{})–(270:[[R]{}]{})–(300:[[R]{}]{})–(330:[[R]{}]{})–(0:[[R]{}]{});
[[R]{}]{} [[R]{}]{}=1.9cm (0:[[R]{}]{}) – cycle (360:[[R]{}]{}) node\[right\] [$w_3$]{} – cycle (30:[[R]{}]{}) node\[above right\] [$u_3$]{} – cycle (60:[[R]{}]{}) node\[above right\] [$v_3$]{} – cycle (90:[[R]{}]{}) node\[above\] [$w_1$]{} – cycle (120:[[R]{}]{}) node\[above left\] [$u_2$]{} – cycle (150:[[R]{}]{}) node\[above left\] [$v_1$]{} – cycle (180:[[R]{}]{}) node\[left\] [$u_1$]{} – cycle (210:[[R]{}]{}) node\[below left\] [$w_2$]{} – cycle (240:[[R]{}]{}) node\[below left\] [$v_4$]{} – cycle (270:[[R]{}]{}) node\[below\] [$u_4$]{} – cycle (300:[[R]{}]{}) node\[below right\] [$w_4$]{} – cycle (330:[[R]{}]{}) node\[below right\] [$v_2$]{}; (150:[[R]{}]{}) – (90:[[R]{}]{}) – (180:[[R]{}]{})–(60:[[R]{}]{}) – (210:[[R]{}]{}) – (30:[[R]{}]{})–(240:[[R]{}]{})–(0:[[R]{}]{}) – (270:[[R]{}]{}) – (330:[[R]{}]{}); (0:[[R]{}]{})–(30:[[R]{}]{}) –(60:[[R]{}]{})– (90:[[R]{}]{})–(120:[[R]{}]{})–(150:[[R]{}]{})–(180:[[R]{}]{})–(210:[[R]{}]{}) –(240:[[R]{}]{})–(270:[[R]{}]{})–(300:[[R]{}]{})–(330:[[R]{}]{})–(0:[[R]{}]{});
[[R]{}]{} [[R]{}]{}=1.9cm (0:[[R]{}]{}) – cycle (360:[[R]{}]{}) node\[right\] [$w_3$]{} – cycle (30:[[R]{}]{}) node\[above right\] [$u_3$]{} – cycle (60:[[R]{}]{}) node\[above right\] [$v_3$]{} – cycle (90:[[R]{}]{}) node\[above\] [$w_1$]{} – cycle (120:[[R]{}]{}) node\[above left\] [$u_2$]{} – cycle (150:[[R]{}]{}) node\[above left\] [$v_1$]{} – cycle (180:[[R]{}]{}) node\[left\] [$u_1$]{} – cycle (210:[[R]{}]{}) node\[below left\] [$w_2$]{} – cycle (240:[[R]{}]{}) node\[below left\] [$v_4$]{} – cycle (270:[[R]{}]{}) node\[below\] [$u_4$]{} – cycle (300:[[R]{}]{}) node\[below right\] [$w_4$]{} – cycle (330:[[R]{}]{}) node\[below right\] [$v_2$]{}; (210:[[R]{}]{}) – (120:[[R]{}]{}) – (240:[[R]{}]{})–(90:[[R]{}]{}) – (270:[[R]{}]{}) – (60:[[R]{}]{})–(300:[[R]{}]{})– (30:[[R]{}]{}); (150:[[R]{}]{})–(30:[[R]{}]{}); (210:[[R]{}]{})–(330:[[R]{}]{}); (0:[[R]{}]{})–(150:[[R]{}]{})–(300:[[R]{}]{}); (120:[[R]{}]{})–(330:[[R]{}]{})–(180:[[R]{}]{}); (0:[[R]{}]{})–(30:[[R]{}]{}) –(60:[[R]{}]{})– (90:[[R]{}]{})–(120:[[R]{}]{})–(150:[[R]{}]{})–(180:[[R]{}]{})–(210:[[R]{}]{}) –(240:[[R]{}]{})–(270:[[R]{}]{})–(300:[[R]{}]{})–(330:[[R]{}]{})–(0:[[R]{}]{});
[[R]{}]{} [[R]{}]{}=1.9cm (0:[[R]{}]{}) – cycle (360:[[R]{}]{}) node\[right\] [$w_3$]{} – cycle (30:[[R]{}]{}) node\[above right\] [$u_3$]{} – cycle (60:[[R]{}]{}) node\[above right\] [$v_3$]{} – cycle (90:[[R]{}]{}) node\[above\] [$w_1$]{} – cycle (120:[[R]{}]{}) node\[above left\] [$u_2$]{} – cycle (150:[[R]{}]{}) node\[above left\] [$v_2$]{} – cycle (180:[[R]{}]{}) node\[left\] [$u_1$]{} – cycle (210:[[R]{}]{}) node\[below left\] [$w_2$]{} – cycle (240:[[R]{}]{}) node\[below left\] [$v_4$]{} – cycle (270:[[R]{}]{}) node\[below\] [$u_4$]{} – cycle (300:[[R]{}]{}) node\[below right\] [$w_4$]{} – cycle (330:[[R]{}]{}) node\[below right\] [$v_1$]{}; (150:[[R]{}]{})–(210:[[R]{}]{}) – (120:[[R]{}]{}) – (240:[[R]{}]{})–(90:[[R]{}]{}) – (270:[[R]{}]{}) – (60:[[R]{}]{})–(300:[[R]{}]{})– (30:[[R]{}]{})–(330:[[R]{}]{}); (0:[[R]{}]{})–(30:[[R]{}]{}) –(60:[[R]{}]{})– (90:[[R]{}]{})–(120:[[R]{}]{})–(150:[[R]{}]{})–(180:[[R]{}]{})–(210:[[R]{}]{}) –(240:[[R]{}]{})–(270:[[R]{}]{})–(300:[[R]{}]{})–(330:[[R]{}]{})–(0:[[R]{}]{});
[[R]{}]{} [[R]{}]{}=1.9cm (0:[[R]{}]{}) – cycle (360:[[R]{}]{}) node\[right\] [$w_3$]{} – cycle (30:[[R]{}]{}) node\[above right\] [$u_3$]{} – cycle (60:[[R]{}]{}) node\[above right\] [$v_3$]{} – cycle (90:[[R]{}]{}) node\[above\] [$w_1$]{} – cycle (120:[[R]{}]{}) node\[above left\] [$u_2$]{} – cycle (150:[[R]{}]{}) node\[above left\] [$v_2$]{} – cycle (180:[[R]{}]{}) node\[left\] [$u_1$]{} – cycle (210:[[R]{}]{}) node\[below left\] [$w_2$]{} – cycle (240:[[R]{}]{}) node\[below left\] [$v_4$]{} – cycle (270:[[R]{}]{}) node\[below\] [$u_4$]{} – cycle (300:[[R]{}]{}) node\[below right\] [$w_4$]{} – cycle (330:[[R]{}]{}) node\[below right\] [$v_1$]{}; (30:[[R]{}]{})–(90:[[R]{}]{})–(150:[[R]{}]{})–cycle; (210:[[R]{}]{})–(270:[[R]{}]{})–(330:[[R]{}]{})–cycle; (150:[[R]{}]{})–(0:[[R]{}]{})–(180:[[R]{}]{})–(330:[[R]{}]{}); (0:[[R]{}]{})–(30:[[R]{}]{}) –(60:[[R]{}]{})– (90:[[R]{}]{})–(120:[[R]{}]{})–(150:[[R]{}]{})–(180:[[R]{}]{})–(210:[[R]{}]{}) –(240:[[R]{}]{})–(270:[[R]{}]{})–(300:[[R]{}]{})–(330:[[R]{}]{})–(0:[[R]{}]{});
[[R]{}]{} [[R]{}]{}=1.9cm (0:[[R]{}]{}) – cycle (360:[[R]{}]{}) node\[right\] [$w_3$]{} – cycle (30:[[R]{}]{}) node\[above right\] [$u_3$]{} – cycle (60:[[R]{}]{}) node\[above right\] [$v_3$]{} – cycle (90:[[R]{}]{}) node\[above\] [$w_1$]{} – cycle (120:[[R]{}]{}) node\[above left\] [$u_2$]{} – cycle (150:[[R]{}]{}) node\[above left\] [$v_2$]{} – cycle (180:[[R]{}]{}) node\[left\] [$u_1$]{} – cycle (210:[[R]{}]{}) node\[below left\] [$w_2$]{} – cycle (240:[[R]{}]{}) node\[below left\] [$v_4$]{} – cycle (270:[[R]{}]{}) node\[below\] [$u_4$]{} – cycle (300:[[R]{}]{}) node\[below right\] [$w_4$]{} – cycle (330:[[R]{}]{}) node\[below right\] [$v_1$]{}; (0:[[R]{}]{})–(60:[[R]{}]{})–(120:[[R]{}]{})–cycle; (180:[[R]{}]{})–(240:[[R]{}]{})–(300:[[R]{}]{})–cycle; (330:[[R]{}]{})–(120:[[R]{}]{})–(300:[[R]{}]{})–(150:[[R]{}]{}); (0:[[R]{}]{})–(30:[[R]{}]{}) –(60:[[R]{}]{})– (90:[[R]{}]{})–(120:[[R]{}]{})–(150:[[R]{}]{})–(180:[[R]{}]{})–(210:[[R]{}]{}) –(240:[[R]{}]{})–(270:[[R]{}]{})–(300:[[R]{}]{})–(330:[[R]{}]{})–(0:[[R]{}]{});
\[figure: A’ and B’\] Assume that $V_1=\{u_1, u_2, u_3, u_4\}$, $V_2=\{v_1,v_2,v_3,v_4\}$, $V_3=\{w_1,w_2,w_3,w_4\}$ and $V_4=\{z_1,z_2,z_3,z_4\}$ are the four color sets of a balanced 3-sphere $\Gamma$. We let $\operatorname{\mathrm{lk}}_\Gamma z_1=A\cup_{\partial A \sim \partial C} C$ and $\operatorname{\mathrm{lk}}_\Gamma z_3=B\cup_{\partial B \sim \partial C} C$, where $A$, $B$ and $C$ are triangulated 2-balls sharing the same boundary as shown in Figure \[fig: patches A,B,C\]. All possible edges that do not appear in $A$, $B$ and $C$ are shown in Figure \[fig: patch D\] as solid red edges in disc $D'$. Notice that the dashed edges in $D'$ are edges in discs $A$ and $B$, so we may rearrange the boundary of $D$ by switching the positions of vertices $v_1$ and $v_2$, and then replacing the edges containing $v_1$ or $v_2$ in $\partial D'$ by the dashed edges. In this way, we obtain a triangulation of a 12-gon $D$ as shown in Figure \[fig: patch D\]. Furthermore, $\partial D\subseteq A\cup B$, and $\partial D$ divides the sphere $=A\cup_{\partial A \sim \partial B} B$ into two discs $A'$ and $B'$ as shown in Figure 6.
We let $\operatorname{\mathrm{lk}}_\Gamma z_2=A'\cup_{\partial A' \sim \partial D} D$ and $\operatorname{\mathrm{lk}}_\Gamma z_4=B'\cup_{\partial B' \sim \partial D} D$. Since both $\operatorname{\mathrm{st}}_\Delta z_1\cap \operatorname{\mathrm{st}}_\Delta z_3=C$ and $\operatorname{\mathrm{st}}_\Delta z_2\cap (\operatorname{\mathrm{st}}_\Delta z_1\cup \operatorname{\mathrm{st}}_\Delta z_3)=A'$ are simplicial 2-balls, it follows that $\Sigma=\cup_{i=1}^{3} \operatorname{\mathrm{st}}_\Delta z_i$ is a simplicial 3-ball. Furthermore, the boundary of $\Sigma$ is exactly $\operatorname{\mathrm{lk}}_\Delta z_4$. Hence $\Gamma=\Sigma\cup\operatorname{\mathrm{st}}_\Delta z_4$ is indeed a balanced 2-neighborly 3-sphere.
Here we provide some properties of $\Gamma$ in Construction \[Second Example\].
1. $(A\cup B, C, D)$ is an ear decomposition of $\Gamma_{[3]}$.
2. The automorphism group of $\Gamma$ has two generators $$(u_1u_3u_2u_4)(v_1z_2v_2z_1)(v_3z_4v_4z_3)(w_1w_4w_2w_3),\: (z_1v_1)(z_2v_2)(z_3v_3)(z_4v_4)(u_1w_1)(u_2w_2)(u_3w_3)(u_4w_4).$$ (The second generator is given by switching vertices of color 1 and 3, and color 2 and 4, but with the same subscript.) Hence ${\rm{Aut}}(\Delta)$ has 8 elements.
3. The complex $\Gamma$ given in Construction \[Second Example\] is shellable. For $\operatorname{\mathrm{lk}}_\Gamma z_1=A\cup_{\partial A\sim \partial C}C$, there exist two shellings $c_1,\ldots,c_{10}, a_1,\ldots, a_{10}$ and $a'_1,\ldots,a'_{10}, c'_1,\ldots,c'_{10}$ such that for any $1\leq i\leq 10$, $c_i, c'_i$ are facets from $C$ and $a_i, a'_i$ are facets from $A$. Similarly, there exist two shellings $c_1,\ldots,c_{10}, b_1,\ldots, b_{10}$ and $b'_1,\ldots,b'_{10},c'_1,\ldots,c'_{10}$ for $\operatorname{\mathrm{lk}}_\Gamma z_3=B\cup_{\partial B\sim \partial C}C$, where $b_i,b'_i$ are facets from $B$. Then $$a'_1*z_1,\ldots,a'_{10}*z_1, c'_1*z_1,\ldots,c'_{10}*{z_1}, c_1*z_3, \ldots, c_{10}*z_3,,b_1*z_3\ldots,b_{10}*z_3$$ gives a shelling of $\operatorname{\mathrm{st}}_\Gamma z_1\cup \operatorname{\mathrm{st}}_\Gamma z_3$. We may extend this shelling into a complete shelling of $\Gamma$ by constructing two similar shellings of $\operatorname{\mathrm{lk}}_\Gamma z_2$ and $\operatorname{\mathrm{lk}}_\Gamma z_4$. However, we tried some computer tests and failed to prove either polytopality or non-polytopality.
It is easy to see that if $\Delta_1$ is a balanced 2-neighborly $(d_1-1)$-sphere and $\Delta_2$ is a balanced 2-neighborly $(d_2-1)$-sphere, then $\Delta_1*\Delta_2$ is a balanced 2-neighborly $(d_1+d_2-1)$-sphere. Hence by taking joins, we find balanced 2-neighborly $(4k-1)$-spheres with $16k$ vertices for any $k\geq 1$.
Let $d\geq 4$ and $m\geq 5$ be arbitrary integers. Is there a balanced 2-neighborly simplicial $(d-1)$-sphere all of whose color sets have the same size $m$? Is there a polytopal sphere with these properties?
Second Construction
===================
In this section we present our first construction of a balanced 2-neighborly lens space $L(3,1)$ with 16 vertices. We denote it by $\Delta$. Each color set of $\Delta$ has four vertices.
\[Figure: four links\]\
\[First Example\] We denote the color sets of $\Delta$ by $V_1=\{u_1,u_2,u_3,u_4\}$, $V_2=\{v_1,v_2,v_3,v_4\}$, $V_3=\{w_1,w_2,w_3,w_4\}$ and $V_4=\{z_1,z_2,z_3,z_4\}$.
In Figure 2 we illustrate the construction of the vertex links $\operatorname{\mathrm{lk}}_\Delta z_i$ for $i=1,\ldots,4$. All these links are realized as cylinders. Two links $\operatorname{\mathrm{lk}}_\Delta z_1$ and $\operatorname{\mathrm{lk}}_\Delta z_2$ share the same top and bottom, which are triangulated hexagons spanned by vertices $\{u_i,v_i,w_i:i=1,3\}$ and $\{u_i,v_i,w_i:i=2,4\}$, respectively. To construct $\operatorname{\mathrm{lk}}_\Delta z_3$ from $\operatorname{\mathrm{lk}}_\Delta z_1$, we switch the positions of vertices $u_3, v_3, w_3$ with vertices $u_4,v_4,w_4$ respectively and form a new cylinder. The new top and bottom hexagons contain the 2-faces $\{u_1,v_1,w_1\}$ and $\{u_2,v_2,w_2\}$. Similarly, we construct the link $\operatorname{\mathrm{lk}}_\Delta z_4$ from $\operatorname{\mathrm{lk}}_\Delta z_2$ by switching the positions of vertices $u_3, v_3, w_3$ with vertices $u_4,v_4,w_4$ and letting $\{u_1,v_1,w_1\}$ and $\{u_2,v_2,w_2\}$ be the 2-faces that appear in the triangulation of the top and bottom hexagons. It follows that $\operatorname{\mathrm{lk}}_\Delta z_3$ and $\operatorname{\mathrm{lk}}_\Delta z_4$ also share the same top and bottom.
Now since $\Delta$ is balanced 2-neighborly, by our construction, it only remains to show that $\Delta$ triangulates the lens space $L(3,1)$. The geometric realizations of $\operatorname{\mathrm{st}}_\Delta z_1$ and $\operatorname{\mathrm{st}}_\Delta z_2$ are filled cylinders that share top and bottom. So their union $A:=\operatorname{\mathrm{st}}_\Delta z_1\cup \operatorname{\mathrm{st}}_\Delta z_2$ is a filled torus (that is, a genus-1 handlebody); so is the union $B:=\operatorname{\mathrm{st}}_\Delta z_3\cup \operatorname{\mathrm{st}}_\Delta z_4$. Note that these two handlebodies have identical boundary complexes, thus they provide a Heegaard splitting of a lens space.
To identify which lens space $\Delta$ triangulates, we need to determine the homeomorphism $\phi: \partial A\to \partial B$. Consider two generators $\gamma, \delta$ of $\pi_1(A\cap B)= \pi_1(\partial A)$, where $\gamma$ is the 6-cycle $(u_3,v_1,w_3,u_1,v_3,w_1)$ and $\delta$ is the 4-cycle $(u_1,w_2,u_4,w_3)$. In particular, $\delta$ is also a generator of $\pi_1(A)$. From the construction we see that $\phi(\gamma)$ is a loop running around the equator of $\partial B$ thrice and the meridian of $\partial B$ once. Also $\phi(\delta)$ runs around the equator of $\partial B$ twice and the meridian of $\partial B$ once. Hence it is indeed the lens space $L(3,1)$.
\[rm: property of construction 1\] Our construction $\Delta$ has the following properties:
1. All vertex links are combinatorially equivalent.
2. From Figure 5 we see $\operatorname{\mathrm{lk}}_\Delta z_i\cap\operatorname{\mathrm{lk}}_\Delta z_j$ has two connected components when $\{i,j\}=\{1,2\}$ or $\{3,4\}$ (they are the top and bottom hexagons as shown in Figure 2); and it has three connected components when $i\in\{1,2\}$ and $j\in\{3,4\}$ (each component is the union of two facets along the side of the cylinders). In general, the intersection of two vertex links, where the vertices are of the same color, always has at least two connected components.
3. There are three group actions on the vertices of $\Delta$:
1. Fix the subscript and rotate the corresponding vertices of color 1, 2 and 3 respectively. The generator is given by $(u_1v_1w_1)(u_2v_2w_2)(u_3v_3w_3)$.
2. Rotate vertices of the same color. The generator is $$(u_1u_3u_2u_4)(v_1v_3v_2v_4)(w_1w_3w_2w_4)(z_1z_3z_2z_4).$$
3. Exchange $\operatorname{\mathrm{lk}}_\Delta z_1$ and $\operatorname{\mathrm{lk}}_\Delta z_2$, $\operatorname{\mathrm{lk}}_\Delta z_3$ and $\operatorname{\mathrm{lk}}_\Delta z_4$, by exchanging $v_i$ and $w_i$ (or $u_i$ and $w_i$, $u_i$ and $v_i$) for all $i\in [4]$. The generators are $(z_1z_2)(z_3z_4)(v_1w_1)(v_2w_2)(v_3w_3)(v_4w_4)$, $(z_1z_2)(z_3z_4)(u_1w_1)(u_2w_2)(u_3w_3)(u_4w_4)$ and $(z_1z_2)(z_3z_4)(u_1v_1)(u_2v_2)(u_3v_3)(u_4v_4)$.
The automorphism group of $\Delta$ is of size 96.
The complex $\Delta$ is a balanced vertex minimal triangulation of $L(3,1)$.
By Proposition 6.1 in [@KN], each color set of $\Delta$ is of size at least 3. If there are exactly three vertices $v_1, v_2, v_3$ of color 1 in $\Delta$, apply the Mayer-Vietoris sequence on the triple $(\operatorname{\mathrm{st}}_\Delta v_1\cup\operatorname{\mathrm{st}}_\Delta v_2, \operatorname{\mathrm{st}}_\Delta v_3, \Delta)$ and we obtain that $$0=H_1(\operatorname{\mathrm{lk}}_\Delta v_3)\to H_1(\operatorname{\mathrm{st}}_\Delta v_1\cup\operatorname{\mathrm{st}}_\Delta v_2)\oplus H_1(\operatorname{\mathrm{st}}_\Delta v_3) \to H_1(\Delta)\to H_0(\operatorname{\mathrm{lk}}_\Delta v_3)=0.$$ Hence $H_1(\operatorname{\mathrm{st}}_\Delta v_1\cup\operatorname{\mathrm{st}}_\Delta v_2)\cong H_1(\Delta)=\mathbb{Z}/3\mathbb{Z}$. However, this is impossible since $H_1(\operatorname{\mathrm{st}}_\Delta v_1\cup\operatorname{\mathrm{st}}_\Delta v_2)\cong H_0(\operatorname{\mathrm{st}}_\Delta v_1\cap\operatorname{\mathrm{st}}_\Delta v_2)$, which cannot be $\mathbb{Z}/3\mathbb{Z}$.
The same argument as above also shows that the balanced triangulation of any lens space $L(p,q)$ with $p>1$ must have at least 16 vertices.
Third Construction
==================
In this section our goal is to construct a balanced 3-sphere whose rank-selected subcomplexes do not have ear decompositions. The motivation is from the balanced 2-neighborly construction of $L(3,1)$ in Section 4. Indeed, we want to construct a balanced 3-dimensional complex $\Delta$ so that 1) each vertex link is a 2-sphere; 2) for a fixed color set $V_4=\{v_1,\cdots, v_k\}$, the intersection of any two vertex links $\operatorname{\mathrm{lk}}_\Delta v_i\cap \operatorname{\mathrm{lk}}_\Delta v_j$ always has at least two connected components (as the property listed in Remark \[rm: property of construction 1\]); and 3) $\cup_{i=1}^{4}\operatorname{\mathrm{st}}_\Delta v_i$ is 3-ball, which together with the condition 1) guarantees that $\Delta$ is a 3-sphere.
In the following we take $k=5$ and give such a construction. Figure \[fig1:links\] illustrates the links $\operatorname{\mathrm{lk}}_\Delta v_1,\cdots,\operatorname{\mathrm{lk}}_\Delta v_4$. Every label represents the color of the vertex. Also each connected component of $\operatorname{\mathrm{lk}}_\Delta v_1\cap \operatorname{\mathrm{lk}}_\Delta v_2$ is colored in green, $\operatorname{\mathrm{lk}}_\Delta v_i\cap \operatorname{\mathrm{lk}}_\Delta v_3$ is colored in blue for $i=1,2$, and $\operatorname{\mathrm{lk}}_\Delta v_j\cap \operatorname{\mathrm{lk}}_\Delta v_4$ is colored in pink for $j=1,2,3$. Immediately we check that all these intersections of vertex links have 2 or 3 connected components.
Figure \[fig2:union of links\] shows how $\Delta\backslash W$ is formed from these links. First we glue $\operatorname{\mathrm{lk}}_\Delta v_1$ and $\operatorname{\mathrm{lk}}_\Delta v_2$ along two green triangles. The resulting complex $\operatorname{\mathrm{lk}}_\Delta v_1\cup \operatorname{\mathrm{lk}}_\Delta v_2$ is shown in Figure \[link12\]. Then we place $\operatorname{\mathrm{lk}}_\Delta v_3$ on top of $\operatorname{\mathrm{lk}}_\Delta v_1\cup\operatorname{\mathrm{lk}}_\Delta v_2$. As we see from Figure \[link123\], the boundary complex of $\cup_{i=1}^{3}\operatorname{\mathrm{st}}_\Delta v_i$ is a triangulated torus. Finally, we place $\operatorname{\mathrm{lk}}_\Delta v_4$ on top of $\cup_{i=1}^{3}\operatorname{\mathrm{lk}}_\Delta v_i$ so that $\operatorname{\mathrm{st}}_\Delta v_4$ “covers the 1-dimensional hole" in $\cup_{i=1}^{3}\operatorname{\mathrm{st}}_\Delta v_i$, see Figure \[link1234\]. We denote the subspace of $\mathbb{R}^3$ enclosed by $\operatorname{\mathrm{lk}}_\Delta v_i$ as $S_i$ for $1\leq i\leq 4$, and let $S_5:=\cup_{i\leq 4} S_i$. From our construction it follows that the boundary complex of $S_5$ is a 2-sphere; we let it be $\operatorname{\mathrm{lk}}_\Delta v_5$. Indeed $\Delta$ is a 3-sphere since $\Delta$ is the union of two 3-balls $S_5$ and $\operatorname{\mathrm{st}}_\Delta v_5$ glued along their common boundary $\operatorname{\mathrm{lk}}_\Delta v_5$.
Since each $\operatorname{\mathrm{lk}}_\Delta v_i\cap \operatorname{\mathrm{lk}}_\Delta v_j$ has at least two connected components for $1\leq i\neq j\leq 4$, the Mayer-Vietoris sequence implies that $S_i\cup S_j$ is not contractible for all $1\leq i\neq j\leq 4$. A similar inspection of $\operatorname{\mathrm{lk}}_\Delta v_i\cup\operatorname{\mathrm{lk}}_\Delta v_j\cup\operatorname{\mathrm{lk}}_\Delta v_k$ also implies that the boundary complexes of $S_i\cup S_j\cup S_k$’s cannot be triangulated 2-spheres for distinct $1\leq i,j,k\leq 4$.
Not all rank-selected subcomplexes of balanced simplicial spheres have ear decompositions.
Consider the complex $\Delta$ constructed above. We denote the union of interior faces of a complex $\tau$ by $\operatorname{\mathrm{int}}\tau$. Suppose $\Delta\backslash V_4$ has an ear decomposition $(\Gamma_1,\Gamma_2,\cdots, \Gamma_k)$. Since $|V_4|=5$ and $\beta_{2}(\Delta\backslash W)=4$, $k$ must be 4. Notice first that $\cup_{i\leq 4} \operatorname{\mathrm{lk}}_\Delta v_i$ divides $\mathbb{R}^3$ into five subspaces, namely, $S_1,\cdots, S_4$ and the complement of $S_5$, each having $\operatorname{\mathrm{lk}}_\Delta v_i$ as the boundary complex for $1\leq i\leq 5$ respectively. Since $\Gamma_1\cup\Gamma_2-\operatorname{\mathrm{int}}(\Gamma_1\cap\Gamma_2)$ must be a triangulated 2-sphere, by the Jordan theorem, it separates $\mathbb{R}^3$ into two connected components, hence the bounded component must be either $S_i\cup S_j$ or $S_i\cup S_j\cup S_k$ for some $1\leq i,j,k\leq 4$. (We may assume that it is not $S_i$, since otherwise we may consider the 2-sphere $\cup_{i\leq 3}\Gamma_i-\cup_{1\leq i\neq j\leq 3}\operatorname{\mathrm{int}}(\Gamma_i\cap\Gamma_j)$ instead of $\Gamma_1\cup\Gamma_2-\operatorname{\mathrm{int}}(\Gamma_1\cap\Gamma_2)$, where the subset enclosed by this sphere in $\mathbb{R}^3$ cannot be $S_i$ anymore.) This contradicts the fact that the boundaries of $S_i\cup S_j$ or $S_i\cup S_j\cup S_k$ are not 2-spheres.
One can think of all the figures illustrated above as projections of a subcomplex of $\Delta -\operatorname{\mathrm{st}}_\Delta v_5$ onto $\mathbb{R}^3$. However, we do not know whether the complex provided in this section can be realized as the boundary of a 4-polytope.
Acknowledgements {#acknowledgements .unnumbered}
================
The author was partially supported by a graduate fellowship from NSF grant DMS-1361423. I thank Moritz Firsching for pointing out the automorphism groups of the constructions in Section 3 and 4 and running some computational tests to decide whether the constructions are polytopal. I also thank Lorenzo Venturello for pointing out mistakes in an earlier version and giving helpful suggestions.
[17]{} G. Burton: The non-neighbourliness of centrally symmetric convex polytopes having many vertices, *J. Combin. Theory Ser. A* **58** (1991), 321-322.
M. K. Chari: Two decompositions in topological combinatorics with applications to matroid complexes, *Trans. Amer. Math. Soc.* **349** (1997), 3925-3943.
D. L. Donoho and J. Tanner: Counting the faces of randomly-projected hypercubes and orthants, with applications, *Discrete Comput. Geom.* **43** (2010), 522-541.
B. Gr$\rm\ddot{u}$nbaum: *Convex polytopes*, Interscience, London, 1967.
I. Izmestiev, S. Klee and I. Novik: Simplicial moves on balanced complexes, arXiv:1512.04384.
W. Jockusch: An infinite family of nearly neighborly centrally symmetric 3-spheres, *J. Combin. Theory. Ser. A* **72** (1995), 318-321.
M. Joswig and T. R$\mathrm{\ddot{o}}$rig: Neighborly cubical polytopes and spheres, *Israel J. Math.* **159** (2007), 221-242.
M. Joswig and G. M. Ziegler: Neighborly cubical polytopes, *Discrete Comput. Geom.* **24** (2000), 325-344.
S. Klee and I. Novik: Lower Bound Theorems and a Generalized Lower Bound Conjecture for balanced simplicial complexes, *Mathematika*, **62** (2016), 441-477.
P. McMullen: The maximum number of faces of a convex polytope, *Mathematika* **17** (1970), 179-184.
M. Juhnke-Kubitzke and S. Murai: Balanced generalized lower bound inequality for simplicial polytopes, Selecta Math., to appear.
N. Linial and I. Novik: How neighborly can a centrally symmetric polytope be?, *Discrete Comput. Geom.* **36** (2006), 273-281.
F. Lutz: Small examples of nonconstructible simplicial balls and spheres, *SIAM J. Discrete Math.*, **18** (2004), 103-109.
A. Padrol: Many neighborly polytopes and oriented matroids, *Discrete Comput. Geom.* **50** (2013), 865-902.
R. Sanyal and G. M. Ziegler: Construction and analysis of projected deformed products, *Discrete Comput. Geom.* **43** (2010), 412-435.
I. Shermer: Neighborly polytopes, *Israel J. Math.* **43** (1982), 291-314.
R. Stanley: The upper bound conjecture and Cohen-Macaulay rings, *Studies in Applied Math.* **54** (1975), 135-142.
R. Stanley: Balanced Cohen-Macaulay complexes, *Trans. Amer. Math. Soc.* **249** (1979), 139-157.
E. Swartz: $g$-elements, finite buildings and higher Cohen-Macaulay connectivity, *J. Combin. Theory. Ser. A* **113** (2006), 1305-1320.
A. Vince: A non-shellable 3-sphere, *European J. Combin.*, (1985), 91-100.
G. Ziegler: Lectures on Polytopes, Volume 152 of Graduate Texts in Math., *Springer-Verlag, New York*, 1995, revised ed., 1998.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In-medium valence-quark distributions of $\pi^+$ and $K^+$ mesons in symmetric nuclear matter are studied by combining the Nambu–Jona-Lasinio model and the quark-meson coupling model. The in-medium properties of the current quarks, which are used as inputs for studying the in-medium pion and kaon properties in the Nambu–Jona-Lasinio model, are calculated within the quark-meson coupling model. The light-quark condensates, light-quark dynamical masses, pion and kaon decay constants, and pion- and kaon-quark coupling constants are found to decrease as nuclear density increases. The obtained valence quark distributions in vacuum for both the $\pi^+$ and $K^+$ could reasonably describe the available experimental data over a wide range of Bjorken-$x$. The in-medium valence $u$-quark distribution in the $\pi^+$ at $Q^2=16~\mbox{GeV}^2$ is found to be almost unchanged compared to the in-vacuum case. However, the in-medium to in-vacuum ratios of both the valence $u$-quark and valence $s$-quark distributions of the $K^+$ meson at $Q^2=16~\mbox{GeV}^2$ increase with nuclear matter density, but show different $x$-dependence. Namely, the ratio for the valence $u$-quark distribution increases with $x$, while that for the valence $s$ quark decreases with $x$. These features are enhanced at higher density regions.'
author:
- 'Parada T. P. Hutauruk'
- 'J. J. Cobos-Martínez'
- Yongseok Oh
- 'K. Tsushima'
title: 'Valence-quark distributions of pions and kaons in a nuclear medium'
---
Introduction {#intro}
============
Pions and kaons, which emerge as a consequence of dynamical chiral symmetry breaking [@NJL61a], play very important and special roles in understanding the nonperturbative features of low-energy Quantum Chromodynamics (QCD) [@Ioffe05]. The nonperturbative aspects of QCD can be accessed through the elastic electromagnetic form factors and quark distribution functions of these mesons [@GPV01; @BR05; @HCT16] as well as hadron distribution functions in the low-$Q^2$ region, where the latter quantities are used as inputs for $Q^2$-evolution to high-$Q^2$ region that the perturbative QCD phenomena are experimentally accessible. They can provide us with important information on the internal structure of pions and kaons as well as the quark and gluon dynamics. However, it is highly nontrivial to calculate the quark distribution functions of these mesons in vacuum at low-energy scale. Several models [@HCT16; @HBCT18; @RA08; @NK07b; @BT02; @Zovko74; @DDEF12; @BM88a; @Hutauruk18; @CCRST13; @FJ79], and lattice QCD simulations [@HPQCD-17; @OKLMM15; @HKLLWZ07; @PACS-17] have been applied to study the internal structure of pions and kaons in vacuum. Despite of these efforts, theoretical understanding of quark distributions of pions and kaons requires more efforts and the situation is worsened by the scarce of experimental data [@NA7-86; @ABBB84; @JLAB_Fpi-08], in particular, for kaons. Attempts to understand quark distribution functions of mesons in a nuclear medium are much more restricted.
However, some recent progress has been reported for understanding the quark distributions of pions and kaons. Several studies were made to uncover the internal structure of pions in a nuclear medium, e.g., pion properties in nuclei [@MSTV90; @DTERF14], and pion photoproduction off nuclei [@BL77]. Furthermore, for the strange quark sector, investigations on kaon photoproduction off nuclei [@LMBW99] and kaon-nucleus Drell-Yan processes [@LLT96] were reported. The first evidence that triggered the interests for hadron properties in a nuclear medium is the EMC effects reported by the European Muon Collaboration (EMC) in Ref. [@EMC83b], where the deep inelastic scattering data revealed that parton distributions of bound nucleons in nuclei are modified and quite different from those of free nucleons [@GST95]. It is, therefore, natural to expect that the internal structure of mesons would also be modified in a nuclear medium and affect the meson production in heavy ion collisions (HIC) [@NA49-07], the reactions involving finite nuclei, and the quark distributions of mesons. A deeper understanding of the in-medium modifications of the quark distributions of mesons can provide us with useful information on the hadronization phenomena in HIC [@NA49-07], pion- and kaon-nuclear Drell-Yan processes [@LLT96; @Saclay-80], as well as the recently proposed experimental measurements of structure functions of pions and kaons via the pion- and kaon-induced Drell-Yan production on polarised and unpolarised proton, deuteron, and nuclear targets at the $M2$ beam line of CERN SPS [@COMPASS18].
Recently, the electromagnetic form factors of pions and kaons [@DTERF14; @YDDTF17] as well as meson distribution amplitudes [@TD16] in a nuclear medium were studied by combining the light-front constituent quark model and the quark-meson coupling (QMC) model. In these publications, the QMC model was used to provide the in-medium light-quark properties, which were then used as inputs for studying the medium modifications of meson properties. However, quark distributions, or parton distribution functions (PDFs) of mesons in a nuclear medium should be explored further in more elaborated manner. For example, consistency with the chiral limit of QCD is an important constraint for a model study, which is not observed by the previous studies. This is one of the main motivations of the present work. An early attempt to investigate the pion structure in a nuclear medium was made around two decades ago, by combining the Nambu–Jona-Lasinio (NJL) model and the operator product expansion [@Suzuki95] to incorporate the nuclear medium effects in the calculations of twist-2 operators. It was found that the in-medium structure function of the pion decreases in the large-Bjorken-$x$ region at the saturation density $\rho_0^{} =0.17~\mbox{fm}^{-3}$. In the present work we study the valence-quark distributions (or valence PDFs) of both the $\pi^+$ and $K^+$ mesons in symmetric nuclear matter based on the NJL model with an improved treatment for the nuclear medium effects. Namely, the in-medium quantities, which are served as inputs in the present study, are consistently constrained by the symmetric nuclear matter properties at saturation density in the QMC model, and the NJL model is used to satisfy the chiral limit.
One of the main focuses of the present work is on the in-medium modifications of the valence $u$-quark distributions in $\pi^{+}$ and $K^{+}$ mesons in symmetric nuclear matter. Our approach is based on the NJL model with the constrained in-medium inputs calculated by the QMC model. The NJL model is a powerful chiral effective quark theory of QCD with numerous successes in the studies of meson properties [@BJM88; @BM88; @VW91; @Klevansky92; @CBT14]. Recently, the NJL model was applied for studying the kaon electromagnetic form factor and valence-quark distributions in vacuum [@HCT16], as well as the in-medium electroweak properties of pions [@HOT18b]. By extending the works of Refs. [@HCT16; @HOT18b], we explore the valence-quark distributions of the $\pi^+$ and $K^+$ mesons, dynamical quark masses, and pion/kaon properties in symmetric nuclear matter.
This paper is organized as follows. In Sec. \[vacnjlprop\] we briefly review the calculation of the meson properties in the NJL model. The in-medium quark properties in the QMC model are described in Sec. \[QMCprop\], which will be used as inputs to study the in-medium pion and kaon properties based on the NJL model in Sec. \[pionNJLmedium\]. Presented in Sec. \[mediumstructurefunction\] are the in-medium modifications of valence-parton distribution functions of $\pi^+$ and $K^+$ mesons calculated in the present work. The implications of our numerical results are also discussed. Section \[summary\] gives the summary.
Pion and Kaon Free-Space Properties in the NJL model {#vacnjlprop}
====================================================
In this section we briefly review how we calculate the pion and kaon free-space properties in the NJL model. (See Refs. [@CBT14; @HCT16] for details.) The NJL model is an effective theory of QCD in the low-energy region, which contains important features of QCD such as chiral symmetry and its breaking pattern. Thus it has been widely used for understanding various low-energy nonperturbative phenomena of QCD [@Klevansky92].
To study kaon properties, one needs to extend the NJL model to three-flavor case of which Lagrangian with four-fermion contact interactions reads $$\begin{aligned}
\label{eq:lagNJL}
\mathscr{L}_{\rm NJL} & = & \bar{\psi} ( i \slashed{\partial} - \hat{m} ) \psi
+ G_\pi \left[ (\bar{\psi} \lambda_a^{} \psi )^2 - (\bar{\psi} \lambda_a^{}
\gamma_5^{} \psi )^2 \right]
\nonumber \\ &&
- G_\rho \left[ (\bar{\psi} \lambda_a^{} \gamma^\mu \psi )^2
+ (\bar{\psi} \lambda_a^{} \gamma^\mu \gamma_5^{} \psi )^2 \right] ,\end{aligned}$$ where a sum over $a=(0, \dots, 8)$ is understood and $\lambda_a$ are the Gell-Mann matrices in flavor space with $\lambda_0 \equiv \sqrt{\frac{2}{3}} \mathbbm{1}$. The quark field $\psi$ represents $\psi = (u, d, s)^T$ and $\hat{m} = \mbox{diag} (m_u, m_d, m_s)$ is the current-quark mass matrix. The four-fermion coupling constants are represented by $G_\pi$ and $G_\rho$.[^1] Throughout this work, we assume $m_l \equiv m_u = m_d$, where $l=( u, d)$ stands for “light”, and thus $m_l$ is the light-quark mass.
The general solution to the standard NJL gap equation has the form of $$S^{-1}_q(k) = \slashed{k} - M_q + i \epsilon,$$ for a quark flavor $q$ ($= u, d, s$) and the dynamically generated constituent quark mass $M_q$ is obtained as $$\begin{aligned}
\label{eq:masNJL}
M_q &=& m_q - 4 \, G_\pi \braket{ \bar{q} q }
\nonumber \\
&=& m_q^{} + 12 i G_\pi \int \frac{d^4 k}{(2\pi)^4} \mbox{Tr}_D^{} [S_q (k)].\end{aligned}$$ Here, the quark condensate is denoted by $\braket{ \bar{q} q }$ and the trace is only for the Dirac-space indices. Then the proper-time regularization scheme leads to $$\begin{aligned}
\label{eq:massNJLProp}
M_q &= m_q^{} + \frac{3G_\pi M_q}{\pi^2}
\int_{1/\Lambda_{\rm UV}^{2}}^{1/\Lambda_{\rm IR}^{2}} \frac{d\tau}{\tau^2}
\exp\left(-\tau M_q^2\right),\end{aligned}$$ where $\Lambda_{\rm UV}^{}$ and $\Lambda_{\rm IR}^{}$ are, respectively, the ultraviolet and infrared cutoff parameters.
Pions and kaons are described in the NJL model as quark-antiquark bound states by solving the corresponding Bethe-Saltpeter equations (BSEs). The solutions to the BSEs are given by a two-body $t$-matrix that depends on the interaction channel. For the pseudoscalar channel $\alpha$ ($= \pi, K$), it is given by $$\begin{aligned}
\label{eq:tmatrix}
\tau_\alpha(p) = \frac{-2i\,G_\pi}{1 + 2\,G_\pi\,\Pi_\alpha (p^2)}, \end{aligned}$$ where the bubble diagrams lead to $$\begin{aligned}
& \Pi_{\pi}(p^2) = 6i \int \frac{d^4k}{(2\pi)^4}\ \mbox{Tr}_D^{} \left[ \gamma_5^{} \,S_{l}(k) \gamma_5^{} \,S_{l} (k+p) \right], \\
& \Pi_{K}(p^2) = 6i \int \frac{d^4k}{(2\pi)^4}\ \mbox{Tr}_D^{} \left[ \gamma_5^{} \,S_{l}(k) \gamma_5^{} \,
S_{s} (k+p) \right].\end{aligned}$$
The meson masses are identified by the pole positions in the corresponding $t$-matrices. Thus, the pion and kaon masses are given, respectively, by the solutions of the equations, $$\begin{aligned}
1 + 2\, G_\pi\, \Pi_{\pi} (p^2 = m_{\pi}^2) &= 0, \nonumber \\
1 + 2\, G_\pi\, \Pi_{K} (p^2 = m_{K}^2) &= 0.\end{aligned}$$ These equations can be rearranged to give the pion and kaon masses as $$\begin{aligned}
m_{\pi}^2 &= \frac{m_l}{M_l} \frac{2}{G_\pi\, \mathcal{I}_{ll}(m_\pi^2)}, \nonumber \\
\label{eqn:mesonmass}
m_{K}^2 &= \left(\frac{m_s}{M_s} + \frac{m_l}{M_l} \right)
\frac{1}{G_\pi \mathcal{I}_{ls} (m_K^2)} + (M_s -M_l)^2, \end{aligned}$$ where $$\begin{aligned}
\mathcal{I}_{ab}(p^2) &= \frac{3}{\pi^2}
\int_0^1 dz \int_{1/\Lambda^2_{\rm UV}}^{1/\Lambda^2_{\rm IR}} \frac{d\tau}{\tau}\,
e^{-\tau[-z(1-z)\,p^2 + z\,M_b^2 + (1-z)\,M_a^2]}\, , \end{aligned}$$ for quark flavors $a$ and $b$. Equation (\[eqn:mesonmass\]) makes it evident that chiral symmetry and its breaking pattern are embedded in the NJL model. The model satisfies the chiral limit and the pions and kaons become massless in the chiral limit being realized as Goldstone bosons. The residue at a pole in the $\bar{q}q$ $t$-matrix defines the meson ($\alpha$)-quark coupling constant $g_{\alpha q q}$ as $$\begin{aligned}
\label{eq:couplinconstant}
g_{\alpha q q}^{-2} &= - \left.\frac{\partial\, \Pi_\alpha (p^2)}{\partial p^2}
\right|_{p^2 = m_\alpha^2} \end{aligned}$$ with $\alpha = \pi, K$.
The pion and kaon weak decay constants are determined from the meson to vacuum transition matrix element $\braket{ 0 | \mathscr{J}_a^{5 \mu} (0) | \alpha (p) }$ with $\mathscr{J}_a^{5 \mu}$ being the quark weak axial-vector current operator for a flavor quantum number $a$. Evaluating the corresponding matrix elements, pion and kaon weak decay constants in the proper-time regularization scheme are expressed as [@Hutauruk16; @NBC14] $$\begin{aligned}
\label{eq:decayconNJL}
f_\pi &= \frac{3 g_{\pi qq}^{} M_q}{4 \pi^2} \int_0^1 dz
\int_{1/\Lambda_{\rm UV}^{2}}^{1/\Lambda_{\rm IR}^{2}}
\frac{d\tau}{\tau} e^{-\tau [ M_q^2 - z(1-z) m_\pi^2 ]}, \nonumber \\
f_K &= \frac{3 g_{K qq}^{}}{4 \pi^2} \int_0^1 dz
\int_{1/\Lambda_{\rm UV}^{2}}^{1/\Lambda_{\rm IR}^{2}} \frac{d\tau}{\tau}
\left[ M_s + z(M_l - M_s) \right] \nonumber \\
&\hspace{18ex} \times e^{-\tau [ M_s^2 - z(M_s^2 -M_l^2)-z(1-z) m_K^2]}.\end{aligned}$$ This completes the formalism for meson masses, decay constants, and meson-quark coupling constants in vacuum.
Following Refs. [@CBT14; @HCT16], we choose $\Lambda_{\rm IR} = 240$ MeV which is set based on $\Lambda_{\rm QCD}$ associated with confinement, and $M_l = 400$ MeV. The model parameters are then fixed to give $m_\pi^{} = 140$ MeV and $m_K^{} = 495$ MeV, together with the pion decay constant $f_\pi = 93$ MeV. This procedure gives $\Lambda_{\rm UV} = 645$ MeV, $G_\pi = 19.0~\mbox{GeV}^{-2}$, and $M_s = 611$ MeV. The corresponding current quark masses are $m_{l} = 16.4$ MeV and $m_s= 356$ MeV. The predicted values for the kaon decay constant and $u$ and $s$ quark condensates in vacuum are $f_{K}=91$ MeV, $\braket{\bar{u} u}^{1/3}= -171$ MeV, and $\braket{ \bar{s} s }^{1/3}= -151$ MeV, respectively.
In-medium quark properties in the QMC model {#QMCprop}
===========================================
In the present approach, the in-medium current-quark properties are provided by the QMC model [@Guichon88], which will be used in the NJL model to explore the in-medium dynamical quark and meson properties. The QMC model has been successfully applied for studying many topics in nuclear and hadron physics such as finite nuclei [@GSRT95; @STT96b; @STT96; @SGRT16; @GST18], hypernuclei [@TSHT98; @GTT07], superheavy nuclei [@SGT17], neutron star properties [@WCTTS13], and nucleon/hadron properties in a nuclear medium [@STT07; @KTT17]. In the QMC model, medium effects are incorporated by the self-consistent exchanges of the scalar ($\sigma$), vector-isoscalar ($\omega$), and vector-isovector ($\rho$) meson fields, which directly couple to the confined light quarks inside the nucleon, rather than to a pointlike nucleon. In the following, we consider symmetric nuclear matter in its rest frame in the Hartree mean-field approximation. (See Ref. [@KTT98] for detailed discussions on the Hartree-Fock treatment.)
The effective Lagrangian for symmetric nuclear matter is given by [@GTT07; @STT07] $$\begin{aligned}
\label{eqintro1}
\mathscr{L}_{\textrm{QMC}} & = \bar{\psi}_N \left[ i\gamma \cdot
\partial - M_{N}^{*}(\sigma)
- g_\omega \omega^{\mu} \gamma_\mu \right] \psi_N + \mathscr{L}_\textrm{meson},\end{aligned}$$ where $\psi_N$, $\sigma$, and $\omega$ are the nucleon, $\sigma$-, and $\omega$-meson fields, respectively. The effective nucleon mass $M_N^{*}$ is defined by $$\begin{aligned}
M_{N}^{*} \left( \sigma \right) &= M_{N} - g_\sigma \left( \sigma \right) \sigma.\end{aligned}$$ Here, $g_{\sigma} ( \sigma )$ and $g_{\omega}$ are the $\sigma$-dependent nucleon-$\sigma$ and nucleon-$\omega$ coupling constants, respectively. We define the nucleon-$\sigma$ coupling constant as $g^N_\sigma \equiv g_\sigma (\sigma=0)$ for later convenience. Because symmetric nuclear matter is isospin saturated, the isospin-dependent $\rho$-meson field vanishes in the Hartree approximation and is not included in Eq. (\[eqintro1\]). The free meson Lagrangian density in Eq. (\[eqintro1\]) is given by $$\begin{aligned}
\label{eqintro3}
\mathscr{L}_\textrm{meson} &= \frac{1}{2} (\partial_\mu \sigma
\partial^\mu \sigma - m_\sigma^2 \sigma^2)
- \frac{1}{2} \partial_\mu \omega_\nu (\partial^\mu \omega^\nu
- \partial^\nu \omega^\mu)\nonumber \\
&+ \frac{1}{2} m_\omega^2 \omega^\mu \omega_\mu. \end{aligned}$$
In the Hartree mean-field approximation the nucleon Fermi momentum $k_F^{}$ is related to the baryon density ($\rho_{B}^{}$) and scalar density ($\rho_{s}^{}$) defined as $$\begin{aligned}
\label{eqintro4}
\rho_{B}^{} &= \frac{\gamma}{(2\pi)^3} \int d\bm{k}\, \Theta \left( k_F^{} - | \bm{k} | \right) =
\frac{\gamma k_F^3}{3 \pi^2}, \nonumber \\
\rho_{s}^{} &= \frac{\gamma}{(2\pi)^3} \int d\bm{k}\, \Theta \left( k_F^{} - | \bm{k} | \right)
\frac{M_N^{*} (\sigma)}{\sqrt{M_N^{*2} (\sigma ) + \bm{k}^2}},\end{aligned}$$ where $\Theta(x)$ is the Heaviside step function and $\gamma =4$ for symmetric nuclear matter. The baryon density $\rho_B$ is given by $\rho_B^{} = \rho_p + \rho_n$, where $\rho_p$ and $\rho_n$ are proton and neutron densities, respectively.
In the QMC model, nuclear matter is described as a collection of nonoverlapping MIT bags of nucleons [@CJJTW74]. The Dirac equations for the quarks and antiquarks in the bag are given by $$\begin{aligned}
\label{eqintro5}
\left[ i \gamma \cdot \partial_{x} - \left( m_l^{} - V_{\sigma}^{q} \right)
\mp \gamma^{0} \left( V_{\omega}^{q}
+ \frac{1}{2} V_{\rho}^{q} \right) \right] \left( \begin{array}{c}\psi_u(x) \\
\psi_{\bar{u}}(x) \\ \end{array} \right) &= 0, \nonumber \\
\left[ i \gamma \cdot \partial_{x} - \left( m_l^{} - V_{\sigma}^{q} \right)
\mp \gamma^{0} \left( V_{\omega}^{q}
- \frac{1}{2} V_{\rho}^{q} \right) \right] \left( \begin{array}{c} \psi_d(x) \\
\psi_{\bar{d}}(x) \\ \end{array} \right) &= 0, \nonumber \\
\left[ i \gamma \cdot \partial_{x} - m_{s} \right] \left( \begin{array}{c} \psi_s(x) \\
\psi_{\bar{s}}(x) \\ \end{array} \right) &= 0,\end{aligned}$$ and we can define the effective in-medium current quark mass $m_l^{*}$ as $$\begin{aligned}
\label{eqintro5a}
m_l^{*} & \equiv m_l^{} - V_{\sigma}^{q},\end{aligned}$$ with $-V_\sigma^{q}$ being the scalar potential, while $m_s^* = m_s^{}$. The scalar and vector mean-field potentials felt by the light quarks in symmetric nuclear matter are, respectively, defined by $$\begin{aligned}
V_{\sigma}^{q} & \equiv g_{\sigma}^{q} \sigma = g_{\sigma}^{q} \braket{\sigma},
\nonumber \\
V_{\omega}^{q} &\equiv g_{\omega}^{q} \omega = g_{\omega}^{q} \, \delta^{\mu , 0} \braket{ \omega^{\mu} },\end{aligned}$$ where $g_{\sigma}^{q}$ and $g_{\omega}^{q}$ are the coupling constants of the light quarks to the mean-field $\sigma$ and $\omega$, respectively. Note that in the present approach the strange quark is decoupled from the scalar and vector mean-field potentials in nuclear medium.
The bag radius of hadron $h$ in a nuclear medium, $R_h^{*}$, is determined through the stability condition of the hadron mass. The eigenenergies of the quarks, in units of $1/ R_h^{*}$, in the MIT bags are obtained as $$\begin{aligned}
\label{eq:kaonmed9}
\left( \begin{array}{c}
\epsilon_u \\ \epsilon_{\bar{u}}
\end{array} \right)
& = \Omega_l^* \pm R_h^* \left( V^q_\omega + \frac{1}{2} V^q_\rho \right), \nonumber \\
\left( \begin{array}{c} \epsilon_d \\ \epsilon_{\bar{d}}
\end{array} \right)
&= \Omega_l^* \pm R_h^* \left( V^q_\omega - \frac{1}{2} V^q_\rho \right), \nonumber \\
\left( \begin{array}{c} \epsilon_s \\ \epsilon_{\bar{s}}
\end{array} \right)
& = \Omega_{s}.\end{aligned}$$ The effective mass of hadron $h$, $ m_h^{*}$, in nuclear medium is calculated by $$\begin{aligned}
\label{eq:kaonmed10}
m_h^{*} &= \sum_{j = l, \bar{l}, s, \bar{s}} \frac{n_j \Omega_j^{*} -z_h}{R^{*}_h}
+ \frac{4}{3} \pi R_h^{* 3} B,\end{aligned}$$ which determines $R_h^{*}$ by the stability condition, i.e., $$\begin{aligned}
\left. \frac{d m_h^{*}}{d R_h} \right \vert_{R_h = R_h^{*}} = 0,\end{aligned}$$ where $\Omega^{*}_l = \Omega^{*}_{\bar{l}} = \left [x_l^2 + \left(R_h^{*} m_l^{*} \right)^2
\right]^{1/2}$ and $\Omega_{\bar{s}}^{*} = \Omega_{s}^{*} = [x_{s}^2
+ (R_h^{*} m_s )^2 ]^{1/2}$. The parameter $z_h$ accounts for the sum of the center-of-mass and gluon fluctuation corrections and is assumed to be independent of density [@SGRT16], and $B$ is the bag constant. For the quark in the bag of hadron $h$, the ground state wave function satisfies the boundary condition at the bag surface, $j_0 (x) = \beta_q j_1 (x)$, where $$\begin{aligned}
\
\label{eq:beta}
\beta_q &= \sqrt{\frac{\Omega_q^{*} -m_q^{*} R_h^{*}}{\Omega_q^{*} + m_q^{*} R_h^{*}}},\end{aligned}$$ and $j_{0,1}$ are spherical Bessel functions. This determines the values of $x_l$ and $x_s$.
Except for the saturation density and the binding energy at the saturation point that are used to fix the quark-meson coupling constants, nuclear matter properties depend on the light-quark current mass in vacuum. In Table \[tab:model1\] we list two sets of the QMC model results corresponding to two different current-quark mass values as well as the corresponding calculated quantities. The first row shows the quantities obtained with the standard QMC model values, $m_l = 5~\mbox{MeV}$ and $m_s = 250~\mbox{MeV}$. In the present work, however, since we use the NJL model, we adopt $m_l = 16.4~\mbox{MeV}$ and $m_s = 356~\mbox{MeV}$ following Ref. [@HCT16]. The bag radius of the nucleon in free space, $R_N = 0.8~\mbox{fm}$, is also used as an input.
---------------- ---------------------------------- ------------------------ ----------- --------- ----------- ----------
$m_l$ $(g^N_{\sigma})^{\, 2} / 4 \pi$ $g_{\omega}^2 / 4 \pi$ $B^{1/4}$ $z_N$ $M_N^{*}$ $K$
\[0.2em\] $5 $ $5.393$ $5.304$ $170.0$ $3.295$ $754.55$ $279.30$
$16.4$ $5.438$ $5.412$ $169.2$ $3.334$ $751.95$ $281.50$
---------------- ---------------------------------- ------------------------ ----------- --------- ----------- ----------
: Current-quark masses of light quarks in vacuum and the corresponding coupling constants, bag constant $B$, the parameter $z_N$, effective nucleon mass $M_N^{*}$, and the nuclear incompressibility $K$, at saturation density, $\rho_0^{} = 0.15~\mbox{fm}^{-3}$ obtained in the QMC model. The units of $m_l$, $M_N^*$, and $B^{1/4}$ are MeV. The standard QMC model quark masses are $m_l = 5$ MeV and $m_s = 250$ MeV, while, in this work, we use $m_l = 6.4$ MeV and $m_s = 356$ MeV to be consistent with the NJL model calculations. []{data-label="tab:model1"}
The scalar and vector meson mean fields at the hadron level can be related with the baryon and scalar densities by $$\begin{aligned}
\label{eq:kaonmed11}
\omega &= \frac{g_\omega \rho_B^{} }{m_\omega^2}, \nonumber \\
\sigma &= \frac{4 g^N_\sigma C_N (\sigma)}{(2\pi)^3m_{\sigma}^2}
\int d\bm{k}\, \Theta (k_F - | \bm{k} | )
\frac{M_N^{*} (\sigma)}{\sqrt{M_N^{*2} (\sigma) + \bm{k}^2}}, \nonumber \\
&= \frac{4 g^N_\sigma C_N (\sigma)}{(2\pi)^3m_{\sigma}^2} \rho_s,\end{aligned}$$ where $$\begin{aligned}
C_N (\sigma) &= \frac{-1}{g^N_\sigma}
\left[ \frac{\partial M_N^{*} (\sigma )}{\partial \sigma } \right],\end{aligned}$$ which yields $C_N (\sigma)=1$ for a pointlike nucleon [@SW86; @SW97]. This is the origin of the novel saturation properties in the QMC model and contains the quark dynamics of nucleons (hadrons). Solving the self-consistent equation for the scalar mean field of Eq. (\[eq:kaonmed11\]), the total energy per nucleon is calculated as $$\begin{aligned}
\label{eq:kaonmed12}
E^{\rm tot}/A &= \frac{4}{(2\pi)^3 \rho_B^{}} \int d\bm{k}\, \Theta (k_F - | \bm{k} |) \sqrt{M_N^{*2} (\sigma) + \bm{k}^2}
\nonumber \\
&+ \frac{m_\sigma^2 \sigma^2}{2\rho_B^{}} + \frac{g_\omega^2 \rho_B^{}}{2 m_\omega^2}.\end{aligned}$$ The quark level coupling constants $g_\sigma^q$ and $g_\omega^q$ are fitted by the binding energy of 15.7 MeV of symmetric nuclear matter at the saturation density $\rho_0$ with $g^N_\sigma = 3 g^q_\sigma S_N (\sigma= 0)$ and $g_\omega = 3g_\omega^q$, where $ g_\sigma^q \simeq 5.6251$ for $m_l = 16.4$ MeV and $S_N(\sigma)$ is given by [@KTT17] $$\begin{aligned}
\dfrac{\partial M_{N}^*(\sigma)}{\partial \sigma}
&=& - 3 g_{\sigma}^q \int_{\rm bag} d^3y \bm{}
\ {\overline \psi}_q(\bm{y})~\psi_q(\bm{y})
\nonumber \\
&\equiv& - 3 g_{\sigma}^q S_{N}(\sigma) = - \dfrac{\partial}{\partial \sigma}
\left[ g_\sigma(\sigma) \sigma \right],
\label{Ssigma}\end{aligned}$$ which gives $S_N(\sigma=0) \approx 0.4899$ for $m_l = 16.4$ MeV. Here, $\psi_q$ is the lowest mode light-quark bag wave function obtained by solving the Dirac equation self-consistently in the scalar-$\sigma$ and vector-$\omega$ mean fields.
Shown in Fig. \[fig1\] is the obtained negative of binding energy per nucleon for symmetric nuclear matter. The calculated effective nucleon mass $M_N^{*}$ is illustrated in Fig. \[fig2\] as a function of baryon density. The corresponding effective light-quark current mass $m_l^*$, scalar mean-field potential $- V^q_\sigma$, and vector mean-field potential $V_\omega^q$ are also presented in Fig. \[fig3\].
![\[fig1\] Negative of binding energy per nucleon ($E^{\rm tot} /A - M_N $) for symmetric nuclear matter obtained in the QMC model with the free-space quark mass $m_l = 16.4~\mbox{MeV}$.](fig1.pdf){width="0.95\columnwidth"}
![\[fig2\] Effective nucleon mass $M^{*}_N$ for symmetric nuclear matter calculated by the QMC model with $m_l = 16.4~\mbox{MeV}$.](fig2.pdf){width="0.95\columnwidth"}
![\[fig3\] Effective current quark mass $m^*_l $ of light quarks (solid line), scalar mean-field potential $-V^q_\sigma$ (dashed line), and vector mean-field potential $V_\omega^q$ (dotted line) calculated by the QMC model with $m_l = 16.4~\mbox{MeV}$.](fig3.pdf){width="0.95\columnwidth"}
In-Medium Pion and Kaon Properties {#pionNJLmedium}
==================================
Based on the combined approach of the NJL-model formalism equipped with in-medium quark properties in the QMC model, we now explore the in-medium pion and kaon properties in this section. The in-medium mean-field potentials felt by the light quarks are computed in the QMC model and are shown in Fig. \[fig3\]. These in-medium properties of current quarks are used to estimate the in-medium dynamical quark masses in the NJL model, and they allow us to study the pion and kaon properties in medium such as the medium modifications of the valence-quark distributions of pions and kaons.
The gap equation for the dynamical quark mass $M^*_q$ for a quark $q$ in medium can be straightforwardly read from Eq. (\[eq:masNJL\]), $$\begin{aligned}
\label{eq:kaonmed13}
M_q^{*} & = m_q^{*} + \frac{3G_\pi M_q^{*}}{\pi^2}
\int_{1/\Lambda_{\rm UV}^2}^{\infty} \frac{d\tau}{\tau^2} e^{-\tau M_q^{*2}},\end{aligned}$$ where $m_q^{*}$ and $M_q^{*}$ are, respectively, the corresponding density-dependent in-medium current- and dynamical-quark masses. We use the free space values for the coupling constant $G_\pi$ and the ultraviolet cutoff $\Lambda_{\rm UV}$, while we use $\Lambda_{\rm IR} \simeq 0$ or $(1/\Lambda_{\rm IR}^2) = \infty$, since we do not have information on $\Lambda_{\rm IR}$ (or $\Lambda_{\rm QCD}$) in medium and the results are not affected much by the value of $\Lambda_{\rm IR}$ when $1/\Lambda_{\rm IR}^2 \to \infty$. In the present approach, the above relation holds only for light quarks, whille the strange quark mass does not change in nuclear matter. In addition, the extra density dependent term introduced in Refs. [@Maedan09; @WMT15], which is proportional to quark chemical potentials to treat finite density systems, is not included because the information of symmetric nuclear matter saturation, and thus the medium effects, are self-consistently included in the in-medium light-quark properties calculated in the QMC model.
The in-medium dressed quark propagators are expressed by $$\begin{aligned}
\label{eq:kaonmed14}
S_l^{*}(k^*) &= \frac{\slashed{k}^* + M_l^{*}}{(k^*)^2 - (M_l^{*})^2
+ i \epsilon},
\\
S_s^{*}(k^*) &= S_s(k) =\frac{\slashed{k} + M_s}{k^2 - M_s^2 + i \epsilon},\end{aligned}$$ where the in-medium modifications enter as the shift of the light-quark momenta through $k^{\mu} \to k^{*\mu}=k^\mu + V^\mu$. Here, $V^\mu = (V^0, \textbf{0})$ is the vector mean-field potential defined in the previous section [@STTS98; @STT04; @CMPR09]. An asterisk over a quantity denotes an in-medium quantity as before.
As in the vacuum case, mesons are described as dressed quark-antiquark bound states that appear as solutions of the BSE in the random phase approximation. The solution to the BSE in each meson channel is given by the $t$-matrix of a two-body scattering that depends on the nature of the interaction channel. The in-medium reduced $t$-matrices for $\pi$ and $K$ mesons take the same form as in vacuum: $$\begin{aligned}
\label{eq:tmatrix_pi}
\tau^{*}_\pi (p^{*}) &= \frac{-2i\,G_\pi}{1 + 2 G_\pi \Pi^{*}_\pi (p^{*2})}, \\
\label{eq:tmatrix_k}
\tau^{*}_K (p^{*}) &= \frac{-2i\,G_\pi}{1 + 2 G_\pi \Pi^{*}_K (p^{*2})}, \end{aligned}$$ where the in-medium bubble diagrams lead to $$\begin{aligned}
\Pi^{*}_{\pi} (p^{*2}) &= 6i \int \frac{d^4k}{(2\pi)^4} \,
\mbox{Tr}_D^{} \left[ \gamma_5^{} S^{*}_{l}(k^*) \gamma_5^{} S^{*}_{l}(k^*+p^*) \right],\\
\Pi^{*}_{K} (p^{*2}) &= 6i \int \frac{d^4k}{(2\pi)^4}\,
\mbox{Tr}_D^{} \left[ \gamma_5^{} S^{*}_{l}(k^*) \gamma_5^{} S^{*}_{s}(k^*+p^*) \right].\end{aligned}$$ These equations show that the medium-modified momentum enters for the momenta of light quarks and the kaon also feels the vector potential, which is in contrast to the pion case where the vector potentials for the light quark and light antiquark cancel out. Thus, although the vector potential effect can be eliminated by the integral variable shift for $\Pi^*_\pi$, it should be explicitly included in the calculation of $\Pi^*_K$.
The meson masses are defined by the poles in the corresponding $t$-matrices as in the vacuum case, and Eqs. (\[eq:tmatrix\_pi\]) and (\[eq:tmatrix\_k\]) lead to $$\begin{aligned}
1 + 2 G_\pi \Pi^{*}_{\pi} (p^{*2} = m_{\pi}^{* 2}) &= 0,\\
1 + 2 G_\pi \Pi^{*}_{K} (p^{*2}= m_{K}^{* 2}) &= 0.\end{aligned}$$ These relations can be rewritten as $$\begin{aligned}
m_{\pi}^{* 2} &= \frac{m^{*}_l}{M^{*}_l} \frac{2}{G_\pi \mathcal{I}_{ll}(m_\pi^{*2})},
\nonumber \\
m_{K}^{* 2} &= \left[ \frac{m^{*}_s}{M^{*}_s}
+ \frac{m^{*}}{M^{*}_l} \right] \frac{1}{G_\pi \mathcal{I}_{l s}(m_K^{* 2})}
+ (M^{*}_s-M^{*}_l)^2,\end{aligned}$$ where $$\begin{aligned}
\mathcal{I}_{\!ab}(p^{* 2}) &= \frac{3}{\pi^2} \int_0^1 dz \int
\frac{d\tau}{\tau}\, \nonumber \\
\times &e^{-\tau \left[ -z(1-z)\,p^{*2} + 2p^{*}V^0 z(1-z) - z(1-z)({V^0})^2
+ z\,M_b^{* 2} + (1-z)\,M_a^{* 2} \right]}.\end{aligned}$$ As in the vacuum case, the residue at a pole in the $\bar{q}q$ $t$-matrix defines the in-medium coupling constant $g^{*}_{\alpha q q}$ as $$\begin{aligned}
\label{eq:couplinconstantmed}
g_{\alpha q q}^{* -2} &= - \left.\frac{\partial\,
\Pi_\alpha (p^{*2})}{\partial p^{*2}} \right|_{p^{*2} = m_\alpha^{*2}}.\end{aligned}$$
------------------- ----------- ----------- ----------- ----------------- ------------- ------------- ------------------- ---------------------------------
$\rho_B / \rho_0$ $M_u^{*}$ $m_K^{*}$ $f_K^{*}$ $g_{K q q}^{*}$ $m_\pi^{*}$ $f_\pi^{*}$ $g_{\pi q q}^{*}$ $-\braket{ \bar{u} u }^{* 1/3}$
\[0.2em\] $0.0$ $0.400$ $0.495$ $0.091$ $4.570$ $0.140$ $0.093$ $4.255$ $0.171$
$0.25$ $0.370$ $0.465$ $0.091$ $4.536$ $0.136$ $0.092$ $3.964$ $0.167$
$0.50$ $0.339$ $0.437$ $0.090$ $4.495$ $0.134$ $0.089$ $3.720$ $0.162$
$0.75$ $0.307$ $0.411$ $0.089$ $4.455$ $0.132$ $0.086$ $3.494$ $0.156$
$1.00$ $0.270$ $0.386$ $0.088$ $4.408$ $0.131$ $0.081$ $3.265$ $0.149$
$1.25$ $0.207$ $0.359$ $0.084$ $4.332$ $0.136$ $0.069$ $2.948$ $0.136$
------------------- ----------- ----------- ----------- ----------------- ------------- ------------- ------------------- ---------------------------------
Results for the in-medium properties of dynamical quarks, pions, and kaons are presented in Table \[tab:model2\]. These results show that the considered in-medium quantities of dynamical quarks and mesons decrease with increasing density. At normal nuclear density $\rho_0^{}$, the dynamical $u$-quark mass is found to decrease by about 30%, while the magnitude of the $u$-quark condensate decreases by about 13%. These results indicate that chiral symmetry is partially restored in the finite density system.
In the case of the pion, we find that its mass decreases by about 7% at normal nuclear density. For the pion-quark coupling constant and pion decay constant we find $g_{\pi qq}^*/g_{\pi qq} \approx 0.77$ and $f_\pi^*/ f_\pi = 0.87$ at normal nuclear density. Our result for $f_\pi^*/ f_\pi$ is in good agreement with that of Refs. [@KY04; @KW97] but is about 10-20% larger than that of Refs. [@MOW01; @TW95]. In the kaon case, the tendency of medium modifications is somehow different as the $s$-quark properties are not modified in the present approach because it decouples from mean fields. As a result, the medium modifications of kaon properties are realized only through the light quark in the kaon. Our results show that the kaon mass decreases by about 20% at normal nuclear matter density, while both the kaon-quark coupling constant and kaon decay constant, decrease only a few percent.
In-Medium Valence-Quark Distributions of Pions and Kaons {#mediumstructurefunction}
========================================================
As quark properties are modified in nuclear medium, it is quite natural to expect that the quark distributions or parton distribution functions (PDFs) of mesons are also modified in medium. In this section, following the PDF calculations of Ref. [@HCT16], we evaluate the in-medium valence PDFs (or valence-quark distributions) of pions and kaons. We start with the twist-2 quark distribution in a hadron $\alpha$ defined by $$\begin{aligned}
\label{eq:valence1}
q_\alpha(x) &= p^{+} \int \frac{d\xi^{-}}{2\pi} e^{ix p^{+} \xi^{-}}
\braket{ \alpha | \bar{\psi}_{q}(0) \gamma^{+} \psi_q (\xi^{-}) | \alpha }_{c},\end{aligned}$$ where $c$ denotes the connected-diagram matrix element and $x = {k^+}/{p^+}$ is the Bjorken scaling variable with $p^+$ ($k^+$) being the plus-component of the hadron (struck quark) momentum. In the NJL model, gluons are “integrated out” and the gauge-link, which should appear in Eq. , is unity.
![\[fig:strucfun1\] Feynman diagrams for the valence-quark distributions in a meson. The operator insertion $\gamma^+ \delta \left( p^+x - k^+ \right) \hat{P}_q$, where $\hat{P}_q$ is the projection operator for a quark $q$, is represented by the red cross.](fig4.pdf){width="\columnwidth"}
The valence-quark distribution functions (or valence PDFs) given by Eq. (\[eq:valence1\]) are calculated based on the two Feynman diagrams depicted in Fig. \[fig:strucfun1\], where the operator insertion is given by $\gamma^+\delta \left( p^+x - k^+ \right) \hat{P}_q$, with $\hat{P}_q$ being the projection operator for a quark $q$ defined as $$\begin{aligned}
\hat{P}_{u/d} &= \textstyle \frac{1}{2} \left( \frac{2}{3}\, \mathbbm{1} \pm \lambda_3
+ \frac{1}{\sqrt{3}}\,\lambda_8 \right), &
\hat{P}_s &= \textstyle \frac{1}{3}\,\mathbbm{1} - \frac{1}{\sqrt{3}}\,\lambda_8.\end{aligned}$$ Using the relation $\bar{q}(x) = -q(-x)$, the valence-quark and valence-antiquark distributions in a meson $\alpha$ are calculated as $$\begin{aligned}
\label{eq:valence31}
q_\alpha (x) &=& i g_{\alpha q q}^{*2} \int \frac{d^4k}{(2\pi)^4}\,
\delta \left( p^{*+}x - k^{*+} \right)
\nonumber \\ && \mbox{}
\hspace{-5ex}
\times \mathrm{Tr} \left[ \gamma_5\lambda_\alpha^\dagger\,S^{*}(k^*)\,\gamma^+
\hat{P}_q\,S^{*}(k^*)\,\gamma_5\lambda_\alpha\,S^{*}(k^*-p^*) \right], \\
\textbf{\label{eq:valence32}}
\bar{q}_\alpha(x) &=& -i\,g_{\alpha q q}^{*2} \int \frac{d^4k}{(2\pi)^4}\
\delta \left( p^{*+}x + k^{*+} \right)
\nonumber \\ && \mbox{}
\hspace{-5ex}
\times \mathrm{Tr} \left[ \gamma_5\lambda_\alpha \,S^{*}(k^*)\,\gamma^+
\hat{P}_q\,S^{*}(k^*)\,\gamma_5\lambda_\alpha^\dagger\,S^{*}(k^*+p^*) \right].\end{aligned}$$ It should be noted that the Bjorken-$x$ variable appearing in the above equations is defined in nuclear medium.
To evaluate these quantities we first take the moments defined by $$\begin{aligned}
\label{eq:valence2}
\mathcal{A}_n &= \int_0^1 dx\, x^{n-1}\, q(x),\end{aligned}$$ where $n = 1,\,2,\dots$ is an integer. Using the Ward-like identity, $S(k) \gamma^+ S(k) = -\partial S(k)/\partial k_+$, and the Feynman parametrization, the quark and antiquark distributions for the $K^+$-meson ($u\bar s$) are obtained as $$\begin{aligned}
\label{eq:valence51}
u_{K^+}^{} (x) &=& \frac{3\,g_{K q q}^{*2}}{4\pi^2} \int d\tau\
e^{-\tau \left[ x(x - 1)\,m_K^{* 2} + x\,M_s^{* 2} + (1-x)\,M_l^{* 2} \right]}
\nonumber \\ && \mbox{}
\times \left[\frac{1}{\tau} + x(1 - x)\left[m_K^{* 2} - (M_l^{*}
- M_s^{*})^2\right] \right], \\
\label{eq:valence52}
\bar{s}_{K^+}^{}(x) &=& \frac{3\,g_{K q q}^{*2}}{4\pi^2} \int d\tau\
e^{-\tau \left[ x(x - 1)\,m_K^{* 2} + x\,M_l^{* 2} + (1-x)\,M_s^{* 2} \right]}
\nonumber \\ && \mbox{}
\times \left[\frac{1}{\tau} + x(1 - x)
\left[m_K^{* 2} - (M_l^{*} - M_s^{*})^2\right]\right].\end{aligned}$$ Valence-quark distributions of the $\pi^+$ can be obtained by replacing $M_s^{*} \to M_l^{*}$ and $g_{K q q}^{*} \to g_{\pi q q}^{*}$, which leads to $u_{\pi^+}(x) = \bar{d}_{\pi^+}(x)$. The in-medium valence-quark distributions of other pseudoscalar mesons can be related by flavor symmetry. The expressions given by Eqs. (\[eq:valence51\]) and (\[eq:valence52\]) are consistent with those given in Ref. [@HCT16] for zero baryon density.
The valence-quark distributions in medium and in vacuum, with the corresponding Bjorken variables $\tilde{x}_{a}$ and $x_a$, respectively, are related by [@MBITY03] $$\begin{aligned}
\label{eq:valence6a}
q_{K^{+}}^{} (x_a) &= \frac{\epsilon_F^{}}{E_F} q_{K^{+}}^{*} ( \tilde{x}_a ), \end{aligned}$$ with $$\tilde{x}_a = \frac{\epsilon_F^{}}{E_F} x_a - \frac{V^0}{E_F}$$ where $\epsilon_F^{} = \sqrt{(k_F^{q})^{2} + (M_q^{*})^{2}} + V^0 \equiv E_F + V^0$, with $V^0$ being the vector potential, is the in-medium quark energy ($\epsilon_F^{} = E_F - V^0$ for an antiquark) and $k_F^q$ is the quark Fermi momentum which is related to the nuclear matter density as $\rho_B^{} = 2(k_F^{q})^{3}/3\pi^2$. The above formulas are valid only for light ($u$, $d$) quarks in the present approach. The values obtained by the QMC model for these quantities for the light quark are given in Table \[tab:model3\].
------------------------- ---------- -------
$\rho_B^{} / \rho_0^{}$ $k_F^q$ $V^0$
\[0.2em\] $0.00$ $0$ $0$
$0.25$ $162.18$ $11$
$0.50$ $204.33$ $21$
$0.75$ $233.90$ $32$
$1.00$ $257.44$ $43$
$1.25$ $277.32$ $53$
------------------------- ---------- -------
: Fermi momentum $k_F^q$ and the time component of the vector potential ($V^0$) for the light quark obtained in the QMC model for various values of ($\rho_{B}^{}/\rho_0$) with the quantities given in Table \[tab:model1\] for $m_l=16.4$ MeV. $k_F$ and $V^0$ are given in units of MeV.[]{data-label="tab:model3"}
The in-medium valence-quark distributions satisfy the baryon number and momentum sum rules as $$\begin{aligned}
\label{eq:valence6}
& \int_0^1\!\! dx \left[ u_{K^{+}}^{} (x) - \bar{u}_{K^+}^{} (x) \right] = 1 , \nonumber \\
& \int_0^1\!\! dx \left[ \bar{s}_{K^{+}}^{} (x) - s_{K^+}^{} (x) \right] = 1, \nonumber \\
& \int_0^1 dx\ x \left[ u_{K^+}^{}(x) + \bar{u}_{K^+}^{}(x)
+ s_{K^+}^{} (x) + \bar{s}_{K^+}^{} (x) \right] = 1. \end{aligned}$$ for the $K^+$. Analogous relations hold for the $\pi^+$ as well.
{width="\textwidth"}
{width="\textwidth"}
{width="\textwidth"}
The results for the valence PDFs of the $\pi^+$ and $K^+$ mesons in symmetric nuclear matter are presented in Figs. \[fig5\]–\[fig7\] together with those in vacuum. The valence-quark distributions have been evolved using the next-to-leading order (NLO) Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations [@MK95; @GL72; @AP77; @Dokshitzer77] from the model scale of $Q_0^2 = 0.16~\mbox{GeV}^2$, which was determined in Ref. [@CBT05] as a typical of valence-dominated models for studying nucleon valence-quark distributions, to $Q^2 = 16~\mbox{GeV}^2$. The $Q^2$ evolution is carried out in order to compare with the experimental data available at $Q^2 = 16$ GeV$^{2}$.
Shown in Fig. \[fig5\] are the results for the valence $u$-quark distributions of the $\pi^+$ and $K^+$ and the valence $s$-quark distribution of the $K^+$ at $Q^2 = 16~\mbox{GeV}^2$. The results are shown in vacuum and for symmetric nuclear matter at $\rho_B^{} / \rho_0^{} = 0.25$, $0.5$, $0.75$, $1.0$, and $1.25$. In Fig. \[fig5\](a) the obtained valence $u$-quark distributions of the $\pi^+$ are compared with the empirical data of Ref. [@CAAA89]. Although our results underestimate the available experimental data by up to 20%, the general behavior of the valence PDF is in reasonable agreement with the data. Comparison of the valence PDFs for several nuclear matter densities shows that the density dependence of the light-quark PDFs of the $\pi^+$ is rather mild, which is consistent with the conclusion of Ref. [@Suzuki95]. This can also be verified in Fig. \[fig6\] which shows the in-medium to in-vacuum ratios of the valence PDFs.
However, the valence PDFs of the $K^+$ show a different density dependence as shown by the dashed and dotted lines in Fig. \[fig5\]. Not only the magnitude but the shape of the valence PDFs of the $K^+$ change with densities. In particular, the peak position of $x \bar{s}(x)$ in Fig. \[fig5\] changes from $x \approx 0.37$ in vacuum to $x \approx 0.45$ at $\rho_B^{} = \rho_0^{}$. The change can easily be verified in Fig. \[fig6\], which shows that the valence $u$-quark distribution of the $K^+$ changes noticeably, in particular, in the small-$x$ region. This feature becomes remarkable at higher densities. The enhancement is almost 50% in the small-$x$ region at normal nuclear density. In contrast to the enhancement of the valence $u$-quark distribution of the $K^+$ in the small-$x$ region, the valence $\bar{s}$-quark distribution of the $K^+$ is mostly enhanced in the large-$x$ region as density increases. This enhancement is even larger than that of the valence-$u$-quark distribution in the small-$x$ region, and experimental measurements are highly required to verify this prediction.
Finally, in Fig. \[fig7\] we present relative strength of the $u$-quark distribution of the $K^+$ with respect to the other quark distributions, i.e., $u_K(x)/u_\pi(x)$ by the solid lines and $u_K(x)/\bar{s}_K(x)$ by the dashed lines. Note that all the distributions in Fig. \[fig7\] are those of quarks (or partons), but not the valence ones. Our results for $u_K^{}(x)/u_\pi^{}(x)$ in vacuum is compared with the available experimental data of Ref. [@Saclay-80] in Fig. \[fig7\](a). One can find that this ratio is enhanced at higher densities and, in particular, in the small-$x$ region. In the large-$x$ region, the ratio $u_K^{}(x) / \bar{s}_K^{} (x)$ is suppressed when nuclear density increases. This behavior is opposite to the case of $u_K^{}(x)/u_\pi^{}(x)$. The deviations of these ratios from unity show the pattern of the flavor symmetry breaking. Our results demonstrate that the flavor symmetry breaking effects become larger as density increases.
Summary
=======
To summarize, we have studied the current-quark properties in symmetric nuclear matter in the QMC model. Then, using the in-medium quark properties obtained in the QMC model as inputs, we have studied the in-medium properties of dynamical quarks, pions, and kaons in symmetric nuclear matter in the NJL model. In particular, the valence-parton (valence-quark) distribution functions of $\pi^+$ and $K^+$ mesons in symmetric nuclear matter in the NJL model were investigated with the proper-time regularization scheme.
In the present study, we estimated the quark condensates, dynamical quark masses, meson decay constants, and meson-quark coupling constants for pions and kaons in symmetric nuclear matter in the NJL model. The valence parton distribution functions of $\pi^+$ and $K^+$ mesons in vacuum as well as in symmetric nuclear matter at $Q^2=16~\mbox{GeV}^2$ were calculated and compared with the available data. We found that the effects of nuclear medium on the valence $u$-quark distribution of the $\pi^{+}$ is rather weak, which supports the observation of Ref. [@Suzuki95]. However, the valence quark distributions of the $K^+$ show appreciable medium effects, namely, the valence $u$-quark distribution of the $K^+$ shows enhancement in the small-$x$ region, while that of the $\bar{s}$ shows enhancement in the large-$x$ region. The ratios, $u_{K} (x) / u_{\pi} (x)$ and $u_{K} (x) / \bar{s}_{K} (x)$, were found to indicate the flavor symmetry breaking pattern. This implies that the quark distributions would depend on the surrounding quarks and on the hadron species they reside. The flavor symmetry breaking effects become larger at higher densities, and the two ratios show the opposite density-dependence in the large-$x$ region, although they show a similar density-dependence in the small-$x$ region. Experimental confirmation is, therefore, highly desired.
For future prospects, a few comments are in order. First, in the present work, we estimated the in-medium dynamical quark mass. This feature can be improved by including momentum-dependent dynamical quark masses generated in nuclear medium, e.g., based on the Schwinger-Dyson equations. Second, it would be interesting to extend the present approach to heavy mesons with charm or bottom flavor. These studies would give us further hints on the dynamical chiral symmetry breaking and the realization of heavy quark spin symmetry through the evolution of heavy systems in nuclear matter. Moreover, for an alternative elaborated study in the future, it would be interesting to apply a bosonized version of the NJL model or the NJL model proposed in Ref. [@BT01] to describe the nuclear matter properties, in particular, the quark distributions of hadrons in a nuclear medium.
K.T. and J.J.C.M thank the Asia Pacific Center for Theoretical Physics (APCTP) and Kyungpook National University for warm hospitality and support during their visits. This work was supported by Kyungpook National University Bokhyeon Research Fund, 2017. P.T.P.H. was supported by the Ministry of Science, ICT and Future Planning, Gyeongsangbuk-do, and Pohang City. The work of K.T. was supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq Process, No. 313063/2018-4 and No. 426150/2018-0, and was also part of the projects, Instituto Nacional de Ciência e Tecnologia - Nuclear Physics and Applications (INCT-FNA), Brazil, Process, No. 464898/2014-5, and FAPESP Temático, Brazil, Process. No. 2017/05660-0.
[10]{}
Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. I, Phys. Rev. **122**, 345 (1961). B. L. Ioffe, QCD (Quantum Chromodynamics) at low energies, Prog. Part. Nucl. Phys. **56**, 232 (2006). K. Goeke, M. V. Polyakov, and M. Vanderhaeghen, Hard exclusive reactions and the structure of hadrons, Prog. Part. Nucl. Phys. **47**, 401 (2001). A. V. Belitsky and A. V. Radyushkin, Unraveling hadron structure with generalized parton distributions, Phys. Rep. **418**, 1 (2005). P. T. P. Hutauruk, I. C. Clo[ë]{}t, and A. W. Thomas, Flavor dependence of the pion and kaon form factors and parton distribution functions, Phys. Rev. C **94**, 035201 (2016). P. T. P. Hutauruk, W. Bentz, I. C. Clo[ë]{}t, and A. W. Thomas, Charge symmetry breaking effects in pion and kaon structure, Phys. Rev. C **97**, 055210 (2018). U. Raha and A. Aste, Electromagnetic pion and kaon form factors in light-cone resummed perturbative QCD, Phys. Rev. D **79**, 034015 (2009). S.-I. Nam and H.-C. Kim, Electromagnetic form factors of the pion and kaon from the instanton vacuum, Phys. Rev. D **77**, 094014 (2008). J. Bijnens and P. Talavera, Pion and kaon electromagnetic form factors, JHEP **03**, 046 (2002). N. Zovko, Pion and kaon form factors and heavy vector mesons, Phys. Lett. **51B**, 54 (1974). E. O. da Silva, J. P. B. C. de Melo, B. El-Bennich, and V. S. Filho, Pion and kaon elastic form factors in a refined light-front model, Phys. Rev. C **86**, 038202 (2012). V. Bernard and U.-G. Meissner, Electromagnetic Structure of the Pion and the Kaon, Phys. Rev. Lett. **61**, 2296 (1988), **61**, 2973(E) (1988). P. T. P. Hutauruk, Kaon structure in the confining Nambu–Jano-Lasinio model, talk at the 45th International Meeting on Fundamental Physics, arXiv:1801.06299. L. Chang, I. C. Clo[ë]{}t, C. D. Roberts, S. M. Schmidt, and P. C. Tandy, Pion Electromagnetic Form Factor at Spacelike Momenta, Phys. Rev. Lett. **111**, 141802 (2013). G. R. Farrar and D. R. Jackson, Pion Form Factor, Phys. Rev. Lett. **43**, 246 (1979). J. Koponen, A. C. Zimermmane-Santos, C. T. H. Davies, G. P. Lepage, and A. T. Lytle (HPQCD Collaboration), Pseudoscalar meson electromagnetic form factor at high $Q^2$ from full lattice QCD, Phys. Rev. D **96**, 054501 (2017). B. J. Owen, W. Kamleh, D. B. Leinweber, M. Selim Mahbub, and B. J. Menadue, Light meson form factors at near physical masses, Phys. Rev. D **91**, 074503 (2015). J. N. Hedditch, W. Kamleh, B. G. Lasscock, D. B. Leinweber, A. G. Williams, and J. M. Zanotti, Pseudoscalar and vector meson form factors from lattice QCD, Phys. Rev. D **75**, 094504 (2007). J. Kakazu, K.-I. Ishikawa, N. Ishizuka, Y. Kuramashi, Y. Nakamura, Y. Namekawa, Y. Taniguchi, N. Ukita, T. Yamazaki, and T. Yoshie (PACS Collaboration), Electromagnetic pion form factor near physical point in $N_f = 2+1$ lattice QCD, PoS **LATTICE2016**, 160 (2017). S. R. Amendolia *et al.* (NA7 Collaboration), A measurement of the space-like pion electromagnetic form factor, Nucl. Phys. B **277**, 168 (1986). S. R. Amendolia *et al.*, A measurement of the pion charge radius, Phys. Lett. **146B**, 116 (1984). G. M. Huber *et al.* (Jefferson Lab $F_\pi$ Collaboration), Charged pion form factor between $Q^2 = 0.60$ and $2.45~\mbox{GeV}^2$. II. Determination of, and results for, the pion form factor, Phys. Rev. C **78**, 045203 (2008). A. B. Migdal, E. E. Saperstein, M. A. Troitsky, and D. N. Voskresensky, Pion degrees of freedom in nuclear matter, Phys. Rep. **192**, 179 (1990). J. P. B. C. de Melo, K. Tsushima, B. El-Bennich, E. Rojas, and T. Frederico, Pion structure in the nuclear medium, Phys. Rev. C **90**, 035201 (2014). I. Blomqvist and J. M. Laget, A non-relativistic operator convenient for analysis of pion photoproduction on nuclei in the $\Delta(1236)$ region, Nucl. Phys. A **280**, 405 (1977). F. X. Lee, T. Mart, C. Bennhold, and L. E. Wright, Quasifree kaon photoproduction on nuclei, Nucl. Phys. A **695**, 237 (2001). J. T. Londergan, G. Q. Liu, and A. W. Thomas, Kaon-nucleus Drell-Yan processes and kaon structure functions, Phys. Lett. B **380**, 393 (1996). J. J. Aubert *et al.*, The ratio of the nucleon structure functions $F_2^N$ for iron and deuterium, Phys. Lett. B **123B**, 275 (1983). D. F. Geesaman, K. Saito, and A. W. Thomas, The nuclear EMC effect, Annu. Rev. Nucl. Part. Sci. **45**, 337 (1995). C. Alt *et al.* (NA49 Collaboration), Pion and kaon production in central $\nuclide{Pb} + \nuclide{Pb}$ collisions at $20A$ and $30A$ GeV: Evidence for the onset of deconfinement, Phys. Rev. C **77**, 024903 (2008). J. Badier *et al.*, Measurement of the $K^-/\pi^-$ structure function ratio using the Drell-Yan process, Phys. Lett. **93B**, 354 (1980). B. Adams *et al.*, Letter of intent: A new QCD facility at the M2 beam line of the CERN SPS (COMPASS++/AMBER), arXiv:1808.00848. G. H. S. Yabusaki, J. P. B. C. de Melo, W. de Paula, K. Tsushima, and T. Frederico, In-medium $K^+$ electromagnetic form factor with a symmetric vertex in a light front approach, Few-Body Syst. **59**, 37 (2018). K. Tsushima and J. P. B. C. de Melo, In-medium pion valence distribution amplitude, Few-Body Syst. **58**, 85 (2017). K. Suzuki, Pion structure function in a nuclear medium, Phys. Lett. B **368**, 1 (1996). V. Bernard, R. L. Jaffe, and U.-G. Meissner, Strangeness mixing and quenching in the Nambu–Jona-Lasinio model, Nucl. Phys. B **308**, 753 (1988). V. Bernard and U.-G. Meissner, Properties of vector and axial-vector mesons from a generalized Nambu–Jona-Lasinio model, Nucl. Phys. A **489**, 647 (1988). U. Vogl and W. Weise, The Nambu and Jona-Lasinio model: Its applications for hadrons and nuclei, Prog. Part. Nucl. Phys. **27**, 195 (1991). S. P. Klevansky, The Nambu–Jona-Lasinio model of Quantum Chromodynamics, Rev. Mod. Phys. **64**, 649 (1992). I. C. Clo[ë]{}t, W. Bentz, and A. W. Thomas, Role of diquark correlations and the pion cloud in nucleon elastic form factors, Phys. Rev. C **90**, 045202 (2014). P. T. P. Hutauruk, Y. Oh, and K. Tsushima, Electroweak properties of pions in a nuclear medium, Phys. Rev. C **99**, 015202 (2019). P. T. P. Hutauruk, *Nonperturbative aspects of kaon structure*, PhD thesis, Univ. of Adelaide, 2016. Y. Ninomiya, W. Bentz, and I. C. Clo[ë]{}t, Dressed quark mass dependence of pion and kaon form factors, Phys. Rev. C **91**, 025202 (2015). P. A. M. Guichon, A possible quark mechanism for the saturation of nuclear matter, Phys. Lett. B **200**, 235 (1988). P. A. M. Guichon, K. Saito, E. Rodionov, and A. W. Thomas, The role of nucleon structure in finite nuclei, Nucl. Phys. A **601**, 349 (1996). K. Saito, K. Tsushima, and A. W. Thomas, Self-consistent description of finite nuclei based on a relativistic quark model, Nucl. Phys. A **609**, 339 (1996). K. Saito, K. Tsushima, and A. W. Thomas, Variation of hadron masses in finite nuclei, Phys. Rev. C **55**, 2637 (1997). J. R. Stone, P. A. M. Guichon, P. G. Reinhard, and A. W. Thomas, Finite Nuclei in the Quark-Meson Coupling Model, Phys. Rev. Lett. **116**, 092501 (2016). P. A. M. Guichon, J. R. Stone, and A. W. Thomas, Quark-meson-coupling (QMC) model for finite nuclei, nuclear matter and beyond, Prog. Part. Nucl. Phys. **100**, 262 (2018). K. Tsushima, K. Saito, J. Haidenbauer, and A. W. Thomas, The quark-meson coupling model for $\Lambda$, $\Sigma$ and $\Xi$ hypernuclei, Nucl. Phys. A **630**, 691 (1998). P. A. M. Guichon, A. W. Thomas, and K. Tsushima, Binding of hypernuclei in the latest quark-meson coupling model, Nucl. Phys. A **814**, 66 (2008). J. Stone, P. Guichon, and A. Thomas, Superheavy nuclei in the quark-meson-coupling model, EPJ Web Conf. **163**, 00057 (2017). D. L. Whittenbury, J. D. Carroll, A. W. Thomas, K. Tsushima, and J. R. Stone, Quark-meson coupling model, nuclear matter constraints, and neutron star properties, Phys. Rev. C **89**, 065801 (2014). O. Scholten, S. Tamenaga, and H. Toki, Loop corrections and the $K$-matrix formalism, Phys. Rev. C **75**, 055203 (2007). G. Krein, A. W. Thomas, and K. Tsushima, Nuclear-bound quarkonia and heavy-flavor hadrons, Prog. Part. Nucl. Phys. **100**, 161 (2018). G. Krein, A. W. Thomas, and K. Tsushima, Fock terms in the quark-meson coupling model, Nucl. Phys. A **650**, 313 (1999). A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V. F. Weisskopf, New extended model of hadrons, Phys. Rev. D **9**, 3471 (1974). B. D. Serot and J. D. Walecka, The relativistic nuclear many body problem, Adv. Nucl. Phys. **16**, 1 (1986). B. D. Serot and J. D. Walecka, Recent progress in Quantum Hadrodynamics, Int. J. Mod. Phys. E **6**, 515 (1997). S. Maedan, Influence of current mass on the spatially inhomogeneous chiral condensate, Prog. Theor. Phys. **123**, 285 (2010). D. L. Whittenbury, H. H. Matevosyan, and A. W. Thomas, Hybrid stars using the quark-meson coupling and proper-time Nambu–Jona-Lasinio models, Phys. Rev. C **93**, 035807 (2016). F. M. Steffens, K. Tsushima, A. W. Thomas, and K. Saito, Spin dependent parton distributions in a bound nucleon, Phys. Lett. B **447**, 233 (1999). H. Shen, S. Tamenaga, and H. Toki, Variation of hadron masses in nuclear matter in the relativistic Hartree approximation, Nucl. Phys. A **745**, 121 (2004). I. C. Clo[ë]{}t, G. A. Miller, E. Piasetzky, and G. Ron, Neutron Properties in the Medium, Phys. Rev. Lett. **103**, 082301 (2009). P. Kienle and T. Yamazaki, Pions in nuclei, a probe of chiral symmetry restoration, Prog. Part. Nucl. Phys. **52**, 85 (2004). M. Kirchbach and H. J. Weber, Is the $\eta$ meson a Goldstone boson?, Comments Part. Nucl. Phys. **22**, 171 (1998). U.-G. Mei[ß]{}ner, J. A. Oller, and A. Wirzba, In-medium chiral perturbation theory beyond the mean-field approximation, Ann. Phys. (N.Y.) **297**, 27 (2002). V. Thorsson and A. Wirzba, S-wave meson-nucleon interactions and the meson mass in nuclear matter from chiral effective Lagrangians, Nucl. Phys. A **589**, 633 (1995). H. Mineo, W. Bentz, N. Ishii, A. W. Thomas, and K. Yazaki, Quark distributions in nuclear matter and the EMC effect, Nucl. Phys. A **735**, 482 (2004). M. Miyama and S. Kumano, Numerical solution of $Q^2$ evolution equations in a brute-force method, Computer Phys. Commun. **94**, 185 (1996). V. N. Gribov and L. N. Lipatov, $e^+ e^-$ pair annihilation and deep inelastic $ep$ scattering in perturbation theory, Yad. Fiz. **15**, 1218 (1972), . G. Altarelli and G. Parisi, Asymptotic freedom in parton language, Nucl. Phys. B **126**, 298 (1977). Y. L. Dokshitzer, Calculation of the structure functions for deep inelastic scattering and $e^+ e^-$ annihilation by perturbation theory in Quantum Chromodynamics, Zh. Eksp. Teor. Fiz. **73**, 1216 (1977), . I. C. Clo[ë]{}t, W. Bentz, and A. W. Thomas, Nucleon quark distributions in a covariant quark-diquark model, Phys. Lett. B **621**, 246 (2005). J. S. Conway *et al.*, Experimental study of muon pairs produced by 252-GeV pions on tungsten, Phys. Rev. D **39**, 92 (1989). W. Bentz and A. W. Thomas, The stability of nuclear matter in the Nambu–Jona-Lasinio model, Nucl. Phys. A [**696**]{}, 138 (2001).
[^1]: In principle, the two pieces in the $G_\rho$ term of Eq. may have different coupling constants since they are individually chiral invariant. Our choice of the common coupling avoids flavor mixing and gives the flavor content of $\omega$-meson as $(u \bar{u} + d \bar{d})/\sqrt2$ and $\phi$-meson as $s\bar{s}$, that is, we assume the “ideal mixing” for these mesons.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Swarm verification and parallel randomised depth-first search are very effective parallel techniques to hunt bugs in large state spaces. In case bugs are absent, however, scalability of the parallelisation is completely lost. In recent work, we proposed a mechanism to inform the workers which parts of the state space to explore. This mechanism is compatible with any action-based formalism, where a state space can be represented by a labelled transition system. With this extension, each worker can be strictly bounded to explore only a small fraction of the state space at a time. In this paper, we present the tool together with two search algorithms which were added to the tool suite to both perform a preprocessing step, and execute a bounded worker search. The new tool is used to coordinate informed swarm explorations, and the two new algorithms are employed for preprocessing a model and performing the individual searches.'
author:
- 'Anton Wijs[^1]'
bibliography:
- 'shortlit.bib'
title: 'The <span style="font-variant:small-caps;">Hive</span> Tool for Informed Swarm State Space Exploration'
---
Introduction {#sec:intro}
============
In explicit-state model checking (MC), it is checked whether a given system specification yields a given temporal property. This is done by exploring the so-called [*state space*]{} of the specification, which is a directed graph describing explicitly all potential behaviour of the system. Since state space exploration algorithms often need to keep track of all explored states in order to efficiently perform the MC task,[^2] and since state spaces can be very large, for many years, the amount of available memory in a computer has been the most important bottleneck for MC.
In recent years, however, the increase of available memory in state-of-the-art computers has continued to follow Moore’s Law [@moore], while the increase of their processors’ speed no longer has. For MC, this means that large state spaces can be stored in memory, but the time needed to explore them is impractically long, hence a [*time explosion problem*]{} has emerged. This can be mitigated by developing [*distributed*]{} exploration algorithms, in which a number of computers in a cluster or grid are used to perform an exploration. Many of those algorithms use a partitioning function to assign states to workers, and require frequent synchronisation between these workers, see e.g. [@divine10; @preach; @ltsmintool; @Garavel2006; @lerda.sista.distributedspin].
[*Swarm verification*]{} [@holzmann.swarm2] (SV) (and [*parallel randomised Depth-First Search*]{} [@dwyer.elbaum.parallelrandomized]) are recent techniques to perform state space exploration in a so-called [*embarrassingly parallel*]{} [@foster.parallel] way, where the individual workers never need to synchronise with each other. In SV, each worker starts at the initial state and performs a search based on Depth-First Search (). The direction of a worker is determined by a given successor ordering strategy. As the direction of a depends on the fact that a stack is used to order successor states (i.e. a Last-In-First-Out strategy), changing this ordering directly influences the direction of the search. By providing each worker a unique strategy, they will explore different parts of the state space first. With this method, some states may be explored multiple times by different workers, but if the property does not hold, any bug states present are likely to be detected very quickly, due to the diversity of the searches, which often means that the whole state space does not have to be explored.
However, if a property holds, each worker will exhaustively explore the whole reachable state space, which means that the benefits of parallelisation are completely lost. Recently, we proposed a mechanism to bound each worker to a particular reachable strict subset of the set of reachable states, in such a way that together, the workers explore the whole state space [@wijs.isv]. This mechanism is compatible with any action-based formalism such as $\mu$CRL [@gp95], where each transition in a state space is labelled with some action name corresponding with system behaviour. In this paper, we explain how the [*Heuristics Instructor for parallel VErification*]{} () tool, which resulted from [@wijs.isv], works in practice. Section \[sec:ISV\] presents the functionality of the tool together with some new algorithms implemented in the tool suite [@ltsmintool]. How all these have been implemented and how the resulting tools can be used is explained in Section \[sec:impl\]. In Section \[sec:exp\], experimental results are discussed. Finally, conclusions and pointers for possible future work are given in Section \[sec:conc\].
The Informed Swarm Exploration Technique {#sec:ISV}
========================================
**<span style="font-variant:small-caps;">LTSmin</span>** **<span style="font-variant:small-caps;">Hive</span>**
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------
[**P1. Trace-counting **<span style="font-variant:small-caps;">Dfs</span>****]{}: Constructs $\sprocess'$ with ${\it tc}(s) = \min(1, \Sigma_{s'\in\Next'} {\it tc}(s'))$. [**F1. Trace selection**]{}: select a swarm trace $\sigma$ for worker
[**F2. Informed Swarm Search (**<span style="font-variant:small-caps;">Iss</span>**)**]{}: Search of $\sprocess$ restricted to $\sigma$ [**F3. Update swarm set:**]{} remove inspected traces
: The four major functionalities of ISV
\[tab:functions\]
#### The Setting {#the-setting .unnumbered}
The so-called [*Informed*]{} SV technique (ISV) implemented in and is applicable if three conditions are met: (1) A system specification ${\mathfrak{P}}$ should be an implicit description of a [*Labelled Transition System*]{} () $\sprocess$. An $\sprocess$ is a quadruple $(\states, \actions, \transitions, {s_{\it in}})$, where ${s_{\it in}}$ is the initial state, $\states$ is the set of states reachable from ${s_{\it in}}$, $\actions$ is a set of transition labels (actions), and $\transitions:\states \times \actions \times \states$ is the set of transitions between states. With $s \step{A} t$, $A \subseteq \actions$, we say that there exists an $\ell \in A$ such that $(s,\ell,t) \in \transitions$. The reflexive transitive closure of $\step{}$ is denoted as $\tstep{}{^*}$. In on-the-fly state space exploration, ${s_{\it in}}$ and $\actions$ are known a priori, but $\states$ and $\transitions$ are not, and a next-state function $\Next:\states \to 2^\states$ provides the set of successors of a given state. A state $t$ is the successor of a state $s$ iff $s\step{\actions} t$. $\Next$ is used to construct $\states$ and $\transitions$, starting at ${s_{\it in}}$. In the following, we use the notation $\Next \mid A$, with $A\subseteq\actions$, to denote $\Next$ restricted to a set of transition labels $A$, i.e. $\Next\mid A (s) = \{ s'\in\states \mid \exists \ell \in A . (s,\ell,s')\in\transitions\}$. Clearly, $\Next \mid \actions = \Next$. Finally, a sequence of actions $\langle \ell_0, \ell_1, \ldots \rangle$ describes all transition sequences (traces) through an $\sprocess$ with ${s_{\it in}}\step{\{\ell_0\}} s_0\step{\{\ell_1\}} s_1 \cdots$ for some $s_0,s_1,\ldots$. If $\sprocess$ is label-deterministic, i.e. for all $s,t\in\states, \ell\in\actions$ with $s\step{\ell} t$, there does not exist a state $t' \neq t$ with $s\step{\ell} t'$, such an action sequence corresponds to a single trace. Here, we assume that all [s]{} are label-deterministic. If this is not the case, relabelling of some transitions can resolve this.
\(2) ${\mathfrak{P}}$ should consist of a [*finite number $n>1$ of process descriptions (e.g. process algebraic terms) in parallel composition*]{}. This is the case for any concurrent system. (3) At least some of these processes in parallel composition, i.e. a subsystem, should yield [*finite behaviour*]{}, hence only finite traces. This is not a strict requirement, but if it is not met, then the method relies on bounded analysis of the subsystem, and it does not automatically guarantee anymore that all reachable states are visited.
#### ISV {#isv .unnumbered}
Say that a specification describes a system of concurrent processes ${\mathfrak{P}}= \{P_0, \ldots, P_n\}$, with $n\in\natdom$. ISV exploits the fact that parallel composition is a major cause for state space explosion, and that $\sprocess$ of ${\mathfrak{P}}$ is the synchronous product of [s]{} $\sprocess_i = (\states^i, \actions^i, \transitions^i, {s_{\it in}}^i)$ of the $P_i$ ($0 \leq i \leq n$), restricted by some synchronisation rules between processes, given by a symmetric function ${\mathfrak{C}}$. E.g. ${\mathfrak{C}}(\ell,\ell')=\ell''$ states that if actions $\ell$ and $\ell'$ can be performed by different processes, then the result is action $\ell''$ in the system. For the formal details, see [@wijs.isv]. We assume that the $\actions^i$ are disjoint (if this is not the case, then some rewriting can resolve this)[^3] and that no action is involved in more than one rule defined by ${\mathfrak{C}}$ (either as an input, or as a result). All this implies that for any $\ell\in\actions$, it can be determined whether or not it stems from some behaviour of a particular process $P_i$. Say that $\actions_c \subseteq \actions$ is the set of actions stemming from synchronisations, and that $\actions^i_c \subseteq \actions^i$ is the set of actions of $\sprocess_i$ which are forced to synchronise with other actions, then $\actions = (\bigcup_{i \leq n} \actions^i \setminus \actions^i_c) \cup \actions_c$. Now, for any $A\subseteq \actions$, we can define ${\mathfrak{M}}(A)$ as $\{ \ell'' \in \actions_c \mid \exists \ell\in A, \ell' \not\in A. {\mathfrak{C}}(\ell,\ell') = \ell'' \}$, which is the set of actions resulting from synchronisation involving one action in $A$. Finally, the assumptions about ${\mathfrak{C}}$ allow us to define a relabelling function ${\mathfrak{R}}$ as follows: ${\mathfrak{R}}(\{\ell\}) = \{\ell''\}$ iff there exists an $\ell'$ such that ${\mathfrak{C}}(\ell,\ell') = \ell''$, and ${\mathfrak{R}}(\{\ell\}) = \{\ell \}$, otherwise.
[l]{}[0.66]{}
Four basic functionalities are required to perform ISV in practice. These are listed in Table \[tab:functions\]; a preprocessing step (P1) involving the analysis of a defined subsystem yielding finite behaviour, and three techniques for the three major phases of ISV (F1-3). P1 and F2 require two new search algorithms, which have been implemented in . F1 and F3 have been implemented in the new stand-alone tool. The general procedure to perform an ISV is as follows: First, the user selects a strict subsystem ${\mathfrak{P}}' \subset {\mathfrak{P}}$ which is guaranteed to yield finite behaviour (again, alternatively, this behaviour is bounded in the ISV, but then, the overall search could be non-exhaustive). In P1, the $\sprocess'=(\states',\actions',\transitions',{s_{\it in}}')$, described by ${\mathfrak{P}}'$, is constructed and saved to disk, together with a weight function ${\it tc}:\states' \to \natdom$. For this, we have extended the implementation in , as described in Alg. \[alg:tracecounting\]. The [*tc*]{} function (see also Table \[tab:functions\]) assigns $1$ to deadlock states, i.e. if $\Next' (s) = \emptyset$, and the sum of all the successor weights to any other state (note that $\Next'$ is the next-state function of $\sprocess'$).
This allows efficient reasoning about the traces through $\sprocess'$; the number of traces represented by a trace prefix from ${s_{\it in}}$ to some $s\in\states'$ equals ${\it tc}(s)$. E.g., Fig. \[fig:ipod\] shows a simplified acyclic [^4] of an iPod process as part of the DRM protocol specification from [@drm], with part of the definition of [*tc*]{}. With this, if we sort the states based on their numbering (which was assigned by the ), then each trace can be uniquely referred to with a natural number: note that the number of possible traces is $14$, which corresponds with ${\it tc}(0)$. Say that we want to identify trace 10 (shown in Fig. \[fig:ipod\]). State $0$ has successors $\{1,\ldots, 4\}$. Sorted by increasing state number, we first consider state $1$; since ${\it tc}(1)=3$, we conclude that trace $\langle \textsf{recv\_C1\_3} \rangle$ represents traces $0$ to $2$, i.e. $3$ traces, starting at trace $0$.
[r]{}[0.35]{}
\[alg:tracecounting\] ${\mathfrak{P}}' \subset {\mathfrak{P}}$, ${s_{\it in}}'$, $\actions'$ $\sprocess'$ and ${\it tc}:\states'\rightarrow\natdom$ are constructed $\Closed \leftarrow \emptyset$ ${\it tc}({s_{\it in}}) \leftarrow {\it dfs}({s_{\it in}})$
[***dfs(s) =***]{}
\[alg:tracedfs\] ${\it tc}(s) \leftarrow 0$ ${\it tc}(s) \leftarrow {\it tc}(s) + {\it dfs}(s')$ ${\it tc}(s) \leftarrow 1$ $\Closed \leftarrow \Closed \cup \{s\}$ ${\it tc}(s)$
We also have ${\it tc}(2)=3$, therefore $\langle \textsf{recv\_C1\_1} \rangle$ represents traces $3$ to $5$. Similarly, $\langle \textsf{recv\_C1\_2} \rangle$ represents traces $6$ to $12$, and $\langle \textsf{off(C1)} \rangle$ represents trace $13$. This means that $\langle \textsf{recv\_C1\_2} \rangle$ is a prefix of trace $10$. Since state $3$ only has state $5$ as a successor, clearly $\langle \textsf{recv\_C1\_2}, \textsf{send} \rangle$ is also a prefix (this agrees with ${\it tc}(5)=7$: all $7$ traces represented by $\langle\textsf{recv\_C1\_2} \rangle$ are also represented by $\langle \textsf{recv\_C1\_2}, \textsf{send} \rangle$). In this fashion, the complete trace can be constructed following the states listed on the right of Fig. \[fig:ipod\]. This principle is used for trace selection in (F1). In ISV, each worker is bounded by a trace through $\sprocess'$, given by (this will be explained next). Therefore, each trace represents a worker job to be performed, and $\sprocess'$ represents the set of jobs. From a trace $\langle \ell_0, \ldots, \ell_n \rangle$ through $\sprocess'$ ($n\in\natdom$), a so-called [*swarm trace*]{} $\sigma=\langle {\mathfrak{R}}(\ell_0), \ldots, {\mathfrak{R}}(\ell_n)\rangle$ can be constructed, taking into account synchronisations with ${\mathfrak{P}}\setminus {\mathfrak{P}}'$. Whenever a worker thread can be launched, selects a swarm trace. When the tool is launched to start an ISV, this is done first.
A launched worker thread performs an informed swarm search (), implemented in (Alg. \[alg:swarmsearch\] and F2). In Alg. \[alg:swarmsearch\], $\sigma$ is the swarm trace assigned by the tool, and $\sigma(i)$ is the singleton set containing the $(i+1)^{\it th}$ element of $\sigma$ (If $\sigma$ contains fewer than $i+1$ elements, we say that $\sigma(i)= \emptyset$). In the , $\sprocess$ is explored, but not exhaustively: the potential behaviour of the subsystem ${\mathfrak{P}}'$ is restricted to $\sigma$, which restricts exploration of $\sprocess$. For each visited state $s$, $\Nextset$ is extended with $\Next \mid (\actions \setminus A) (s)$, i.e.all successor states reachable via behaviour of ${\mathfrak{P}}\setminus {\mathfrak{P}}'$, and $\Step$ is extended with $\Next \mid \sigma(i) (s)$, i.e. all successor states reachable via the current behaviour in $\sigma$.
[l]{}[0.32]{}
\[alg:swarmsearch\] ${\mathfrak{P}}$, ${s_{\it in}}$, $\actions$, $A = \actions' \cup {\mathfrak{M}}(\actions')$, $\sigma$ $\sprocess$ restricted to $\sigma$ is explored $i\leftarrow 0$ $\Open\leftarrow {s_{\it in}}$; $\Closed, \Nextset, \Step, \feedback_i \leftarrow \emptyset$ $i \leftarrow i+1$ $\Open\leftarrow \Step \setminus \Closed$; $\Step, \feedback_i \leftarrow \emptyset$ $\Nextset \leftarrow \Nextset \cup \Next \mid (\actions \setminus A) (s)$ $\Step \leftarrow \Step \cup \Next \mid \sigma(i) (s)$ $\feedback_i \leftarrow \feedback_i \cup \{\ell \in A \mid \Next \mid \{\ell\} (s) \neq \emptyset \}$ $\Closed \leftarrow \Closed \cup \Open$ $\Open \leftarrow \Nextset \setminus \Closed$; $\Nextset \leftarrow \emptyset$
When all states in $\Open$ are explored, the contents of $\Nextset$ is moved to $\Open$, after duplicate detection (for which the search history $\Closed$ is used). Note that when all reachable states have been explored in this manner, $i$ is increased, by which the moves to the next step in $\sigma$, and new states become available. The main idea of ISV is to construct the set of all possible traces through $\sprocess'$, and to perform an through $\sprocess$ for each of those traces. This means that eventually $\sprocess$ is completely explored. A proof of correctness can be sketched as follows: say that all traces through $\sprocess'$ have been used by workers to explore $\sprocess$, and that after this, some reachable state $s\in\states$ has never been visited. We will show that this leads to a contradiction. It follows from Alg. \[alg:swarmsearch\] that for each state $t$ to be explored, all new states $t' \in \Next \mid (\actions \setminus A) (t)$ are going to be explored as well, and for some $i$, $t'' \in \Next \mid \sigma(i) (t)$ is going to be added to $\Step$. This implies that all states ${\hat t} \in \Next \mid (A \setminus \sigma(i)) (t)$ are going to be ignored. From this and the fact that ${s_{\it in}}$ is explored, it follows that a state $s$ is ignored iff for all traces through $\sprocess$ from ${s_{\it in}}$ to $s$, there exist $t, {\hat t} \in \states$ such that ${s_{\it in}}\tstep{\actions}{^*} t \step{\ell} {\hat t} \tstep{\actions}{^*} s$, with $\ell \in A \setminus \sigma(i)$, $i$ being the current position in $\sigma$ when exploring $t$. Let us consider one of those traces. We call $\sigma'$ the swarm trace followed to reach $t$ from ${s_{\it in}}$ over that trace. Note that this is a prefix of $\sigma$. Let us assume that by following $\sigma'$ extended with $\ell$, $s$ can be reached from ${s_{\it in}}$.[^5] Since $\sigma'$ has been derived from a trace through $\sprocess'$ and $\ell \in A$, the extended trace must also be derivable from a trace through $\sprocess'$. But then, since all traces through $\sprocess'$ have been used in the ISV, $s$ must have been visited by some other worker that followed $\sigma'$, and we have a contradiction.
In case ${\mathfrak{P}}$ and ${\mathfrak{P}}'$ sometimes synchronise, the trace counting will produce an over-approximation of the possible set of traces of ${\mathfrak{P}}'$ [*in the context of*]{} ${\mathfrak{P}}$. This is because in the trace counting, it is always assumed that whenever ${\mathfrak{P}}'$ needs to synchronise, this can happen in ${\mathfrak{P}}$. The result is that some swarm traces may not correspond with actual potential behaviour in $\sprocess$. To deal with this, includes a feedback procedure: For every position $i$ in $\sigma$, it is recorded in $\feedback_i$ which potential behaviour of ${\mathfrak{P}}'$ has actually been observed. When finished, returns the $\feedback_i$, and using these, can prune away both $\sigma$ and other, invalid, traces (F3). Since each trace prefix represents a set of traces with [*consecutive numbers*]{} (see e.g. the ranges for states in trace 10 in Fig. \[fig:ipod\]), the set of explored and pruned swarm traces can be represented in a relatively small list of ranges. Initially, the set of swarm traces is empty. Say we explore the $\sprocess$ of the DRM specification, and $\sprocess'$ is as displayed in Fig. \[fig:ipod\], and say it is detected that synchronisation with at state $0$ (to state $3$) can actually not happen in $\sprocess$. As already mentioned, $\langle \textsf{recv\_C1\_2} \rangle$ represents $[6, 13\rangle$. So after pruning, $[6,13\rangle$ represents the new set of explored traces. Furthermore, elements in the list can often be merged. E.g., if ranges $[0,5\rangle$ and $[8,14\rangle$ have been explored earlier, and range $[5,8\rangle$ is to be added, the result can again be described using a single range $[0,14\rangle$. At all times, is ready to launch another worker (F1) and to prune more traces (F3). When there are no more swarm traces left to explore, the ISV is finished.
Implementation and Using the Tools {#sec:impl}
==================================
#### Implementation {#implementation .unnumbered}
The trace-counting and have both been implemented in an unofficial extension of the toolset version 1.6-19, which has been written in $C$. Since already contains a whole range of exploration algorithms (both explicit-state and symbolic), there was no need to implement new data structures. ISV is very light-weight in terms of communication between the and the workers, the only information sent to launch an being a swarm trace in the form of a list of actions. This list is being stored in in a linked list, and a pointer traverses this list when exploring, to keep track of the current swarm trace position. In addition to this, a bit set is used to keep track of the encountered actions stemming from ${\mathfrak{P}}'$ since the last move along the swarm trace. This is done to construct the $\feedback_i$. A bit set implementation using a tree data structure is available in the toolset.
Unfortunately, it is currently not possible to automatically extract ${\mathfrak{P}}'$ of a given subsystem from ${\mathfrak{P}}$, meaning that ${\mathfrak{P}}'$ must manually be derived from ${\mathfrak{P}}$. At times, this requires quite some inside knowledge of the description, therefore it is at the moment the main reason that we have not yet performed more experiments. Automatic construction of the ${\mathfrak{P}}'$ description is listed as future work (see Section \[sec:conc\]).
The tool consists of about 1,200 lines of $C$-code. Because of the communication being light-weight, and because interactions between the workers and either involve asking for a new trace and receiving it, or sending the results of an , we decided to implement all communication in the request-response method using TCP/IP sockets. During an ISV, frequently needs to extract traces from the $\sprocess'$, which is kept in memory together with the [*tc*]{}-function. Besides this, a linked list $L$ of nodes containing trace ranges (the $[i,j\rangle$ mentioned in Section \[sec:ISV\]) is maintained, representing the set of explored traces. Currently, when a trace is selected (F1), an ID is chosen randomly from $L$, but one can imagine other selection strategies (see Section \[sec:conc\]). Then, the corresponding trace is extracted from $\sprocess'$.
When launching many workers, frequent requests to are to be expected. Therefore, ${\mbox{\sc Hive}}$ has been implemented with *pthreads*; whenever a worker sends a request, a new thread is launched in to handle the request. If a new swarm trace is required, F1 is performed, and if feedback is given, F3 is performed. The $\sprocess'$ is never changed, hence no race conditions can occur when multiple threads read it, but $L$ is frequently accessed and updated, when selecting a trace and processing feedback, the latter involving writing. For this reason, we introduced a data lock on $L$. We plan to use more fine-grained locking in the future, but we have not yet experienced a real slowdown when using one lock.
During an ISV, keeps accepting new requests until $L$ has one node containing the range $[0, {\it tc}({s_{\it in}}') \rangle$. From that moment on, any requests are answered with the command to terminate, effectively ending all worker executions. The same is done if a worker reports in its feedback that a counter-example to a property to check has been found, because the ISV can stop immediately in that case.
Finally, all has been tested on <span style="font-variant:small-caps;">Linux</span> (<span style="font-variant:small-caps;">Red Hat</span> 4.3.2-7 and <span style="font-variant:small-caps;">Debian</span> 6.0.1) and <span style="font-variant:small-caps;">Mac OS X</span> 10.6.8.
#### Setting up and launching an ISV {#setting-up-and-launching-an-isv .unnumbered}
In the following, we assume that we have a $\mu$CRL specification named describing ${\mathfrak{P}}$, and a specification named describing ${\mathfrak{P}}'$. Actually, any action-based modelling language compatible with is suitable for ISV as well. A $\mu$CRL specification is usually first linearised to a file, using the $\mu$CRL toolset [@blom.fokkink.groote.mcrltoolset], which is subsequently used as the actual input of . Having and , the weighted $\sprocess'$ is saved to disk as follows, with $\langle \textsf{sub} \rangle$ being the chosen base name for the files storing the weighted $\sprocess'$:[^6]
$$\textsf{lpo2lts-grey --getswarm=}\langle \textsf{sub}\rangle\ \textsf{specsub.tbf}$$
This produces the files , , and , containing the actions in $\actions'$, the transitions in $\transitions'$ (with actions and states represented by numbers), and the weights of the states, respectively. Actually, if yields infinite behaviour, this can be bounded by a depth $n$ using the option $\textsf{--swbound=}\langle \textsf{n}\rangle$.
The can now be launched on the same machine by invoking the following, with $\langle \textsf{portnr}\rangle$ being the port number it is supposed to listen at for incoming requests:
$$\textsf{hive } \langle \textsf{portnr}\rangle\ \langle \textsf{sub}\rangle$$
An can be started as follows, $\langle \textsf{server} \rangle$ being the IP address of the machine running :
$$\textsf{lpo2lts-grey --swarm=}\langle \textsf{sub}\rangle \textsf{ --hiveserver=}\langle \textsf{server}\rangle \textsf{ --hiveport=}\langle \textsf{port}\rangle \textsf{ spec.tbf}$$
Note that each also needs information on $\sprocess'$. Actually, only is read from disk, to learn $\actions'$. Therefore, this file should be available on all machines where ${\mbox{\sc Iss}s}$ are started. Finally, in practice, one often wants to start many [s]{} simultaneously, and start a new every time one terminates. This whole procedure can be launched using the shell script .
Experimental Results {#sec:exp}
====================
Table \[tab:results\] shows experimental results using the $\mu$CRL [@gp95] specifications of a DRM procotol [@drm] and the Link Layer Protocol of the IEEE-1394 Serial Bus (Firewire) [@1394link]. We were not yet able to perform more experiments using other specifications, mainly because subsystem specifications still need to be constructed manually, which requires a deep understanding of the system specifications. The experiments were performed on a machine with two dual-core <span style="font-variant:small-caps;">amd opteron</span> (tm) processors 885 2.6 GHz, 126 GB RAM, running <span style="font-variant:small-caps;">Red Hat</span> 4.3.2-7. We simulated the presence of 10 and 100 workers for each experiment (the fully independent worker threads can also be run in sequence). This has some effect on the results: in order to simulate 10 and 100 parallel [s]{}, postponed the processing of feedback until 10 and 100 of them had been accumulated, respectively. When the [s]{} truly run in parallel, this feedback processing is done continuously, and redundant work can therefore be avoided at an earlier stage. In the DRM case, we selected both one and two iPod processes for ${\mathfrak{P}}'$, and in the Firewire case, a bounded analysis of one of the link protocol entities resulted in $\sprocess'$. The SV runs have been performed with the of . Since the specifications are correct, there is no early termination for the explorations, meaning that in SV, all reachable states are explored 10 times. In the DRM case, ISV based on one iPod process leads to an initial swarm set with 5,124 traces, 45 of which were actually needed for different runs. Each run needed to explore [*at most*]{} 18% of $\sprocess$, and in total, the number of states explored was smaller than in the SV. ISV based on two iPod processes leads to a much larger swarm set, and clearly, feedback information is essential. ISV with 10 parallel workers explored in total $2.5$ times more states than SV, but each covered at most $\frac{1}{2}\%$ of $\sprocess$, meaning that they needed a small amount of memory. This demonstrates that ISV is useful in a network where the machines do not have large amounts of RAM. In the Firewire case with 10 parallel workers, each explored at most only $\frac{1}{6}\%$ of $\sprocess$, and in total, the SV explored $83\%$ more states than the ISV. In terms of scalability related to the number of parallel workers, the results with 100 workers show that the overall execution times can be drastically reduced when increasing the number of workers: compared to having 10 workers, 100 workers reduce the time by $86\%$ in the DRM case, and $88\%$ in the Firewire case. The number of [s]{} has actually increased, but we expect this to be an effect of the simulations of parallel workers, as explained before.
A full experimental analysis of the algorithms would also incorporate cases with bugs, to test the speed of detection. This is future work, but since ISV has practically no overhead compared to SV, and the [s]{} are embarrassingly parallel and explore very different parts of a state space, we expect ISV and SV to be comparable in their bug-hunting capabilities. Finally, we chose not to compare ISV experimentally with other distributed techniques (e.g. those using frequent synchronisations), because there are too many undesired factors playing a role when doing that (e.g. implementation language, modelling language, level of expertise of the user with the model checker).
------------------ ------------------------ -------------------- ------------------------ ------------------- -------------------------- --------------------- --------------------------- ----------------------
[[**case**]{}]{} [[**\# workers**]{}]{} [[**search**]{}]{}
[[*\# est. runs*]{}]{} [[*\# runs*]{}]{} [[*max. \# states*]{}]{} [[*max. time*]{}]{} [[*total \# states*]{}]{} [[*total time*]{}]{}
10 [SV]{} [10]{} [10]{} [13,246,976]{} [19,477 s]{} [132,469,760]{} [19,477 s]{}
10 [ISV, 1 iPod]{} [$5,124$]{} [45]{} [2,352,315]{} [2,832 s]{} [85,966,540]{} [14,157 s]{}
10 [ISV, 2 iPods]{} [$1.31*10^{13}$]{} [7,070]{} [70,211]{} [177 s]{} [353,591,910]{} [125,139 s]{}
100 [ISV, 2 iPods]{} [$1.31*10^{13}$]{} [9,900]{} [70,211]{} [175 s]{} [361,050,900]{} [17,325 s]{}
10 [SV]{} [10]{} [10]{} [137,935,402]{} [105,020 s]{} [1,379,354,020]{} [105,020 s]{}
10 [ISV]{} [$3.01*10^9$]{} [1,160]{} [236,823]{} [524 s]{} [235,114,520]{} [60,784 s]{}
100 [ISV]{} [$3.01*10^9$]{} [1,400]{} [236,823]{} [521 s]{} [252,206,430]{} [7,294 s]{}
------------------ ------------------------ -------------------- ------------------------ ------------------- -------------------------- --------------------- --------------------------- ----------------------
: Results for bug-free cases with SV and ISV, 10 and 100 workers.[]{data-label="tab:results"}
Conclusions and Future Work {#sec:conc}
===========================
We presented the functionality of the tool and two new algorithms for ISVs. ISV is an SV method for action-based formalisms to bound the embarrassingly parallel workers to different parts. Worst case, if the system under verification is correct, no worker needs to perform an exhaustive exploration, and memory and time requirements for a single worker can remain low.
[**Tool availability**]{} Both the ISV extended version of and are available at <http://www.win.tue.nl/~awijs/suppls/hive_ltsmin.html>.
[**Future work**]{} We plan to further develop the tool such that a description ${\mathfrak{P}}'$ of a given subsystem can automatically be derived from a given description ${\mathfrak{P}}$. We also wish to investigate which kind of subsystems are particulary effective for the work distribution in ISV, and which are not, so that an automatic subsystem selection method can be derived. If ${\mathfrak{P}}'$ yields infinite behaviour, we want to support its full behaviour automatically in the future. As long as $\sprocess$ is finite-state, this should be possible. Furthermore, we want to investigate different strategies to select swarm traces and to guide individual [s]{}. Finally, we will perform more experiments with much larger state spaces, using a computer cluster.
[^1]: Supported by the Netherlands Organisation for Scientific Research (NWO) project 612.063.816 [*Efficient Multi-Core Model Checking.*]{}
[^2]: A Depth-First Search can in principle be performed by just using a stack, but this means that the MC task can often not be performed in linear time (depending on the structure of the state space).
[^3]: Strictly speaking, a weaker requirement suffices [@wijs.isv].
[^4]: The actual of this example, in which the actions are extended with some data parameters, consists of $547$ states.
[^5]: This is not true if there are multiple transitions stemming from ${\mathfrak{P}}'$ on the trace to $s$ not agreeing with $\sigma$, but then, we can repeat the reasoning in the proof sketch until there is only one left.
[^6]: For $\mu$CRL specifications, is the explicit-state space generator of . For other modelling languages, another appropriate tool should be used.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The explicit form of the next-to-next-to-leading order (N$^2$LO) of the Skyrme effective pseudopotential compatible with all required symmetries and especially with gauge invariance is presented in Cartesian basis. It is shown in particular that for such a particular pseudopotential there is no spin-orbit contribution and that the D-wave term suggested in the original Skyrme’s formulation does not satisfy the invariance properties. The six new N$^2$LO terms contribute to both the equation of state and Landau parameters. These contributions to symmetric nuclear matter are explicitly given and discussed.'
address:
- |
Universit[é]{} de Lyon, F-69622 Lyon, France,\
Université Lyon 1, Villeurbanne; CNRS/IN2P3, UMR5822, IPNL
- 'Institut d’Astronomie et d’Astrophysique, CP 226, Université Libre de Bruxelles, B-1050 Bruxelles, Belgium'
- 'IFIC (CSIC-Universidad de Valencia), Apartado Postal 22085, E-46.071-Valencia, Spain'
author:
- 'D. Davesne'
- 'A. Pastore'
- 'J. Navarro'
title: 'Skyrme effective pseudopotential up to next-to-next-to leading order'
---
Introduction
============
Since the implementation of Vautherin and Brink [@vau72], the Skyrme interaction [@sky56; @sky59] has become a popular tool for the description of nuclear properties based on a self-consistent mean-field approach. In its standard form it consists of zero-range central, spin-orbit and density-dependent terms, involving up to 11 parameters, which are fitted to properties of infinite nuclear matter and some selected nuclei. Systematic studies of binding energies and one-body properties of nuclei can thus be successfully performed in a very wide region of the nuclear chart [@ben03]. However, there are some nuclear properties which cannot correctly be reproduced with the standard Skyrme terms, especially as one moves away from the valley of $\beta$-stability [@erl12]. Intense work is currently being devoted to improve the existing parametrizations, in particular considering additional terms to the standard form [@rai11].
In the original proposal, an effective pseudopotential was constructed as an expansion of the nuclear interaction in relative momenta, thus simulating finite-range effects in a zero-range interaction. It actually contained more terms than the current standard form, which is limited to second order in momenta plus a density dependent term which comes from a zero-range three-body force. These additional terms have received special attention in the last years, in particular the second order tensor term [@les07], a D-wave term [@ben13] as well as some three- and four-body terms [@sad12; @sad13].
As an alternative route, the underlying energy density functional (EDF) kernel has been considered [@per04], with no direct references to the effective interaction neither to the Hartree-Fock approximation. In principle, dealing with functional coupling constants instead of pseudopotential parameters introduces more degrees of freedom in the fitting process. It is nevertheless interesting to keep in mind the interplay between the EDF and the pseudopotential descriptions [@lac09; @dug09]. The EDF related to the standard Skyrme interaction contains up to second-order derivatives of nuclear densities. In the search for a universal EDF, a connection with underlying nucleon interactions has been investigated [@car08], aiming at obtaining the general EDF structure guided solely by symmetry principles. An expansion of the EDF kernel has been given including up to sixth-order derivatives, that is up to next-to-next-to-next leading order (N$^3$LO). In an analogous way, it has been shown how to construct the associated pseudopotential up to sixth order in the relative momenta [@rai11]. The number of coupling constants and interaction parameters increases dramatically as the order increases. However, imposing gauge invariance introduces specific relations between the different terms that considerably reduce that number.
In this work we start from the general N$^{2}$LO pseudopotential derived in [@rai11] and give its explicit form in the more familiar Cartesian basis, constraining it to be gauge invariant. Although this symmetry is not explicitly required from basic principles, there is some current discussion about the necessity of imposing it in general, since it has been shown [@rai11b] that gauge invariance is equivalent to continuity equation for local potentials. The continuity equation is of particular interest in view of using such a pseudopotential for calculations of the time evolution of a quantal system. For this reason, and from the fact that a local potential is automatically gauge invariant [@doba95; @bla86], we restrict ourselves to pseudopotentials which are gauge invariant at all orders.
The article is organized as follows. In section \[building:n2lo\] we write the N$^{2}$LO pseudopotential in Cartesian basis and we discuss some properties related to gauge invariance. In section \[inm\] we derive the contribution of the new terms to the equation of state and Landau parameters of symmetric nuclear matter. The last section will give a brief summary and draw some conclusions.
Construction of the N$^{2}$LO pseudopotential {#building:n2lo}
=============================================
For consistency we will briefly sketch now the notation and general results of [@rai11]. The Cartesian representation is usually employed to write the Skyrme interaction. However, the general pseudo-potential is more conveniently constructed in the spherical-tensor representation. With the conventions of [@rai11], the three components of a rank-1 tensor as the Pauli matrices are written as $$\begin{aligned}
\sigma^{(i)}_{ 1,\mu=\left\{-1,0,1\right\}} &=&
-i \left\{\frac{ 1}{\sqrt{2}}\left(\sigma^{(i)}_{ x} -i\sigma^{(i)}_{ y}\right),
\sigma^{(i)}_{ z},
\frac{-1}{\sqrt{2}}\left(\sigma^{(i)}_{ x} +i\sigma^{(i)}_{ y}\right)\right\}.\end{aligned}$$
The mathematical construction of a pseudopotential is equivalent to the construction of scalars with relative momenta ${\bf k}$, ${\bf k'}$, and $\bsigma^{(1,2)}$ as basic ingredients, with the usual definitions ${\bf k} = ( \overrightarrow{\nabla}_1 - \overrightarrow{\nabla}_2)/2i$ and $ {\bf k}' = - ( \overleftarrow{\nabla}_1 - \overleftarrow{\nabla}_2)/2i$. Thus we have to consider, up to a given order, all possible tensors built with these quantities and couple them to get a scalar. In the so-called LS-coupling, the general potential reads: $$\hat{V}=\sum_{{\tilde{n}' \tilde{L}', \\ \tilde{n} \tilde{L},v_{12} S}}
C_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{L}'}
\hat{V}_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{L}'} \, ,
\label{hatV}$$ with $C_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{L}'}$ the coupling constants and $$\begin{aligned}
\nonumber \hat{V}_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{L}'}=
&&\frac{1}{2}i^{v_{12}} \left( \left[ \left[K'_{\tilde{n}'\tilde{L}'}
K_ {\tilde{n }\tilde{L}}\right]_{S}
\hat{S}_{v_{12} S }\right]_{0}
+ (-1)^{v_{12}+S} \left[ \left[K'_{\tilde{n} \tilde{L}}
K_ {\tilde{n}'\tilde{L}'}\right]_{S}
\hat{S}_{v_{12} S}\right]_{0} \right) \\
&&\times \left(1-\hat{P}^{M}\hat{P}^{\sigma}\hat{P}^{\tau}\right)
\delta({\bf r'}_1-{\bf r}_1) \delta({\bf r'}_2-{\bf r}_2) \delta({\bf r}_1-{\bf r}_2).
\label{VV}\end{aligned}$$ By construction, this potential is invariant under space rotation, space inversion, time reversal and hermitian conjugation. $K_{\tilde{n} \tilde{L}}$ and $K'_{\tilde{n} \tilde{L}}$ are tensors of order $\tilde n$ and rank $\tilde L$ in the relative momenta ${\bf k}$ and ${\bf k'}$, respectively (see [@rai11] for explicit expressions). They are coupled to a total angular momentum $S$, which is coupled to the spin part to get a zero total angular momentum. The spin part is: $$\begin{aligned}
\hat{S}_{v_{12} S} =\left(1-\frac{1}{2}\delta_{v_1,v_2}\right)\left(
[\sigma^{(1)}_{v_1}\sigma^{(2)}_{v_2}]_S +
[\sigma^{(1)}_{v_2}\sigma^{(2)}_{v_1}]_S \right),\end{aligned}$$ where $v_{12}=v_1+v_2$. In the last line of Eq. (\[VV\]), $\hat{P}^{M}$, $\hat{P}^{\sigma}$ and $\hat{P}^{\tau}$ are exchange operators in direct, spin and isospin spaces, respectively, and the sum of indices $\tilde{n}$ and $\tilde{n}'$, which must be even so that the potential is invariant under space inversion, fixes the order of the expansion.
To obtain the pseudopotential one has to constrain it with general symmetry properties, in particular Galilean invariance in the non-relativistic case. In doing so we get 2, 7 and 15 independent parameters at 0th, 2nd and 4th order, respectively. A possible choice to reduce these numbers consists in imposing some other additional symmetry, namely the gauge invariance. It does not introduce any further simplification at 0th and 2nd order, while at 4th order we are left with only 6 independent parameters.
Even if gauge invariance is imposed in a different way on the energy functional density and on the pseudo-potential [@rai11; @car08], the consequence is always to provide some relations between the different coupling constants so that only specific combinations with $\bf k'$, $\bf k$ and $\bsigma^{(1,2)}$ can survive. More specifically, in the case under consideration, the $U(1)$-type gauge transformation acts on a wave function as a multiplicative phase $$|\Psi'\rangle = \exp \left( i \sum_{j=1}^{A}\phi({\bf r}_j) \right)|\Psi\rangle,
\label{transfo}$$ where $\phi({\bf r}_j)$ is an arbitrary real function depending on the position. To see the practical consequences of this invariance on the pseudopotential, it is convenient to consider a general 2-body potential $V({\bf r}'_1, {\bf r'}_2, {\bf r}_1, {\bf r}_2)$, where for simplicity spin and isospin indices are dropped since they are not relevant for the argument. The interaction energy can then be written as: $$E = \int d{\bf r}'_1 d{\bf r}'_2 d{\bf r}_1 d{\bf r}_2 V({\bf r}'_1, {\bf r}'_2, {\bf r}_1, {\bf r}_2)
\left[ \rho({\bf r}_1, {\bf r}'_1) \rho({\bf r}_2, {\bf r}'_2) -\rho({\bf r}_2, {\bf r}'_1) \rho({\bf r}_1 {\bf r}'_2)\right]$$ The gauge tranformation on the $A$-body wave function defined above implies that the density matrix transforms as $\rho'({\bf r}, {\bf r}') = \exp(-i [\phi({\bf r})-\phi({\bf r}') ] )\rho({\bf r},{\bf r}')$. In the case of a local interaction, the presence of two delta functions $\delta({\bf r}'_1 - {\bf r}_1)\delta({\bf r}'_2 - {\bf r}_2) $ in the above integral guarantees automatically the gauge invariance [@doba95].
Concerning the pseudopotential, the transformation (\[transfo\]) translates into: $$\hat{V'}= e^{-i\phi({\bf r}'_2)} e^{-i\phi({\bf r}'_1)} \hat{V} e^{i\phi({\bf r}_1)} e^{i\phi({\bf r}_2)}.$$ The invariance $\hat{V'}=\hat{V}$ then leads to the following equation $$[\phi({\bf r}_1),\hat{V}] + [\phi({\bf r}_2),\hat{V}] = 0,
\label{gauge}$$ which has to be imposed order by order because of the velocity-dependent terms entering (\[hatV\])-(\[VV\]).
As mentioned before, up to second order there are only nine independent parameters. As shown in [@rai11], all together define the parameters of the standard Skyrme interaction, including tensor and spin-orbit terms, but excluding density-dependent terms, which are originated from the three- and four-body contributions not considered here. As gauge invariance is automatically satisfied up to second order, the standard interaction leads to a coherent approach. From the 15 different terms of the general pseudopotential (\[hatV\]) at fourth order, we have identified the combinations satisfying condition (\[gauge\]) and discarded the remaining non-invariant terms.
As a result, we are left with only 6 independent parameters, thus confirming the findings of [@rai11]. We chose them as $C_{22,00}^{22}$, $C_{22,20}^{22}$, $C_{11,00}^{31}$, $C_{11,20}^{31}$, $C_{22,22}^{22}$, and $C_{11,22}^{33}$. Afterwards we realized that it is more convenient to deal with the following linear combinations $$\begin{aligned}
\frac{1}{4} t_1^{(4)} = \frac{3 C_{22,00}^{22}+\sqrt{3} C_{22,20}^{22}}{12\sqrt{5}}, &&
\frac{1}{4} t_1^{(4)} x_1^{(4)} = -\frac{C_{22,20}^{22}}{2\sqrt{15}}, \\
t_2^{(4)} = \frac{3 C_{11,00}^{31} + \sqrt{3} C_{11,20}^{31}}{18}, &&
t_2^{(4)} x_2^{(4)} = -\frac{ \sqrt{3}C_{11,20}^{31}}{9}, \\
t_e^{(4)} = - \frac{C_{22,22}^{22} }{2\sqrt{105}} , &&
t_o^{(4)} = -\frac{ C_{11,22}^{33}}{30\sqrt{7}}. \end{aligned}$$ We can thus write the fourth order pseudopotential in a Skyrme-like form as $$\begin{aligned}
\hat V ^{(4)}_{\rm Sk} &=&
\frac{1}{4} t_1^{(4)} (1+x_1^{(4)} P_{\sigma}) \left[({\bf k}^2 + {\bf k'}^2)^2 + 4 ({\bf k'} \cdot {\bf k})^2\right] \nonumber \\
&+& t_2^{(4)} (1+x_2^{(4)} P_{\sigma}) ({\bf k} \cdot {\bf k'}) ({\bf k}^2 + {\bf k'}^2) \nonumber \\
&+& t_e^{(4)} \left[ ({\bf k}^2+{\bf k'}^2) T_e({\bf k'},{\bf k})
+ 2 ({\bf k} \cdot {\bf k'}) T_o({\bf k'},{\bf k}) \right] \nonumber \\
&+& t_o^{(4)} \left[ 5 ({\bf k} \cdot {\bf k'}) T_e({\bf k'},{\bf k})
- \frac{1}{2} ({\bf k}^2+{\bf k'}^2) T_o({\bf k'},{\bf k}) \right] ,
\label{V-four}\end{aligned}$$ where we have defined two operators involving momenta and spins $$\begin{aligned}
T_e({\bf k'},{\bf k}) &=& 3 (\vec \sigma_1 \cdot {\bf k'}) (\vec \sigma_2 \cdot {\bf k'})
+ 3 (\vec \sigma_1 \cdot {\bf k}) (\vec \sigma_2 \cdot {\bf k})
- ({\bf k'}^2 + {\bf k}^2) (\vec \sigma_1 \cdot \vec \sigma_2), \\
T_o({\bf k'},{\bf k}) &=& 3 (\vec \sigma_1 \cdot {\bf k'}) (\vec \sigma_2 \cdot {\bf k})
+ 3 (\vec \sigma_1 \cdot {\bf k}) (\vec \sigma_2 \cdot {\bf k'})
- 2 ({\bf k'} \cdot {\bf k}) (\vec \sigma_1 \cdot \vec \sigma_2),\end{aligned}$$ which are even and odd under parity transformation, respectively. In all these expressions, a $\delta({\bf r}_1-{\bf r}_2)$ function is to be understood, which nevertheless we have omitted for the sake of simplicity.
The definition of 4th order parameters has been chosen such that their contributions to the equation of state and Landau parameters of symmetric nuclear matter maintain a close analogy with those of 2nd order, as shown below. These six parameters are actually the obvious extension to 4th order of the standard $t_{1,2}$, $x_{1,2}$ and $t_{e,o}$ parameters.
Two important remarks are now in order concerning the form of $\hat V ^{(4)}_{\rm Sk}$. The first one concerns the so-called D-wave term contained in Skyrme’s original proposal [@sky59] in the form: $$\hat V ^{(D)} = \frac{t_D}{2} \left[ {\bf k'}^2 {\bf k}^2 - ({\bf k'} \cdot {\bf k})^2 \right] .
\label{d-wave}$$ Actually this term is not a pure D-wave but also contains a S-wave contribution. It has received some recent attention as a possible improvement of the standard Skyrme interaction [@ben13]. It has been wrongly identified in [@rai11] as the contribution $\hat{V}_{20,00}^{20}$ (see equation (10) in that article). However it does not and can not appear alone in $\hat{V}^{(4)}_{\rm Sk}$ because it is not gauge invariant. By inspecting (\[V-four\]) one can see that the bilinear combination related to $t_1^{(4)}$ includes both terms, ${\bf k'}^2 {\bf k}^2$ and $({\bf k'} \cdot {\bf k})^2$, although with different weight each and combined with other powers of momenta. One can also easily verify that the D-wave (\[d-wave\]) does not contribute to the equation of state of symmetric nuclear matter, neither to the $\ell =2$ Landau parameters, contrarily to the $t_1^{(4)}$ term, whose explicit contribution will be given later on.
The second remark is the absence of a spin-orbit contribution in the form suggested by Bell and Skyrme [@bell56] : $$\label{so-skyrme}
i ( \bsigma_1+ \bsigma_2) ({\bf k'} \wedge {\bf k}) F\left({\bf k'}^2, {\bf k}^2, ({\bf k'} \cdot {\bf k}) \right) ,$$ where $F$ is a scalar bilinear function of momenta. In the process of constructing the pseudopotential (\[hatV\]) we have obtained two terms of this type, containing the scalar contributions ${\bf k'}^2 +{\bf k}^2$ and $({\bf k'} \cdot {\bf k})$, which [*a priori*]{} could be guessed on general grounds. However, neither one of them nor a combination of them fulfill the condition (\[gauge\]) and thus have been discarded. Nevertheless $\hat{V}^{(4)}_{\rm Sk}$ does contain tensor terms which induce spin-orbit effects. Some attempts have been made in the past to explain the spin-orbit coupling in terms of higher order effects of the two-body tensor force [@ter60; @ari60]. More recently, in a mean-field study of exotic nuclei [@otsu06] it has been shown that tensor and two-body spin-orbit interactions produces effects of the same order of magnitude. To this respect, it is interesting to observe that the spin-dependent parts proportional to $t_e^{(4)}$ and $t_o^{(4)}$ can actually be rewritten with terms like $(\bsigma_1 . ({\bf k'} \wedge {\bf k}))(\bsigma_2 . ({\bf k'} \wedge {\bf k}))$ which can be interpreted as higher-order spin-orbit contributions [@ring80]. It is worth mentioning that the close connection between tensor and spin-orbit forces has also been pointed out in a different physical system, as dipolar Fermi gases [@sog12].
Symmetric nuclear matter properties {#inm}
===================================
The introduction of a fourth order pseudopotential, or some of its terms, could in principle improve the description of nuclear properties. Considering the recent work about finite size instabilities [@pas12a; @pas12b; @pas12c] one can be worried about the occurrence of such instabilities with an increasing number of terms, in particular with increasing powers of momenta. In that sense, the present $\hat{V}^{(4)}_{\rm Sk}$ is not ready for practical applications as its parameters are not yet fixed nor constrained. At present the only application of the N$^{2}$LO has been presented within the context of the density matrix expansion. A converging expansion in power of momenta has been obtained in [@car10], thus allowing to convert the interaction energies characteristic to finite and short-range nuclear effective forces into quasi-local density functionals. The parameters obtained in this promising way can be considered as a starting point for a complete minimization procedure [@kor10].
We now give explicitly the contribution of $\hat{V}^{(4)}_{\rm Sk}$ to some properties of symmetric nuclear matter namely, the equation of state and the Landau parameters. The contribution to the energy per particle reads $$\label{contrib}
(E/A)^{(4)} = \frac{9}{280} \left[ 3 t_1^{(4)} + (5 + 4 x_2^{(4)}) t_2^{(4)} \right] \rho k_F^4,$$ where $\rho$ is the density and $k_F=(3 \pi^2 \rho /2)^{1/3}$ the Fermi momentum. This term has to be added to the standard equation of state obtained up to second order given for instance in [@cha97]. Notice that the above combination of 4th order parameters is the same as its analogous 2nd order one, apart from a numerical factor and the additional $k_F^2$ power. We thus conclude that $\hat{V}^{(4)}_{\rm Sk}$ modifies the effective mass, introducing an additional density dependence in it. In the case of finite nuclei one would expect additional density gradient terms, which can be of fundamental importance to have a peaked value at the surface, which is a crucial element for spectroscopy [@ma83]. For completeness we also give the contribution of $\hat{V}^{(4)}_{\rm Sk}$ to the incompressibility at the saturation density $\rho_0$: $$K_{v}^{(4)}
=\frac{9}{10} \left[ 3 t_1^{(4)} + (5 + 4 x_2^{(4)}) t_2^{(4)} \right] \left( \frac{3 \pi^2}{2} \right)^{4/3} \rho_0^{7/3}.$$
Finite size instabilities in nuclear matter are detected using the linear response, which in general is not an easy task [@gar92; @dav09; @pas13]. A first insight can nevertheless be obtained in the Landau limit. To this purpose we have calculated the contribution of $\hat{V}^{(4)}_{\rm Sk}$ to the Landau parameters of symmetric nuclear matter, in terms of which the particle-hole interaction is written as [@mig67] $$\begin{aligned}
\label{landau:expr}
\label{landau}
&&
\sum_{l} \,
\bigg\{ f_{l} + f_{l}' \, (\btau_1 \cdot \btau_2) + \left[ g_{l}
+ g_{l}' (\btau_1 \cdot \btau_2 )\right] (\bsigma_1 \cdot \bsigma_2 ) \nonumber \\
&& \quad
+ \left[ h_{l} + h_{l}' (\btau_1 \cdot \btau_2 ) \right] \frac{k_{12}^{2}}{k_{F}^{2}} \,\, S_{12}
\bigg\} \, P_{l} ( \cos \theta )\end{aligned}$$ For the tensor parameters we have followed the definition of [@dab76; @bac79]. The 4th order pseudopotential only contributes to $l \le 2$ central parameters and $\l \le 1$ tensor parameters. We write them in the following way: $$\begin{aligned}
f_0 &=& \frac{1}{4} L_0[f] + \frac{1}{8} k_F^2 L_2[f] + \frac{1}{6} k_F^4 L_4[f] \label{fgh:param1} \\
f_1 &=& - \frac{1}{8} k_F^2 L_2[f] - \frac{1}{4} k_F^4 L_4[f] \label{fgh:param2} \\
f_2 &=& \frac{1}{12} k_F^4 L_4[f] \label{fgh:param3} \\
h_0 &=& \frac{1}{4} k_F^2 L_2[h] + \frac{1}{2} k_F^4 L_4[h] \label{fgh:param4} \\
h_1 &=& - \frac{1}{2} k_F^4 L_4[h] \label{fgh:param5}
\end{aligned}$$ and analogously for $f', g, g'$ and $h'$. In this notation the power of $k_F$ reflects the contribution coming from each order. The explicit expressions of the $L_{n=0,2,4}$ functions entering (\[fgh:param1\]-\[fgh:param5\]) are the following. At zeroth order we have $$\begin{aligned}
L_0[f] &=& 3 t_0, \\
L_0[g] &=& - t_0 (1-2 x_0) , \\
L_0[f'] &=& - t_0 (1+2 x_0) , \\
L_0[g'] &=& - t_0,
\end{aligned}$$ with no density-dependent contribution as previously explained. At second order, $$\begin{aligned}
L_2[f] &=& 3 t_1 + (5+4x_2) t_2, \\
L_2[g] &=& - (1-2x_1) t_1 + (1+2x_2) t_2 , \\
L_2[f'] &=& -(1+2x_2) t_1 + (1+2x_2) t_2 , \\
L_2[g'] &=& - t_1 + t_2, \\
L_2[h] &=& t_e + 3 t_o, \\
L_2[h'] &=& - t_e+t_o.
\end{aligned}$$ Finally, at fourth order we have $$\begin{aligned}
L_4[f] &=& 3 t_1^{(4)} + (5+4x_2^{(4)}) t_2^{(4)}, \\
L_4[g] &=& - (1-2x_1^{(4)}) t_1^{(4)} + (1+2x_2^{(4)}) t_2^{(4)}, \\
L_4[f'] &=& -(1+2x_2^{(4)}) t_1^{(4)} + (1+2x_2^{(4)}) t_2^{(4)}, \\
L_4[g'] &=& - t_1^{(4)} + t_2^{(4)}, \\
L_4[h] &=&t_e^{(4)} +3t_o^{(4)}, \\
L_4[h'] &=& - t_e^{(4)} +t_o^{(4)}.
\end{aligned}$$ Discarding the 4th order contributions $L_{4}$, these results agree with the ones given in [@cao10] for a standard Skyrme interaction including tensor terms.
For purely central interactions, the stability of the spherical Fermi surface of nuclear matter against small deformations can be expressed in terms of the Landau parameters, which should fulfill some well-known inequalities [@mig67]. The inclusion of tensor terms produces a coupling between the spin-dependent parts, leading to generalized stability criteria, which have been given in a compact form in [@bac79] and recently discussed for current effective and microscopic interactions in [@nav13]. These generalized criteria put additional constraints on the interaction parameters, which could be of great help during the optimization procedure to avoid regions of instabilities.
Conclusions {#conclusion}
===========
In summary, we have presented the explicit expression in Cartesian basis of the most general 4th-order contributions to a Skyrme-type pseudopotential $\hat{V}^{(4)}_{\rm Sk}$ fulfilling gauge invariance. Although the requirement of this symmetry is still under debate in the general case, we have chosen to impose it order by order to this particular type of pseudopotential because it ensures the validity of the continuity equation. Besides, this choice significantly reduces to six the number of independent pseudopotential parameters, which are an obvious extension to 4th order of the standard $t_{1,2}$, $x_{1,2}$ and $t_{e,o}$ parameters.
We have observed that $\hat{V}^{(4)}_{\rm Sk}$ contains neither the original D-wave term suggested by Skyrme nor higher order spin-orbit terms. We have used $\hat{V}^{(4)}_{\rm Sk}$ in the context of symmetric nuclear matter to obtain its explicit contribution to the equation of state and related quantities. We have also derived the explicit expression of the Landau parameters, interesting in themselves due to their universal character. Indeed, numerical values obtained within some microscopic approach could be used to put explicitly constraints on the values of the coupling constants. Moreover, they could be of some help during the minimization procedure of fixing parameters to avoid unstable regions. Finally, the close connection between higher-order spin-orbit and tensor terms has been pointed out and is believed to hold not only at N$^2$LO but also at N$^3$LO. Work along these lines is in progress.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by NESQ project (ANR-BLANC 0407, France) and Mineco (Spain), grant FIS2011-28617-C02-02. The authors thank M. Bender, K. Bennaceur, B.G. Carlsson, J. Dobaczewski, T. Duguet, J. Meyer and F. Raimondi for stimulating and encouraging discussions and for useful comments.
References {#references .unnumbered}
==========
[99]{}
D. Vautherin and D.M. Brink, Phys. Rev. C [**5**]{}, 626 (1972). T.H.R. Skyrme, Philos. Mag. [**1**]{}, 1043 (1956). T.H.R. Skyrme, Nucl. Phys. [**9**]{}, 615 (1959). M. Bender, P.-H. Heenen, and P.-G. Reinhard, Rev. Mod. Phys. [**75**]{}, 121 (2003). J. Erler, N. Birge, M. Kortelainen, W. Nazarewicz, E. Olsen, A. Perhac and M. Stoitsov, Nature [**486**]{} 509 (2012). F. Raimondi, B. G. Carlsson and J. Dobaczewski, Phys. Rev. C [**83**]{}, 054311 (2011). T. Lesinski, M. Bender, K. Bennaceur, T. Duguet, and J. Meyer, Phys. Rev. C [**76**]{}, 014312 (2007). K. Bennaceur, private communication. J. Sadoudi, PhD Thesis, Université Paris XI, 2011. J. Sadoudi, M. Bender, K. Bennaceur, D Davesne, R Jodon and T Duguet Phys. Scr. 2013 ,014013, (2013). E. Perlinska, S. G. Rohozinski, J. Dobaczewski and W. Nazarewicz, Phys. Rev. C [**69**]{}, 014316 (2004). D. Lacroix, T. Duguet, and M. Bender Phys. Rev. C [**79**]{}, 044318 (2009). T. Duguet, M. Bender, K. Bennaceur, D. Lacroix, and T. Lesinski, Phys. Rev. C [**79**]{}, 044320 (2009). B.G. Carlsson, J. Dobaczewski and M. Kortelainen, Phys. Rev. C [**78**]{}, 044326 (2008); Phys. Rev. C 81, 029904(E) (2010). F. Raimondi, B. G. Carlsson, J. Dobaczewski, and J. Toivanen, Phys. Rev. C [**84**]{}, 064303 (2011). J. Dobaczewski, J. Dudek, Phys. Rev. C[**52**]{} (1995) 1827; Phys. Rev. C 55, 3177Ð3177 (1997). J.P. Blaizot and G. Ripka, [*Quantum Theory of Finite Systems*]{}, MIT Press, Cambridge (1980). J.S. Bell and T.H.R. Skyrme, Philos. Mag. [**1**]{}, 1055 (1956). T. Terasawa, Progr. Theor. Phys. [**23**]{}, 87 (1960). A. Arima and T. Terasawa, Progr. Theor. Phys. [**23**]{}, 115 (1960). T. Otsuka, T. Matsuo and D. Abe, Phys. Rev. Lett. [**97**]{}, 162501 (2006). P. Ring and P. Schuck, [*The Nuclear Many-Body Problem*]{}, Springer-Verlag, New York (1980). T. Sogo, M. Urban, P. Schuck, and T. Miyakawa, Phys. Rev. A [**85**]{}, 031601 (R) (2012). A. Pastore, D. Davesne, Y. Lallouet, M. Martini, K. Bennaceur, and J. Meyer, Phys. Rev. C [**85**]{}, 054317 (2012). A. Pastore, M. Martini, V. Buridon, D. Davesne, K. Bennaceur, and J. Meyer, Phys. Rev. C [**86**]{}, 044308 (2012). A. Pastore, K. Bennaceur, D. Davesne and J. Meyer, Int. J. Mod. Phys. E [**21**]{}, 1250040 (2012). B. G. Carlsson and J. Dobaczewski, Phys. Rev. Lett. **105**, 122501, (2010). M. Kortelainen, T. Lesinski, J. Moré, W. Nazarwicz, J. Sarich, N. Scunck, M.V. Stoitsov and S. Wild, Phys. Rev. C **82**, 024313, (2010). E. Chabanat, P. Bonche, P. Haensel, J. Meyer, R. Scheffer, Nucl. Phys. A [**627**]{}, 710 (1997). Z.Y.Ma and J. Wambach, Nucl. Phys A [**402**]{}, 275 (1983). C. García-Recio, J. Navarro, Van Giai Nguyen, and L.L. Salcedo, Ann. Phys. (NY) [**214**]{}, 293 (1992). D. Davesne, M. Martini, K. Bennaceur, and J. Meyer, Phys. Rev. C [**80**]{}, 024314 (2009); [*ibid.*]{}[**84**]{} 059904 (2011). A. Pastore, D. Davesne, K. Bennaceur, J. Meyer and V. Hellemans, Phys. Scr. **2013**, 014014, (2013). A. B. Migdal, *The Theory of Finite Fermi Systems*, Wiley, New York, (1967). J. Dabrowski and P. Haensel, Ann. Phys. (NY) [**97**]{}, 452 (1976). S. O. Bäckman, O. Sjöberg, and A. D. Jackson, Nucl. Phys. A [**321**]{}, 10 (1979). Li-Gang Cao, G. Colò, and H. Sagawa, Phys. Rev. C [**81**]{}, 044302 (2010). J. Navarro and A. Polls, Phys. Rev. C [**87**]{}, 044329 (2013).
| {
"pile_set_name": "ArXiv"
} |
[**Comment on “Anomalous heat conduction and anomalous diffusion in one-dimensional systems”**]{}
A relation between anomalous diffusion, in which the mean squared displacement grows in time like $\langle (\Delta x)^2\rangle=2D_{\alpha}
t^{\alpha}$ ($0<\alpha\le 2$), and anomalous heat conduction was recently derived through a scaling approach by Li and Wang [@li] (hereafter LW). In this model it is assumed that heat transport in a 1D channel is solely due to the flow of non-interacting particles: those entering the channel from the left have a different average kinetic energy than those entering from the right. The energies of the particles at both ends of the channel are defined through the Boltzmann distributions that correspond to the temperatures of two heat baths coupled to either end. The authors are correct in stating that different billiard models discussed in literature belong to this class of processes. However, in this Comment we point out certain crucial inconsistencies of the LW model with the physical picture of random processes leading to normal and anomalous diffusion.
Firstly, consider the collision-free heat transport between the two heat baths. This situation corresponds to ballistic transport, $\alpha=2$, and the mean first passage time acquires the scaling $T\propto
L/v$. In this case, the model of LW reproduces the original result [@lebowitz] for the heat conductivity, $\kappa\propto L$. The first inconsistency becomes apparent already in this limiting case: since the typical velocities of particles entering the channel from the left and from the right are different, the corresponding left and right mean first passage time necessarily differ, as well. The equality of both first passage times invoked in LW can only be fulfilled if the particles are thermalized [*within*]{} the channel; however, under this assumption, the ballistic nature is lost and the whole model no longer holds. Partially, this problem may be circumvented by taking the limiting transition $T_L-T_R\to 0$ in Eq. (4).
The crucial flaw in LW, as we are going to show now, is the fact that Eq. (1) does not necessarily imply Eq. (2) in the range $0<\alpha\le 2$, and vice versa. Thus, although it is tempting to argue that if the typical displacement of the particle grows like $\langle (\Delta x)^2\rangle^{1/2}\propto t^{
\alpha/2}$ then the [*typical*]{} time for traveling a distance $L$ will scale like $\tau\propto L^{2/\alpha}$, one cannot conclude what exactly this time $\tau$ defines: it may well differ from the mean first passage time, $T$. In particular, the latter may even diverge while $\tau$ exists.
To explain this need for caution let us first address subdiffusion, which corresponds to a long-tailed waiting time distribution of the form $\psi
(t)\sim(t/t_0)^{-1-\alpha}/t_0$ ($0<\alpha <1$) [@klablushle]. In this case, it was shown in Ref. [@bvp] that the temporal eigenfunctions for a finite geometry are given by Mittag-Leffler functions, and therefore the survival probability decays like $t^{-\alpha}$. Thus, the associated mean first passage time diverges: $T=\infty$, corresponding to the dominance of the probability of [*not*]{} moving in subdiffusion [@klablushle]. (We should note that in one of the three references, Ref. [@gittermann], cited in LW to support their scaling relation, the result for the first passage time distribution is based on an integral expression, which diverges for a waiting time distribution of long-tailed nature, and is therefore wrong.) Without an external bias, the conductivity of a subdiffusive system vanishes [@bvp; @scher]. The other case, in which the approach presented in LW fails are L[é]{}vy flights [@klablushle]. Their mean first passage time exists and is finite, as can be shown using the methods described in Ref. [@sokolov]; however, their mean squared displacement [*diverges*]{} [@rem].
It must therefore be concluded that the model proposed in LW is by far less general than assumed there, and due to the combination of two a priori unrelated equations contains a crucial flaw in the foundations such that erroneous results ensue for both subdiffusion and L[é]{}vy flights. We also note that in a related context a model developed in Ref. [@denisov] provides analytical and numerical results for the heat conductivity consistent with our objections.
Finally, we point out that the interpretation in terms of the finite-time measurement in the case of subdiffusion, brought forth in the Reply of LW [@reply], would lead to a correct result. However, it would cause an explicitly cutoff time-dependent mean first passage time, and would therefore be different from the original model developed in Ref. [@li].
We acknowledge discussions with A. Chechkin, S. Denisov, J. Klafter and N. Korabel.
Ralf Metzler\
NORDITA – Nordic Institute for Theoretical Physics\
Blegdamsvej 17, 2100 Copenhagen [Ø]{}, Denmark
Igor M. Sokolov\
Institut f[ü]{}r Physik, Humboldt-Universit[ä]{}t zu Berlin\
Newtonstra[ß]{}e 15, 12489 Berlin, Germany
[99]{}
B. Li and J. Wang, Phys. Rev. Lett. [**91**]{}, 044301 (2003)
Z. Rieder, J. L. Lebowitz and E. Lieb, J. Math. Phys. [**8**]{}, 1073 (1967).
J. Klafter, A. Blumen and M. F. Shlesinger, Phys. Rev. A [**35**]{}, 3081 (1987).
R. Metzler and J. Klafter, Physica A [**278**]{}, 107 (2000); compare to Phys. Rep. [**339**]{}, 1 (2000).
M. Gittermann, Phys. Rev. E [**62**]{}, 6065 (2000).
H. Scher and E. W. Montroll, Phys. Rev. B [**12**]{}, 2455 (1975).
I. M. Sokolov, J. Klafter and A. Blumen, Phys. Rev. E [**64**]{}, 021107 (2001).
We note that for L[é]{}vy walks, which are often used to describe sub-ballistic superdiffusion [@klablushle], it is not known what is the exact scaling of the mean first passage time. Thus, also for superdiffusion it is not clear whether the result in LW holds.
S. Denisov, J. Klafter and M. Urbakh, eprint cond-mat/0306393. Phys. Rev. Lett., at press
B. Li and J. Wang, Reply
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study the problem of routing Connected and Automated Vehicles (CAVs) in the presence of mixed traffic (coexistence of regular vehicles and CAVs). In this setting, we assume that all CAVs belong to the same fleet, and can be routed using a centralized controller. The routing objective is to minimize a given overall fleet traveling cost (travel time or energy consumption). We assume that regular vehicles (non-CAVs) choose their routing decisions selfishly to minimize their traveling time. We propose an algorithm that deals with the routing interaction between CAVs and regular uncontrolled vehicles. We investigate the effect of assigning system-centric routes under different penetration rates (fractions) of CAVs. To validate our method, we apply the proposed routing algorithms to the Braess Network and to a sub-network of the Eastern Massachusetts (EMA) transportation network using actual traffic data provided by the Boston Region Metropolitan Planning Organization. The results suggest that collaborative routing decisions of CAVs improve not only the cost of CAVs, but also that of the non-CAVs. Furthermore, even a small CAV penetration rate can ease congestion for the entire network.'
author:
- 'Arian Houshmand$^{1}$, Salomón Wollenstein-Betech$^{1}$, and Christos G. Cassandras$^{1}$[^1][^2]'
bibliography:
- 'NEXTCAR.bib'
title: '[**The Penetration Rate Effect of Connected and Automated Vehicles in Mixed Traffic Routing** ]{}'
---
INTRODUCTION
============
Every year Americans face more than 6.9 billion hours of delay in traffic which costs the US more than 160 billion dollars in urban congestion costs [@david_schrank_bill_eisele_tim_lomax_jim_bak_2015_2015]. In addition, due to heavy traffic congestion, an annual amount of 3.1 billion gallons of fuel is being wasted in traffic [@david_schrank_bill_eisele_tim_lomax_jim_bak_2015_2015].
The advent of Connected and Automated Vehicles (CAVs) has been facilitated by the emergence of vehicle automation technologies, as well as new forms of telecommunication technologies, such as Dedicated Short-Range Communication (DSRC) [@kenney_dedicated_2011] and 5G [@andrews_what_2014]. The latter has enabled Vehicle-to-Vehicle (V2V) and Vehicle-to-Infrastructure (V2I) communication capabilities. Therefore, CAVs can help reduce traffic congestion and environmental impacts of our daily commute, as well as improve safety through collaborative decisions.
![The centralized controller is assigning routes to CAVs entering the network. Red, blue, and green links show three different routes for CAVs.[]{data-label="fig: CAV schematic"}](Figs/CAVs.png){width="0.80\linewidth"}
Many studies have been performed to investigate how CAVs can transform the future of cities [@guanetti_control_2018]. For instance, we may be able to eliminate traffic lights and create unsignalized intersections to reduce congestion and energy consumption [@malikopoulos_decentralized_2018]. Another interesting area of focus is cooperative adaptive cruise control (CACC) [@van_arem_impact_2006; @rajamani_experimental_2001], an extension of adaptive cruise control (ACC). By exploiting V2V communication, CACC can reduce the headway between vehicles. As a result, CACC can reduce drag forces and increase energy efficiency [@al_alam_experimental_2010], as well as road capacity and throughput [@lazar_capacity_2017; @van_arem_impact_2006].
Additionally, an increasing number of autonomous fleet operated businesses are emerging, namely autonomous mobility on demand (AMoD) [@gibbs_google_2017], delivery services, and freight shipping. Coming up with strategies to minimize the travel time and energy cost of these fleets not only reduces traffic congestion and carbon emission impacts, but also increases profitability.
In this paper, we seek to find how optimizing routing decisions of CAVs affects the overall energy consumption costs and travel times of all vehicles. We investigate the interaction between CAVs and regular vehicles and their effects on total travel time and energy consumption in traffic networks. We assume that all CAVs belong to the same fleet (e.g., AMoD), and the fleet operator is trying to minimize their costs (energy or time) by systematically routing the fleet given their origin-destination (O-D) demand.
Similar to this work, Mehr et al. [@mehr_can_nodate] studied how the presence of CAVs can affect mobility in traffic networks. They assumed CAVs can benefit from CACC by creating shorter headway which increases the road capacity. In this context, they adopted the mixed traffic road capacity model from [@lazar_capacity_2017]. Their results show that if all vehicles (CAVs and non-CAVs) make selfish routing decisions, the presence of CAVs might worsen traffic conditions. In contrast to [@mehr_can_nodate], we adopt the viewpoint that there is a centralized controller capable of routing all CAVs given their origin and destination. Moreover, we do not assume the shorter headway for CAVs which was considered in [@mehr_can_nodate]. We show that optimal routing of CAVs under these assumptions can not only benefit CAVs, but also help non-CAVs to save time and energy.
The contributions of this paper are summarized as follows. We first review a system-centric (socially optimal) routing algorithm that minimizes the total travel time assuming 100% CAVs in the system. We then propose algorithms which assign system-centric time-optimal or energy-optimal routes (eco-routes) to CAVs in the presence of mixed traffic (both CAVs and non-CAVs in the system). The eco-routing algorithm can handle different vehicle classes including electric vehicles (EVs), hybrid electric vehicles (HEVs), plug-in hybrid electric vehicles (PHEVs), and conventional vehicles (CVs). Additionally, using the notion of *Wardrop equilibrium* [@beckmann_studies_1955], we model the user-centric (selfish) routing decisions of non-CAVs.
Our results indicate that optimal routing of CAVs can benefit both CAVs and non-CAVs in energy savings and travel times. In addition, we study the performance of the routing algorithms for various CAV penetration rates. We provide evidence that even under small CAV penetration rates, CAVs and non-CAVs benefit.
The remainder of this paper is organized as follows. In Section \[sec: time optimal routing\] we propose an algorithm which assigns system-centric time-optimal routes to CAVs in the presence of mixed traffic. In Section \[sec: mixed traffic energy\], after reviewing different energy models for various vehicle classes, we formulate the system-centric eco-routing problem for CAVs to minimize their overall energy consumption costs in mixed traffic. In Section \[sec: non-CAV flow modeling\], we review the Traffic Assignment Problem and propose a framework to model non-CAV flow. In Section \[sec: numerical results\], we use both a simple example and actual historical data to validate the performance of our routing algorithms. Finally, conclusions and further research directions are outlined in Section \[sec: conclusions\].
Time-Optimal Routing {#sec: time optimal routing}
====================
The objective of the Time-Optimal routing problem is to minimize the overall travel time of CAVs. To achieve this goal, we assume (1) the central controller for CAVs has full information on the Origin-Destination (O-D) demand of both CAVs and non-CAVs and (2) non-CAVs route themselves selfishly (i.e., use the route that minimizes their individual travel time). In order to build a model that captures the effect of mixed traffic on the optimal routing of CAVs, we first consider all vehicles to be CAVs and that they can be controlled centrally. In this respect, in Section \[sec: system centric time optimal\], we calculate the system-centric (social) optimal solution for the 100% CAV penetration rate. Subsequently, in Section \[sec: mixed traffic time\], we generalize the routing model to find optimal routes for CAVs in mixed traffic scenarios.
System-Centric Time-Optimal Routing {#sec: system centric time optimal}
------------------------------------
First we assume an all-CAV network, and we can route them using a centralized controller. The system-centric objective is to minimize total traveling time (delay) of CAVs in the network. In particular, we seek to find the route occupancy matrix (probabilities) for allocating vehicles to routes. In other words, find the probability matrix $\textbf{P}$ where its elements denote the probability that a vehicle traveling from an origin *O* to destination *D* uses route $r$.
### Problem Formulation {#sec: problem formulation 100 CAV time}
As in [@zhang_price_2018], we model the traffic network as a directed graph $G=(\mathcal{V},\mathcal{A},\mathcal{W})$ where $\mathcal{V}$ is the set of nodes, $\mathcal{A}$ is the set of links, and $\mathcal{W}=\{\textbf{w}_{i}:\textbf{w}_{i}=(w_{si},w_{ti}),i \in [\![\mathcal{W}]\!]\}$ is the set of all O-D pairs. We assume that all O-D pairs start and end at one of the network’s nodes. Let the node-link incidence matrix for the strongly-connected and directed graph $G$ be denoted by $\textbf{N}\in \{0,1,-1\}^{|\mathcal{V}|\times|\mathcal{A}|}$, and let the link-route incidence matrix be denoted by **A**. Let us define $d^{w}\geq0$ as the flow demand from $w_{s}$ to $w_{t}$ for any O-D pair $\textbf{w}=(w_{s},w_{t})$. Moreover, the route choice probability matrix is defined as $\textbf{P}=[p_{ir}]$, where $p_{ir}$ is the probability of taking route *r* while traveling through O-D pair *i*. Let $\textbf{g}=(g_{i}; i\in [\![\mathcal{W}]\!])$ be the O-D demand vector.
Let us define the power-set of routes $\mathcal{\textbf{R}}=\{\mathcal{R}_{i} ;i\in [\![\mathcal{W}]\!]\}$, where $\mathcal{R}_{i}$ is the set of allowable routes for each O-D pair *i*. Finally, the link-route incidence matrix is denoted by $\textbf{A}=\{\alpha_{a,r}^{i}; i\in [\![\mathcal{W}]\!], r\in \mathcal{R}_{i}, a\in \mathcal{A} \}$ in which: $$\alpha_{a,r}^{i}=\begin{cases}
1; & \text{if route \(r\in \mathcal{R}_{i}\) uses link a}\\
0; & \text{otherwise.}
\end{cases}$$ Additionally, let $A_{i}$ be the sub-matrix of $\textbf{A}$ which includes the columns of $\textbf{A}$ where $r\in \mathcal{R}_{i}$. The total flow is denoted by $\textbf{x}=\{x_{a};a\in \mathcal{A}\}$ where $x_{a}$ is the flow on each link $a\in \mathcal{A}$.
Considering $a\in \mathcal{A}, i\in [\![\mathcal{W}]\!], r\in \mathcal{R}_{i}$ we can formulate the system-centric time-optimal problem as follows:
\[eqn: 100 CAV main problem\] $$\begin{gathered}
\label{eqn: system-centric time cost}
\min_{\textbf{P}}\sum_{a\in \mathcal{A}}t_{a}(x_{a})x_{a}
\\
\label{eqn: total flow on each link}
\textbf{x}=\textbf{AP}^{T}\textbf{g}
\\
\label{eqn: travel time func}
t_{a}(x_{a})=t_{a}^{0}\sum _{i=1}^{n}\beta_{i}(\frac{x_{a}}{m_{a}})^{(i-1)}
\\
\label{eqn: probability const. Social time}
\begin{array}
[c]{lr}
\sum_{r\in \mathcal{R}_{i}}p_{ir}=1;& \forall i\in [\![\mathcal{W}]\!]
\end{array}
\\
\begin{array}
[c]{lr}
p_{ir}\in [0,1];&\forall i\in [\![\mathcal{W}]\!],\forall r\in \mathcal{R}_{i}
\end{array}
$$
where $t_{a}(x_{a})$ is the traveling time of link $a$ as a function of its corresponding traffic flow $x_{a}$, which can be modeled as an increasing polynomial function using . $t_{a}^{0}$, and $m_{a}$ are the free flow travel time and flow capacity of link $a \in \mathcal{A}$ respectively. Moreover, $\boldsymbol{\beta}=(\beta_{i}, i=1,2,...,n) $ is the vector of coefficient factors for calculating traveling time in . A common value is $\boldsymbol{\beta}=\{1,0,0,0,0.15\} $ which is the US Bureau of Public Roads (BPR) travel time function [@manual_bureau_1964; @huang_combined_1995]. The constraint enforces the requirement that the sum of all the fractions of vehicles traveling through an O-D pair is 1. The decision variable is the routing-probability matrix $\textbf{P}=[p_{ir}]$ that for each O-D pair $ i\in [\![\mathcal{W}_i]\!]$ assigns fractions of vehicles to allowable existing routes $r\in \mathcal{R}_{i}$ between any given O-D pair. The inputs to the problem are the link-route incidence matrix ($\textbf{A}$), and the O-D demand vector $\textbf{g}$. Note that instead of solving for individual links to follow for each vehicle, we are here assigning CAVs routes to follow between each OD pair. This transformation helps us reduce the decision space to select routes between OD pairs rather than finding link-based decisions. The solution of Problem \[eqn: 100 CAV main problem\] is often referred to as the system-centric or social optimal solution in the transportation literature.
System-Centric Time-Optimal Routing in the Presence of Mixed Traffic {#sec: mixed traffic time}
--------------------------------------------------------------------
In this section, we address the system-centric time-optimal routing of CAVs in the presence of mixed traffic (CAVs and non-CAVs). In this case, only a portion of vehicles are CAVs and can be controlled through a centralized controller. As a result, instead of finding a routing scheme that minimizes total costs for all vehicles in the system, we focus on minimizing travel time for the CAV portion of traffic. We consider all CAVs as belonging to the same fleet (e.g., AMoD) and that the fleet management company is trying to minimize total traveling time of the fleet. In order to solve this problem we make four assumptions: (1) The non-CAV traffic flow equilibrium is inferred from data (more details in Sec. \[sec: non-CAV flow modeling\]). (2) There exists a centralized controller which can route the CAV portion of traffic. (3) Up to $m$ number of routes are chosen for every O-D pair. (4) Travel time functions $t(\cdot)$ are strongly monotone and continuously differentiable.
### Problem Formulation {#sec: mixed CAV time}
Let us define the CAV penetration rate $\gamma$, as the portion of traffic that consists of CAVs (fraction of CAVs in the system). As in the system-centric case for 100% CAV, we define $\textbf{P}_{c}=[p_{ir}^{c}]$ to be the route choice probability matrix for the CAV portion of traffic. Moreover, $\textbf{g}_{c}=\{g_{i}^{c}; i\in [\![\mathcal{W}]\!]\}$ is the O-D demand vector for CAVs. As mentioned before, we assume that the non-CAV traffic flow equilibrium is inferred from data, and is known. Let us define $\textbf{x}^{c}=\{x_{a}^{c};a\in \mathcal{A}\}$ and $\textbf{x}^{nc}=\{x_{a}^{nc};a\in \mathcal{A}\}$ as the flow of CAVs and non-CAVs in the system respectively, where $x_{a}^{c}$ and $x_{a}^{nc}$ are the CAV and non-CAV flow on each link $a\in \mathcal{A}$. As a result, using the same notation as in Section \[sec: problem formulation 100 CAV time\], the optimization problem can be written as:
\[eqn: mixed time main problem\] $$\begin{gathered}
\label{eqn: mixed CAV time objective}
\min_{\textbf{P}_c}\sum_{a\in \mathcal{A}}t_{a}(x_{a})x_{a}^{c}
\\
\label{eqn: total flow mixed time}
\textbf{x}=\textbf{x}^{c}+\textbf{x}^{nc}
\\
\label{eqn: total flow CAV on each link}
\textbf{x}^{c}=\textbf{A}\textbf{P}_{c}^{T}\textbf{g}^{c}
\\
\label{eqn: travel time func mixed}
t_{a}(x_{a})=t_{a}^{0}\sum _{i=1}^{n}\beta_{i}(\frac{x_{a}}{m_{a}})^{(i-1)}
\\
\label{eqn: probability constraint mixed CAV time}
\begin{array}
[c]{lr}
\sum_{r\in \mathcal{R}_{i}}p_{ir}^{c}=1;& \forall i\in [\![\mathcal{W}]\!]
\end{array}
\\
\begin{array}
[c]{lr}
p_{ir}^{c}\in [0,1];&;\forall i\in [\![\mathcal{W}]\!],\forall r\in \mathcal{R}_{i}
\end{array}
$$
Constraint states that the total flow in the network is the summation of the CAV flow ($\textbf{x}^c$) and non-CAV flow ($\textbf{x}^{nc}$). Notice that we are minimizing the travel time for the CAV share of traffic. As a result, in the traveling time of each link which is a function of both CAV flow and non-CAV flow ($t_a(x_a)$), is multiplied by the flow of CAVs only ($x_{a}^{c}$). The inputs to the optimization problem are the link-route incidence matrix $\textbf{A}$, O-D demand vector $\textbf{g}$, and non-CAV flow $\textbf{x}^{nc}$.
By solving Problem \[eqn: mixed time main problem\], we find optimal flows of CAVs over each O-D pair (route-probability matrix $\textbf{P}_c$). In other words, when a CAV enters the network at an origin *O* given its destination *D*, the algorithm gives it the desired socially optimal route to follow in terms of a sequence of links.
As stated in Section 2.4 of [@patriksson_traffic_2015] the system-centric problem can be reformulated as a user-centric problem by slightly changing the travel cost function. Therefore, the results on the existence and uniqueness of the solution for the user-centric problem (Section \[subsec: wardprop\]) extend to the system-centric case. As a requirement for such a result we need positive and strictly increasing travel time functions on $\textbf{P}$ which is achieved by having increasing polynomial functions.
System-Centric Eco-Routing in the Presence of Mixed Traffic {#sec: mixed traffic energy}
===========================================================
In this section we solve the eco-routing problem for a fleet of CAVs in the presence of mixed traffic. Eco-routing refers to the procedure of finding the optimal route for a vehicle to travel between two points which utilizes the least amount of energy costs. This problem shares similar properties to , with the difference that we minimize energy instead of time. As a result, we need an energy model to calculate energy consumption on each link. In this section, we first review an energy model for conventional vehicles and then formulate the system-centric eco-routing problem for CAVs.
Energy Consumption Modeling {#sec: Empirical energy model}
---------------------------
Energy consumption of vehicles depends on many different factors including velocity and acceleration [@kamal_ecological_2011] of the vehicle, as well as the power-train’s architecture. Since in eco-routing we are making high-level decisions that can affect the energy consumption, a low-fidelity model can be sufficient for our needs. Moreover, when solving the eco-routing problem, we are dealing with a large number of decision variables. Having a model with a simple mathematical function would allow us to speed up the calculation for practical purposes. Hence, we are looking for an energy model which can estimate the energy consumption as a function of the average speed of a vehicle. We adopt the empirical energy model for conventional vehicles proposed by Boriboonsomsin et al. [@boriboonsomsin_eco-routing_2012]. This model is a polynomial function of of the average speeds of links. According to this empirical model (which is calibrated for an Audi A8), the average fuel consumption in grams per mile for every link $a\in \mathcal{A}$ can be calculated as follows: $$\label{eqn: energy model CV}
ln(e_{a})=\sum _{i=0}^{4}\theta_{i}(v_{a})^{i}+\theta_{5}R_{a}$$ in which $e_{a}$ is the average energy consumption on link $a$ in $g/mi$, $v_{a}$ is the average speed of the link in *mph*, $R_{a}$ is the road grade (in percentage), and $\boldsymbol{\theta}=(\theta_{i}, i=0,1,...,5) $ is the vector of coefficients for calculating the energy cost. Typical values of $\boldsymbol{\theta}$ are given in Table \[tab: conversion factors energy CV\]. Average fuel consumption per mile using and $\theta$ values in Table \[tab: conversion factors energy CV\] is shown in Fig. \[fig: fuel consumption CV\].
![Average fuel consumption of conventional vehicles using Boriboonsomsin model ($R_a=0$)[]{data-label="fig: fuel consumption CV"}](Figs/EnergyCostModelCV.png){width="0.8\linewidth"}
In Appendix \[sec: CD/CS energy model\] we also review a charge depleting(CD)/charge sustaining(CS) energy model [@karabasoglu_influence_2013] which can be used for PHEVs, HEVs, and EVs.
Eco-routing Problem Formulation for Conventional Vehicles {#sec: eco-rputing conventional}
---------------------------------------------------------
In order to formulate the eco-routing problem for conventional vehicles, we use energy model . This problem is almost the same as , with the only difference that $t_a(x_a)$ should be replaced with $e_a(x_a)$, which is the average fuel consumption per mile for traveling link $a\in \mathcal{A}$ as seen in . Considering this, we rewrite the eco-routing problem of CAVs for conventional vehicles as follows:
\[eqn: mixed CAV main problem energy CV\] $$\begin{gathered}
\label{eqn: mixed CAV energy CV objective}
\min_{\textbf{P}_c}\sum_{a\in \mathcal{A}}c_{gas}l_{a}e_{a}(v_{a}(x_a))x_{a}^{c}
\\
\label{eqn: total flow mixed energy CV}
\textbf{x}=\textbf{x}^{c}+\textbf{x}^{nc}
\\
\label{eqn: total flow CAV on each link energy CV}
\textbf{x}^{c}=\textbf{A}\textbf{P}_{c}^{T}\textbf{g}^{c}
\\
\label{eqn: travel time func mixed energy CV}
t_{a}(x_{a})=t_{a}^{0}\sum _{i=0}^{n}\beta_{i}(\frac{x_{a}}{m_{a}})^{i}
\\
\label{eqn: average speed energy CV}
v_{a}(x_{a})=\frac{l_{a}}{t_{a}(x_{a})}
\\
\label{eqn: energy model CV mixed CAV}
ln(e_{a})=\sum _{i=0}^{4}\theta_{i}(v_{a})^{i}+\theta_{5}R_{a}
\\
\label{eqn: probability constraint mixed CAV time}
\begin{array}
[c]{lr}
\sum_{r\in \mathcal{R}_{i}}p_{ir}^{c}=1;& \forall i\in [\![\mathcal{W}]\!]
\end{array}
\\
\begin{array}
[c]{lr}
p_{ir}^{c}\in [0,1];&\forall i\in [\![\mathcal{W}]\!],\forall r\in \mathcal{R}_{i}
\end{array}
\end{gathered}$$
where $c_{gas}$ is the cost of gas (\$/gal), and $l_a$ is the length of link $\ a\in \mathcal{A}$. Moreover, $e_a$ is the average energy consumption per link’s length $\forall a\in \mathcal{A}$, and $\boldsymbol{\theta}=(\theta_{i}, i=0,1,...,5) $ is the energy cost coefficient (Table \[tab: conversion factors energy CV\]).
The eco-routing problem formulation for PHEVs is formulated in Appendix \[sec: eco-route PHEV\].
Non-CAV Flow Modeling {#sec: non-CAV flow modeling}
=====================
One of our assumptions is that the non-CAV flow is an input to our models, and can be inferred from actual traffic data. However, since we currently do not have CAVs in cities, we model the non-CAV flow by considering how non-CAVs react to the optimal decisions made by CAVs. To achieve this task, we assume non-CAVs act selfishly by minimizing their travel time. This modeling framework has been extensively studied and is often referred to as the *Traffic Assignment Problem* (TAP)[@patriksson_traffic_2015]. As a result, we propose an iterative method for finding non-CAV flow considering the routing decisions of CAVs. The basis of this methodology is that whenever CAVs change their routing decisions, non-CAVs adjust theirs and vice versa. This process is well-known in game theory and is referred to as a Stackelberg game [@roughgarden_stackelberg_2004]. For this particular problem, we consider an iterative procedure to find an equilibrium for mixed traffic flow of CAVs and non-CAVs.
In order to obtain the non-CAV flow for a given CAV penetration rate $\gamma$, we first consider only non-CAVs in the network and the O-D demand of non-CAVs is given by: $$\textbf{g}^{nc}=(1-\gamma)\textbf{g}$$ Even though we choose a uniform demand distribution for non-CAVs between O-D pairs, without loss of generality, we can use any other given demand for both CAVs and non-CAVs.
Considering a non-CAV demand $\textbf{g}^{nc}$, we solve the selfish (user-centric) routing problem which minimizes their travel time. In this respect, we use the *Method of Successive Averages* (MSA)[@sheffi_equilibrium_1985]. After finding $\textbf{x}^{nc}$ using the MSA, we solve the time optimal (\[eqn: mixed time main problem\]) or energy optimal (\[eqn: mixed CAV main problem energy CV\]) routing problem for the CAV portion of traffic considering its demand to be: $$\textbf{g}^{c}=\gamma\textbf{g}$$ Since non-CAVs were unaware of CAVs in the system while solving the TAP, we re-solve the problem considering CAV flow on each link. Hence, we re-iterate by considering the CAV solution $\textbf{x}^{c}$. Furthermore, the TAP is solved again for non-CAVs. Re-iteration of this process continuous until convergence (Figs. \[fig: non-CAV flow iter process\]).
[0.24]{} ![ (a) Procedure for solving the system-centric routing problem; (b) Convergence plot for iterating through TAP and social problem []{data-label="fig: non-CAV flow iter process"}](Figs/CAV-nonCAV-Iter.png "fig:"){width="1\columnwidth"}
[0.24]{} ![ (a) Procedure for solving the system-centric routing problem; (b) Convergence plot for iterating through TAP and social problem []{data-label="fig: non-CAV flow iter process"}](Figs/CAV-Time-Iteration.png "fig:"){width="1\columnwidth"}
Traffic Assignment Problem (TAP) and Wardrop equilibrium {#subsec: wardprop}
--------------------------------------------------------
The objective of the Traffic Assignment Problem is to find link flows in a transportation network given the O-D demands and cost functions. A standard solution to this problem is to find travel flows that minimize their travel times. Such a solution individually optimizes every vehicle’s travel time based on network conditions. This leads to a *Nash Equilibrium* that in transportation networks is known as the *Wardrop Equilibrium* [@beckmann_studies_1955]. The resulting flows $\mathbf{x^{*}}$ (equilibrium flows) require that for every O-D pair $\mathbf{w}$, and any route $r$ connecting $(w_{s},w_{t})$, the associated travel time is not greater than the traveling time from any other route. Formally $$t_a(x_a^*) \leq t_a(x_{a'}^*) \ \ \ \forall a,a' \in \mathcal{A}$$ or equivalently $$t_a(x_r^*) \leq t_r(x_{r'}^*) \ \ \ \forall r,r' \in \mathcal{R}_i, \ \ \ \forall i \in[\![\mathcal{W}]\!] $$ To obtain such flows, we can solve $$\label{eqn: TAP original}
\min_{\mathbf{x} \in \mathcal{F}} \ \ \Phi(\mathbf{x}) = \sum\limits_{a \in \mathcal{A}}{\int\limits_{0}^{x_a}{t_a(s)ds}}$$ where $\mathcal{F}$ is the set of feasible flow vectors defined by $$\mathcal{F} = \Big\{ \mathbf{x}:\exists {\mathbf{x}^{\mathbf{w}}} \in \mathbb{R}_ +
^{\left| \mathcal{A} \right|} ~\text{s.t.}~\mathbf{x} =
\sum\limits_{\mathbf{w} \in \mathcal{W}} {{\mathbf{x}^{\mathbf{w}}}},\, \mathbf{N}{\mathbf{x}^{\mathbf{w}}} = {\mathbf{d}^{\mathbf{w}}},\,\forall \mathbf{w} \in \mathcal{W} \Big\},
$$ and where $\mathbf{x}^{\mathbf{w}}$ is the flow vector attributed to O-D pair $\mathbf{w}$. Recall that $t_{a}(\cdot)$ in is continuous. Since $\mathcal{F}$ is a compact set, the Weierstrass Theorem [@beckmann_studies_1955] implies that there exists a solution to this minimization problem. Moreover, since cost functions are non-decreasing (by assumption), then $\Phi(\cdot)$ is convex and therefore a unique solution exists [@beckmann_studies_1955].
Now, let us write the TAP in terms of non-CAV flows and take into account the presence of the CAV flow in the network.
\[eqn: TAP\] $$\begin{gathered}
\label{eqn: TAP objective}
\min_{\mathbf{x}^{nc}} \sum\limits_{a \in \mathcal{A}}{\int\limits_{x^c_a}^{x^c_a + x_a^{nc}}{t_a(s)ds}}
\\
s.t \ \ \mathbf{x}^{nc} = \sum\limits_{\mathbf{w} \in \mathcal{W}} {{\mathbf{x}^{nc, w}}}
\\
\hspace{2,7cm} \mathbf{N}{\mathbf{x}^{nc, w}} = {\mathbf{d}^{nc, w}}, \ \ \forall \mathbf{w} \in \mathcal{W} \\
\mathbf{x}^{nc, w} \geq \textbf{0} \\ \notag\end{gathered}$$
Numerical Results {#sec: numerical results}
=================
In order to validate the proposed routing algorithms we perform two case studies. First we analyze the widely explored Braess network (Fig. \[fig: Braess’s Network\]), and study the effect of the CAV penetration rate on the total time savings and energy savings in this network. As an alternative benchmark, we applied the algorithms to a sub-network of the Eastern Massachusetts interstate highways (Fig. \[fig:EMA-small-subnet\]). For finding the energy optimal routes, we assume the road grade is zero ($R_{a}=0$ in ), and we assume the cost of gas is 2.75 \$/gal. We solve the NLP problems using IPOPT [@wachter_implementation_2006] in Julia [@bezanson_julia:_2017].
For the eco-routing case, we only show the results for solving . In other words, we only solve the eco-routing problem for conventional vehicles using the energy model discussed in Section \[sec: Empirical energy model\]. As mentioned before, eco-routing results are extremely sensitive to the energy model. Given a more accurate energy model which is convex, smooth and differentiable we may get different results. Hence, the eco-routing results shown in this paper should only be considered as preliminary results which show the potential of saving energy using centralized routing of CAVs.
Braess Network Example
----------------------
To demonstrate how optimal routing of CAVs under different penetration rates can affect both CAVs and non-CAVs, we first apply algorithm \[eqn: mixed time main problem\] to the well-known Braess network (Fig. \[fig: Braess’s Network\]). Note that in this case instead of using the BPR function , we use the travel time functions shown on each link of the Braess network in Fig. \[fig: Braess’s Network\]. We consider a demand of 4000 veh/hr travels from node 1 to node 4, the lengths of links 1,2,3 and 5 equal to 30.5 miles and the length of link 4 equal to zero. First we solve the time-optimal routing of CAVs under different penetration rates. Using the obtained flows, and energy model we calculate energy costs for traveling through the network (all cars are assumed to be conventional vehicles). Time-optimal results are shown in Figs. \[fig: Braess graph penetration rate time opt- Time\], in which we compare traveling time of CAVs with non-CAVs under different penetration rates. The energy cost for traveling through the optimal routes are also shown in Fig. \[fig: Braess graph penetration rate time opt-energy\]. In addition, we compared the traveling time of CAVs and non-CAVs under different penetration rates using the case of 0% CAV as a baseline and reported the time savings in Fig. \[fig: Braess graph penetration rate time opt-improvements\]. As shown in Fig. \[fig: Braess energy optimal plots\], introducing CAVs into the system not only improves the time saving of CAVs, but also helps non-CAVs to save time. This is because smart routing decisions of CAVs reduce the traffic intensity in the highly congested roads, which consequently helps non-CAVs to travel faster. Note that the baseline is the 0 % CAV case (uncontrolled traffic), in which all vehicles act selfishly. As we inject CAVs into the system, we see that travel time (as well as energy cost) per vehicle of CAVs starts decreasing compared with the uncontrolled traffic. Moreover, the traveling time of commuting through the fastest route decreases as we inject more CAVs to the system. Typically, we expect a trade off between time saving and energy saving in routing problems [@sun_save_2016; @houshmand_eco-routing_2018]. However, in Fig. \[fig: Braess energy optimal plots\] we see that time and energy follow the same trend. In other words, time savings result in energy savings. The reason for this behavior is the energy model used in the eco-routing problem \[eqn: energy model CV\]. As we can see in Fig. \[fig: fuel consumption CV\], the higher the speed, the better fuel efficiency. Hence, for conventional vehicles, and based on this energy model [@boriboonsomsin_eco-routing_2012], we get similar results for energy and time.
It is interesting to see that when a small percentage of CAVs are in the system, there is no improvement for anyone. This happens because CAVs are optimizing over their own small fraction of overall traffic and this fraction is not sufficient to change the network conditions. However, as we increase the penetration rate, CAVs create a more balanced flow distribution in the network from which both CAVs and non-CAVs can benefit. In the Braess network example, it can be seen that if all the cars in the system are replaced with CAVs, we can save 18.9% in terms of travel time. This value is often referred to as the Price of Anarchy (PoA) [@zhang_price_2018].
In addition to the time-optimal case, we also solve the eco-routing (energy-optimal) problem for CAVs using the Braess network. As mentioned earlier, there are many different models to calculate energy costs of vehicles which depend on the vehicle type. In Section \[sec: mixed traffic time\] we formulated the eco-routing routing problem using an empirical energy model for conventional vehicles in . In Appendix \[sec: eco-route PHEV\], we formulate the system-centric eco-routing problem for PHEVs as shown in . However, due to the non-convexity of and issues with local optima, we are only showing the energy-optimal results by solving . As we can see in Figs. \[fig: Braess graph penetration rate energy opt- Time\], \[fig: Braess graph penetration rate energy opt-energy\], and \[fig: Braess graph penetration rate energy opt-improbements\], energy-optimal results follow the same trend as time-optimal result. In other words, centralized eco-routing of CAVs can benefit both CAVs and non-CAVs. The maximum energy savings happens at the 100% CAV penetration rate (19.1%).
\(1) [$1$]{}; (2) \[above right =1.3cm and 3.6cm of 1\] [$2$]{}; (3) \[below right =1.3cm and 3.6cm of 1\] [$3$]{}; (4) \[below right =1.3cm and 3.6cm of 2\] [$4$]{}; (2) edge \[mystyle\] node [$t(x_{3})=45$]{} (4) (1) edge \[mystyle\] node [$t(x_{1})=\frac{x}{100}$]{} (2); (1) edge \[mystyle\] node [$t(x_{2})=45$]{} (3) (3) edge \[mystyle\] node [$t(x_{5})=\frac{x}{100}$]{} (4) (2) edge \[mystyle\] node [$t(x_{4})=0$]{} (3) ;
[.49]{} ![Braess network routing results under different penetration rates by solving system-centric time-optimal problem (a), (c), (e) and, system-centric energy-optimal problem (b), (d), (f).[]{data-label="fig: Braess energy optimal plots"}](Figs/Results/Braess_TimeOpt_Time.png "fig:"){width="1\columnwidth"}
[.49]{} ![Braess network routing results under different penetration rates by solving system-centric time-optimal problem (a), (c), (e) and, system-centric energy-optimal problem (b), (d), (f).[]{data-label="fig: Braess energy optimal plots"}](Figs/Results/Braess_EnergyOpt_Time.png "fig:"){width="1\columnwidth"}
[.49]{} ![Braess network routing results under different penetration rates by solving system-centric time-optimal problem (a), (c), (e) and, system-centric energy-optimal problem (b), (d), (f).[]{data-label="fig: Braess energy optimal plots"}](Figs/Results/Braess_TimeOpt_Energy.png "fig:"){width="1\columnwidth"}
[.49]{} ![Braess network routing results under different penetration rates by solving system-centric time-optimal problem (a), (c), (e) and, system-centric energy-optimal problem (b), (d), (f).[]{data-label="fig: Braess energy optimal plots"}](Figs/Results/Braess_EnergyOpt_Energy.png "fig:"){width="1\columnwidth"}
[.49]{} ![Braess network routing results under different penetration rates by solving system-centric time-optimal problem (a), (c), (e) and, system-centric energy-optimal problem (b), (d), (f).[]{data-label="fig: Braess energy optimal plots"}](Figs/Results/Braess_TimeOpt_Time_Improvement.png "fig:"){width="1\columnwidth"}
[.49]{} ![Braess network routing results under different penetration rates by solving system-centric time-optimal problem (a), (c), (e) and, system-centric energy-optimal problem (b), (d), (f).[]{data-label="fig: Braess energy optimal plots"}](Figs/Results/Braess_EnergyOpt_Energy_Improvement.png "fig:"){width="1\columnwidth"}
EMA Interstate Highway Network
-------------------------------
In order to obtain more realistic results, we perform a data-driven case study using the actual traffic data from the Eastern Massachusetts (EMA) road network. These data were collected by *INRIX* and provided to us by the Boston Region Metropolitan Planning Organization. The sub-network including the interstate highways of EMA (Fig. \[fig:EMA-small-subnet\]) is chosen for the case study. For this network, we use the O-D demand which has been estimated using an inverse optimization framework in [@zhang_price_2018]. In order to solve the problem we consider 56 O-D pairs, and allow up to 3 routes between each origin and destination (top 3 shortest routes). We then solve and in order to find the time optimal and energy optimal paths for CAVs respectively. Time optimal results are shown in Fig. \[fig: EMA time optimal plots\], and the energy optimal results are shown in Fig. \[fig: Braess energy optimal plots\]. The results follow the same behavior as the results of the Braess example. We again see that as the CAV penetration rate increases, both CAVs and non-CAVs benefit from optimal routing decisions of non-CAVs. In Fig. \[fig: imrovement time 50 and 100\], we show the time improvement of different O-D pairs with their corresponding O-D demand for 100% and 50% CAV penetration rates. It is interesting to see in Fig. \[fig: imrovement time 50 and 100\] that two OD pairs with relatively high demand are being affected by -0.8% and -1.3% for 50% and 100% $\gamma$’s respectively. However, we see improvements on most of the OD pairs. In this manner, we are able to identify which OD pairs are getting worse and which ones are improving. This gives the opportunity to better understand the dynamics of the network.
[0.2]{} ![ (a) All available road segments in the road map of Eastern Massachusetts [@zhang_price_2018] ; (b) Interstate highway sub-network of eastern Massachusetts[]{data-label="C"}](Figs/EMA-Network.png "fig:"){width="1\columnwidth"}
[0.2]{} ![ (a) All available road segments in the road map of Eastern Massachusetts [@zhang_price_2018] ; (b) Interstate highway sub-network of eastern Massachusetts[]{data-label="C"}](Figs/EMA-sub.png "fig:"){width="1\columnwidth"}
[.49]{} ![EMA network routing results under different penetration rates by solving system-centric time-optimal problem (a), (c), (e) and system-centric energy-optimal problem (b), (d), (f).[]{data-label="fig: EMA time optimal plots"}](Figs/Results/EMA_Apr_AM_TimeOpt_Time.png "fig:"){width="1\columnwidth"}
[.49]{} ![EMA network routing results under different penetration rates by solving system-centric time-optimal problem (a), (c), (e) and system-centric energy-optimal problem (b), (d), (f).[]{data-label="fig: EMA time optimal plots"}](Figs/Results/EMA_Apr_AM_EnergyOpt_Time.png "fig:"){width="1\columnwidth"}
[.49]{} ![EMA network routing results under different penetration rates by solving system-centric time-optimal problem (a), (c), (e) and system-centric energy-optimal problem (b), (d), (f).[]{data-label="fig: EMA time optimal plots"}](Figs/Results/EMA_Apr_AM_TimeOpt_Energy.png "fig:"){width="1\columnwidth"}
[.49]{} ![EMA network routing results under different penetration rates by solving system-centric time-optimal problem (a), (c), (e) and system-centric energy-optimal problem (b), (d), (f).[]{data-label="fig: EMA time optimal plots"}](Figs/Results/EMA_Apr_AM_EnergyOpt_Energy.png "fig:"){width="1\columnwidth"}
[.49]{} ![EMA network routing results under different penetration rates by solving system-centric time-optimal problem (a), (c), (e) and system-centric energy-optimal problem (b), (d), (f).[]{data-label="fig: EMA time optimal plots"}](Figs/Results/EMA_Apr_AM_TimeOpt_Time_Improvement.png "fig:"){width="1\columnwidth"}
[.49]{} ![EMA network routing results under different penetration rates by solving system-centric time-optimal problem (a), (c), (e) and system-centric energy-optimal problem (b), (d), (f).[]{data-label="fig: EMA time optimal plots"}](Figs/Results/EMA_Apr_AM_EnergyOpt_Energy_Improvement.png "fig:"){width="1\columnwidth"}
[0.24]{} ![ (a) Time saving improvement of different O-D pairs for 50% and 100% CAV time optimal routing (EMA network) ; (b) Travel time improvement of different O-D pairs as a function of CAV penetration rate (EMA network)[]{data-label="fig:EMA Interstate improvements"}](Figs/Results/EMA_Apr_AM_OD_improvement_TimeOpt.png "fig:"){width="1\columnwidth"}
[0.24]{} ![ (a) Time saving improvement of different O-D pairs for 50% and 100% CAV time optimal routing (EMA network) ; (b) Travel time improvement of different O-D pairs as a function of CAV penetration rate (EMA network)[]{data-label="fig:EMA Interstate improvements"}](Figs/OD_vs_penetration_Time.png "fig:"){width="1\columnwidth"}
Conclusions and Future Work {#sec: conclusions}
===========================
In this paper we proposed system-centric optimal routing algorithms for a fleet of CAVs in the presence of mixed traffic. We consider two objectives for routing: (1) minimizing travel time (2) minimizing energy consumption cost. Moreover, in order to model the routing behavior of regular vehicles, we assume that they make selfish decisions by minimizing their own travel time. Then, by iteratively solving the TAP, and finding optimal routes for CAVs we estimate the non-CAV flow in the network. Historical traffic data and a simple illustrative example were used to validate the models. The results indicate that optimal routing of CAVs can not only benefit CAVs, but the smart routing decision of CAVs helps ease traffic congestion in the network which helps regular vehicles as well. Additionally, we empirically showed that even a small CAV penetration rate has significant impact on the overall traveling cost of the network.
So far, we assumed that all CAVs belong to the same fleet and we can route all of them. In ongoing work, we are considering multiple fleets of CAVs, in which each fleet is trying to minimize its own cost. Moreover, as pointed out earlier, eco-routing of CAVs is highly dependent on the energy model. We are actively looking for other energy models which can predict the cost for different vehicle types (e.g., EVs, PHEVS) with reasonable levels of accuracy.
CD/CS Energy Model {#sec: CD/CS energy model}
------------------
Karabasoglu et al. [@karabasoglu_influence_2013] proposed an approximate energy model for different classes of vehicles including EVs, HEVs, PHEVs, and CVs. They considered two operational modes for vehicles: charge depleting (CD), and charge sustaining (CS). The CD mode refers to the case in which the main propulsion energy for driving the car comes from the battery pack (electricity). In addition, the CS mode occurs when the vehicle uses the internal combustion engine (gas) to drive the vehicle. PHEVs use both CD and CS modes since they have both engine and electric motor. HEVs only use the CS mode, and EVs only operate on the CD mode. Even though conventional vehicles do not have a battery pack and the CS mode does not apply to them, for the sake of consistency, we use the CS mode for conventional vehicles to refer to their gas operational mode. Karabasoglu et al. calculated the average $mi/gal$ and $mi/kWh$ that a vehicle can travel through different standard drive cycles (NYC, UDDS, HWFET, etc.) under CD or CS operational modes. They made these calculations using *PSAT*, a commercially available software package simulating different power-train architectures using high-fidelity models. Calculated average energy consumption values under CD and CS modes are referred to as $\mu_{CD}$ and $\mu_{CS}$ respectively and are reported in Table \[tab: conversion factors\]. Qiao et al. [@qiao_vehicle_2016] and Houshmand et al. [@houshmand_eco-routing_2018] solved the eco-routing problem for PHEVs by benefiting from the values calculated in [@karabasoglu_influence_2013] (Table \[tab: conversion factors\]).
Eco-routing Problem Formulation for PHEVs {#sec: eco-route PHEV}
-----------------------------------------
In order to solve the eco-routing problem for CAVs, we follow the same formulation used in [@houshmand_eco-routing_2018]. The essence of this eco-routing model is that it categorizes each link based on its average speed into 3 different modes: heavy traffic, medium traffic, and low traffic links (note that we can have more than 3 modes as well). We then assign a drive cycle to each link based on its average speed (Table \[tab: Drive cycle assignment\]).
We consider two sets of decision variables: the CAV route-probability matrix, $\textbf{P}_c=\{p_{ir}^c, i\in [\![\mathcal{W}]\!], r\in \mathcal{R}\}$ and the CD/CS switching strategy on each link, $\textbf{Y}=\{y_{a,r}^{i}, i\in [\![\mathcal{W}]\!], r\in \mathcal{R}, a\in \mathcal{A} \}$. Here, $y_{a,r}^{i}$ represents the fraction of link length ($l_a$) over which we use the CD mode. Thus, if we only use the CD mode over link $a\in \mathcal{A}$, then $y_{a,r}^{i}=1$. Note that in order to solve this problem for EVs, we should set $\textbf{Y}=1$; and for HEVs and CVs we let $\textbf{Y}=0$. Consequently we remove the dependency of the problem on $\textbf{Y}$ in these cases.
This problem shares similarities with , with the only difference that here we are minimizing energy instead of time. Using the same notation introduced in sections \[sec: problem formulation 100 CAV time\], and \[sec: mixed traffic time\], for a given CAV penetration rate $\gamma$, we can formulate the problem as follows:
\[eqn: mixed CAV energy PHEV\] $$\begin{gathered}
\label{eqn: mixed CAV system-centric energy}
\begin{split}
\min_{\textbf{P}_{c},\textbf{Y}} & \sum_{i\in [\![\mathcal{W}]\!]}\sum_{r\in \mathcal{R}_{i}}\sum_{a\in \mathcal{A}}(c_{gas}\frac{l_{a}}{\mu_{cs}^{a}(v_a(x_a))}(1-y_{a,r}^{i}) \\
& \hspace{3cm}+ c_{ele}\frac{l_{a}}{\mu_{CD}^{a}(v_a(x_a))}y_{a,r}^{i})x_{a}^{c}
\end{split}
\\
\label{eqn: battery constraint}
\begin{array}
[c]{lccc}s.t. & \sum_{a\in \mathcal{A}}\frac{\alpha_{a,r}^{i}y_{a,r}^{i}l_{a}}{\mu_{cd}^{a}}\leq E_{0}^{r,i}
& ;\forall r\in \mathcal{R}_{i}, & \forall i\in [\![\mathcal{W}]\!]
\end{array}
\\
\label{eqn: total flow mixed energy}
\textbf{x}=\textbf{x}^{c} +\textbf{x}^{nc}
\\
\textbf{x}^{c}=\textbf{A}\textbf{P}_{c}^{T}\textbf{g}^{c}
\\
\label{eqn: average speed energy CD/CS}
v_{a}(x_{a})=\frac{l_{a}}{t_{a}(x_{a})}
\\
\begin{array}
[c]{lrr}
\textbf{P}_{c}=[p_{ir}^{c}] &, p_{ir}^{c}\in [0,1];&\forall i\in [\![\mathcal{W}]\!],\forall r\in \mathcal{R}_{i}
\end{array}
\\
\label{eqn: probability const. Social energy}
\begin{array}
[c]{lr}
\sum_{r\in \mathcal{R}_{i}}p_{ir}^{c}=1;& \forall i\in [\![\mathcal{W}]\!]
\end{array}
\\
\label{eqn: Y constraint mixed energy}
\begin{array}
[c]{lr}
y_{a,r}^{i}\in [0,1];&\forall a\in \mathcal{A}, \forall i\in [\![\mathcal{W}]\!],\forall r\in \mathcal{R}_{i}
\end{array}\end{gathered}$$
where $c_{gas}$ and $c_{ele}$ are the cost of gas (\$/gal), and electricity (\$/kWh) respectively. Constraint , is the energy constraint stating that the total electrical energy used on each path and O-D pair should not be more than the available energy in the battery pack at the start of that path ($E_{0}^{r,i}$). $mu_{CS}$ and $mu_{CD}$ are functions of velocity (Table \[tab: conversion factors\], and \[tab: Drive cycle assignment\]), and velocity is a function of flow . As mentioned in Sec. \[sec: mixed traffic time\], we assume that the non-CAV traffic flow equilibrium is inferred from data, and is known ($\textbf{x}_{nc}$). By solving , when a vehicle enters the network at an origin *O* given its destination *D*, the algorithm gives it the desired socially optimal route to follow in terms of a sequence of links, and the optimal CD/CS switching strategy on each link.
This model finds the eco-route for PHEVs, as well as HEVs, and EVs by setting $Y=0$ or $1$ respectively. We should note that the energy model used in is a piece-wise constant function of velocity, and can cause non-convexity and differentiability issues in the problem solution.
[^1]: \*This work was supported in part by NSF under grants ECCS-1509084, DMS-1664644, CNS-1645681, by AFOSR under grant FA9550-15-1-0471, by ARPA-E’s NEXTCAR program under grant DE-AR0000796 and by the MathWorks.
[^2]: $^{1}$The authors are with the Division of Systems Engineering, Boston University, Brookline, MA 02446 USA `arianh@bu.edu; salomonw@bu.edu; cgc@bu.edu`
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The spectral properties of a multilevel atomic system interacting with multiple electromagnetic fields, a modified inverted-Y system, have been theoretically investigated. In this study, a numerical matrix propagation method has been employed to study the spectral characteristics beyond the validity regime of the rotating wave approximations. The studied atomic system, comprising of several basic sub-systems *i.e.* lambda, ladder, vee, N and inverted-Y, is useful to study the interdependence among these basic sub-systems. The key features of the obtained probe spectra as a function of coupling strength and detuning of the associated electromagnetic fields show inter-conversion, splitting and shifting of the transparency and absorption peaks. The dressed and doubly dressed state formalism have been utilized to explain the numerically obtained results. This study has application in design of novel optical devices capable of multi-channel optical communication along with switching.'
author:
- Charu Mishra
- 'A. Chakraborty'
- 'S. R. Mishra'
title: 'Spectral characteristics of a modified inverted-Y system beyond rotating wave approximation'
---
Introduction
============
Studies on atomic systems interacting with electromagnetic fields has a long and prosperous history. However, in the recent past, multi-level atomic systems have been investigated rigorously either to gain an insight into the fundamental features of the atom-field interactions or to implement the physical behaviors in optical devices. A plethora of cutting edge technologies has already been attributed to the optical phenomena discovered due to the advancement of spectroscopy techniques. The three level atomic systems which facilitated the first observation of quantum interference effects in atomic systems are one of the most basic building blocks for the optical switching devices using Electromagnetically induced transparency (EIT) [@Harris:1990; @Harris:1st] and electromagnetically induced absorption (EIA) [@akulshin:1998]. Apart from these three level systems, namely $\Lambda$ [@Li:4:1995; @Li:6:1995; @Charu:65:2018], ladder [@MOSELEY:1995; @Anil:2009; @Dinh:2016] and vee [@Zhao:2002; @Kang:2014] atomic systems, there are atomic systems consisting higher number of atomic levels which have already been proven to display wonderful physical properties which not only have enriched physics but the application value is enormous too. Two promising four level atomic systems, *i.e.* inverted Y [@Gao:2000; @Yan:64:2001; @Qi:2009; @Dong:2012; @Sabir:2016; @Kavita:2017] and N configuration [@Goren:2004; @Kong:2007; @Abi:2011; @Somia:2013; @Phillips:2013; @Kang:2015; @Islam:2017; @Tuan:2018], have already been earmarked for their application in non-linear spectroscopy for producing large Kerr non-linearities in an optical medium [@Kou:2010; @Yang:2015]. These third order non-linear effects relies on large field strengths to obtain considerably enhanced third order optical susceptibility. Historically, increasing the number of atomic levels interacting with increased number of externally applied electromagnetic fields increases the diversity of observed physical phenomena but also the complexity to a many fold both theoretically and experimentally.
In this article, a modified inverted-Y system, hereafter denoted as $IY^+$ system, is considered which comprises of all the three basic three-level sub-systems ($\Lambda$, ladder and Vee) and two basic four-level sub-systems (inverted-Y and N). This system is investigated with variation in external field parameters like field strength and detuning. The aim of this study is to investigate this atomic system for cold atoms to control the probe absorption in a desired manner for device applications. Here, we have used a numerical matrix propagation method in which a complete density matrix has been propagated through time in order to obtain the transient characteristics as well as the steady state condition for a probe field absorption. The rotating wave approximations (RWA) are routinely employed in the spectroscopy for finding pertinent informations regarding the interaction of atomic systems with electromagnetic fields in steady-state. Though extremely successful, RWA suffers severely when the atom-field interactions are far from the resonance condition and the applied electromagnetic fields are strong enough to be treated perturbatively. The numerical approach utilized in this article, to solve the Liouville equation is described in detail in [@masayoshi:1994; @masayoshi:1995] and a complex algebraic form of this method is numerically implemented in this article. The obtained results employing this approach, have also been explained using dressed and doubly dressed state formalism.
The article is organized as follows. In section \[sec:NMP\], the numerical matrix propagation (NMP) technique is discussed. In section \[sec:transient\], the study of transient characteristics of inverted-Y system and equivalence of NMP method with the conventional RWA method is established. In section \[sec:result\], the probe absorption in $IY^+$ system is explored in steady state regime and the obtained results are discussed. The conclusion of work is presented in section \[sec:conc\].
Numerical Matrix Propagation {#sec:NMP}
============================
An in depth description of the numerical solution technique employed in this article has been provided in [@masayoshi:1994; @masayoshi:1995]. A brief outline of the theoretical background and numerical implementation is provided here for completeness. The generalized electromagnetic radiation composed of M individual field components can be written as, $$\mathcal{E}(t)=\sum_{p}^{M}\varepsilon_p(\omega_p)\cos(\omega_pt),$$ where $\varepsilon_p$ and $\omega_p$ represent the field strength and angular frequency for $p^{th}$ field component. The Hamiltonian of the composite atom-field system can be written as, $$H(t)=H_0+H_{I}(t),$$ where $H_0$ is the atomic Hamiltonian described in the atomic basis states $\{|\alpha\rangle\}$ as, $$H_0|\alpha\rangle=\epsilon_i|\alpha\rangle.$$ The electromagnetic interaction Hamiltonian $H_I$ can not be written exactly in the atomic basis states due to the infinite sequence of multipole interactions that can originate from the interaction of the atom with the electromagnetic fields. Under the long wavelength approximation (*i.e.* dipole approximation), the interaction can be written as, $$H_I(t)=-\mu.\mathcal{E}(t).$$ The evolution of the density matrix ($\rho=\sum_{\alpha}|\alpha\rangle\langle\alpha|$) for all the atomic levels can be described by the Liouville equation as, $$\label{eq:liouville}
\frac{\partial}{\partial t}\rho(t)=-\frac{i}{\hbar}\left[H(t),\rho(t)\right]-\frac{1}{\hbar}[R,\rho(t)].$$ The first part in right-hand side of the Liouville equation describes the unitary evolution whereas the second part describes the decay of the composite atom-field system. The second part can be written explicitly in terms of the individual decay rates between different states as, $$\left[R,\rho(t)\right]_{\alpha\alpha}=-\Gamma_{\alpha\alpha}+\sum_{\beta\neq\alpha}\gamma_{\beta\alpha}\rho_{\beta\beta}$$ and, $$\left[R,\rho(t)\right]_{\alpha\beta}=-\Gamma_{\alpha\beta}\ (\alpha\neq\beta)$$ The decay rates can also be written as, $$\Gamma_{\alpha\beta}=\frac{1}{2}\left(\Gamma_{\alpha\alpha}+\Gamma_{\beta\beta}\right)+\Gamma'_{\alpha\beta}$$ with the property $$\Gamma_{\alpha\beta}=\Gamma_{\beta\alpha}$$ and $$\Gamma_{\alpha\alpha}=\sum_{\beta\neq\alpha}^{N}\gamma_{\alpha\beta},$$ where $\gamma_{\alpha\beta}$ are the feeding parameters and $\Gamma'_{\alpha\beta}$ is dephasing factor due to phase changing collisions which is considered zero in this study. To solve Liouville equation \[eq:liouville\], density matrix elements are divided into the diagonal and off-diagonal elements denoted by complex variables $\rho_{ii}$ and $\rho_{ij}$.
The evolution of population of state $i$ can be obtained by solving following equation,
$$\begin{split}
\Re(\dot{\rho_{ii}})&=\frac{1}{\hbar}\sum_{p}^{M}\sum_{k}^{N}(\Im(\rho_{ik})\mu_{ki}-\mu_{ik}\Im(\rho_{ki}))\varepsilon_p\cos(\omega_pt)\\&+\sum_{k\neq i}^{N}\gamma_{ki}\Im(\rho_{kk})-\Gamma_{ii}\Re(\rho_{ii}).
\end{split}$$
and the equations from which time dependent off-diagonal density matrix elements can be evaluated, are written as,
$$\begin{split}
\Re(\dot{\rho_{ij}})&=\frac{\epsilon_{ij}}{\hbar}\Im(\rho_{ij})\\&+\frac{1}{\hbar}\sum_{p}^{M}\sum_{k}^{N}(\Im(\rho_{ik})\mu_{kj}-\mu_{ik}\Im(\rho_{kj}))\varepsilon_p\cos(\omega_pt)\\&+\Gamma_{ij}\Re(\rho_{ij})
\end{split}$$
and $$\begin{split}
\Im(\dot{\rho_{ij}})&=\frac{\epsilon_{ij}}{\hbar}\Re(\rho_{ij})\\&+\frac{1}{\hbar}\sum_{p}^{M}\sum_{k}^{N}(\mu_{ik}\Re(\rho_{kj})-\Re(\rho_{ik})\mu_{kj})\varepsilon_p\cos(\omega_pt)\\&+\Gamma_{ij}\Re(\rho_{ij}).
\end{split}$$ The resulting time series of the propagated matrix $\rho$ can be utilized to obtain the populations of individual states $\Re(\rho_{ii})$ or the coherence of individual transitions $\Im(\rho_{ij})$ which gives absorption between states $i$ and $j$.
Transient characteristics and Comparison with RWA {#sec:transient}
=================================================
![\[figure:level\_diagram\] The level diagrams of the (a) inverted-Y (IY) atomic system and (b) modified inverted-Y system (described as $IY^+$ system throughout the article). The atomic energy levels are denoted as $\alpha$ with $\alpha \in 0,1,2,3,4$. The coupling strength of individual field connecting states $i$ and $j$ are denoted as $\Omega_{ij}$ and the detuning from corresponding transitions are denoted as $\Delta_{ij}$. ](level_diagram_combined.pdf){width="8.5"}
In order to study the numerical stability of the utilized algorithm, we characterize the transient response of an inverted-Y (IY) system consisting four levels $|\alpha\rangle,\ \alpha\in \{0,1,2,3\}$ along with three electromagnetic fields with coupling strengths $\Omega_{ij}$ connecting states $i$ and $j$ with $i,j \in 0,1,2,3$, as shown in figure \[figure:level\_diagram\](a). The population of the ground state $|0\rangle$ and the coherence of the $|0\rangle\longleftrightarrow|2\rangle$ transition of the inverted-Y system as a function of scaled time $\gamma t$ is shown in figure \[figure:transient\] (a) and (b) respectively. As expected, after initial rapid variations, the steady state is achieved at $\gamma t=\ 4\pi$ and without any external perturbation the atomic system continues to be in the steady state.
In order to establish the equivalence of the numerical matrix propagation (NMP) method with the RWA method in the low field strength and small detuning regime, the evolution of the density matrix equations has been studied using both the formalisms in the atomic system described above. The density matrix equations of an inverted-Y system (figure. \[figure:level\_diagram\](a)) can be explicitly written in the rotating wave approximations as,
$$\begin{aligned}
\begin{aligned}
\frac{\partial\rho_{00}}{\partial t}&=2\gamma_2\rho_{22}-2\gamma_0\rho_{00}+i\Omega_{02}(\rho_{02}-\rho_{20})
\nonumber\\
\frac{\partial\rho_{11}}{\partial t}&=2\gamma_1\rho_{22}+2\gamma_0\rho_{00}+i\Omega_{12}(\rho_{12}-\rho_{21})\nonumber\\
\frac{\partial\rho_{22}}{\partial t}&=-2(\gamma_1+\gamma_2)\rho_{22}+2\gamma_3\rho_{33}-i\Omega_{12}(\rho_{12}-\rho_{21})\\&-i\Omega_{02}(\rho_{02}-\rho_{20})+i\Omega_{23}(\rho_{23}-\rho_{32})\nonumber\\
\frac{\partial\rho_{33}}{\partial t}&=-2\gamma_3\rho_{33}-i\Omega_{23}(\rho_{23}-\rho_{32})\nonumber\\
\frac{\partial\rho_{01}}{\partial t}&=-(\gamma_0+i(\Delta_{12}-\Delta_{02}))\rho_{01}+i\Omega_1\rho_{02}-i\Omega_{02}\rho_{21}\nonumber\\
\frac{\partial\rho_{02}}{\partial t}&=-(\gamma_1+\gamma_2-i\Delta_{02})\rho_{02}+i\Omega_{12}\rho_{01}\\&+i\Omega_{23}\rho_{03}+i\Omega_{02}(\rho_{00}-\rho_{22})\nonumber\\
\end{aligned}\end{aligned}$$
$$\begin{aligned}
\begin{aligned}
\frac{\partial\rho_{03}}{\partial t}&=-(\gamma_0+\gamma_3-i(\Delta_{02}+\Delta_{23}))\rho_{03}+i\Omega_{23}\rho_{02}\\&-i\Omega_{02}\rho_{23}\nonumber\\
\frac{\partial\rho_{12}}{\partial t}&=-(\gamma_1+\gamma_2-i\Delta_{12})\rho_{12}+i\Omega_{02}\rho_{10}\\&+i\Omega_{23}\rho_{13}+i\Omega_{12}(\rho_{11}-\rho_{22})\nonumber\\
\frac{\partial\rho_{13}}{\partial t}&=-(\gamma_3-i(\Delta_{12}+\Delta_{23}))\rho_{13}+i\Omega_{23}\rho_{12}-i\Omega_{12}\rho_{23}\nonumber\\
\frac{\partial\rho_{23}}{\partial t}&=-(\gamma_1+\gamma_2+\gamma_3-i\Delta_{23})\rho_{23}-i\Omega_{02}\rho_{03}-\\&i\Omega_{12}\rho_{13}+i\Omega_{23}(\rho_{22}-\rho_{33})\nonumber\\
\end{aligned}\end{aligned}$$
and $$\begin{aligned}
\begin{aligned}
\frac{\partial\rho_{ij}}{\partial t}&=\frac{\partial\rho_{ji}^*}{\partial t},
\end{aligned}\end{aligned}$$
![\[figure:transient\] (Color online) Amalgamated spectrum of (a) population of the ground state $\Re(\rho_{00})$ and coherence of the probe transition $\Im(\rho_{02})$ as a function of the dimensionless time $\gamma t$. The observed transient characteristics show the atomic system achieving the steady state at $\gamma t\sim\ 4\pi$ with $\Omega_{02}/\gamma=1.0$, $\Omega_{12}/\gamma=5.0$, $\Omega_{23}/\gamma=5.0$, $\Delta_{12}/\gamma=-20$, $\Delta_{23}/\gamma=0$.](transient.pdf){width="8.5"}
where $\Delta_{ij}$ is detuning of the applied electromagnetic fields from the corresponding atomic transitions between states $i$ and $j$. $\gamma_1$, $\gamma_2$ and $\gamma_3$ are the decay rates of the excited states $|1\rangle$, $|2\rangle$ and $|3\rangle$ respectively. The total population is conserved by $\sum_{i}\rho_{ii}=1$. The non-radiative relaxation rate of the ground state, i.e. $\gamma_0$, is zero for this closed system. The steady-state of the above set of equation can be obtained using the matrix method described as, $$\dot{\rho}=\mathcal{L}\rho=0.$$ The eigenvector corresponding to the ‘zero’ eigenvalue of the Liouvillian super-operator ($\mathcal{L}$) correspond to the steady-state value of the density matrix $\rho$.
![\[figure:rwa\_comparison\] The probe absorption $\Im(\rho_{02})$ as a function of scaled detuning $\Delta_{02}/\gamma$ with (a) $\Omega_{02}/\gamma=0.01$, $\Omega_{12}/\gamma=0.5$, $\Omega_{23}/\gamma=1.5$, $\Delta_{12}/\gamma=-2$, $\Delta_{23}/\gamma=2$ and (b) $\Omega_{02}/\gamma=1.0$, $\Omega_{12}/\gamma=5.0$, $\Omega_{23}/\gamma=5.0$, $\Delta_{12}/\gamma=-20$, $\Delta_{23}/\gamma=0$. ](invY_comp_rwa.pdf){width="8.5"}
In order to perform the comparison, the steady state value of the density matrix is obtained from both the method for two different coupling strength and detuning regime and the results are shown in figure \[figure:rwa\_comparison\]. Figure \[figure:rwa\_comparison\] (a) correspond to the low coupling strength and low detuning regime with Rabi frequencies $\Omega_{02}/\gamma=0.01$, $\Omega_{12}/\gamma=0.5$ and $\Omega_{23}/\gamma=1.5$ with detuning values $\Delta_{12}/\gamma=-2$ and $\Delta_{23}/\gamma=2$. The additional parameters utilized in the numerical matrix propagation method is $\gamma \delta t=10^{-5}$ propagated upto $\gamma t_{max}=2\pi\times 15$. The equivalence of the RWA (dashed line) and NMP (solid line) methods in low field strength is clearly visible in the graph. The slight shift in the peak of the $\Im(\rho_{02})$ value can be attributed to the counter rotating terms neglected in the RWA method.
However, in the high field strength and large detuning regime, though the spectral features remains same, the difference between the outcomes of both the methods grow further. In this regime, the field strengths are $\Omega_{02}/\gamma=1.0$, $\Omega_{12}/\gamma=5.0$ and $\Omega_{23}/\gamma=5.0$ with detuning values $\Delta_{12}/\gamma=-20$ and $\Delta_{23}/\gamma=0$. The time sequence parameters of the NMP method were kept same.
Results and discussions {#sec:result}
=======================
In the previous section, the stability of the numerical implementation has been presented along with the comparison with the rotating wave approximations. The established validity of the numerical matrix propagation method enables one to employ the same in field strength and frequency regimes commonly not considered in case of RWA.
In this section, a modified inverted-Y system (figure. \[figure:level\_diagram\] (b)) has been studied after incorporating an additional coupling field connecting the ground state to another excited state. This particular choice of an atomic system serves the purpose of integrating all the primitive three level atomic systems namely $\Lambda$, ladder and vee along with an option to study the effect of the merger of two of the basic four level atomic system, *i.e.* the inverted-Y system and N system. The level diagram of the proposed atomic system is shown in figure. \[figure:level\_diagram\] (b) where the unperturbed atomic states are denoted as $|\alpha\rangle$, $\alpha \in \{0,1,2,3,4\}$. The applied electromagnetic fields are described by coupling strength $\Omega_{ij}$, and detuning $\Delta_{ij}$, where subscript $ij$ represents the state $i$ and $j$ through which electromagnetic field couples. The atomic system consist a $\Lambda$ sub-system $|0\rangle\longleftrightarrow|2\rangle\longleftrightarrow|1\rangle$, a ladder sub-system $|0\rangle\longleftrightarrow|2\rangle\longleftrightarrow|3\rangle$ and a vee system $|2\rangle\longleftrightarrow|0\rangle\longleftrightarrow|4\rangle$. The inverted-Y sub-system is formed by the transition paths $|0\rangle\longleftrightarrow|2\rangle\longleftrightarrow|3\rangle$ and $|1\rangle\longleftrightarrow|2\rangle\longleftrightarrow|3\rangle$. Finally the N sub-system is formed via the transitions $|1\rangle\longleftrightarrow|2\rangle\longleftrightarrow|0\rangle\longleftrightarrow|4\rangle$. Assuming the field $\Omega_{02}$ and detuning $\Delta_{02}$ for the transition $|0\rangle\longleftrightarrow|2\rangle$ as a probe field, response of all the above atomic sub-systems can be simultaneously probed. In the proceeding section, the coherence of the density matrix element (*i.e.* $\Im(\rho_{02})$) is analyzed to identify the key spectral features of the $IY^+$ system. If not explicitly described, the field values are the following: $\Omega_{02}/\gamma=1.0$, $\Omega_{12}/\gamma=5.0$, $\Omega_{23}/\gamma=5.0$ and $\Omega_{04}/\gamma=5.0$. The probe beam detuning $\Delta_{02}/\gamma$ has been varied in a wide range $-100\longleftrightarrow 100$ and the relevant part of the spectral response are depicted in the various plots.
![\[figure:d12\_variation\] (Color online) Amalgamated spectrum (*i.e.* probe absorption versus probe detuning $\Delta_{02}/\gamma$) as a function of the scaled detuning $\Delta_{12}/\gamma$ for (a) inverted-Y system with $\Omega_{02}/\gamma=1.0$, $\Omega_{12}/\gamma=5.0$, $\Omega_{23}/\gamma=5.0$, $\Delta_{23}/\gamma=0$ and (b) $IY^+$ system with parameters $\Omega_{02}/\gamma=1.0$, $\Omega_{12}/\gamma=5.0$, $\Omega_{23}/\gamma=5.0$, $\Omega_{04}/\gamma=5.0$, $\Delta_{23}/\gamma=0$, $\Delta_{04}/\gamma=0$.](d12_variation_spectra.pdf){width="8.5"}
To begin with, the detuning of the coupling field $\Delta_{12}/ \gamma$ has been varied from $- 40 $ to $ 40 $ for the inverted-Y system as well as the $IY^+$system and the obtained results are shown in figure \[figure:d12\_variation\] . Figure \[figure:d12\_variation\] (a) and (b) show the amalgamated spectrum for inverted-Y and $IY^+$ system respectively and the individual spectrum corresponding to the white lines drawn in plots (a) and (b) are presented in plots (c) and (d) respectively. For inverted-Y system (figure \[figure:level\_diagram\] (a)), as reported in the earlier studies [@Qi:2010], two sharp transparencies (*i.e* EIT) have been obtained at the specific detuning positions where the two photon resonance condition in the $\Lambda$ system $|0\rangle\longleftrightarrow|2\rangle\longleftrightarrow|1\rangle$ and ladder system $|0\rangle\longleftrightarrow|2\rangle\longleftrightarrow|3\rangle$ are satisfied. For both the resonant coupling fields, two EIT dips merge to give a single sharp EIT at zero probe field detuning. The coupling of ground state $|0\rangle$ to excited state $|4\rangle$ via field of strength $\Omega_{04}$ in the $IY^+$ system (figure \[figure:level\_diagram\] (b)) has shown different spectral feature than that of inverted-Y system. At zero detuning of $\Delta_{12}$, figure \[figure:d12\_variation\] (b)) shows higher $\Im(\rho_{02})$ at resonant probe (encoded by red color) implying the existence of an absorption peak rather than a transparency as obtained for the case of inverted-Y system.
![\[figure:dressed\] The energy level diagrams of inverted-Y (IY) and $IY^+$ systems. (a) and (c) show bare states with resonant coupling fields, whereas, (b) and (d) show their corresponding dressed states.](dressed_new.pdf){width="7.5"}
A qualitative understanding of such change of spectral behavior from inverted-Y to $IY^+$ system can be obtained using a semi-classical dressed state formalism as explained in the following discussion. In this formalism, the bare states coupled with strong field are transformed into a new atom-photon basis state which is known as dressed state. In the case of resonant driving fields, the dressed states can be determined by diagonalizing the interaction Hamiltonian. The interaction Hamiltonian of the inverted-Y system, after RWA and dipole approximation, can be written as,
$$\label{eq:hamiltonian}
H_1=\frac{1}{2} \begin{bmatrix}
0 &0& &\Omega_{02}& &0 \\
0 &0& &\Omega_{12}& &0 \\
\Omega_{02} &\Omega_{12}& &0& &\Omega_{23} \\
0 &0& &\Omega_{23}& &0 \\
\end{bmatrix}.$$
As the strength $\Omega_{02}$ of the probe field is weaker than the strength of all other coupling fields, the probe field can be neglected while evaluating the eigen dressed states corresponding to the Hamiltonian $H_I$ in equation \[eq:hamiltonian\]. The resulting three eigen-frequencies and their corresponding eigen dressed states are\
$\lambda_{0,\pm}=0, \pm\frac{1}{2} \sqrt{\Omega_{12}^2+\Omega_{23}^2}$\
$|0\rangle_D =\frac{\Omega_{23}}{\sqrt{\Omega_{12}^2 + \Omega_{23}^2}} |1\rangle - \frac{\Omega_{12}}{\sqrt{\Omega_{12}^2 + \Omega_{23}^2}} |3\rangle$\
$|+\rangle_1 =\frac{\Omega_{12}}{\sqrt{2 (\Omega_{12}^2 + \Omega_{23}^2)}} |1\rangle + \frac{1}{\sqrt{2}} |2\rangle + \frac{\Omega_{23}}{\sqrt{2 (\Omega_{12}^2 + \Omega_{23}^2)}} |3\rangle$\
$|-\rangle_1 =\frac{\Omega_{12}}{\sqrt{2 (\Omega_{12}^2 + \Omega_{23}^2)}} |1\rangle - \frac{1}{\sqrt{2}} |2\rangle + \frac{\Omega_{23}}{\sqrt{2 (\Omega_{12}^2 + \Omega_{23}^2)}} |3\rangle$.\
The absorption process in this formalism can be understood by determining the transition probabilities between bare state $|0\rangle$ and upper dressed states, which can be expressed as,
$$T_{\alpha \rightarrow \beta} = |\langle \alpha|\vec{d}\cdot \vec{E_0}|\beta \rangle|^2 ,$$
where $\alpha$ is ground state, and $\beta \in \{|-\rangle, |0\rangle_D, |+\rangle\}$.
The transition between $|0\rangle$ and $|0\rangle_D$ corresponds to the transition at line center (*i.e.* $\Delta_{02} = 0$) does not exist due to its zero transition probability (*i.e.* $|\langle 0|\vec{d}\cdot \vec{E_0}|0 \rangle_D|^2 = 0$). The other transitions (*i.e.* between $|0\rangle$ and $|\pm\rangle_1$) exhibit non-zero transition probability. As a consequence, the inverted-Y system exhibits EIT at line center along with the two absorption peaks surrounding the EIT. This is schematically shown in figure \[figure:dressed\] (a) and (b).
In the $IY^+$ system, the ground state $|0\rangle$ also couples with a state $|4\rangle$ via a strong field $\Omega_{04}$ which results in conversion of the bare ground state into dressed state. Thus, the dressed states for $IY^+$ system include three upper dressed states due to bare states $|1\rangle$, $|2\rangle$, $|3\rangle$ as evaluated for the case of inverted-Y system, and two lower dressed states due to coupling of states $|0\rangle$ and $|4\rangle$ through strong field $\Omega_{04}$. The lower dressed states can be evaluated by diagonalizing two-level interaction Hamiltonian $H_{2}$ after RWA and dipole approximation,\
$H_{2}=\frac{1}{2}\Omega_{04} |0\rangle \langle 4| + h.c.,$\
and the eigen-energies and corresponding eigen dressed states are given as,
$\lambda_2= \pm\frac{1}{2} \Omega_{04}$\
$|+\rangle_2 =\frac{1}{\sqrt{2}} |0\rangle + \frac{1}{\sqrt{2}} |4\rangle$.\
$|-\rangle_2 =\frac{1}{\sqrt{2}} |0\rangle - \frac{1}{\sqrt{2}} |4\rangle$.\
As the upper dressed state $|0\rangle_D$ has zero component of bare state $|2 \rangle$, the transition probabilities $T_{|+\rangle_2 \rightarrow |0\rangle_D}$ and $T_{|-\rangle_2 \rightarrow |0 \rangle_D}$ are zero, while the rest of the transitions have non-zero transition probabilities. This is depicted in figure \[figure:dressed\] (d). The calculated location of each transitions and hence the existence of absorption peaks in frequency space are at $\Delta_{02} / \gamma = $ -6.0, -1.0, 1.0, 6.0.
The absorption peaks at $\Delta_{02}/ \gamma =$ -1 and 1 merge with each other and result in a broad single peak. Thus, there are total three peaks observable. The non-zero detuning $\Delta_{12}$ leads to a detuned $\Lambda$ system where the off-resonant two-photon Raman resonance gives rise to an additional absorption peak near zero probe detuning along with the EIT dip at position where two-photon resonance condition of detuned $\Lambda$ system is satisfies [@Hemmer:1989]. Consequently, for non-zero $\Delta_{12}$, four absorption peaks at resonance and a EIT at off resonant probe detuning exist. Also, since the eigenvalues and dressed states are actually function of detuning and strengths of all coupling fields, their variations can result in a change in location of spectral features and their strength as well.
![\[figure:d04\_variation\] (Color online) Amalgamated probe absorption spectrum (*i.e.* probe absorption versus detuning $\Delta_0/\gamma$) as a function of detuning $\Delta_{04}/\gamma$ for (a) $\Delta_{12}/\gamma=-20$ and (b) $\Delta_{12}/\gamma=-30$. The other common parameters are $\Omega_{02}/\gamma=1.0$, $\Omega_{12}/\gamma=5.0$, $\Omega_{23}/\gamma=5.0$, $\Omega_{04}/\gamma=5.0$, $\Delta_{23}/\gamma=0$. The plots (c) and (d) show the probe absorption spectra for values of $\Delta_{04}/\gamma =$ -6.0 and 6.0.](d04_variation.pdf){width="8.5"}
The obtained spectral features in above study can further be tailored by varying the detuning $\Delta_{04}$ of another coupling field. For the rest of the studies, the $\Lambda$ system was made far detuned so that its effect can be separated while studying the effect of other systems. The spectra of the probe absorption as a function of detuning $\Delta_{04}$ for $\Delta_{12}/ \gamma =-20$ and $-30$ are shown in left and right column of figure \[figure:d04\_variation\] respectively. Since the electromagnetic field with strength $\Omega_{04}$ directly couples the ground state $|0\rangle$ and an excited state $|4\rangle$, its detuning may affect the coherence between the two ground states $|0\rangle$ and $|1\rangle$. This results in shift in the EIT feature of far-detuned $\Lambda$ system as shown in the figure \[figure:d04\_variation\] (as indicated by an arrow in figure \[figure:d04\_variation\] (c)). Similarly, the resonant spectral features in this figure also show the dependence on the field detuning $\Delta_{04}$. It can be noted here that, in contrast to resonant $\Delta_{04}$ case, where four absorption peaks were observed near the probe resonance frequency, the resonance features get modified for non-zero $\Delta_{04}$ and only show three absorption peaks with sharp transparencies between the peaks. The effect of detuned $\Lambda$ system can also be noted by comparing figures \[figure:d04\_variation\] (a) and (b) or (c) and (d). The detuned $\Lambda$ system affected the spectrum in terms of strength only. This study summarizes that by varying $\Delta_{04}$, the absorption features of probe can be considerably tailored and transparency can be obtained.
In order to investigate the dependence of the detuned EIT and the resonant spectral features further on other parameters of the externally applied electromagnetic fields, the detuning of another coupling field $\Delta_{23}$ is varied in three different conditions and the obtained results are shown in figure \[figure:d23\_variation\]. The three different conditions chosen are $\Delta_{04}/ \gamma=-6, 0$ and $6$, as contrasting spectral features have been obtained for these detuning values in previous study. In all the configurations of the detuning $\Delta_{23}$, the detuned EIT shows negligible change and hence is not shown in figure \[figure:d23\_variation\], whereas the resonant spectra is modified in all the three conditions. This negligible change of EIT may be because of absence of direct decay channel from excited state $|3\rangle$ to ground state $|0\rangle$ resulting in no change in the coherence between ground states $|0\rangle$ and $|1\rangle$.
![\[figure:d23\_variation\] (Color online) Amalgamated spectrum ( *i.e.* probe absorption versus the probe detuning $\Delta_{02}/\gamma$) as a function of the scaled detuning $\Delta_{23}/\gamma$ for (a) $\Delta_{04}/\gamma=-6$, (b) $\Delta_{04}/\gamma=0$ and (c) $\Delta_{04}/\gamma=6$ with other parameters $\Omega_{02}/\gamma=1.0$, $\Omega_{12}/\gamma=5.0$, $\Omega_{23}/\gamma=5.0$, $\Omega_{04}/\gamma=5.0$ and $\Delta_{12}/\gamma=-20$.](d23_variation.pdf){width="8.5"}
Figure \[figure:d23\_variation\] (a), (b) and (c) correspond to values of $\Delta_{04}/ \gamma=-6, 0$ and $6$. Previously it has been observed that non-zero $\Delta_{04}$ gives rise to three absorption peaks. Among these three absorption peaks, two of the peaks show variation in its strength with the variation in $\Delta_{23}$, while the third peak is independent of $\Delta_{23}$. The position of these peaks depend on positive or negative values of $\Delta_{04}$. This is clearly visible from figure \[figure:d23\_variation\] (a) and the position of these peaks are inter changed in figure \[figure:d23\_variation\] (c). Along with this, for the case of far detuned $\Delta_{04}$ values *i.e.* $\Delta_{04}/ \gamma= \pm 6$ (figure \[figure:d23\_variation\] (a) and (c)), a single transparency exist at the same probe field frequency for all the values of detuning $\Delta_{23}$. For the resonant case *i.e.* $\Delta_{04}\ / \gamma=\Delta_{23}\ /\gamma=0$ (figure \[figure:d23\_variation\] (b)), there exist four absorption peaks in which the central peak shows a shift in its position with the detuning $\Delta_{23}$. Another central peak shows the change in its strength as well as position with the variation in detuning $\Delta_{23}$ (figure \[figure:d23\_variation\] (b)). In addition to this, a sharp transparency window begins to appear as the detuning $\Delta_{23}$ value is increased in either positive or negative side. Thus, the strength of the absorption peaks can be tuned by applying appropriate detuning $\Delta_{23}$ and to attain a large transparency, $\Delta_{04}$ should be kept non-zero. These spectral characteristics of the system $IY^+$ can be useful for developing optical switching devices.
![\[figure:O12\_variation\] (Color online) Amalgamated spectrum (*i.e.* the probe absorption versus probe detuning $\Delta_{02}/\gamma$) for different values of coupling strength $\Omega_{12}/\gamma$. The other parameters are $\Omega_{02}/\gamma=1.0$, $\Omega_{23}/\gamma=5.0$, $\Omega_{04}/\gamma=5.0$, $\Delta_{12}/\gamma=-20$, $\Delta_{23}/\gamma=\Delta_{04}/\gamma=0$. The left and right panels correspond to the negative detuning and near resonant case of probe field respectively. Plots (c) and (d) show individual spectra corresponding to the specific field strengths marked by the white line in plots (a) and (b).](O12_variation.pdf){width="8.5"}
Subsequent to the studies on effect of detuning, the spectral features of the probe field have also been studied by varying the strength of all the coupling fields. The variation in probe absorption with the variation in coupling strength $\Omega_{12}$ is shown in figure \[figure:O12\_variation\]. For lower field strength $\Omega_{12}$, the EIT feature (at $\Delta_{02}/\gamma=-20$) corresponding to the detuned $\Lambda$ system ($\Delta_{12}/\gamma=-20$) is not observable. It begins to appear as the field strength $\Omega_{12}$ is increased. The resonant spectral feature also shows dependence on the coupling field strength $\Omega_{12}$. With increase in $\Omega_{12}$ value, the small central absorption peak (shown by an arrow in figure) that appear due to the detuned $\Lambda$ system, merges with the other peaks as shown in figure 8 (b) and (d).
![\[figure:O1\_dressed\]Pictorial representation of doubly dressed approach. (a) Five level $IY^+$-system. (b) The formation of primary dressed states due to strong coupling of $\Omega_{12}$ with states $|1\rangle$ and $|2\rangle$. (c) The picture representing all possible dressed states with the allowed transitions between them.](Omega_dressed.pdf){width="8.5"}
These obtained results from the NMP method (as discussed above) can be explained using the doubly dressed approach [@Yan:2001], for the case of $\Omega_{ij} > \Omega_{lm}$ where $ij,lm \in \{12,23\}$. In this approach, initially, the bare atomic states coupled with stronger coupling field form the primary dressed states. One of these primary dressed states again gets dressed due to its coupling with another bare state. We considered a particular case for $\Omega_{12}/ \gamma =9, \Omega_{23}/ \gamma=\Omega_{04}/ \gamma = 5, \Omega_{02}/ \gamma = 1, \Delta_{12}/ \gamma=-20, \Delta_{23}/ \gamma=\Delta_{04}/ \gamma = 0$ (figure \[figure:O12\_variation\] (d)). The states $|1\rangle$ and $|2\rangle$ coupled with the strong field of strength $\Omega_{12}$ form primary dressed states $|-\rangle$ and $|+\rangle$ with energies $\frac{1}{2} (\Delta_{12} - \Omega_{12}^{'})$ and $\frac{1}{2} (\Delta_{12} + \Omega_{12}^{'})$ respectively, where $\Omega_{12}^{'}=\sqrt{\Delta_{12}^2 + \Omega_{12}^2}$. The dressed state $|+\rangle$ is again coupled with the bare state $|3\rangle$ through the field $\Omega_{23}$ which creates doubly dressed states $|+, -\rangle$ and $|+, +\rangle$ with energy $\frac{1}{2} ( \Delta^{'} -\sqrt{\Delta^{'^2} + \Omega_{23}^2})$ and $\frac{1}{2} ( \Delta^{'} +\sqrt{\Delta^{'^2} + \Omega_{23}^2})$ respectively, where $\Delta^{'}=\frac{1}{2}(\Delta_{12} + \Omega_{12}^{'})$. The formation of these dressed states are depicted in figure \[figure:O1\_dressed\]. The transition of lower dressed states with three upper dressed states via probe field $\Omega_{02}$ give rise to six possible transitions. The calculated location of all these possible transitions in frequency domain are $\Delta_{02}/ \gamma =$ -23.4, -18.4, -3.7, 1.3, 1.4 and 6.4. These calculated locations of transition peaks are in agreement with the obtained results through NMP method as shown in figure \[figure:O12\_variation\] (d).
![\[figure:O04\_variation\] (Color online) Amalgamated spectrum (*i.e.* the probe absorption as a function of the probe detuning $\Delta_{02}/\gamma$) for the different values of scaled coupling strength $\Omega_{04}/\gamma$ (a) in far detuned condition and (b) around resonance. Plots (c) and (d) show individual spectra corresponding to the specific field strengths marked by the white lines in (a) and (b). The other parameters are $\Omega_{02}/\gamma=1.0$, $\Omega_{12}/\gamma=5.0$, $\Omega_{23}/\gamma=5.0$, $\Delta_{04}/\gamma=\Delta_{23}/\gamma=0$ and $\Delta_{12}/\gamma=-20$.](O04_variation.pdf){width="8.5"}
The study on the effect of variation in coupling strength $\Omega_{04}$ on probe absorption characteristics has also been performed and the corresponding spectrum is plotted in figure \[figure:O04\_variation\]. For a lower value of the coupling strength, *i.e.* $\Omega_{04}=1.8$, one large dispersive EIT feature appears at the detuning position fixed by the $\Delta_{12}$ value. As $\Omega_{04}$ is increased, this large EIT feature splits into two smaller EIT features. With further increase in $\Omega_{04}$, the separation between two EIT peaks increases (figure \[figure:O04\_variation\] (a) and (c)). This shows that the strength of the coupling field $\Omega_{04}$ modifies the coherence created between two ground states $|0\rangle$ and $|1\rangle$ due to its direct coupling with the state $|0\rangle$ to $|4\rangle$. Using the dressed state approach, this can be attributed to shift in location of the allowed transitions due to increase in the coupling strength $\Omega_{04}$. When we look into resonance spectra of the probe absorption, it is observed that these are also considerably modified as the strength of the coupling field is varied. A large transparency at lower coupling strength $\Omega_{04}$ gets converted into the absorption peak at higher coupling strength $\Omega_{04}$. The observed broad transparency, within which this absorption peak appears, could be a result of depletion in the population from the ground state $|0\rangle$ due to strong coupling $\Omega_{04}$.
![\[figure:O23\_variation\] (Color online) Amalgamated spectrum (*i.e.* probe absorption as a function of probe detuning $\Delta_{02}/\gamma$) for different values of scaled coupling strength $\Omega_{23}/\gamma$ for (a) $\Delta_{04}/\gamma=0$ and (b) $\Delta_{04}/\gamma=10$ with other parameters $\Omega_{02}/\gamma=1.0$, $\Omega_{12}/\gamma=5.0$, $\Omega_{04}/\gamma=5.0$, $\Delta_{12}/\gamma=-20$ and $\Delta_{23}/\gamma=0$.](O23_variation_comp.pdf){width="8.5"}
![\[figure:O23\_dressed\]Pictorial representation of doubly dressed approach. (a) Five level $IY^+$ system. (b) The formation of primary dressed states due to strong coupling of $\Omega_{23}$ with states $|2\rangle$ and $|3\rangle$. (c) The picture representing all possible dressed states with the allowed transitions between them.](Omega_23_dressed.pdf){width="8.5"}
The effect of variation in coupling strength $\Omega_{23}$ on the probe absorption has been studied for resonant and off resonant detuning values of $\Delta_{04}$. The obtained results are shown in figure \[figure:O23\_variation\]. For the case of resonant condition *i.e.* $\Delta_{04}=0$, one can obtain a coupling strength $\Omega_{23}$ dependent disappearance and reappearance of the absorption peaks in the central region. For a weak coupling strength $\Omega_{23}$, the upper transition between states $|2\rangle$ and $|3\rangle$ acts as a perturbation and the system can be considered as a perturbed N-system. The black continuous curve in figure \[figure:O23\_variation\] (c) shows the corresponding spectrum which is similar to the earlier reported spectrum for the case of N-system [@Salloum:2009]. From the figure \[figure:O23\_variation\] (a), it is clear that as strength $\Omega_{23}$ becomes comparable to other coupling strengths, this perturbed N-system becomes equivalent to $IY^+$ system resulting in four absorption peaks as observed earlier (shown by blue curve in figure \[figure:d12\_variation\] (d)).
For $\Omega_{23}$ greater than the strength of the other coupling fields, the obtained results can be again explained by using the doubly dressed state formalism. The method is same as described earlier. Here, the primary dressed states are created by field $\Omega_{23}$ coupling the bare states $|2\rangle$ and $|3\rangle$. The newly formed dressed states have eigen-energies $\pm \Omega_{23} /2$. The one of the resonantly close primary dressed state (one with energy $-\Omega_{23} /2$) gets doubly dressed due to its interaction with the field $\Omega_{12}$. The energy of these doubly dressed states are $\frac{\Delta_{12} + \Omega_{23}/2}{2} \pm \frac{\sqrt{(\Delta_{12} + \Omega_{23}/2)^2 + \Omega_{12}^2}}{2}$ (see figure \[figure:O23\_dressed\]). The location of all allowed transitions between two lower dressed states and three upper dressed states are expected to be at $\Delta_{02}/ \gamma=$ -22.8, -17.8, -6.6, -1.5, 2.0, 7.0, which is consistent with the results obtained using NMP method (figure \[figure:O23\_variation\] (c)). In case of a far off resonant detuning condition $\Delta_{04}/\gamma=10$, the spectral structure of the central region remains almost independent of the coupling strength $\Omega_{23}$, while amplitude of absorption depends on $\Omega_{23}$.
Conclusion {#sec:conc}
==========
A five-level modified inverted-Y system, *i.e.* $IY^+$ system, comprising of basic three-level sub-systems, *i.e.* $\Lambda$, ladder and vee systems, and basic four-level sub-systems, *i.e.* N and inverted-Y, has been investigated for probe absorption characteristics using a numerical matrix propagation method. The superiority of this method over the well known RWA method has been established by investigating an inverted-Y system within and beyond the validity regime of RWA method. The presence of a strong coupling field connecting the ground state to another state in the $IY^+$ atomic system leads to conversion of resonant probe transparency (obtained in the inverted-Y system) into absorption. The transparency in $IY^+$ system is recovered when the aforementioned coupling field is kept off-resonant. Apart from this, the coupling field detuning dependent splitting of the transparency and coupling field strength dependent shifting of the transparency and absorption have also been obtained for this $IY^+$ system. The numerically obtained results are also found consistent with the dressed and doubly dressed state formalism. This study shows that the $IY^+$ system can be used to design optical devices for switching and multi-channel optical communication.
ACKNOWLEDGMENTS
===============
Charu Mishra is grateful for financial support from RRCAT, Indore under HBNI, Mumbai program.
[33]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevLett.64.1107>.
, , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevLett.66.2593>.
, , , ****, ().
, ****, (), <http://link.aps.org/doi/10.1103/PhysRevA.51.R2703>.
, ****, (), <https://link.aps.org/doi/10.1103/PhysRevA.51.4959>.
, , , , , , , ****, (), , <https://doi.org/10.1080/09500340.2018.1502824>.
, , , , , ****, (), ISSN , <http://www.sciencedirect.com/science/article/pii/003040189500316Z>.
, ****, (), <https://link.aps.org/doi/10.1103/PhysRevA.79.063821>.
, , , , ****, (), <http://stacks.iop.org/1402-4896/91/i=3/a=035401>.
, , , , , , ****, (), ISSN , <http://www.sciencedirect.com/science/article/pii/S0030401802014165>.
, , , , , , ****, (), , <http://dx.doi.org/10.1080/09500340.2014.904019>.
, , , , , , , ****, (), <https://link.aps.org/doi/10.1103/PhysRevA.61.023401>.
, , , ****, (), <https://link.aps.org/doi/10.1103/PhysRevA.64.043807>.
, ****, (), <http://stacks.iop.org/1402-4896/81/i=1/a=015402>.
, , , , , ****, (), <https://link.aps.org/doi/10.1103/PhysRevA.86.023828>.
, , , ****, (), <https://doi.org/10.1140/epjd/e2015-60533-5>.
, ****, (), ISSN , <http://www.sciencedirect.com/science/article/pii/S037596011730734X>.
, , , , ****, (), <https://link.aps.org/doi/10.1103/PhysRevA.69.053818>.
, , , , , ****, (), ISSN , <http://www.sciencedirect.com/science/article/pii/S0030401806008522>.
, , , , ****, (), , <https://doi.org/10.1080/09500340.2011.603441>.
, ****, ().
, , , , , ****, (), , <https://doi.org/10.1080/09500340.2012.733433>.
, , , , , , ****, (), ISSN , <http://www.sciencedirect.com/science/article/pii/S0030401814012449>.
, , , , , ****, (), <http://stacks.iop.org/0953-4075/50/i=21/a=215401>.
, , , ****, (), <http://josab.osa.org/abstract.cfm?URI=josab-35-6-1233>.
, , , , , , , , ****, (), <http://josab.osa.org/abstract.cfm?URI=josab-27-10-2035>.
, , , , ****, (), <http://stacks.iop.org/2040-8986/17/i=4/a=045505>.
, ****, (), <https://link.aps.org/doi/10.1103/PhysRevA.50.2989>.
, ****, (), ISSN , <http://www.sciencedirect.com/science/article/pii/000926149500058C>.
, ****, (), <http://stacks.iop.org/1402-4896/81/i=1/a=015402>.
, , , , ****, (), <http://josab.osa.org/abstract.cfm?URI=josab-6-8-1519>.
, , , ****, (), <https://link.aps.org/doi/10.1103/PhysRevA.64.013412>.
, , , , ****, (), , <https://doi.org/10.1080/09500340903199939>.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We investigate the design challenges of constructing effective and efficient neural sequence labeling systems, by reproducing twelve neural sequence labeling models, which include most of the state-of-the-art structures, and conduct a systematic model comparison on three benchmarks (i.e. NER, Chunking, and POS tagging). Misconceptions and inconsistent conclusions in existing literature are examined and clarified under statistical experiments. In the comparison and analysis process, we reach several practical conclusions which can be useful to practitioners.'
author:
- |
Jie Yang, Shuailong Liang, Yue Zhang\
Singapore University of Technology and Design\
[{jie\_yang, shuailong\_liang}@mymail.sutd.edu.sg]{}\
[yue\_zhang@sutd.edu.sg]{}\
bibliography:
- 'coling2018.bib'
title: Design Challenges and Misconceptions in Neural Sequence Labeling
---
Introduction {#intro}
============
Sequence labeling models have been used for fundamental NLP tasks such as POS tagging, chunking and named entity recognition (NER). Traditional work uses statistical approaches such as Hidden Markov Models (HMM) and Conditional Random Fields (CRF) [@ratinov2009design; @passos2014lexicon; @luo2015joint] with handcrafted features and task-specific resources. With advances in deep learning, neural models have given state-of-the-art results on many sequence labeling tasks [@ling2015finding; @lample2016neural; @ma2016end]. Words and characters are encoded in distributed representations [@mikolov2013distributed] and sentence-level features are learned automatically during end-to-end training. Many existing state-of-the-art neural sequence labeling models utilize word-level Long Short-Term Memory (LSTM) structures to represent global sequence information and a CRF layer to capture dependencies between neighboring labels [@huang2015bidirectional; @lample2016neural; @ma2016end; @peters2017semi]. As an alternative, Convolution Neural Network (CNN) [@lecun1989backpropagation] has also been used for its ability of parallel computing, leading to an efficient training and decoding process.
Despite them being dominant in the research literature, reproducing published results for neural models can be challenging, even if the codes are available open source. For example, conduct a large number of experiments using the code of , but cannot obtain comparable results as reported in the paper. report lower average F-scores on NER when reproducing the structure of , and on POS tagging when reproducing . Most literature compares results with others by citing the scores directly [@huang2015bidirectional; @lample2016neural] without re-implementing them under the same setting, resulting in less persuasiveness on the advantage of their models. In addition, conclusions from different reports can be contradictory. For example, most work observes that stochastic gradient descent (SGD) gives best performance on NER task [@chiu2015named; @lample2016neural; @ma2016end], while report that SGD is the worst optimizer on the same datasets.
The comparison between different deep neural models is challenging due to sensitivity on experimental settings. We list six inconsistent configurations in literature, which lead to difficulties for fair comparison.
$\;$ **Dataset**. Most work reports sequence labeling results on both CoNLL 2003 English NER [@tjong2003introduction] and PTB POS [@marcus1993building] datasets [@collobert2011natural; @huang2015bidirectional; @ma2016end]. give results only on POS dataset, while some papers [@chiu2015named; @lample2016neural; @strubell2017fast] report results on the NER dataset only. conducts experiments on NER for Portuguese and Spanish.
Most work uses the development set to select hyperparameters [@lample2016neural; @ma2016end], while others add development set into training set [@chiu2015named; @peters2017semi]. use a smaller dataset (13862 vs 14987 sentences). Different from and , choose a different data split on the POS dataset. and use different development sets for chunking.
$\;$ **Preprocessing**. A typical data preprocessing step is to normize digit characters [@chiu2015named; @lample2016neural; @yang2016multi; @strubell2017fast]. use fine-grained representations for less frequent words. do not use preprocessing.
$\;$ **Features**. and apply word spelling features and further integrate context features. and use neural features to represent external gazetteer information. Besides, and use end-to-end structure without handcrafted features.
$\;$ **Hyperparameters** including learning rate, dropout rate [@srivastava2014dropout], number of layers, hidden size etc. can strongly affect the model performance. search for the hyperparameters for each task and show that the system performance is sensitive to the choice of hyperparameters. However, existing models use different parameter settings, which affects the fair comparison.
$\;$ **Evaluation**. Some literature reports results using mean and standard deviation under different random seeds [@chiu2015named; @peters2017semi; @liu2017empower]. Others report the best result among different trials [@ma2016end], which cannot be compared directly.
$\;$ **Hardware environment** can also affect system accuracy. observes that the system gives better accuracy on NER task when trained using GPU as compared to using CPU. Besides, the running speeds are highly affected by the hardware environment.
To address the above concerns, we systematically analyze neural sequence labeling models on three benchmarks: CoNLL 2003 NER [@tjong2003introduction], CoNLL 2000 chunking [@tjong2000introduction] and PTB POS tagging [@marcus1993building]. Table \[tab:models\] shows a summary of the models we investigate, which can be categorized under three settings: (i) character sequence representations ; (ii) word sequence representations; (iii) inference layer. Although various combinations of these three settings have been proposed in the literature, others have not been examined. We compare all models in Table \[tab:models\], which includes most state-of-the-art methods. To make fair comparisons, we build a unified framework[^1] to reproduce the twelve neural sequence labeling architectures in Table \[tab:models\]. Experiments show that our framework gives comparable or better results on reproducing existing works, showing the practicability and reliability of our analysis for practitioners. The detailed comparison and analysis show that (i) Character information provides a significant improvement on accuracy; (ii) Word-based LSTM models outperforms CNN models in most cases; (iii) CRF can improve model accuracy on NER and chunking but does not on POS tagging. Our framework is based on PyTorch with batched implementation, which is highly efficient, facilitating quick configurations for new tasks.
![Neural sequence labeling architecture for sentence “COLING is held in New Mexico”. Green, red, yellow and blue circles represent character embeddings, word embeddings, character sequence representations and word sequence representations, respectively.[]{data-label="fig:architecture"}](allfigure.pdf){width="5in"}
Related Work
============
proposed a seminal neural architecture for sequence labeling. It captures word sequence information with a one-layer CNN based on pretrained word embeddings and handcrafted neural features, followed with a CRF output layer. extended this model by integrating character-level CNN features. built a deeper dilated CNN architecture to capture larger local features. was the first to exploit LSTM for sequence labeling. built a BiLSTM-CRF structure, which has been extended by adding character-level LSTM [@lample2016neural; @liu2017empower], GRU [@yang2016multi], and CNN [@chiu2015named; @ma2016end] features. proposed a neural reranking model to improve NER models. These models achieve state-of-the-art results in the literature.
compared several word-based LSTM models for several sequence labeling tasks, reporting the score distributions over multiple runs rather than single value. They investigated the influence of various hyperparameters and configurations. Our work is similar in comparing different neural architectures under unified settings, but differs in four main aspects: 1) Their experiments are based on a BiLSTM with handcrafted word features, while our experiments are based on end-to-end neural models without human knowledge. 2) Their system gives relatively low performances on standard benchmarks[^2], while ours can give comparable or better results with state-of-the-art models, rendering our observations more informative for practitioners. 3) Our findings are more consistent with most previous work on configurations such as usefulness of character information [@lample2016neural; @ma2016end], optimizer [@chiu2015named; @lample2016neural; @ma2016end] and tag scheme [@ratinov2009design; @dai2015enhancing]. In contrast, many results of contradict existing reports. 4) We conduct a wider range of comparison for word sequence representations, including all combinations of character CNN/LSTM and word CNN/LSTM structures, while studied the word LSTM models only.
Neural Sequence Labeling Models
===============================
Our neural sequence labeling framework contains three layers, i.e., a character sequence representation layer, a word sequence representation layer and an inference layer, as shown in Figure \[fig:architecture\]. Character information has been proven to be critical for sequence labeling tasks [@chiu2015named; @lample2016neural; @ma2016end], with LSTM and CNN being used to model character sequence information (“Char Rep.”). Similarly, on the word level, LSTM or CNN structures can be leveraged to capture long-term information or local features (“Word Rep.”), respectively. Subsequently, the inference layer assigns labels to each word using the hidden states of word sequence representations.
Character Sequence Representations
----------------------------------
Character features such as prefix, suffix and capitalization can be represented with embeddings through a feature-based lookup table [@collobert2011natural; @huang2015bidirectional; @strubell2017fast], or neural networks without human-defined features [@lample2016neural; @ma2016end]. In this work, we focus on neural character sequence representations without hand-engineered features.
**Character CNN**. Using a CNN structure to encode character sequences was firstly proposed by , and followed by many subsequent investigations [@dos2015boosting; @chiu2015named; @ma2016end]. In our experiments, we take the same structure as , using one layer CNN structure with *max-pooling* to capture character-level representations. Figure \[fig:charcnn\] shows the CNN structure on representing word “Mexico”.
**Character LSTM**. Shown as Figure \[fig:charlstm\], in order to model the global character sequence information of a word “Mexico”, we utilize a bidirectional LSTM on the character sequence of each word and concatenate the *left-to-right* final state $F_{LSTM}$ and the *right-to-left* final state $B_{LSTM}$ as character sequence representations. applied one bidirectional LSTM for the character sequence over a sentence rather than each word individually. We examined both structures and found that they give comparable accuracies on sequence labeling tasks. We choose ’s structure as its character LSTMs can be calculated in parallel, making the system more efficient.
Word Sequence Representations {#ssec:wordrep}
-----------------------------
Similar to character sequences in words, we can model word sequence information through LSTM or CNN structures. LSTM has been widely used in sequence labeling [@lample2016neural; @ma2016end; @chiu2015named; @huang2015bidirectional; @liu2017empower]. CNN can be much faster than LSTM due to the fact that convolution calculation can be parallel on the input sequence [@collobert2011natural; @dos2015boosting; @strubell2017fast].
**Word CNN**. Figure \[fig:wordcnn\] shows the multi-layer CNN on word sequence, where words are represented by embeddings. If a character sequence representation layer is used, then word embeddings and character sequence representations are concatenated for word representations. For each CNN layer, a window of size 3 slides along the sequence, extracting local features on the word inputs and a ReLU function [@glorot2011deep] is followed. We follow by using 4 CNN layers. Batch normalization [@ioffe2015batch] and dropout [@srivastava2014dropout] are applied following each CNN layer.
**Word LSTM**. Shown in Figure \[fig:wordlstm\], word representations are fed into a forward LSTM and backward LSTM, respectively. The forward LSTM captures the word sequence information from left to right, while the backward LSTM extracts information in a reversed direction. The hidden states of the forward and backward LSTMs are concatenated at each word to give global information of the whole sequence.
Inference Layer {#ssec:crf}
---------------
The inference layer takes the extracted word sequence representations as features and assigns labels to the word sequence. Independent local decoding with a linear layer mapping word sequence representations to label vocabulary and performing softmax can be quite effective [@ling2015finding], while for tasks that with strong output label dependency, such as NER, CRF is a more appropriate choice. In this work, we examine both softmax and CRF as inference layer on three sequence labeling tasks.
Experiments
===========
We investigate the main influencing factors to system accuracy, including the character sequence representations, word sequence representations, inference algorithm, pretrained embeddings, tag scheme, running environment and optimizer; analyzing system performances from the perspective of decoding speed and accuracies on in-vocabulary (IV) and out-of-vocabulary (OOV) entities/chunks/words.
Settings
--------
**Data**. The NER dataset has been standardly split in . It contains four named entity types: <span style="font-variant:small-caps;">Person, Location, Organization,</span> and <span style="font-variant:small-caps;">Misc</span>. The chunking task is evaluated on CoNLL 2000 shared task [@tjong2000introduction]. We follow by using sections 15-18 of Wall Street Journal (WSJ) for training, section 19 as development set and section 20 as test set. For the POS tagging task, we use the WSJ portion of Peen Treebank, which has 45 POS tags. Following previous works [@toutanova2003feature; @santos2014learning; @ma2016end; @liu2017empower], we adopt the standard splits by using sections 0-18 as training set, sections 19-21 as development set and sections 22-24 as test set. No preprocessing is performed on either dataset except for normalizing digits. The dataset statistics are listed in Table \[tab:statistics\].
**Hyperparameters**. Table \[tab:hyperparameter\] shows the hyperparameters used in our experiments, which mostly follow , including the learning rate $\eta=0.015$ for word LSTM models. For word CNN based models, a large $\eta$ leads to convergence problem. We take $\eta=0.005$ with more epochs (200) instead. GloVe 100-dimension [@pennington2014glove] is used to initialize word embeddings and character embeddings are randomly initialized. We use mini-batch stochastic gradient descent (SGD) with a decayed learning rate to update parameters. For NER and chunking, we the BIOES tag scheme.
**Evaluation**. Standard precision (P), recall (R) and F1-score (F) are used as the evaluation metrics for NER and chunking; token accuracy is used to evaluate the performance of POS tagger. Development datasets are used to select the optimal model among all epochs, and we report scores of the selected model on the test dataset. To reduce the volatility of the system, we conduct each experiment 5 times under different random seeds, and report the *max, mean,* and *standard deviation* for each model.
Results
-------
Tables \[tab:nerresult\], \[tab:chunkresult\] and \[tab:posresult\] show the results of the twelve models on NER, chunking and POS datasets, respectively. Existing work has also been listed in the tables for comparison. To simplify the description, we use “CLSTM” and “CCNN” to represent character LSTM and character CNN encoder, respectively. Similarly, “WLSTM” and “WCNN” represent word LSTM and word CNN structure, respectively.
As shown in Table \[tab:nerresult\], most NER work focuses on WLSTM+CRF structures with different character sequence representations. We re-implement the structure of several reports [@chiu2015named; @ma2016end; @peters2017semi], which take the CCNN+WLSTM+CRF architecture. Our reproduced models give slightly better performances. The results of can be reproduced by our CLSTM+WLSTM+CRF. In most cases, our “Nochar” based models underperform their corresponding prototypes [@huang2015bidirectional; @strubell2017fast], which utilize the hand-crafted features.
Table \[tab:chunkresult\] shows the results of the chunking task. give the best reported single model results (95.00$\pm$0.08), and our CLSTM+WLSTM+CRF gives a comparable performance (94.93$\pm$0.05). We re-implement ’s model in our Nochar+WLSTM but cannot reproduce their results, this may because that they use grid search for hyperparameter selection. Our Nochar+WCNN+CRF can give comparable results with , even ours does not include character information.
The results of the POS tagging task is shown in Table \[tab:posresult\]. The results of , and can be reproduced by our CLSTM+WLSTM+CRF and CCNN+WLSTM+CRF models. Our WLSTM based models give better results than WLSTM+CRF based models, this is consistent with the fact that take CLSTM+WLSTM without CRF layer but achieve the best POS accuracy. build a pure CNN structure on both character and word level, which can be reproduced by our CCNN+WCNN models.
Based on above observations, most results in the literature are reproducible. Our implementations can achieve the comparable or better results with state-of-the-art models. We do not fine-tune any hyperparameter to fit the specific task. Results on Table \[tab:nerresult\], \[tab:chunkresult\] and \[tab:posresult\] are all under the same hyperparameters, which demonstrates the generalization ability of our framework.
Network settings {#ssc:archi}
----------------
**Character LSTM vs Character CNN**. Unlike the observations of , in our experiments, character information can significantly ($p<0.01$)[^3] improve sequence labeling models (by comparing the row of Nochar with CLSTM or CCNN on Table \[tab:nerresult\], \[tab:chunkresult\] and \[tab:posresult\]), while the difference between CLSTM and CCNN is not significant. In most cases, CLSTM and CCNN can give comparable results under different frameworks and different tasks. CCNN gives the best NER result under the WLSTM+CRF framework, while CLSTM gets better NER results in all other configurations. For chunking and POS tagging, CLSTM consistently outperforms CCNN under all settings, while the difference is statistically insignificant ($p>0.2$). We conclude that the difference between CLSTM and CCNN is small, which is consistent with the observation of .
**Word LSTM vs Word CNN**. By comparing the performances of WLSTM+CRF, WLSTM with WCNN+CRF, WCNN on the three benchmarks, we conclude that word-based LSTM models are significantly ($p<0.01$) better than word-based CNN models in most cases. It demonstrates that global word context information is necessary for sequence labeling.
**Softmax vs CRF**. Models with CRF inference layer can consistently outperform the models with softmax layer under all configurations on NER and chunking tasks, proving the effectiveness of label dependency information. While for POS tagging, the local softmax based models give slightly better accuracies while the difference is insignificant ($p>0.2$).
External factors {#sc:influencefactor}
----------------
In addition to model structures, external factors such as pretrained embeddings, tag scheme, and optimizer can significantly influence system performance. We investigate a set of external factors on the NER dataset with the two best models: CLSTM+WLSTM+CRF and CCNN+WLSTM+CRF.
**Pretrained embedding**. Figure \[fig:embcompare\] shows the F1-scores of the two best models on the NER test set with two different pretrained embeddings, as well as the random initialization. Compared with the random initialization, models using pretrained embeddings give significant improvements ($p<0.01$). The GloVe 100-dimension embeddings give higher F1-scores than SENNA [@collobert2011natural] on both models, which is consistent with the observation of .
**Tag scheme**. We examine two different tag schemes: BIO and BIOES [@ratinov2009design]. The results are shown in Figure \[fig:allsetting\]. In our experiments, models using BIOES are significantly ($p<0.05$) better than BIO. Our observation is consistent with most literature [@ratinov2009design; @dai2015enhancing]. report that the difference between the schemes is insignificant.
**Running environment**. observe that neural sequence labeling models can give better results on GPU rather than CPU. We conduct repeated experiments on both GPU and CPU environments. The results are shown in Figure \[fig:allsetting\]. Models run on CPU give a lower mean F1-score than models run on GPU, while the difference is insignificant ($p>0.2$).
**Optimizer**. We compare different optimizers including SGD, Adagrad [@duchi2011adaptive], Adadelta [@zeiler2012adadelta], RMSProp [@tieleman2012lecture] and Adam [@kingma2014adam]. The results are shown in Figure \[fig:optisetting\][^4]. In contrast to , who reported that SGD is the worst optimizer, our results show that SGD outperforms all other optimizers significantly ($p<0.01$), with a slower convergence process during training. Our observation is consistent with most literature [@chiu2015named; @lample2016neural; @ma2016end].
Analysis
--------
**Decoding speed**. We test the decoding speeds of the twelve models on the NER dataset using a Nvidia GTX 1080 GPU. Figure \[fig:decodespeed\] shows the decoding times on 10000 NER sentences. The CRF inference layer severely limits the decoding speed due to the left-to-right inference process, which disables the parallel decoding. Character LSTM significantly slows down the system. Compared with models without character information, adding character CNN representations does not affect the decoding speed too much but can give significant accuracy improvements (shown in Section \[ssc:archi\]). Due to the support of parallel computing, word-based CNN models are consistently faster than word-based LSTM models, with close accuracies, leading to large utilization potential in practice.
![Decoding times (10000 NER sentences).[]{data-label="fig:decodespeed"}](optimsetting.pdf){width="3.1in"}
![Decoding times (10000 NER sentences).[]{data-label="fig:decodespeed"}](decodespeed.pdf){width="3in"}
**OOV error**. We conduct error analysis on in-vocabulary and out-of-vocabulary words with the CRF based models[^5]. Following , words in the test set are divided into four subsets: in-vocabulary words, out-of-training-vocabulary words (OOTV), out-of-embedding-vocabulary words (OOEV) and out-of-both-vocabulary words (OOBV). For NER and chunking, we consider entities or chunks rather than words. The OOV entities and chunks are categorized following .
Table \[tab:OOV\] shows the performances of different OOV splits on three benchmarks. The top three rows list the performances of word-based LSTM CRF models, followed by the word-based CNN CRF models. The results of OOEV in NER keep 100% because of there exist only 8 OOEV entities and all are recognized correctly. It is obvious that character LSTM or CNN representations improve OOTV and OOBV the most on both WLSTM+CRF and WCNN+CRF models across all three datasets, proving that the main contribution of neural character sequence representations is to disambiguate the OOV words. Models with character LSTM representations give the best IV scores across all configurations, which may be because character LSTM can be well trained on IV data, bringing the useful global character sequence information. On the OOVs, character LSTM and CNN gives comparable results.
Conclusion
==========
We built a unified neural sequence labeling framework to reproduce and compare recent state-of-the-art models with different configurations. We explored three neural model design decisions: character sequence representations, word sequence representations, and inference layer. Experiments show that character information helps to improve model performances, especially on disambiguating OOV words. Character-level LSTM and CNN structures give comparable improvements, with the latter being more efficient. In most cases, models with word-level LSTM encoders outperform those with CNN, at the expense of longer decoding time. We observed that the CRF inference algorithm is effective on NER and chunking tasks, but does not have the advantage on POS tagging. With controlled experiments on the NER dataset, we showed that BIOES tags are better than BIO. Besides, pretrained GloVe 100d embedding and SGD optimizer give significantly better performances compared to their competitors.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank the anonymous reviewers for their useful comments. Yue Zhang is the corresponding author.
[^1]: Our code has been released at <https://github.com/jiesutd/NCRFpp>.
[^2]: Based on their detailed experiment report [@reimers2017optimal], the F1-scores on CoNLL 2003 NER task are generally less than 90%, while *state-of-the-art* results are around 91%.
[^3]: We use *t-test* to calculate the $p$ value, reporting the highest $p$ value when giving the conclusions on multiple configurations.
[^4]: We fine-tune the learning rates for Adagrad, Adadelta, RMSProp and Adam, and report the best results in the figure.
[^5]: We choose the models that give the median performance on the test set for conducting result analysis.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study Diophantine arithmetic properties of birational divisors in conjunction with concepts that surround ${\mathrm{K}}$-stability for Fano varieties. There is also an interpretation in terms of the barycentres of Newton-Okounkov bodies. Our main results show how the notion of divisorial instability, in the sense of K. Fujita, implies instances of Vojta’s Main Conjecture for Fano varieties. A main tool in the proof of these results is an arithmetic form of Cartan’s Second Main Theorem that has been obtained by M. Ru and P. Vojta.'
address: 'Department of Mathematics, Michigan State University, East Lansing, MI, USA, 48844'
author:
- Nathan Grieve
title: 'Divisorial instability and Vojta’s Main Conjecture for [${\mathbb{Q}}$]{}-Fano varieties'
---
[^1]
Introduction {#Introduction}
============
[[**.** ]{}]{}Our main purpose here, is to indicate how the barycentres of Newton-Okounkov bodies, together with the concept of *divisorial instability* for a given ${\mathbb{Q}}$-Fano variety, relate to Vojta’s Main Conjecture. We then discuss examples which are made possible by insights of K. Fujita [@Fujita:2016a]. Finally, we mention analogous implications for ${\mathrm{K}}$-instability of Fano varieties.
[[**.** ]{}]{}A key theme in this article is that ideas from toric geometry, K-stability and measures of singularities influence results in the direction of Diophantine approximation. We refer to [@Grieve:2018:autissier], and the references therein, for a more detailed discussion on these and other topics. Here, we find it convenient to express our Diophantine approximation results using the language of birational divisors. The concept of birational divisor appears in work of V. V. Shokurov (for example [@Shokurov:1996]). It is closely related to the theory of divisorial valuations and the classification problem in general.
[[**.** ]{}]{}That this birational divisor language is important for arithmetic purposes was noted by P. Vojta [@Vojta:1996]. This birational viewpoint from [@Vojta:1996] has recently been developed further [@Ru:Vojta:2016].
[[**.** ]{}]{}In what follows, we state our arithmetic results in the number field setting. Similar results hold true in the (characteristic zero) function field setting. Indeed, these viewpoints are well developed in [@Grieve:Function:Fields] and [@Grieve:2018:autissier], for example, building on, and applying, earlier results from [@Wang:2004] and [@Ru:Wang:2012].
[[**.** ]{}]{} In particular, we establish the following result which shows how the concept of divisorial instability, in the sense of [@Fujita:2016a], implies instances of Vojta’s Main Conjecture. At the same time, it shows how these topics are related to the Newton-Okounkov body, of a given Fano variety, with respect to a suitably defined (divisorial) admissible flag.
\[Vojta:canonical:Fano:not:D:stable:barycentre\] Let ${\mathbf{K}}$ be a number field and fix a finite set of places $S$ of ${\mathbf{K}}$. Let $X$ be a ${\mathbb{Q}}$-Fano variety defined over ${\mathbf{K}}$ and let $E$ be an irreducible and reduced Cartier divisor over $X$ and defined over ${\mathbf{F}}/ {\mathbf{K}}$, a finite extension of ${\mathbf{K}}$, with ${\mathbf{K}}\subseteq {\mathbf{F}}\subseteq \overline{{\mathbf{K}}}$. Let $$\pi \colon Y \rightarrow X_{\mathbf{F}}$$ be a proper normal model with the property that $E$ is supported on $Y$ and fix an admissible flag $$Y_\bullet
\colon
Y = Y_0 \supsetneq Y_1 \supsetneq \hdots \supsetneq Y_d = \{\mathrm{pt}\}$$ of subvarieties of $Y$ with divisorial component $$Y_1 = E.$$ Let $$\Delta = \Delta_{Y_\bullet}(-{\mathrm{K}}_X)$$ be the Okounkov body of $- \pi^*{\mathrm{K}}_X$ computed with respect to the flag $Y_\bullet$ and let $\mathbf{b}_1$ be the first coordinate of the barycentre of $\Delta$. Assume that $$\mathbf{b}_1 {\geqslant}1.$$ Then the inequalities predicted by Vojta’s Main Conjecture hold true for $\mathbb{E}$, the birational divisor that is determined by $E$. Precisely, if $B$ is a big line bundle on $X$ and $\epsilon > 0$, then $$\label{Vojta:Inequality:Eqn:intro:a}
\sum_{v \in S} \lambda_{\mathbb{E},v}(x) + h_{{\mathrm{K}}_X}(x) {\leqslant}\epsilon h_B(x) + \mathrm{O}(1)$$ for all ${\mathbf{K}}$-rational points $x \in X \setminus Z$ and $Z \subsetneq X$ some proper Zariski closed subset defined over ${\mathbf{K}}$.
We prove Theorem \[Vojta:canonical:Fano:not:D:stable:barycentre\] in Section \[Vojta:Instability:Section\].
[[**.** ]{}]{}Next, we turn to the question of Vojta’s Main Conjecture within the context of ${\mathrm{K}}$-stability for Fano varieties. In doing so, we obtain Theorem \[Vojta:canonical:Fano:not:K:stable\] below which extends and improves upon existing recent related results (for example [@Grieve:2018:autissier Corollary 1.2] and [@Ru:Vojta:2016 Corollary 1.12]). For a brief summary of the results from ${\mathrm{K}}$-stability for Fano varieties, we refer to Section \[K:stable:Fano\].
\[Vojta:canonical:Fano:not:K:stable\] Let ${\mathbf{K}}$ be a number field and fix a finite set of places $S$ of ${\mathbf{K}}$. Suppose that $X$ is a ${\mathbb{Q}}$-Fano variety with canonical singularities, defined over ${\mathbf{K}}$, and which is not ${\mathrm{K}}$-stable. Then $X$ admits an irreducible and reduced Cartier divisor $E$, which is defined over some finite extension field ${\mathbf{F}}/{\mathbf{K}}$, with ${\mathbf{K}}\subseteq {\mathbf{F}}\subseteq \overline{{\mathbf{K}}}$, for which the inequalities predicted by Vojta’s Main Conjecture hold true. Precisely, if $B$ is a big line bundle on $X$ and $\epsilon > 0$, then $$\label{Vojta:Inequality:Eqn:intro:b}
\sum_{v \in S} \lambda_{\mathbb{E},v}(x) + h_{{\mathrm{K}}_X}(x) {\leqslant}\epsilon h_B(x) + \mathrm{O}(1)$$ for all ${\mathbf{K}}$-rational points $x \in X \setminus Z$ and $Z \subsetneq X$ some proper Zariski closed subset defined over ${\mathbf{K}}$.
In and , $\mathbb{E}$ denotes the birational divisor determined by $E$ and $\lambda_{\mathbb{E},v}(\cdot)$ is the *birational Weil function* of $\mathbb{E}$ with respect to $v$. Theorem \[Vojta:canonical:Fano:not:K:stable\] is also proved in Section \[Vojta:Instability:Section\].
[[**.** ]{}]{}[**Notations and other conventions.**]{} Unless explicitly stated otherwise, we let ${\mathbf{K}}$ be a number field and ${\mathbf{F}}/ {\mathbf{K}}$ a finite field extension, with ${\mathbf{K}}\subseteq {\mathbf{F}}\subseteq {\overline{{\mathbf{K}}}}$, for $\overline{{\mathbf{K}}}$ a fixed algebraic closure of ${\mathbf{K}}$. By *variety* over a fixed base field ${\mathbf{k}}$, we mean a reduced projective scheme over $\operatorname{Spec}({\mathbf{k}})$. By a *model* of a normal variety $X$, we mean a proper birational morphism $\pi' \colon X' \rightarrow X$ from a normal variety $X'$. When no confusion is likely, we omit explicit mention of fields of definition for such models $X'$. By a ${\mathbb{Q}}$-Fano variety $X$ over ${\mathbf{K}}$, we mean that $X$ is a geometrically normal, geometrically irreducible projective variety, defined over ${\mathbf{K}}$, having the property that $X_{\overline{{\mathbf{K}}}}$ has at most *log-terminal singularities* and, finally, having the property that the *anti-canonical divisor* $-{\mathrm{K}}_X$ is an ample ${\mathbb{Q}}$-Cartier divisor.
[[**.** ]{}]{}[**Acknowledgements.**]{} I thank Steven Lu, Mike Roth and colleagues for discussions on related topics. This work benefited from visits to CIRGET, Montreal, and NCTS, Taipei, during the Summer of 2018. It was written while I was postdoctoral fellow at Michigan State University.
Construction of Okounkov bodies {#Construction:Okounkov:Bodies}
===============================
[[**.** ]{}]{}Recall the construction of Okounkov bodies [@Lazarsfeld:Mustata:2009]. In this section, we implicitly work over a fixed algebraically closed base field. Similar considerations apply for the case of non-algebraically closed based fields; we then must keep track of fields of definition.
[[**.** ]{}]{}Let $X$ be an irreducible projective variety, of dimension $d$, and fix a flag of irreducible subvarieties $$\label{flag:subvarieties}
Y_\bullet : X = Y_0 \supsetneq Y_1 \supsetneq \hdots \supsetneq Y_{d-1} \supsetneq Y_d = \{ \mathrm{pt} \}.$$ We assume that this flag satisfies the two properties that
1. [each $Y_i$ has codimension $i$ in $X$; and]{}
2. [each $Y_i$ is nonsingular at $Y_d$.]{}
We call such flags *admissible*.
[[**.** ]{}]{}Let $D$ be a Cartier divisor on $X$. Fixing such an admissible flag , we may discuss the *Okounkov body* $$\Delta(D) = \Delta_{Y_\bullet}(D)$$ of $D$ with respect to $Y_\bullet$. In particular, for each $m {\geqslant}0$, the given fixed admissible flag determines a *valuation like function*. We denote such functions as $$\nu_{Y_\bullet} = \nu_{Y_\bullet, mD} : {\mathrm{H}}^0(X, {{\mathcal O}}_X(mD)) \rightarrow {\mathbb{Z}}^d \bigcup \{\infty \}\text{;}$$ the *graded semigroup* of $D$ is then $$\Gamma(D) := \{ (\nu_{Y_{\bullet}}(s),m) : \text{ $0 \not = s \in {\mathrm{H}}^0(X,{{\mathcal O}}_X(mD))$} \text{, for $m {\geqslant}0$}
\} \subseteq {\mathbb{N}}^d \times {\mathbb{N}}= {\mathbb{N}}^{d+1} \text{.}$$
[[**.** ]{}]{}To define the Okounkov body of $D$ with respect to $Y_\bullet$, let $$\Sigma(\Gamma) \subseteq {\mathbb{R}}^{d+1}$$ be the closed convex cone (with vertex at the origin) spanned by $\Gamma$. The *Okounkov body* $$\Delta(D) = \Delta_{Y_\bullet}(D) \text{, }$$ with respect to the fixed flag $Y_\bullet$, is the compact convex set $$\Delta(D) = \Delta_{Y_\bullet}(D) = \Sigma(\Gamma) \bigcap \left({\mathbb{R}}^d \times \{1 \} \right).$$
[[**.** ]{}]{}The starting point for the theory of Okounkov bodies is the fact that, for big Cartier divisors $D$ on $X$, the Euclidean volume of $\Delta(D)$ is related to the volume of ${{\mathcal O}}_X(D)$ via $$\label{OK:body:ThmA}
{\operatorname{Vol}}_{{\mathbb{R}}^d}(\Delta(D)) = \frac{1}{d!}{\operatorname{Vol}}_X({{\mathcal O}}_X(D)).$$
Slices of Okounkov bodies and restricted volume functions {#Slices:Okounkov:Bodies}
=========================================================
[[**.** ]{}]{}Fix $X$, a geometrically irreducible and geometrically normal projective variety over ${\mathbf{K}}$ and let $L$ be a big and nef line bundle on $X$, defined over ${\mathbf{K}}$.
[[**.** ]{}]{}Let $E$ be an irreducible and reduced Cartier divisor over $X$ and defined over ${\mathbf{F}}$. We regard $E$ as a Cartier divisor on $Y$ a proper normal model of $$X_{\mathbf{F}}:= X \times_{{\operatorname{Spec}}({\mathbf{K}})} {\operatorname{Spec}}({\mathbf{F}}).$$ In particular, we fix a proper birational morphism $$\pi \colon Y \rightarrow X_{\mathbf{F}}$$ and, by abuse of notation, denote $$\pi^* L = \pi^*L_{\mathbf{F}}\text{, }$$ for the pullback of $L_{\mathbf{F}}$ to $Y$.
[[**.** ]{}]{}Henceforth, we want to study the Okounkov body $$\Delta = \Delta(\pi^*L) = \Delta_{Y_\bullet}(\pi^* L)$$ which is computed with respect to an admissible flag $$\label{admissible:flag:field:defn}
Y_\bullet \colon Y = Y_0 \supsetneq Y_1 \supsetneq \hdots \supsetneq Y_{d-1} \supsetneq Y_d = \{\mathrm{pt}\}$$ with divisorial component $$Y_1 = E.$$
[[**.** ]{}]{}Let $$\operatorname{pr}_1 \colon \Delta \rightarrow {\mathbb{R}}$$ be the projection onto the first component and set $$\Delta(\pi^*L)|_{\nu_1 = t} = \Delta(\pi^*L)_{\nu_1 = t} := \operatorname{pr}^{-1}_1(t) \subseteq \{t \} \times {\mathbb{R}}^{d-1} = {\mathbb{R}}^{d-1}.$$ For later use, put $$\mu = \mu(L,E) := \sup \{ t > 0 : \pi^*L - t E \in \operatorname{Big}(Y) \} \text{;}$$ we mention the following important fact from [@Lazarsfeld:Mustata:2009].
\[Okounkov:Restricted:Volume:Equation\] With the notations and assumptions of this section, the Euclidean volume of $$\Delta(\pi^*L)|_{\nu_1 = t} = \Delta(\pi^*L)_{\nu_1 = t}$$ and the restricted volume of the line bundle $$\pi^*L - t E \text{,}$$ along $E$, for $0 {\leqslant}t < \mu$, are related by $$\operatorname{Vol}_{{\mathbb{R}}^{d-1}}(\Delta(\pi^*L)|_{\nu_1 = t}) = \frac{1}{(d-1)!} \operatorname{Vol}_{Y | E}(\pi^*L - t E).$$
This is implied by statement (ii) of [@Lazarsfeld:Mustata:2009 Corollary 4.27].
The barycentre of Okounkov bodies {#Slices:Okounkov:Barycentre}
=================================
[[**.** ]{}]{}In this section, we continue with the context of Section \[Slices:Okounkov:Bodies\]. To this end, let $X$ be a geometrically irreducible and geometrically normal projective variety over ${\mathbf{K}}$ and let $L$ be a big and nef line bundle on $X$, defined over ${\mathbf{K}}$. Let $E$ be an irreducible and reduced Cartier divisor over $X$, defined over ${\mathbf{F}}$, and fix a proper normal model $$\pi \colon Y \rightarrow X_{\mathbf{F}}$$ for which $E$ is realized as a Cartier divisor.
[[**.** ]{}]{}Our main goal is to indicate the manner in which, on the one hand, the *asymptotic volume constant* of $(X,L)$ along $E$, namely the quantity $$\label{asym:vol:constant}
\beta(L,E) := \int_0^\infty \frac{\operatorname{Vol}_Y( \pi^*L - t E ) }{\operatorname{Vol}_X(L) } \mathrm{d}t \text{, }$$ is related to, on the other hand, the *centre of mass* of the Okounkov body $$\Delta(L) = \Delta_{Y_\bullet}(L) = \Delta(\pi^*L),$$ with respect to an admissible flag $Y_\bullet$ of the form with divisorial component $$Y_1 = E.$$
[[**.** ]{}]{}Such observations are essentially due to K. Fujita and are implicit, for example, in [@Fujita:2016a Proof of Theorem 7.1]. Here, we isolate a form of that statement which is suitable for our purposes.
\[barycentre:volume:constant:prop\] Suppose that $(X,L)$ is a polarized variety over a number field ${\mathbf{K}}$, let $E$ be an irreducible and reduced Cartier divisor over $X$, defined over a finite extension ${\mathbf{F}}/ {\mathbf{K}}$, ${\mathbf{K}}\subseteq {\mathbf{F}}\subseteq \overline{{\mathbf{K}}}$, and fix an admissible flag $Y_\bullet$ of the form , with divisorial component $$Y_1 = E,$$ and supported on some normal proper model $$\pi \colon Y \rightarrow X_{\mathbf{F}}\text{.}$$ Let $\mathbf{b}_1$ be the first coordinate of the barycentre of the Okounkov body $$\Delta = \Delta(L) = \Delta_{Y_\bullet}(L)$$ of $L$ with respect to $Y_\bullet$. Then $\mathbf{b}_1$ and $\beta(L,E)$ coincide $$\beta(L,E) = \mathbf{b}_1.$$
As mentioned, in Proposition \[Okounkov:Restricted:Volume:Equation\], we have, for all $t \in [0,\mu]$, that $$\operatorname{Vol}_{{\mathbb{R}}^{d-1}}(\Delta|_{\nu_1 = t}) = \frac{1}{(d-1)!} \operatorname{Vol}_{Y|E}(\pi^*L-tE).$$ In particular, the first coordinate of the barycentre of $\Delta(L)$ is $$\label{barycentre:first:coord:eqn}
\mathbf{b}_1 := \int_0^{\mu} \frac{ t \cdot \operatorname{Vol}_{{\mathbb{R}}^{d-1}}(\Delta|_{\nu_1 = t}) }{ \operatorname{Vol}_{{\mathbb{R}}^d}(\Delta) } \mathrm{d}t
= d \int_0^{\mu} \frac{ t \cdot \operatorname{Vol}_{Y|E}(\pi^*L-tE) }{{\operatorname{Vol}}_X(L)} \mathrm{d} t.$$ It is known, for example as proven in [@Grieve:chow:approx], that the rightmost quantity which appears in is equal to $\beta(L,E)$.
Divisorial stability for ${\mathbb{Q}}$-Fano varieties {#divisorial:stability:QQ:Fano}
======================================================
[[**.** ]{}]{}In this section, we discuss the concept of *divisorial stability* for ${\mathbb{Q}}$-Fano varieties, which was introduced by K. Fujita [@Fujita:2016a].
[[**.** ]{}]{}Let $X$ be a ${\mathbb{Q}}$-Fano variety over ${\mathbf{K}}$. By this, we mean that $X$ is a geometrically normal, geometrically irreducible projective variety, defined over ${\mathbf{K}}$, having the property that $X_{\overline{{\mathbf{K}}}}$ has at most *log-terminal singularities* and, finally, having the property that the *anti-canonical divisor* $-{\mathrm{K}}_X$ is an ample ${\mathbb{Q}}$-Cartier divisor.
[[**.** ]{}]{}Let $E$ be an irreducible and reduced Cartier divisor over $X$, defined over ${\mathbf{F}}$, and fix a proper normal model $$\pi \colon Y \rightarrow X_{\mathbf{F}}$$ for which $E$ is realized as a Cartier divisor. Similar to [@Fujita:2016a], we set $$\label{divisorial:threshold:stability:eqn}
\eta(E) := {\operatorname{Vol}}_X(-{\mathrm{K}}_X) - \int_0^\infty {\operatorname{Vol}}_Y(- \pi^* {\mathrm{K}}_X - t E) \mathrm{d}t.$$ We say that $\eta(E)$ is the *divisorial stability threshold* for $X$ with respect to $E$.
[[**.** ]{}]{}We mention the concept of *divisorial stability for $X$ along $E$* as is defined in [@Fujita:2016a]. Similarly, we may discuss the concept of *divisorial stability* for $X$ in general. This concept is similar to but, strictly speaking, different than the concept of K-stability. We refer, for example, to [@Fujita:2016a] for a more detailed discussion of these topics. In particular, the concepts of K-stability and divisorial stability coincide in dimension $2$ [@Fujita:2016a Proposition 2.6].
[[**.** ]{}]{}To begin with, the concept of divisorial stability along a divisor is made precise in the following way.
\[divisorial:stab:div:defn\] We say that a ${\mathbb{Q}}$-Fano variety $X$ is *divisorially stable* along $E$ if $\eta(E) > 0$ for all reduced, irreducible Cartier divisors $E$ over $X$, and having field of definition some finite extension ${\mathbf{F}}/ {\mathbf{K}}$, ${\mathbf{K}}\subseteq {\mathbf{F}}\subseteq \overline{{\mathbf{K}}}$. We say that $X$ is *divisorially semi-stable* along $E$ if $\eta(E) {\geqslant}0$ for all such reduced and irreducible Cartier divisors $E$ over $X$.
[[**.** ]{}]{}Similarly, we formulate the concept of divisorial stability in general.
We say that a ${\mathbb{Q}}$-Fano variety $X$ is *divisorially stable* if $\eta(E) > 0$ for all reduced, irreducible Cartier divisors $E$ over $X$, and having field of definition some finite extension ${\mathbf{F}}/ {\mathbf{K}}$, ${\mathbf{K}}\subseteq {\mathbf{F}}\subseteq \overline{{\mathbf{K}}}$. We also say that $X$ is *divisorially semi-stable* if $\eta(E) {\geqslant}0$ for all such reduced and irreducible Cartier divisors $E$ over $X$. We use the term *unstable*, or *instability*, to indicate that a condition of being stable, or of stability, does not hold true.
[[**.** ]{}]{}Before stating Theorem \[characteriztion:instability\] below, we make precise the (evident) relation amongst the *signatures* of $\beta(-{\mathrm{K}}_X,E)$ and $\eta(E)$ along $E$, a Cartier divisor over $X$ and having field of definition ${\mathbf{F}}$.
\[stability:lemma\] With the notations and assumptions of this section, we have that $$\label{eta:eqn1}
\eta(E) < 0 \text{ (resp. ${\leqslant}0$)}$$ if and only if $$\label{eta:eqn2}
\beta(-{\mathrm{K}}_X, E) > 1 \text{ (resp. ${\geqslant}1$).}$$ In particular, $$\label{eta:eqn3}
\beta(-{\mathrm{K}}_X, E) {\geqslant}1$$ if and only if $X$ is not divisorially stable along $E$.
The desired inequalities and follow from the definitions and . Indeed, note that $\eta(E)$ and $\beta(-{\mathrm{K}}_X,E)$ are related by $$\label{eta:eqn:eqn1}
\frac{\eta(E)}{{\operatorname{Vol}}_X(-{\mathrm{K}}_X)} = 1 - \beta(-{\mathrm{K}}_X,E).$$ It follows from that $$\text{
$\eta(E) < 0$ (resp. ${\leqslant}0$)
}$$ if and only if $$\text{$\beta(-{\mathrm{K}}_X,E) > 1$ (resp. ${\geqslant}1$).
}$$ Finally, to obtain the instability conclusion which is implied by the inequality , use Definition \[divisorial:stab:div:defn\] together with the equivalence of and .
[[**.** ]{}]{} We conclude this section by describing the characterization of divisorial instability along a given Cartier divisor. This observation is an extended complementary form of [@Fujita:2016a Theorem 7.1].
\[characteriztion:instability\] Let $X$ be a ${\mathbb{Q}}$-Fano variety defined over ${\mathbf{K}}$, and fix an admissible flag of the form , with divisorial component $$Y_1 = E$$ an irreducible and reduced Cartier divisor over $X$, defined over ${\mathbf{F}}$, and supported on a normal proper model $$\pi \colon Y \rightarrow X_{\mathbf{F}}.$$ Let $$\Delta = \Delta(-{\mathrm{K}}_X) = \Delta_{Y_\bullet}(-{\mathrm{K}}_X)$$ be the Okounkov body of $-{\mathrm{K}}_X$, computed with respect to the flag $Y_\bullet$, and let $\mathbf{b}_1$ be the first coordinate of its barycentre. The following conditions are equivalent.
1. [$X$ is not divisorially stable along $E$;]{}
2. [$\mathbf{b}_1 {\geqslant}1$; and]{}
3. [$\beta(-{\mathrm{K}}_X,E) {\geqslant}1$.]{}
The equivalence of all three assertions follows by combining Lemma \[stability:lemma\] and Proposition \[barycentre:volume:constant:prop\], applied to $L = - {\mathrm{K}}_X$.
Examples of divisorial unstable ${\mathbb{Q}}$-Fano varieties {#examples:divisorial:instability:QQ:Fano}
=============================================================
[[**.** ]{}]{}In this section, we briefly describe examples, including several from [@Fujita:2016a], which pertain to divisorial instability for ${\mathbb{Q}}$-Fano varieties along divisors. The fact that these examples are divisorially unstable along a given divisor follows, for example, from Theorem \[characteriztion:instability\] together with a calculation of the asymptotic volume constant.
[[**.** ]{}]{}[**Examples.**]{} To begin with, we mention three well known examples, which may be obtained by direct methods.
1. [ If $X = {\mathbb{P}}^n$ and $H$ is a hyperplane, then it follows directly from the Definition \[asym:vol:constant\] that $$\beta(-{\mathrm{K}}_{X}, H) = 1.$$ Thus, $X = {\mathbb{P}}^n$ is a ${\mathbb{Q}}$-Fano variety which is not divisorially stable along $H$. ]{}
2. [ If $X = {\mathbb{P}}^1 \times {\mathbb{P}}^1$ and $E$ an exceptional divisor that is obtained by blowing up a closed point, then $$-{\mathrm{K}}_X = {{\mathcal O}}_X(2,2)$$ and so $$\beta(-{\mathrm{K}}_X, E) = 4/2$$ by [@McKinnon-Roth Example before Lemma 4.1]. ]{}
3. [ Let $E_1,\dots, E_q$ be effective Cartier divisors on a $d$-dimensional ${\mathbb{Q}}$-Fano variety $X$, defined over ${\mathbf{K}}$, and in general position. Suppose further, that each of the $E_i$ are linearly equivalent to a fixed ample divisor on $X$ and that $$-{\mathrm{K}}_X = E_1 + \hdots + E_q.$$ Then, combining the discussion that follows the Analytic General Theorem of [@Ru:Vojta:2016], see also [@Autissier:2011], together with results from [@Grieve:2018:autissier] and [@Grieve:chow:approx], we obtain that $$\beta(-{\mathrm{K}}_X,E_i) = \frac{q}{d + 1}.$$ ]{}
[[**.** ]{}]{}[**Examples.**]{} The case of divisorial stability for ${\mathbb{Q}}$-Fano varieties is studied in detail in [@Fujita:2016a]. Here, we summarize some representative examples of divisorial instability.
1. [ It follows from [@Fujita:2016a Corollary 6.3] that each toric ${\mathbb{Q}}$-Fano variety $X$ is divisorially unstable along some torus invariant divisor. ]{}
2. [ As two more explicit examples which include instances of (i), we mention that if $\widetilde{X}$ is the blowing-up of ${\mathbb{P}}^2$ at a closed point with exceptional divisor $E$, then $\widetilde{X}$ is not divisorially stable along $E$, see for instance [@Fujita:2016a Example 1]. Further, if $\widetilde{X}$ is the blowing-up of ${\mathbb{P}}^2$ along two distinct points with exceptional divisors $E_1$ and $E_2$ and if $E_0$ denotes the strict transform of the line passing through the centres of the blowing-up, then $\widetilde{X}$ is not divisorially stable along $E_0$. ]{}
3. [ Divisorial stability for Fano $3$-folds is studied in detail in [@Fujita:2016a Theorem 10.1]. For example, there it is proven that each Fano threefold with Picard number at least $6$ is not divisorially stable. ]{}
On ${\mathrm{K}}$-stability for Fano varieties and Vojta’s Main Conjecture {#K:stable:Fano}
============================================================================
[[**.** ]{}]{}In this section, we show how the property that a ${\mathbb{Q}}$-Fano variety $X$, with *canonical singularities*, is not ${\mathrm{K}}$-stable, is related to validity of Vojta’s Main Conjecture, when applied to a certain collection of divisorial valuations.
[[**.** ]{}]{}In general, the extent to which such a ${\mathbb{Q}}$-Fano variety is ${\mathrm{K}}$-stable is not obvious. Fix a divisorial valuation $\nu$ over $X$ with field of definition ${\mathbf{F}}$ a finite extension of ${\mathbf{K}}$ contained in $\overline{{\mathbf{K}}}$. In particular, the valuation $\nu$ determines an irreducible Cartier divisor over $X$ together with a birational divisor as well as a divisorial filtration of the anti-canonical ring. Denote these respective objects by $E$, $\mathbb{D}(\nu)$ and $\mathcal{F}^\bullet R(X, - {\mathrm{K}}_X)$.
[[**.** ]{}]{}Recall, that the filtration $\mathcal{F}^\bullet R(-{\mathrm{K}}_X)$ is defined over ${\mathbf{F}}$. It turns out that $$\label{divisorial:asymptotic}
\beta_\nu(-{\mathrm{K}}_X) = \beta(-{\mathrm{K}}_X,E) \text{,}$$ the *asymptotic volume constant* of $(X, - {\mathrm{K}}_X)$ with respect to $\nu$, is related to the property that $X$ is ${\mathrm{K}}$-stable. Furthermore, by work of K. Fujita [@Fujita:2016] (and others), it is known that ${\mathrm{K}}$-stability for the polarized variety $(X, - {\mathrm{K}}_X)$ can be characterized in terms of the *log discrepancies* together with the quantity .
[[**.** ]{}]{}Let $\mathbb{D}(\nu)$ be the (prime) birational divisor that is determined by $\nu$. Fix a normal proper model $$\pi \colon X' \rightarrow X$$ for which $\mathbb{D}(\nu)$ has trace a Cartier divisor $$\mathbb{D}(\nu)_{X'} = E \text{.}$$ The *discrepancy* of $\nu$ is the well-defined quantity $$\label{discrepancy:eqn:defn}
a(X,E) = a(X, \nu) := \operatorname{ord}_{\mathbb{D}(\nu)_{X'}}({\mathrm{K}}_{X'/ X}).$$ In , the divisor ${\mathrm{K}}_{X' / X}$ denotes the *relative canonical divisor* $${\mathrm{K}}_{X' / X} := \pi^*{\mathrm{K}}_X - {\mathrm{K}}_{X'} \text{.}$$ Recall, that $X$ having *log-terminal singularities* means that all such discrepancies are strictly greater than minus one; the condition that $X$ has *canonical singularities* means that these quantities are all nonnegative.
[[**.** ]{}]{}By reformulating results which were established by Fujita, [@Fujita:2016], the condition that a ${\mathbb{Q}}$-Fano variety $X$ be ${\mathrm{K}}$-stable can be expressed as Theorem \[Fano:K:stability\] below. Before stating this result, using terminology that is consistent with [@Fujita:2016 Definition 1.3], we say that a Cartier divisor $E$ over $X$, supported on some normal proper model $$\pi \colon Y \rightarrow X_{\mathbf{F}}\text{, }$$ and having field of definition ${\mathbf{F}}$, is *dreamy* if the bigraded algebra $$\bigoplus_{m,\ell \in {\mathbb{Z}}_{{\geqslant}0}} {\mathrm{H}}^0\left(Y, - \pi^* rm {\mathrm{K}}_X - \ell E\right)$$ is finitely generated for some $r \in {\mathbb{Z}}_{{\geqslant}0}$, which has the property that $-r {\mathrm{K}}_X$ is a Cartier divisor.
[[**.** ]{}]{}Having fixed such terminology, the characterization of ${\mathrm{K}}$-stability that was established in [@Fujita:2016] may be formulated in the following manner.
\[Fano:K:stability\] A ${\mathbb{Q}}$-Fano variety $(X,-{\mathrm{K}}_X)$ is not ${\mathrm{K}}$-stable if and only if the inequality $$\label{Fano:K:unstable:eqn}
a(X,\nu) {\leqslant}\beta_\nu(-{\mathrm{K}}_X) - 1$$ holds true for at least one such dreamy divisorial valuation $\nu$ over $X$, defined over some finite extension of the base number field ${\mathbf{K}}$.
In [@Fujita:2016 Theorem 1.6], it is shown that such a ${\mathbb{Q}}$-Fano variety $X$ is ${\mathrm{K}}$-stable if and only if $$\beta_\nu(-{\mathrm{K}}_X) > a(X, \nu) + 1$$ for all such dreamy divisorial valuations $\nu$. (Note that the condition there is expressed in terms of the log discrepancies as opposed to the traditional concept of discrepancies, [@Kollar:Mori:1998], which we adopt here.) In particular, Theorem \[Fano:K:stability\], as stated here, is simply an equivalent formulation of [@Fujita:2016 Theorem 1.6].
[[**.** ]{}]{}[**Remark.**]{} Conditions of ${\mathrm{K}}$-stability for a ${\mathbb{Q}}$-Fano variety $X$ may be expressed in terms of its $\delta$-invariant $$\label{delta:invariant}
\delta(X) := \delta(-{\mathrm{K}}_X) = \inf_E
\frac{a(X,E) + 1}{\beta(-{\mathrm{K}}_X,E) } \text{.}$$ Here, in , we take the infimum over all prime Cartier divisors $E$ over $X_{\overline{{\mathbf{K}}}}$. We refer to [@Park:Won:2018], and the references therein, for a more detailed discussion about these topics. For example, it is conjectured that a ${\mathbb{Q}}$-Fano variety $X$ is ${\mathrm{K}}$-stable if and only if $\delta(X) > 1$ [@Park:Won:2018 Conjecture 1.6]. It is known that the condition of ${\mathrm{K}}$-semi-stability is characterized by the condition that $\delta(X) {\geqslant}1$, whereas the concept of *uniform ${\mathrm{K}}$-stability*, in the sense of [@Boucksom-Hisamoto-Jonsson:2016], is characterized by the condition that $\delta(X) > 1$. Establishing such results is the subject of [@Blum:Jonnson:2017].
[[**.** ]{}]{}[**Examples.**]{} The work [@Park:Won:2018] studies the $\delta$-invariant of del Pezzo surfaces.
1. [ In particular, [@Park:Won:2018 Section 6], gives examples of del Pezzo surfaces with $\delta$-invariant at most equal to one. More precisely, there it is shown that if $S$ is the del Pezzo surface of degree seven, obtained by blowing-up ${\mathbb{P}}^2$ at the two points $p_1 = [0:0:1]$ and $p_2 = [0:1:0]$, then $$\delta(S) {\leqslant}21/25.$$ ]{}
2. [ It is also shown in [@Park:Won:2018 Section 6] that the Hirzebruch surface $\mathbb{F}_1$, which is obtained by blowing-up $\mathbb{P}^2$ at the point $p_1 = [0:0:1]$, has the property that $$\delta(\mathbb{F}_1) {\leqslant}6/7.$$ ]{}
[[**.** ]{}]{}Theorem \[Fano:K:stability\] has the following consequence which we use in the course of proving Theorems [\[Vojta:canonical:Fano:not:K:stable\]]{} and \[Vojta:canonical:Fano:not:K:stable:b\].
\[canonical:Fano:Not:K:Stable\] Suppose that $X$ is a ${\mathbb{Q}}$-Fano variety with canonical singularities. If $X$ is not ${\mathrm{K}}$-stable, then $$\label{canonical:Fano:K:unstable:eqn}
\beta_\nu(-{\mathrm{K}}_X) {\geqslant}1$$ for at least one divisorial valuation $\nu$ over $X$ and having field of definition some finite extension of the base number field ${\mathbf{K}}$.
If $X$ has canonical singularities, then $$\label{canonical:singularities:hypothesis}
a(X,\nu) {\geqslant}0$$ and so Corollary \[canonical:Fano:Not:K:Stable\] follows from Theorem \[Fano:K:stability\] because combining with , we obtain the desired inequality $$\beta_\nu(-{\mathrm{K}}_X) {\geqslant}1\text{.}$$
Birational Weil function preliminaries
======================================
[[**.** ]{}]{}In this article, our main interest in divisorial instability for ${\mathbb{Q}}$-Fano varieties is because of its relation to Vojta’s Main Conjecture (see Theorems \[Vojta:canonical:Fano:not:D:stable:barycentre\] and \[Vojta:canonical:Fano:not:D:stable\]). The key points to obtaining these observations, are results that were proved in [@Grieve:2018:autissier Corollary 1.3] and, independently, in [@Ru:Vojta:2016 Corollary 1.12].
[[**.** ]{}]{}We state our result in Section \[Vojta:Instability:Section\] and, there, we find it convenient to express our result using the birational Weil function formalism. In particular, in this section, we fix our arithmetic preliminaries and develop further the basic theory of birational Weil functions that has been given by Ru and Vojta, in [@Ru:Vojta:2016], and extending earlier insights of Vojta [@Vojta:1996].
[ **Absolute values, product formula and height functions.** ]{}
[[**.** ]{}]{}\[prod:form:assump\] Let ${\mathbf{K}}$ be a number field with algebraic closure $\overline{{\mathbf{K}}}$. We let $M_{{\mathbf{K}}}$ be the set of absolute values on ${\mathbf{K}}$, which is constructed following the conventions of [@Bombieri:Gubler]. In particular, $M_{\mathbf{K}}$ satisfies the product formula with multiplicities equal to one $$\prod_{v \in M_{\mathbf{K}}} |x|_v = 1 \text{, }$$ for all $x \in {\mathbf{K}}^\times$. At times, we find it convenient to identify elements of $M_{{\mathbf{K}}}$ with the places of ${\mathbf{K}}$ that they determine.
[[**.** ]{}]{}\[finite:extension:field:conventions\] Let ${\mathbf{F}}/ {\mathbf{K}}$ be a finite extension, with ${\mathbf{K}}\subseteq {\mathbf{F}}\subseteq \overline{{\mathbf{K}}}$, and $w$ a place of ${\mathbf{F}}$ that lies over $v \in M_{\mathbf{K}}$. Put $$||\cdot||_w := |\mathrm{N}_{{\mathbf{F}}_w / {\mathbf{K}}_v}(\cdot)|_v$$ for $$\operatorname{N}_{{\mathbf{F}}_w / {\mathbf{K}}_v}(\cdot) \colon {\mathbf{F}}_w \rightarrow {\mathbf{K}}_v$$ the norm from ${\mathbf{F}}_w$, the completion of ${\mathbf{F}}$ with respect to $w$, to ${\mathbf{K}}_v$ the completion of ${\mathbf{K}}$ with respect to $v$. In particular, setting $$|\cdot|_{w,{\mathbf{K}}} := ||\cdot||_w^{\frac{1}{[{\mathbf{F}}_w : {\mathbf{K}}_v ]}} \text{,}$$ we obtain an absolute value on ${\mathbf{F}}$ that extends $|\cdot|_v$.
[[**.** ]{}]{}Our conventions about height functions are similar to those of [@Grieve:2018:autissier], [@Bombieri:Gubler] and [@Lang:Diophantine]. For example, the logarithmic height of a ${\mathbf{K}}$-rational point $$\mathbf{x} = [x_0:\dots:x_n] \in {\mathbb{P}}^n({\mathbf{K}})\text{, }$$ of projective $n$-space, with respect to the tautological line bundle ${{\mathcal O}}_{{\mathbb{P}}^n}(1)$, is defined to be $$\label{log:height:functions}
h_{{{\mathcal O}}_{{\mathbb{P}}^n}(1)}(\mathbf{x}) = \sum_{v \in M_{\mathbf{K}}} \log \max_i |x_i|_v.$$
[[**.** ]{}]{}By pulling back the quantity , we obtain a concept of (logarithmic) height functions for polarized varieties $(X,L)$ over ${\mathbf{K}}$. In general, by writing an arbitrary line bundle, on $X$ and defined over ${\mathbf{K}}$, as the difference of two ample line bundles, we obtain a concept of logarithmic height for each line bundle on $X$, defined over ${\mathbf{K}}$.
[ **Birational divisors and fields of definition.** ]{}
[[**.** ]{}]{}Let $X$ be a geometrically integral, geometrically normal projective variety over ${\mathbf{K}}$. By a *model* of $X$, we mean a proper birational morphism $Y \rightarrow X$, defined over ${\mathbf{K}}$, where $Y$ is a geometrically integral, geometrically normal variety over ${\mathbf{K}}$. More generally, we may also consider models of $X$, or more precisely models of $X_{\mathbf{F}}$, that are defined over ${\mathbf{F}}$. We let ${\mathbf{K}}(X)$ denote the function field of $X$ and ${\mathbf{F}}(X)$ the function field of $X_{\mathbf{F}}$.
[[**.** ]{}]{}We represent birational ${\mathbb{Q}}$-divisors $\mathbb{D}$ over $X$ as an equivalence class of pairs $(Y,D)$, where $Y$ is a model of $X$ and where $D$ is a ${\mathbb{Q}}$-Cartier divisor on $Y$. In particular, $D$ is the *trace* of $\mathbb{D}$ on $Y$. Furthermore, the equivalence classes are those for which the equivalence relation is that generated by $$(Y_1,D_1) \sim (Y_2,D_2)$$ whenever $Y_1$ dominates $Y_2$ via a morphism $$\phi \colon Y_1 \rightarrow Y_2$$ and $$D_1 = \phi^* D_2.$$ More generally, we may also consider birational divisors $\mathbb{D}$ over $X$, defined over ${\mathbf{F}}$.
[ **Birational Weil functions.** ]{}
[[**.** ]{}]{}We represent the *local birational Weil function* of a birational divisor $\mathbb{D}$, with respect to a place $v$ of ${\mathbf{K}}$, by $\lambda_{\mathbb{D},v}(\cdot)$. Such functions may also be realized as equivalence classes of pairs $$(Y,\lambda_{D,v}(\cdot))\text{,}$$ for $(Y,D)$ a pair corresponding to $D$, and $\lambda_{D,v}(\cdot)$ a local Weil function for $D$ with respect to $v$.
[[**.** ]{}]{}Following the approach of [@Bombieri:Gubler], the idea is to define local birational Weil functions by using equivalence classes of presentations. By a *presentation* of a b-Cartier divisor $$\mathbb{D} = (Y, D)$$ on $X$, we mean an equivalence class of presentations $$\mathbf{D} = (Y, \mathcal{D}).$$ Here, $$\mathcal{D} = (s_D;L;\mathbf{s};M;\mathbf{t})$$ is a presentation for $D$ considered as a Cartier divisor on $Y$.
[[**.** ]{}]{}The equivalence classes of presentations are those for the equivalence relation that is generated by $$(Y_1,\mathcal{D}_1) \sim (Y_2,\mathcal{D}_2)$$ whenever $Y_1$ dominates $Y_2$ via a morphism $$\phi \colon Y_1 \rightarrow Y_2$$ and where $\mathcal{D}_1$ is related to $\mathcal{D}_2$ via $$\mathcal{D}_1 = \phi^* \mathcal{D}_2 \text{.}$$
[[**.** ]{}]{}Fix a birational equivalence class of presentations denoted as $$[\mathcal{D}] = (Y, \mathcal{D}).$$ We then may form $$[\lambda_{\mathcal{D}}(\cdot,v)] = (Y,\lambda_{\mathcal{D}}(\cdot,v)),$$ the birational equivalence class of local Weil functions. Here, $\lambda_{\mathcal{D}}(\cdot,v)$ is the Weil function on $Y$ determined by $\mathcal{D}$. Again, the equivalence relation is governed by $$(Y_1,\lambda_{\mathcal{D}_1}(\cdot,v)) \sim (Y_2, \lambda_{\mathcal{D}_2}(\cdot,v))$$ whenever $$\lambda_{\mathcal{D}_1}(\cdot,v) = \phi^* \lambda_{\mathcal{D}_2}(\cdot, v) = \lambda_{\phi^* \mathcal{D}_2}(\cdot,v)$$ for $Y_1$ and $Y_2$ models of $X$ with $Y_1$ dominating $Y_2$ via a morphism $$\phi \colon Y_1 \rightarrow Y_2$$ and, again, the presentations $\mathcal{D}_1$ and $\mathcal{D}_2$ are related by $$\mathcal{D}_1 = \phi^* \mathcal{D}_2.$$
[[**.** ]{}]{}Finally, similar to [@Grieve:2018:autissier], we may construct birational Weil functions $\lambda_{\mathbb{D},v}(\cdot)$ for $\mathbb{D}$ a birational divisor over $X$ and defined over ${\mathbf{F}}$. Indeed, let $w$ be place of ${\mathbf{F}}$ over $v$. Furthermore, write $$\mathbf{D} = (Y, \mathcal{D})$$ for an equivalence class of presentations defined over ${\mathbf{F}}$. Again $$\mathcal{D} = (s; M, \mathbf{s}; N, \mathbf{t})$$ is a presentation of ${{\mathcal O}}_Y(D)$, for $Y$ a model of $X$ (defined over ${\mathbf{F}}$). From this perspective, the birational Weil function $\lambda_{\mathbb{D}, v}(\cdot)$ has shape $$\lambda_{\mathbb{D}}(\cdot,v) = \max_k \min_\ell \log
\left|
\frac{s_k}{t_\ell s} (\cdot)
\right|_{v,{\mathbf{K}}}
= \max_k \min_\ell \log \left| \left| \frac{s_k}{t_\ell s} (\cdot) \right| \right|_w^{\frac{1}{[{\mathbf{F}}_w : {\mathbf{K}}_v ] }} \text{.}$$
Instability and consequences for Vojta’s Main Conjecture {#Vojta:Instability:Section}
========================================================
[[**.** ]{}]{}The purpose of this section, is to discuss Vojta’s Main Conjecture within the context of divisorial instability for Fano varieties. In doing so, we extend and clarify previous results on this topic, for example [@Grieve:2018:autissier] and [@Ru:Vojta:2016]. Throughout this section, we fix a finite set of places $S$ of ${\mathbf{K}}$. Again, ${\mathbf{F}}/ {\mathbf{K}}$, denotes a finite extension of fields, ${\mathbf{K}}\subseteq {\mathbf{F}}\subseteq \overline{{\mathbf{K}}}$.
[[**.** ]{}]{}To begin with, we observe how the asymptotic volume constant may be used to establish instances of Vojta’s Main Conjecture. Such observations have already been made in [@Grieve:2018:autissier] and, independently, in [@Ru:Vojta:2016]. Here, we establish a slight variant of those results which are adept for our purposes here.
\[Vojta:canonical:Fano:not:D:stable:Prop\] Let ${\mathbf{K}}$ be a number field and fix a finite set of places $S$ of ${\mathbf{K}}$. Let $X$ be a ${\mathbb{Q}}$-Fano variety defined over ${\mathbf{K}}$ and let $E$ be a reduced, irreducible Cartier divisor over $X$ and defined over ${\mathbf{F}}/ {\mathbf{K}}$, a finite extension of ${\mathbf{K}}$, with the property that ${\mathbf{K}}\subseteq {\mathbf{F}}\subseteq \overline{{\mathbf{K}}}$. Let $\mathbb{E}$ be the birational divisor that is determined by $E$ and assume that the asymptotic volume constant $\beta(-{\mathrm{K}}_X,E)$ is at least equal to $1$. Then the inequalities predicted by Vojta’s Main Conjecture hold true for $\mathbb{E}$. Precisely, if $B$ is a big line bundle on $X$ and $\epsilon > 0$, then $$\label{Vojta:Inequality:Eqn:Prop}
\sum_{v \in S} \lambda_{\mathbb{E},v}(x) + h_{{\mathrm{K}}_X}(x) {\leqslant}\epsilon h_B(x) + \mathrm{O}(1)$$ for all ${\mathbf{K}}$-rational points $x \in X \setminus Z$ and $Z \subsetneq X$ some proper Zariski closed subset defined over ${\mathbf{K}}$.
[[**.** ]{}]{}In particular, we now use results from [@Grieve:2018:autissier] to prove Theorem \[Vojta:canonical:Fano:not:D:stable\].
Let $$\pi \colon X' \rightarrow X_{\mathbf{F}}$$ be a normal proper model of $X$ with the property that $E$ is supported on $X'$. By assumption $$\label{K:unstable:Fano:b}
\beta(-{\mathrm{K}}_X,E) {\geqslant}1.$$
We now note that, without loss of generality, by arguing as in [@Grieve:2018:autissier Proof of Theorem 5.1], to establish the desired inequality , it suffices to establish the inequality $$\label{Vojta:Inequality:Eqn:reduction}
\sum_{v \in S} \lambda_{\mathbb{E}, v}(x) + h_{{\mathrm{K}}_X}(x) {\leqslant}\epsilon h_{-{\mathrm{K}}_X}(x) + \mathrm{O}(1)$$ for all ${\mathbf{K}}$-rational points $x \in X$ outside of some proper Zariski closed subset $Z \subsetneq X$.
To establish , note that the inequality is implied by the inequality $$\label{Vojta:Inequality:Eqn:reduction'}
\sum_{v \in S} \lambda_{\mathbb{E}_{X'},v}(x') {\leqslant}(\epsilon + 1)h_{- \pi^* {\mathrm{K}}_{X'}}(x') + \mathrm{O}(1)$$ for all ${\mathbf{K}}$-rational points $x' \in X'$ outside of some proper Zariski closed subset $Z' \subsetneq X'$.
Recall that the asymptotic volume constant $\beta(-{\mathrm{K}}_X, E)$ is birationally invariant. In particular $$\label{asymptotic:volume:birational:invariant}
\beta(-{\mathrm{K}}_X, E) = \beta(- \pi^* {\mathrm{K}}_X, E);$$ it follows, by applying the Main Arithmetic General Theorem of [@Grieve:2018:autissier], compare also with [@Ru:Vojta:2016], that the inequality $$\label{Main:Arithmetic:General:Theorem}
\sum_{v \in S} \lambda_{\mathbb{E}_{X'},v}(x') {\leqslant}\left( \frac{1}{\beta(-{\mathrm{K}}_X, E)} + \epsilon \right) h_{- \pi^* {\mathrm{K}}_X}(x') + \mathrm{O}(1)$$ holds true for all $x' \in X'$ outside of some proper Zariski closed subset $Z' \subsetneq X'$, and defined over ${\mathbf{K}}$.
By assumption, we have the inequality ; it follows that the righthand side of the inequality is at most the righthand side of the inequality . In particular, by combining these inequalities and we obtain the inequality $$\label{Vojta:Inequality:Eqn:reduction''}
\sum_{v \in S} \lambda_{\mathbb{E}_{X'},v}(x')
{\leqslant}\left( \frac{1}{\beta(-{\mathrm{K}}_X, E)} + \epsilon \right) h_{- \pi^* {\mathrm{K}}_X}(x') + \mathrm{O}(1)
{\leqslant}(1 + \epsilon) h_{-\pi^* {\mathrm{K}}_X}(x') + \mathrm{O}(1)$$ for all ${\mathbf{K}}$-rational points $x' \in X'$ outside of some proper Zariski closed subset $Z' \subsetneq X'$. The inequality establishes the inequality and implies the desired inequality .
[[**.** ]{}]{}Using Proposition \[Vojta:canonical:Fano:not:D:stable:Prop\], we are able to establish Theorem \[Vojta:canonical:Fano:not:D:stable\] below.
\[Vojta:canonical:Fano:not:D:stable\] Let ${\mathbf{K}}$ be a number field and fix a finite set of places $S$ of ${\mathbf{K}}$. Let $X$ be a ${\mathbb{Q}}$-Fano variety defined over ${\mathbf{K}}$ and let $E$ be a reduced, irreducible Cartier divisor over $X$ and defined over ${\mathbf{F}}/ {\mathbf{K}}$, a finite extension of ${\mathbf{K}}$, with the property that ${\mathbf{K}}\subseteq {\mathbf{F}}\subseteq \overline{{\mathbf{K}}}$. Let $\mathbb{E}$ be the birational divisor that is determined by $E$ and assume that $X$ is not divisorially stable along $E$. In particular, if $B$ is a line bundle on $X$ and if $\epsilon > 0$, then $$\label{Vojta:Inequality:Eqn}
\sum_{v \in S} \lambda_{\mathbb{E},v}(x) + h_{{\mathrm{K}}_X}(x) {\leqslant}\epsilon h_B(x) + \mathrm{O}(1)$$ for all ${\mathbf{K}}$-rational points $x \in X \setminus Z$ and $Z \subsetneq X$ some proper Zariski closed subset defined over ${\mathbf{K}}$.
Let $$\pi \colon X' \rightarrow X_{\mathbf{F}}$$ be a normal proper model of $X$ with the property that $E$ is supported on $X'$. By assumption, $X$ is not divisorialy stable along $E$. By Theorem \[characteriztion:instability\], this is equivalent to the condition that $$\label{K:unstable:Fano}
\beta(-{\mathrm{K}}_X, E) {\geqslant}1 \text{.}$$ Because of , the hypothesis of Proposition \[Vojta:canonical:Fano:not:D:stable:Prop\] are satisfied. The conclusion of Proposition \[Vojta:canonical:Fano:not:D:stable:Prop\] is that which is desired by Theorem \[Vojta:canonical:Fano:not:D:stable\].
[[**.** ]{}]{}We are now able to prove Theorem \[Vojta:canonical:Fano:not:D:stable:barycentre\].
By Proposition \[barycentre:volume:constant:prop\], the condition that $$\beta(-{\mathrm{K}}_X, E) {\geqslant}1$$ is equivalent to the condition that $$\mathbf{b}_1 {\geqslant}1\text{.}$$ Thus, the conclusion desired by Theorem \[Vojta:canonical:Fano:not:D:stable:barycentre\] is implied by that of Theorem \[Vojta:canonical:Fano:not:D:stable\].
[[**.** ]{}]{}Next, we establish our main result in the direction of Vojta’s Main Conjecture within the context of ${\mathrm{K}}$-stability for Fano varieties. Recall that we stated that result as Theorem \[Vojta:canonical:Fano:not:K:stable\]. In particular, this result may also be seen as an improvement to the known previous existing results on this topic, for example [@McKinnon-Roth Theorem 10.1], [@Grieve:2018:autissier Corollary 1.3] and [@Ru:Vojta:2016 Corollary 1.12]. We restate this result as Theorem \[Vojta:canonical:Fano:not:K:stable:b\] below.
\[Vojta:canonical:Fano:not:K:stable:b\] Suppose that $X$ is a ${\mathbb{Q}}$-Fano variety with canonical singularities, defined over ${\mathbf{K}}$, and which is not ${\mathrm{K}}$-stable. Then $X$ admits an irreducible and reduced Cartier divisor $\mathbb{E}$, with field of definition some finite field extension ${\mathbf{F}}/{\mathbf{K}}$, with ${\mathbf{K}}\subseteq {\mathbf{F}}\subseteq \overline{{\mathbf{K}}}$, for which the inequalities predicted by Vojta’s Main Conjecture hold true.
By assumption, $X$ is a ${\mathbb{Q}}$-Fano variety with canonical singularities and is not ${\mathrm{K}}$-stable. By Corollary \[canonical:Fano:Not:K:Stable\], it follows that $$\beta(-{\mathrm{K}}_X, E) {\geqslant}1$$ for at least one irreducible and reduced Cartier divisor $E$ over $X$ with field of definition some finite field extension ${\mathbf{F}}/ {\mathbf{K}}$, ${\mathbf{K}}\subseteq {\mathbf{F}}\subseteq \overline{{\mathbf{K}}}$. Again, the hypothesis of Proposition \[Vojta:canonical:Fano:not:D:stable:Prop\] is satisfied and the conclusion desired by Theorems \[Vojta:canonical:Fano:not:K:stable\] and \[Vojta:canonical:Fano:not:K:stable:b\] hold true.
[10]{}
P. Autissier, *Sur la non-densité des points entiers*, Duke Math. J. **158** (2011), no. 1, 13–27.
H. Blum and M. Jonsson, *Thresholds, valuations, and [$\mathrm{K}$]{}-stability*, arXiv:1706.04548v1.
E. Bombieri and W. Gubler, *Heights in [D]{}iophantine geometry*, Cambridge University Press, Cambridge, 2006.
S. Boucksom, T. Hisamoto, and M. Jonsson, *Uniform [$\mathrm{K}$]{}-stability, [D]{}uistermaat-[H]{}eckman measures and singularities of pairs*, Ann. Inst. Fourier (Grenoble) **67** (2017), no. 2, 743–841.
K. Fujita, *On [$\mathrm{K}$]{}-stability and the volume functions of [$\Bbb{Q}$]{}-[F]{}ano varieties*, Proc. Lond. Math. Soc. (3) **113** (2016), no. 5, 541–582.
[to3em]{}, *A valuative criterion for uniform [$\mathrm{K}$]{}-stability of [$\mathbb{Q}$]{}-[F]{}ano varieties*, J. reine angew. Math. (Ahead of Print).
N. Grieve, *Diophantine approximation constants for varieties over function fields*, Michigan Math. J. **67** (2018), no. 2, 371–404.
[to3em]{}, *Expectations, concave transforms, [C]{}how weights, and [R]{}oth’s theorem for varieties*, Submitted (2018).
[to3em]{}, *On arithmetic general theorems for polarized varieties*, Houston J. Math. **44** (2018), no. 4, 1181–1202.
J. Kollár and S. Mori, *Birational geometry of algebraic varieties*, Cambridge University Press, Cambridge, 1998.
S. Lang, *Fundamentals of [D]{}iophantine geometry*, Springer-Verlag, New York, 1983.
R. Lazarsfeld and M. Musta[ţ]{}[ǎ]{}, *Convex bodies associated to linear series*, Ann. Sci. Éc. Norm. Supér. (4) **42** (2009), no. 5, 783–835.
D. McKinnon and M. Roth, *Seshadri constants, diophantine approximation, and [R]{}oth’s theorem for arbitrary varieties*, Invent. Math. **200** (2015), no. 2, 513–583.
J. Park and J. Won, *[$\mathrm{K}$]{}-stability of smooth del [P]{}ezzo surfaces*, [Math. Ann.]{} **372** (2018), no. 3-4, 1339–1276.
M. Ru and P. Vojta, *A birational [N]{}evanlinna constant and its consequences*, Amer. J. Math. (To appear).
M. Ru and J. T.-Y. Wang, *An effective [S]{}chmidt’s subspace theorem for projective varieties over function fields*, Int. Math. Res. Not. IMRN (2012), no. 3, 651–684.
V. V. Shokurov, *[$3$]{}-fold log models*, J. Math. Sci. **81** (1996), no. 3, 2667–2699.
P. Vojta, *Integral points on subvarieties of semiabelian varieties, [I]{}*, Invent. Math. **126** (1996), 133–181.
J. T.-Y. Wang, *An effective [S]{}chmidt’s subspace theorem for function fields*, Math. Z. (2004), no. 246, 811–844.
[^1]: *Mathematics Subject Classification (2010):* 14G05; 14G40; 11G50; 11J97; 14C20.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Hyperspectral image(HSI) classification has been improved with convolutional neural network(CNN) in very recent years. Being different from the RGB datasets, different HSI datasets are generally captured by various remote sensors and have different spectral configurations. Moreover, each HSI dataset only contains very limited training samples and thus it is prone to overfitting when using deep CNNs. In this paper, we first deliver a 3D asymmetric inception network, AINet, to overcome the overfitting problem. With the emphasis on spectral signatures over spatial contexts of HSI data, AINet can convey and classify the features effectively. In addition, the proposed data fusion transfer learning strategy is beneficial in boosting the classification performance. Extensive experiments show that the proposed approach beat all of the state-of-art methods on several HSI benchmarks, including Pavia University, Indian Pines and Kennedy Space Center(KSC). Code can be found at: https://github.com/UniLauX/AINet [^1].'
author:
- 'Haokui Zhang$^{1,2^\dag}$, Yu Liu$^{2^\dag}$, Bei Fang$^{1}$, Ying Li$^{1}$, Lingqiao Liu$^{2}$ and Ian Reid$^{2}$ [^2]'
- First Author$^1$
- Second Author$^2$
- |
Third Author$^{2,3}$Fourth Author$^4$ $^1$First Affiliation\
$^2$Second Affiliation\
$^3$Third Affiliation\
$^4$Fourth Affiliation {first, second}@example.com, third@other.example.com, fourth@example.com
bibliography:
- 'reference.bib'
title: Hyperspectral Classification Based on 3D Asymmetric Inception Network with Data Fusion Transfer Learning
---
Introduction
============
For a long time, how to extract the useful information from HSI dataset itself is a very challenging task. At first, researchers mainly focus on extracting spectral signatures while totally ignoring the spatial contexts. Later on, there are two categories of methods to extract the spectral-spatial information from HSI dataset. The first category of methods are to extract the spectral signatures and the spatial contexts separately [@jia2015spectral; @benediktsson2005classification]. The second category of methods are to fuse the spectral signatures and the spatial contexts first, and then extract the concatenated information later [@qian2013hyperspectral]. Between those two categories of methods, it proves that the second category of methods are better in improving the performance of HSI classification.
Among traditional methods, handcrafted features usually are used, and they are expected to be discriminative and representative for capturing the characteristics of the HSI datasets. While in most cases, the extracted features are domain knowledge oriented, thus may lose some valuable details. For feature classification, support vector machine(SVM) [@melgani2004classification] is employed because SVM is robust at representing the high-dimensional vectors, but the capacity for representation is still limited to finite dimensions.
Since 2012, with the renaissance of deep learning, the performance of many vision tasks have been dramatically improved. The reason for the improvements can be mainly summarized to two aspects, that are deep neural network structures and huge training datasets. One of the most absorbing and significant advantage of deep learning is the ability to use CNN to extract features in different granularities. The potential of CNN has also been utilized in HSI classification, and it has been proved works better than the traditional methods. There are several different network structures have been proposed to combine CNN into HSI classification, including residual network in 2D and 3D [@zhong2018spectral], deep belief network(DBN) [@chen2015spectral], stacked autoencoder(SAE) [@chen2014deep] *el al*. All of the networks beat the traditional methods a bit, and which obviously point out that the irresistible trend of deep learning is also the right direction that HSI classification should be toward to.
Generally speaking, there are two constraints which limit the usage of state-of-the-art deep CNNs, which are already employed in vision tasks, directly into HSI classification. The first one is the different data formats between RGB image and HSI. In particular, comparing to the RGB image, which can be well represented by a 2D CNN to extract the features, it is much sensible to utilize a 3D CNN to preserve the abundant information being extracted from the spectral signatures and the spatial contexts of HSI. Moreover, all of those existing networks in HSI classification are quite shallow, almost less than five layers regarding the depth of the network layers, which is eligible comparing with the one thousand layers’ ResNet [@he2016deep] be used in other vision tasks. The reason behind that is the very limited HSI datasets. Specifically, both the capture and the annotation are cost-expensive and labor-exhaustive. Therefore, when deep CNNs are used in HSI classification, over-fitting is likely to occur.
In this paper, we propose a AINet and use two strategies to improve the HSI classification. Our contributions can be summarized as follows:
1. Firstly, a novel deep light-weight 3D CNN with asymmetric structure is delivered to handle HSI classification, which make it possible to use the existing small volume of HSI datasets to train the very deep neural network and fully exert the potential of CNN.
2. Secondly, data fusion transfer learning is exploited to conduct a better model initialization. Which is compensated for the data limitation and dramatically boost the training efficiency and classification performance.
Related Works {#sec:relatedwork}
=============
HSI classification
------------------
Classifying each pixel into its corresponding category is a vital problem for hyperspectral image analysis, and the applications of HSI classification covers object recognition [@zhang2016deep], mineral exploitation [@yokoya2016potential] and other relevant research fields.
### Conventional HSI Classification
Generally speaking, conventional pixel-wise HSI classification mainly focus on two aspects, that are feature extraction and feature classification. Since spectral signatures usually composed of at least hundreds of bands and carry rich information about HSI, it is important to acquire the features that are discriminative and representative for HSI dataset. Regarding to feature extraction, there are two kinds of approaches for the purpose. The first kind of approaches use linear algorithms to select the representative spectral bands, which including mainford ranking [@wang2016salient], multitask joint sparse representation [@yuan2016hyperspectral] *et al*. To sum up, features are usually the extracted spectral bands from the raw HSI dataset and the physical meaning can be retained. The second kind of approaches employ non-linear algorithms to extract the discriminative features, which including principle component analysis (PCA), identity component analysis (ICA) [et al]{}. The step after feature extraction is feature classification, support vector machine (SVM) is usually utilized because it is robust to the high-dimensional vectors [@zhang2012combining]. However, the limited capacity of SVM to simulate the distribution of spectral bands become a bottleneck for HSI classification.
Later on, with the development of technologies in remote sensing (RS), some works propose to integrate the spatial context into HSI classification. Considering the order for extracting spatial contexts, there are two branches: post-processing and spatial extraction. As its name, the post-processing approaches first extracting the spectral and spatial information sequentially, and then using the nearby spectral pixels as the smooth priors for the target spatial pixel, and a graphical model is used to conduct the classification [@tarabalka2010svm]. On the contrary, the spatial extraction methods emphasize to extract a 3D cube from both the spectral and spatial dimension, and SVM is used to execute the classification. Comparing with the post-processing approaches, spatial extraction approaches can achieve better performance, which may due to the reason that post-processing approaches often lose important information during the feature extraction. In addition, both methods share two drawbacks, one is the engineering features just preserve part of the information and can not fully represent the HSI dataset, the other is the traditional methods have limited capacity to fit the abundant information hied in HSI dataset.
### DL Based HSI Classification
Usually, a CNN model is composed of at least three convolutional layers for extracting features both in low-level and high level. In specific, with bottom-layers extracting the textures and edge details, and with the top-layers extracting the abstract shape information. However, traditional methods can only extract limited low-level features. The second advantage of CNN based methods compared with traditional methods for HSI classification is that, instead of separating the feature extraction and feature classification as two steps, the CNN structure integrating the feature extraction and feature classification into one framework through back-propagation, and since the extracted features directly contribute the final classification performance, deep learning methods achieve better performance than the traditional kernel based methods.
From the network structure perspective, there are several representative works for HSI classification. Stacked autoencoder(SAE) stack the extracted spectral and spatial features using layer-wise pretraining models [@chen2014deep]. Deep belief work(DBN) is also explored for HSI classification [@chen2015spectral]. However, both of the methods require to compress the spatial contexts as the 1D vector, which inevitably lose the important information about HSI. 2D-CNN network is directly transfered from the one used for vision tasks to HSI classification, and some researchers [@li2017hyperspectral; @tarabalka2009spectral] propose to use two parallel 2D-CNNs to extract the spectral signatures and the spatial contexts, both of the feature extraction for the spatial contexts and the separation between spectral and spatial channels will cause information lose. Moreover, 3D-CNN [@chen2016deep; @li2017spectral] is proposed to extract the spectral signatures and spatial contexts simultaneously from the HSI dataset, and conducting the classification on the 3D cube, but the 3D CNN models will suffer from over-fitting when the layers of the network become deeper, which mainly due to the very limited training dataset of HSI.
3D CNN Architectures
--------------------
The first 3D CNN network for HSI classification is proposed by Chen *et al* in 2016 [@chen2016deep], the ${L}_{2}$-norm regularization and dropout are used. However, the layers of the network is very shallow and it still come across the over-fitting problem when the available annotated datasets are scarce. A similar work is the spectral-spatial residual network(SSRN) [@zhong2018spectral], which bring the widely used residual blocks from other vision tasks into HSI classification, and combining the batch normalization to achieve a better performance. The disadvantage of SSRN is that it do not explicitly considering the contribution difference between spectral signatures and spatial contexts. our proposed asymmetric residual network beat SSRN with huge gains, which not only benefit from the much deeper and light-weight network design, but also due to the AI unit we come up with that is tailored for HSI dataset.
Transfer Learning
-----------------
Comparing with the thousands of million annotated datasets used in some vision tasks, the existing available HSI annotated datasets are quite scarce. Moreover, the serve unbalance among HSI datasets of intra-class and captured from different sensors also make it challenging to train the neural network for HSI classification. In vision community, one common solution for this problem is the transfer learning. The work principle behind transfer learning is that, in deep neural network, the bottom-level and middle-level features taking up the majority of the parameters stored in the CNN model, and usually capturing the textures and edges of the objects. And those low-level features designed for simple task like detection can be reused by much complex tasks such as segmentation and tracking. The most significant benefit of the usage of transfer learning is obtaining a better model initialization, which is very important for training the model with limited samples.
Our proposed network taking use of data fusion transfer learning strategy. In specific, the designed model is pretrained on HSI datasets captured by different sensors with 3D pyramid pooling, and then fine-tuned on the target datasets to achieve a better performance.
Methodology
===========
Among the deep learning models employed in the HSI literatures, 3D-CNNs are much suitable than 2D-CNNs for HSI classification due to fact that HSI with the 3D data format. In fact, different objects in HSI generally have different spectral structures. Convolving along the spectral dimension is very important. In addition, there are also some different objects which have similar spectral structures. For these objects, convolving along the spatial dimension to capture features is also beneficial.
For 2D-CNNs based methods, on the one hand, without spectral dimension reduction, the number of parameters of 2D-CNNs will be extremely large due to that HSIs generally have hundreds of bands. On the other hand, if the dimension reduction is conducted, it may destroy the information of spectral structure which is important for discriminating different objects.
Generally speaking, 3D-CNN based approaches have better performance than 2D-CNN based approaches [@chen2016deep; @zhong2018spectral]. However, despite being much accurate, the existing 3D-CNN based approaches still have two shortcomings. 1) Comparing with 2D convolutions, 3D convolutions have more parameters and 3D CNN models are computation-intensive. 2) Being limited by the training samples in HSI datasets, 3D-CNN models employed in HSI classification almost consist of less than five convolution layers. Although a large number of experiments in computer vision have proved that deep depth of CNN is very important for improving the performance of tasks related to image processing.
In this section, we start with the description of the proposed AINet, and end with a introduction to the proposed data fusion transfer learning strategy.
AINet for HSI Classification
----------------------------
![Framework of AINet. On the left, the $L\times S\times S$-sized samples from the neighborhood window centered around each target pixel are extracted first, and then the samples are fed to AINet to extract deep spectral-spatial features. Finally, the classification scores are calculated by the classifier.[]{data-label="fig:AINet"}](Fig2.eps){height="4cm"}
**Network Structure** Figure \[fig:AINet\] shows the overall framework of the proposed AINet for HSI classification. In order to fully utilize the spectral and spatial informations contained in HSI, we extract $L\times S\times S$-sized cubes from raw HSI as samples, where $L$ and $S$ respect the number of spectrum bands and the spatial size accordingly (Following [@chen2016deep], we set $S$ to 27 in this paper). Then the samples are fed to AINet to extract deep spectral-spatial features and to calculate classification scores. Inspired by the design of ResNet [@he2016deep], AINet employs a similar basic structure and introduces some key modifications for tailoring on HSI dataset. AINet starts with a 3D convolution layer, then stacks six AI units with increasing widths, and connects one 3D spatial pyramid pooling and one fully connected layer at the end. Specifically, the channels for the six AI units are 32, 64, 64, 128, 128 and 256 accordingly. In order to reduce the dimension of features, four Max pooling layers are added with kernel=\[3, 3, 3\], stride=\[2, 2, 2\] within the six AI units.
**3D Pyramid Pooling** Before the fully connected layer, there is a 3D pyramid pooling for mapping features with different sizes to vectors with fixed dimensions. Different HSI datasets are usually captured by different sensors and with various numbers of spectrum bands, for example, the Pavia University dataset has 103 bands and the Indian Pines dataset contains 200 bands. With 3D pyramid pooling layer, the same network can be applied on different HSI datasets without any modification. In this paper, 3D spatial pyramid pooling layer is composed of three level pooling ($1\times 1 \times 1$, $2\times 1\times 1$, $3\times 1\times 1$). As the last AI unit has 256 channels, the outputs of 3D spatial pyramid pooling layer are $256\times 6\times 1\times 1$-sized cubes.
**Training and Loss** We employ log\_softmax [@de2015exploration] as the activation function in the fully connected layer. During training, we take negative log likelihood as the loss function, and add ${L}_{2}$ regularization term with weight $1e-5$ to the loss function for alleviating over-fitting. And the optimizer is stochastic gradient descent (SGD) with momentum [@krizhevsky2012imagenet]. For all of the experiments, the same setting is adopted, where momentum, weight decay, batch size, epochs and learning rate are 0.9, 1e-5, 20, 60 and 0.01 respectively. During the last 12 epochs, the learning rate is decreased to 0.001.
AI Unit
-------
![Illustration of a AI unit. In AI unit, 3D convolution layer is replaced with two asymmetric inception units, that are space inception unit and spectrum inception unit. In space inception unit, the input cube is fed to three different paths. With path one has a point wise convolution layer only, path two consists of one point wise convolution layer and one 2D convolution layer, and path three has one point wise convolution layer and two 2D convolution layers. The outputs of each path are concatenated in channel, and are added to the output of the shortcut connection. The structure of spectrum inception unit is similar with the space inception unit, except that $1\times 3\times 3$-sized convolution layers are replaced with $3\times 1\times 1$-sized convolution layers in spectrum inception unit.[]{data-label="fig:AIUnit"}](Fig3.eps){height="2.4cm"}
Due to 3D convolution can learn spectral and spatial information from raw HSI dataset, 3D-CNN based methods can obtain the state-of-the-art performance for HSI classification. However, 3D convolutions are prone to over-fitting and computation-intensive compared with the 2D convolutions. In order to address these problem, we propose the asymmetric inception unit (AI unit), which is consist of the space inception unit and the spectrum inception unit. The structure of AI unit is illustrated in Figure \[fig:AIUnit\]. In the space inception unit, there are three space convolution paths. Path one has one point wise convolution layer only, path two consists of one point wise convolution layer and one 2D convolution layer with $1\times 3 \times 3$-sized kernels, and path three has one point wise convolution layers and two 2D convolution layers. The outputs of each path are concatenated in channel, and are added to the output of the shortcut connection. Inspired by the Inception networks [@szegedy2017inception], we set the three paths with different widths. For each unit, we set the widths of three paths with a split ratio 1:2:1. In the last two paths, the width of point wise convolution layer is half of that of the other convolution layers. For instance, in the AI unit with 32 channels, the width of the first path is 8, and for the second path, the widths of the point wise convolution layer and $1\times 3\times 3$-sized convolution layer are 8 and 16 respectively , the widths of the three layers of the last path are 4, 8 and 8 accordingly. In the overall structure, the structure of spectrum inception unit is similar with the space inception unit, except that $1\times 3\times 3$-sized 2D convolution layers in space inception unit are replaced with $3\times 1\times 1$-sized 1D convolution layers.
{height="4.6cm"}
\[fig:AIunit\_2\]
In HSI datasets, the resolution of spectrum is much higher than that of space and the information of spectrum is much richer. Therefore, during spectral-spatial features extraction, we are paying more attention on spectral feature extraction. In the proposed AINet, there are six AI units. The four units located in the middle can be divided into two groups and each group stacks two units with equal width. Here, instead of stack two same AI units in each group, we stack one space inception unit and two spectrum inception units. This is different from some popular networks, such as ResNet[@he2016deep] and MobileNet[@howard2017mobilenets], which build the whole model by stacking same units. Figure \[fig:AIunit\_2\] shows the difference between one AI unit and two AI units.
Transfer Learning with Data Fusion
----------------------------------
![Data fusion transfer learning strategy. (a) Data fusion pretraining. During pertraining, the proposed network are trained on two different HSI datasets for improving the diversity of samples and obtaining a good initialized model. (b) Fine tuning. After acquiring the pretrained model, a new model is initialized with the parameters of pretrained model for the target HSI dataset.[]{data-label="fig:transfer"}](Fig5.eps){height="8cm"}
In RGB image classification, pretraining large-scale network on ImageNet dataset which has over 14 millions of hand-annotated images and over 20 thousands of categories is common, and it is very useful for improving the performance and overcoming the problem of limited training samples. During transfer learning, the diversity of the dataset used for pretraining is the key factor. For example, pretraining the same model on the dataset which has million images with one thousand categories always achieving better performance than the one pretraining on the dataset which has 10 million images with ten categories. We suspect that the model pertraining with the more diverse samples may acquire a better generalization ability.
In HSI classification, the labeled samples are quite limited. However, all of the HSI datasets just contain a few categories. For further improving the performance of HSI classification, we propose a data fusion transfer learning strategy. As shown in Figure \[fig:transfer\], the strategy is composed of pretraining and fine tuning. During pretraining, the proposed network are trained on two source HSI datasets. Here, Pavia Center dataset and Salinas dataset are used as source HSI datasets for pretraining. Among the several public HSI datasets, those two datasets have the largest number of labeled samples. To be more specific, the model is initialized with Gaussian distribution on one source HSI dataset and pretrained for $N$ epochs, and then the feature extraction part is fixed and the classifier is reinitialized with Gaussian distribution. Later on, the feature extraction part and classifier on the other source HSI dataset are pretrained for $\frac{N}{2}$ epochs with a different learning rate. In this paper, $N$ is set to 10 and the learning rate used for the feature extraction part is tenth of that used for the second pretraining HSI dataset.
After the model is pretrained on the two source HSI datasets, we transfer the whole mode except the classifier to the fine-tuning model built for the target HSI dataset as initialization, and then fine tuning the transfer part and new classifier with the same learning rate as the one used for training the second source HSI dataset.
Experiments
-----------
Datasets and Experiments Setting
--------------------------------
![False-color composites (first row) and ground truths (second row) of experimental HSI datasets. Each color represents one kind of object. (a) Pavia University; (b) Indian Pines; (c) Kennedy Space Center; (d) Pavia Center; (e) Salinas.[]{data-label="fig:datasets"}](Fig6.eps){height="5.6cm"}
In this paper, we compare the proposed AINet with other four CNN based approaches for HSI classification on three public HSI datasets, including Pavia University, Indian Pines and KSC. For the experiments with transfer learning, Pavia Center dataset and Salinas dataset are employed as the source datasets. The false-color composite and ground truth of each dataset are shown in Figure \[fig:datasets\]. A brief introduction of each dataset is given in the following part and more information can be found on the website[^3].
Pavia University and Pavia Center datasets are captured by Reflective Optics System Imaging Spectrometer (ROSIS) sensor in 2001. After several noisiest bands being removed, Pavia University has 103 bands and Pavia Center has 102 bands. Both datasets are divided into 9 classes. Indian Pines and Salinas datasets are acquired by the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) sensor in 1992. After correction, each dataset has 200 bands and contains 16 classes. KSC is acquired by the AVIRIS sensor in 1996, and after removing water absorption and low SNR bands, 176 bands were used for analysis. For classification purposes, 13 classes are defined.
For the three target HSI datasets, samples are divided into training samples and testing samples. For comparison purposes, we follow [@chen2016deep] to set the samples distribution for Indian Pines and KSC datasets. And for the Pavia University dataset, we extract 200 samples in a random way from each class as training samples.
In the experiments using transfer learning, we randomly extract 200 samples from each class of Pavia Center dataset, 100 samples from each category of Salinas dataset as test samples, and taking the rest as training samples.
Performance Comparison of different network structures
------------------------------------------------------
-----------------------------------------------------------------------------------------------------------
models 1D-CNN 2D-CNN 3D-CNN SSRN AINet
---------- -------- -------- -------- ------------------------------------------------------------- -------
\# train 3930 3930 3930 **[1800]{} & **[1800]{}\
\# param. & **[2898]{} & 0.183M & 5.849M & 0.453M & 0.487M\
depth & 4 & 4 & 4 & 12 & **[32]{}\
OA & 92.28 & 94.04 & **[99.54]{} & 98.98 & 99.42\
AA & 92.55 & 97.52 & **[99.66]{} & 99.07 & 99.51\
$K$ & 90.37 & 92.43 & **[99.41]{} & 98.64 & 99.22\
**************
-----------------------------------------------------------------------------------------------------------
: Classification results for the Pavia University dataset. []{data-label="table:headings_1"}
-------------------------------------------------------------------------------------------------
models 1D-CNN 2D-CNN 3D-CNN SSRN AINet
----------- ---------------------------------------------------- -------- -------- ------ -------
\# train 1765 1765 1765 1765 1765
\# param. **[25920]{} & 0.183M & 44.893M & 0.453M & 0.487M\
depth &6 & 4 & 4 & 12 & **[32]{}\
OA & 87.81 & 89.99 & 97.56 & 98.40 & **[99.14]{}\
AA & 93.12 & 97.19 & 99.23 & 98.52 & **[99.47]{}\
$K$ & 85.30 & 87.95 & 97.02 & 98.14 & **[99.00]{}\
**********
-------------------------------------------------------------------------------------------------
: Classification results for the Indian Pines dataset. []{data-label="table:headings_2"}
-------------------------------------------------------------------------------------------------
models 1D-CNN 2D-CNN 3D-CNN SSRN AINet
----------- ---------------------------------------------------- -------- -------- ------ -------
\# train 459 459 459 459 459
\# param. **[14904]{} & 0.183M & 5.849M & 0.453M & 0.487M\
depth & 5 & 4 & 4 & 12 & **[32]{}\
OA & 89.23 & 94.11 & 96.31 & 98.65 & **[99.01]{}\
AA & 83.32 & 91.98 & 94.68 & 97.78 & **[98.65]{}\
$K$ & 86.91 & 93.44 & 95.90 & 98.54 & **[98.90]{}\
**********
-------------------------------------------------------------------------------------------------
: Classification results for the KSC dataset. []{data-label="table:headings_3"}
In this section, we compare the proposed AINet with other four CNN based HSI classification methods, that are 1D-CNN, 2D-CNN, 3D-CNN [@chen2016deep], SSRN [@zhong2018spectral], on three aforementioned HSI datasets. The experiments with same settings are running for 5 times to acquire the average performance. The experimental results are listed in Tables 1$\sim$3, where the number of training samples, the number of parameters used in the convolution layers, the depth of CNN models, overall accuracy (OA), average accuracy (AA) and kappa coefficient ($K$) are reported. OA is the ratio between the number of correctly classified samples in test set and the total number of test set. AA is the mean of the OA of all the categories. $K$ is a coefficient which measures inter-rater agreement for qualitative items [@thompson1988reappraisal].
From tables 1$\sim$3, we can see that, the proposed AINet achieves best classification performance on all of the datasets. For instance, in Indian Pines dataset, OA of AINet is 99.14, which is 9.15% better than that of 2D-CNN, 1.58% better than that of 3D-CNN and 0.74 better than that of SSRN. Through the experiments, it is easy to find that all of the 3D-CNN based HSI classification methods obtained better performance than 2D-CNN. From 3D-CNN, SSRN to AINet, the depth of the models are increasing and the classification accuracy goes up. In specific, the depths of the three models are 4, 12, 32 accordingly. Although AINet is much deeper than SSRN in depth, the parameters of AINet is slightly more than that of SSRN and much less than that of 3D-CNN.
Results of Transfer Learning
----------------------------
In this part, we combine the proposed AINet with data fusion transfer learning to further improve the classification performance. For each dataset, we choose $\{15, 30\}$ samples from each class as training samples to test the affect brought by the number of the training samples. The experiment results using transfer learning are shown in Figure \[fig:15samples\] and Figure \[fig:30samples\], where AINet represents our model trained only on the target dataset, AINet+T1 represents our model pretrained on Pavia Center, which captured from the same sensor as Pavia University dataset, and fine-tuning on the target dataset. AINet+T2 represents our model pretrained on Salinas dataset, which captured from the same sensor as the Indian Pines dataset, and fine-tuning on the target dataset. AINet+T3 and AINet+T4 are our models pretrained on both Pavia Center dataset and Salina datasets, with different orders accordingly, and fine-tuning on the target dataset.
![Transfer learning experiments with 15 training samples per class. (a) Pavia University; (b) Indian Pines; (c) Kennedy Space Center.[]{data-label="fig:15samples"}](Fig7.eps){height="2.8cm"}
![Transfer learning experiments with 30 training samples per class. (a) Pavia University; (b) Indian Pines; (c) Kennedy Space Center.[]{data-label="fig:30samples"}](Fig8.eps){height="2.8cm"}
From the overall trends of Figure \[fig:15samples\] and Figure \[fig:30samples\], we can draw two conclusions. Firstly, the performance of our AINet is benefit from the transfer learning, especially when the available training samples are relatively small, the performance gains are huge. Secondly, pretrained on two different datasets achieve much gains than the ones just pretrained on a single dataset. Which we infer the benefits mainly come from the various classes and categories of the pretrain datasets.
From Figure \[fig:15samples\], there are two points that can be concluded: Firstly, from the performance of AINet+T, AINet+T1, AINet+T2 in (a),(b),(c), we can conclude that AINet pretrained on other HSI dataset, will lead performance increasing for the target HSI dataset, no matter if the source and target dataset are captured by the same sensor or not. Secondly, from the performance of AINet+T, AINet+T1, AINet+T2 in (a),(b), we can conclude that when pretraining on other HSI dataset, the performance gains brought by the same sensor is much larger than the one brought by different sensors for small training samples.
From Figure \[fig:30samples\], when the training samples become larger, the performance of HSI classification is still benefit from transfer learning. But notice that the pink bars in Figure \[fig:30samples\], which have a decreasing in performance, the reason behind that is still unclear and need the further investigation. One possible reason we suspect is that on the pretrained dataset, a different learning rate is used comparing with the one used for fine-tuning on the target dataset, and which may not lead to the exactly same converge direction for the training process between the fine-tuning dataset and target dataset.
Conclusions {#sec:conclusions}
===========
This paper proposes a 3D asymmetric inception network for hyperspectral image classification named AINet. Firstly, AINet using a 3D CNN light-weight while still very deep, which can exert the potential of the deep learning for extracting the representative features, and meanwhile alleviate the problem brought by the limited annotation datasets. Secondly, considering the property of hyperspectral image, spectral signatures are emphasized than spatial contexts. Moreover, data fusion transfer learning strategy is utilized for a better model initialization and saving the training time. In the future, there are two topics we are keen to pursue, to investigate the reduction of the training time brought by transfer learning is the first one, and another one is taking use of some policies to overcome the data imbalance in HSI classification.
[^1]: Code is avaliable at: https://github.com/UniLauX/AINet
[^2]: *Haokui Zhang and Yu Liu contributed equally to this work. $^{1}$ H. Zhang, B. Fang and Y. Li are with School of Computer Science and Engineering, Northwestern Polytechnical University, Xi’an, 710072, China [hkzhang1991@mail.nwpu.edu.cn]{} $^{2}$Y. Liu L. Liu and I. Reid are with School of Computer Science, The University of Adelaide, 5005, North Terrace, SA [yu.liu04@adelaide.edu.au]{}*
[^3]: <http://www.ehu.eus/ccwintco/index.php>
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Recently a second type of spicules was discovered at the solar limb with the Solar Optical Telescope (SOT) onboard the Japanese Hinode spacecraft. These previously unrecognized type II spicules are thin chromospheric jets that are shorter-lived (10–60 s) and that show much higher apparent upward velocities (of order 50–100 km s$^{-1}$) than the classical spicules. Since they have been implicated in providing hot plasma to coronal loops, their formation, evolution and properties are important ingredients for a better understanding of the mass and energy balance of the low solar atmosphere. Here we report on the discovery of the disk counterparts of type II spicules using spectral imaging data in the Ca II 854.2 nm and [H$\alpha$]{} lines with the CRisp Imaging SpectroPolarimeter (CRISP) at the Swedish Solar Telescope (SST) in La Palma. We find rapid blueward excursions in the line profiles of both chromospheric lines that correspond to thin, jet-like features that show apparent velocities of order 50 km s$^{-1}$. These blueward excursions seem to form a separate absorbing component with Doppler shifts of order 20 and 50 km s$^{-1}$ for the Ca II 854.2 nm and [H$\alpha$]{} line respectively. We show that the appearance, lifetimes, longitudinal and transverse velocities and occurrence rate of these rapid blue excursions on the disk are very similar to those of the type II spicules at the limb. A detailed study of the spectral line profiles in these events suggests that plasma is accelerated along the jet, and plasma is being heated throughout the short lifetime of the event.'
author:
- 'L. Rouppe van der Voort$^1$'
- 'J. Leenaarts$^{1,2}$'
- 'B. de Pontieu$^3$'
- 'M.Carlsson$^{1,2}$'
- 'G.Vissers$^1$'
bibliography:
- 'spics2.bib'
title: 'On-disk counterparts of type II spicules in the Ca II 854.2 nm and [H$\alpha$]{} lines'
---
Introduction
============
Spicules are slender features protruding from the solar limb when observed in for example the [Ca II H]{} and [H$\alpha$]{} lines. Classically, spicules have been reported to show velocities of order 20–30 km s$^{-1}$, lifetimes of order 5-10 minutes and heights up to 5,000–10,000 km above the photosphere (see @1968SoPh....3..367B and @2000SoPh..196...79S for reviews on the older literature on spicules).
The high spatio-temporal resolution of the Solar Optical Telescope [SOT, @2008SoPh..249..167T] onboard Hinode [@2007SoPh..243....3K] have revolutionized our view of spicules. @2007PASJ...59S.655D showed that spicules can be grouped in two categories based on Hinode/SOT [Ca II H]{} observations of the solar limb. Type I spicules appear to rise up from the limb and fall back again. These structures show a similar dynamical evolution as active region dynamic fibrils [@2006ApJ...647L..73H; @2007ApJ...655..624D] and a subset of quiet sun mottles [@2007ApJ...660L.169R], so they can be interpreted as their off-limb counterparts. Numerical simulations based on radiative MHD indicate that the motion of type I spicules is driven by magneto-acoustic shocks that can be generated by a variety of processes, such as convective buffeting, granular collapse, dissipation of magnetic energy, and leakage of waves into the chromosphere [e.g., @2006ApJ...647L..73H; @2007ApJ...655..624D; @2007ApJ...666.1277H; @2009arXiv0906.4446M]. Type II spicules show different behavior. They exhibit upward motion, followed by rapid fading from the Hinode [Ca II H]{} passband, without a downward moving phase. Sometimes they accelerate while they rise. They have lifetimes between 10 and 100 s, apparent velocities between 50 and 150 km s$^{-1}$ and widths between 150 and 700 km and undergo a swaying motion caused by the upward propagation of Alfvénic waves [@2007Sci...318.1574D]. These spicules are longest in coronal holes, with heights up to 10 Mm, while they appear shorter in active regions, where they rarely extend more than 2 Mm in height.
Many questions about type II spicules remain. For example, it is unclear whether the apparent motions that are measured at the limb (which are based on the temporal evolution of [Ca II H]{} intensity in the plane of the sky) are asssociated with bulk mass motion. Previous studies have found plentiful evidence for line of sight motions of order 20–30 km s$^{-1}$, but little evidence for flows of order 50–100 km s$^{-1}$. Clearly, measuring Doppler shifts on the disk could shed light on this issue. However, while type II spicules are ubiquitous at the limb, both in coronal holes and in quiet sun regions [@2007PASJ...59S.655D], it has been unclear what their counterparts are on the solar disk. Finding such a disk counterpart is crucial for several other reasons. It can help reveal what the formation mechanism of these jets is (which is difficult to study at the limb because of the line of sight superpositon). Type II spicules have also been implicated in providing the corona with hot plasma by @2009arXiv0906.5434D who established a correlation between faint upflowing signals at coronal temperatures (from asymmetries in EUV spectral lines) and the highly dynamic signals in the Hinode/SOT [Ca II H]{} passband. The latter signals are thought to be associated with type II spicules, but unequivocally establishing the disk counterpart of type II spicules would help reduce the uncertainties involved in establishing the importance of type II spicules in the mass and energy balance of the corona.
In this paper we will focus on establishing what type II spicules look like on the disk. This is not a straightforward task, given the high spatial and temporal resolution required to observe these features. @1998SoPh..183...91W and @1998ApJ...504L.123C looked for [H$\alpha$]{} jets on the disk with a spatial resolution of 1 arcsec. They found roundish darkenings in the blue wing of the line, with typical sizes of 3–5 arcsec and lifetimes of 2 minutes, without subsequent redshift. These darkening are often associated with converging magnetic dipoles at supergranular boundaries in the photosphere, even though they also observe some jets in unipolar regions. Using cloud modeling they find typical upflow velocities of 30 km s$^{-1}$. While some properties fit with the observed behavior of off-limb [Ca II H]{} type II spicules, their sizes and lifetimes indicate that these are not the same phenomenon.
More recently, @2008ApJ...679L.167L investigated so called “rapid blueshifted events” (RBEs) in on-disk [Ca II 854.2 nm]{} data obtained by the Interferometric BIdimensional Spectrometer (IBIS). RBEs are a sudden widening of the line profile on the blue side of the line, without an associated redshift. The RBEs are located around the network – but not directly on top of individual network elements, show blueshifts of 15–20 km s$^{-1}$, and have an average lifetime of 45 s. The authors interpret these RBEs as chromospheric upflow events without subsequent downflow and suggest that these might be the on-disk counterparts of Type II spicules. Using Monte Carlo simulations they show that the low observed blueshift can be explained by a wide range of spicule orientations combined with a lack of opacity in the upper chromosphere.
Full 3D non-LTE radiative transfer calculations on a model atmosphere based on a snapshot of radiation-MHD simulation by @2009ApJ...694L.128L show that darkenings in the blue wing of the [Ca II 854.2 nm]{} line indeed correspond to chromospheric upflows.
The evidence for the connection between RBEs and type II spicules presented by @2008ApJ...679L.167L is tantalizing but not conclusive. With their fast evolution and small spatial dimensions, type II spicules are an elusive phenomenon that are on the limit of what is observable with present day telescopes. In this study we improve on the @2008ApJ...679L.167L observations in terms of spatial and temporal resolution, and employ a spectral sampling that is better suited to cover high Doppler shifts. In addition, we observed both the [Ca II 854.2 nm]{} and the [H$\alpha$]{}spectral lines.
Observations and data reduction
===============================
![Details from one RBE observed in [H$\alpha$]{}. The top panel shows the spectral evolution ($\lambda t$-dataslice) in one spatial location in the RBE structure. The location is marked with a cross in the [H$\alpha$]{} blue-wing image ($\Delta\,v=-59.4$ kms$^{-1}$) shown in the bottom panel. A detailed line profile is shown in the middle panel. The profile with open diamonds is the [H$\alpha$]{} profile averaged over the whole observed FOV, with the diamonds marking the sampling positions. The dashed line is the difference of the average and RBE profiles. []{data-label="fig:ha-lamt"}](rouppe_fig01s.pdf){width="\columnwidth"}
{width="\textwidth"}
The observations were obtained with the CRisp Imaging SpectroPolarimeter [CRISP, @2008ApJ...689L..69S] installed at the Swedish 1-m Solar Telescope [SST, @2003SPIE.4853..341S] on La Palma (Spain). CRISP is a spectropolarimeter that includes a dual Fabry-P[é]{}rot interferometer (FPI) system similar to that described by . CRISP is equipped with 3 high-speed low-noise CCD cameras that operate at a frame rate of 35 frames per second and an exposure time of 17 ms. The 3 cameras are synchronized by means of an optical chopper, 2 of these cameras are positioned behind the FPI after a polarizing beam splitter, and the 3rd camera is positioned before the FPI but after the CRISP pre-filter. The latter camera is used as anchor channel for image processing and is referred to as the wide-band channel. The image scale is 0071/pixel and the field of view (FOV) $\sim$71$\times$71 arcseconds. CRISP allows for fast wavelength tuning ($\lesssim$50 ms) within a spectral region and is ideally suited for spectroscopic imaging of the chromosphere where the dynamical evolution time can be on the order of a few seconds, sometimes even faster than 1 s [@2006ApJ...648L..67V]. For [H$\alpha$]{} the transmission FWHM of CRISP is 6.6 pm, and the pre-filter FWHM is 0.49 nm. For [Ca II 854.2 nm]{} the transmission FWHM is 11.1 pm, and the pre-filter FWHM is 0.93 nm. In this paper, we analyze two data sets from 2008 June 15, from an area at the edge of a coronal hole close to disk center at $(x,y)\approx(150,-28)$ ($\mu=0.99$): a 40 min [Ca II 854.2 nm]{} time sequence started at 08:24 UT, and a 24 min [H$\alpha$]{} sequence started at 09:15 UT. The following wavelength sequences were used. For the [Ca II 854.2 nm]{} data set we used a 29 line position sequence, ranging from $-$200 pm to $+$200 pm from line center ($\pm$68 km s$^{-1}$), with 10 pm steps in the core and 20 pm steps in the wings. 12 exposures per wavelength position were recorded, resulting in a time of 11 s to complete a full wavelength scan. For the [H$\alpha$]{} data set we used a 25 line position sequence, with 10 pm steps ranging from $-$160 pm to $+$80 pm from line center ($-$73 to $+$37 km s$^{-1}$ Doppler shift). At each wavelength position, 8 exposures were recorded. The time to complete a full wavelength scan was 6.7 s.
The image quality of the time sequences benefited from the SST adaptive optics system [@2003SPIE.4853..370S] and the image restoration technique Multi-Object Multi-Frame Blind Deconvolution [MOMFBD, @2005SoPh..228..191V]. All images from all wavelength positions in a scan were divided in overlapping 64$\times$64 pixel subfields and processed as a single MOMFBD restoration. In such a restoration, the wide-band channel served as anchor for the narrow-band CRISP exposures to ensure precise alignment between the restored narrow-band images. We refer to for more details on the MOMFBD processing strategies on similar extensive multi-wavelength scans.
After MOMFBD reconstruction of the individual line scans, the images from the different time steps were combined to form time series. The images were de-rotated to account for diurnal field rotations, and aligned and de-stretched to account for translation and warping due to seeing effects. (Local) offsets were determined on the wide band images and applied to the corresponding CRISP images.
The [Ca II 854.2 nm]{} dataset is of excellent quality, with most images close to the diffraction limit of 0.23 arcsec. The [H$\alpha$]{} dataset is not as good, as the seeing was degrading in quality and there were more frequent short moments of blurring, but still at least half of the images are close to the diffraction limit of 0.17 arcsec.
We obtained a context Stokes V magnetogram from a CRISP scan of the Fe 630.25 nm line recorded at 09:39 UT, after the [Ca II 854.2 nm]{} and [H$\alpha$]{}sequences. We followed the procedures of @2008ApJ...689L..69S except that we did not use MOMFBD restoration on the polarimetric observations. In the lower left panel of Fig. \[fig:fov\], a blue wing ($-$4.8 pm) Stokes V magnetogram is shown. The magnetogram is the result of flatfielding and adding of in total 32 exposures – 8 exposures per liquid crystal state. We estimate the noise level to be $2.6 \cdot 10^{-3}$ for Stokes V relative to I$_\textrm{continuum}$, and the spatial resolution better than 05. The magnetogram was aligned to the time sequences using the wide-band channels. As the magnetogram was observed after the time sequences, no detailed information on the individual magnetic elements can be obtained but it serves the purpose of giving general information on the distribution of the different magnetic polarities over the FOV.
We searched both datasets for RBEs. In blue-wing images they appear as roundish or elongated dark structures on top of the photospherically formed background . In order to efficiently explore the datasets, we used widget based analysis tools programmed in the Interactive Data Language (IDL): CRIsp SPectral EXplorer (CRISPEX) and Timeslice ANAlysis Tool (TANAT). These tools allow simultaneous display of spectra, images and $xt$-slices of arbitrarily shaped curves. This way it is easy to distinguish RBEs from intergranular lanes, dark reversed granulation and, by visual inspection of their spectral signature, of shockwaves. Our method to find and study RBEs in our data is different from the @2008ApJ...679L.167L approach. They identified RBEs purely on a spectroscopic basis, searching for specific spectral signatures in $\lambda t$-dataslices. We find numerous examples of short-lived, narrow, and elongated features in our data at wavelength offsets corresponding to significant Doppler velocities. These features emerge from regions with magnetic field concentrations and have a similar spectral signature in $\lambda t$-dataslices as the @2008ApJ...679L.167L RBEs. To expand on the statistical properties of the RBEs we augmented the above described manual detection scheme with an automated algorithm. This algorithm finds RBEs by isolating the location and time of long thin features at high blueshifts (respectively 30 and 60 km s$^{-1}$ for the [Ca II 854.2 nm]{} and [H$\alpha$]{} datasets). To reject intergranular lanes and other non-RBE associated features, we impose that the features not be too curved. This automated procedure identified 413 features in the [Ca II 854.2 nm]{} dataset and 608 in the [H$\alpha$]{} dataset. We confirmed the validity of the automated search results by visually inspecting blue wing images and $\lambda t$-dataslices for the flagged RBEs. This inspection confirmed that the automatically detected features have spectral characteristics similar to the RBEs of @2008ApJ...679L.167L.
In addition to these events, we also (manually) found a number of small, roundish features of high blueshift that the algorithm failed to detect (see Sec. \[sec:bb\]).
Results
=======
Figure \[fig:ha-lamt\] shows details of one selected RBE observed in [H$\alpha$]{}. The bottom panel shows part of a blue-wing [H$\alpha$]{} image at an offset from line center equivalent to a Doppler shift of $-$59.4 kms$^{-1}$. The narrow dark streak in the center is identified as an RBE. At this wavelength position, the first sign of this RBE appeared 40 s earlier, close to the photospheric bright point at the lower left of this feature. During its lifetime, the streak moves away from that bright point and vanishes after 54 s. The white cross marks the spatial position where a detailed spectrum is showed in the middle panel, and the spectral/temporal evolution in the top panel. The detailed spectrum is compared to the average [H$\alpha$]{} profile: the dashed line represents the difference of the two profiles. The RBE spectrum displays significant asymmetry towards the blue wing, differing from the average profile even beyond a Doppler velocity of 72 kms$^{-1}$.
The top panel shows the temporal evolution of the spectra in the central part of an RBE in a $\lambda t$-dataslice. This neatly illustrates why these events are called “rapid blue excursions” as they suddenly develop a highly asymmetric line profile towards the blue. At this spatial location, there is no spectral signature pointing to any down-fall phase associated with this event (i.e., features occurring on the red side of the line).
Figure \[fig:fov\] shows the full field-of-view of the datasets of June 15. The upper-left panel shows a blue-wing [H$\alpha$]{} image at a Dopplershift of $-$45 km s$^{-1}$. It shows bright network elements and many blueshifted chromospheric structures. At this Dopplershift, one can identify many narrow elongated features. However, some of these features are not identified as RBEs as they have different spectral signatures (i.e., not asymmetric to the blue wing) and/or display different temporal behavior (i.e., no sudden, or rapid, appearance).
The upper-right panel shows the simultaneous [H$\alpha$]{} line-core image. Green curves indicate the trajectories of the [H$\alpha$]{} RBEs we found in an automated fashion – a dense concentration was found in the region of interest (ROI) marked ‘1’. The RBEs found in ROI3 appear to have different behavior than the others, with movements perpendicular to the magnetic field structure as outlined by the fibrils in the line core image.
The lower-left panel shows the context magnetogram. It shows two regions of unipolar network in the lower left quadrant, and a small bipolar region at $(x,y) {\ensuremath{\!=\!}}(35,35)$ arcsec. The right side of the field shows scattered patches of unipolar field with opposite polarity from the two network regions. The lower-right panel shows the [Ca II 854.2 nm]{} line-core, with the [Ca II 854.2 nm]{} RBE trajectories overplotted in yellow.
The RBEs are predominantly found around the two patches of unipolar network in the lower-right quadrant of the FOV with a minority of the RBEs scattered throughout the FOV. The [H$\alpha$]{}-core image shows much more fibrilar structure than the [Ca II 854.2 nm]{} one. The latter shows an internetwork acoustic shock pattern except very close to the network. This indicates that [H$\alpha$]{} has a much higher chromospheric opacity, and is likely to show structures at larger height above the solar surface.
Rapid blue excursions in [H$\alpha$]{} {#sec:rbe-ha}
--------------------------------------
![Example line profiles of the [H$\alpha$]{} RBEs. Black: average spectrum over the field of view, with diamonds indicating the different wavelength positions of the filter. Red: profiles of RBEs; blue: profiles of “black beads”. \[fig:ha-detsp\]](rouppe_fig05.pdf){width="\columnwidth"}
Figure \[fig:rbe-cuts-ha\] shows blowups of ROI1 and ROI2, following the detailed evolution of two RBEs whose trajectories are colored red and blue in the top right panel of Fig. \[fig:fov\]. They appear as thin elongated dark streaks in the $\Delta v {\ensuremath{\!=\!}}-73.1$ km s$^{-1}$ column.
We strongly encourage the reader to view the time-lapse animation of this Figure that is available on-line. Presenting the temporal evolution of these narrow, short-lived features on paper as a series of images cannot rival the visual impression of an animation.
In the left-hand panels the RBE from ROI1 is shown. It appears in the lower-right corner of frame 2 (numbers in the lower left corner). It then increases in visibility, becoming darker and longer while moving diagonally towards the upper-left corner of the ROI. While it traverses it starts to fade and is almost invisible in frame 10. The structure is slightly curved and follows a curved path, which suggests movement along a magnetic field line.
The left-hand panels of Fig. \[fig:xt-slice-ha\] show the time evolution of the brightness along the trajectory of the RBE. It appears as a diagonal streak in the $\Delta v {\ensuremath{\!=\!}}-73.1$ km s$^{-1}$ panel. From this panel its apparent horizontal speed can be determined as $84$ km s$^{-1}$.
The RBE from ROI2 is shown in the right-hand panels of Fig. \[fig:rbe-cuts-ha\] . It is present from frame 2, sticking out from the lower-left corner towards the center of the ROI. It does not appear to move until frame 4. It then moves towards the upper-right in frames 5–8. It fades and is completely gone in frame 10. The right-hand panels of Fig. \[fig:xt-slice-ha\] show an $xt$-slice of the brightness along the trajectory of the RBE. There seems to be a structure with similar horizontal speed in the $-$32 km s$^{-1}$ panel (the black structure between $x {\ensuremath{\!=\!}}0$ and $x{\ensuremath{\!=\!}}5$ arcsec). Close inspection of the image sequence in Fig. \[fig:rbe-cuts-ha\] shows that it is caused by a number of different fibrils and most of the absorption is unrelated to the RBE. The apparent horizontal speed of the RBE is $114$ km s$^{-1}$.
The trajectories of both RBEs are roughly aligned with the fibrils and mottles visible in the wavelengths closer to the line core, but there is no one-to-one relationship between the RBEs and the dark structures in the $\Delta v {\ensuremath{\!=\!}}-32$ km s$^{-1}$ indicating that the RBEs are indeed structures with a large line-of-sight velocity component, and not the damping wing of an optically thick chromospheric structure with a small velocity. There is no corresponding absorption in the red wing during or after the RBE. This rules out that our RBEs are high-velocity type I spicules or a related shock phenomenon.
The two red curves in Fig. \[fig:ha-detsp\] show typical line profiles of RBEs. They are strongly asymmetric, with an extended absorption wing on the blue side of the core relative to the average profile. The largest excess absorption occurs between Doppler shifts of $-$60 and $-$30 km s$^{-1}$.
“Black beads” in [H$\alpha$]{} {#sec:bb}
------------------------------
![Detailed cutouts of ROI3, showing the time-evolution of [H$\alpha$]{} “black beads” in different line positions as indicated above the top row of panels. A negative velocity means a blueshift. Time increases downward, with 6.7 s between successive images in a column. The white arrows indicate the black beads when they are most visible. A time-lapse animation of this figure is provided electronically. \[fig:rbe-cuts-bp\]](rouppe_fig06s.pdf){width="\columnwidth"}
We found a large number of RBEs in ROI3. They are different in several respects from the RBEs discussed in Sec. \[sec:rbe-ha\]. Figure \[fig:rbe-cuts-bp\] shows an image sequence of ROI3 with several of these special RBEs, which we dub “black beads”. They appear as roundish darkenings, either individually or in short strings of several beads and are located along a neutral line between two photospheric magnetic field concentrations of opposite polarity (see the magnetogram in Fig. \[fig:fov\]). Their horizontal speeds are relatively low ($15$ km s$^{-1}$) and they move parallel to the neutral line, but perpendicular to the large scale fibril orientation (see the [H$\alpha$]{}-core panel of Fig. \[fig:fov\]). The small scale magnetic field configuration close to the neutral line is complex, as indicated by the appearance of the line-core images. It is unclear how the horizontal motion of the black beads relates to this configuration.
The line profiles of the black beads (see the blue curves in Fig. \[fig:ha-detsp\]) are even more asymmetric than the line profiles of the RBEs associated with the unipolar network, with nearly flat line profiles between $\Delta v {\ensuremath{\!=\!}}-74$ and $\Delta v {\ensuremath{\!=\!}}-50$ km s$^{-1}$. They show strong absorption all the way up to the most blueward position of our line scans, which suggests that the black beads are structures that, at least partly, move with an upward velocity well in excess of 70 km s$^{-1}$.
Typically, these black beads have diameters between 0.15 and 0.3 Mm, as measured at $\Delta v=-74$ km s$^{-1}$.
Rapid blue excursions in [Ca II]{} IR
-------------------------------------
![Example line profiles of the [Ca II 854.2 nm]{} RBEs. Black: average spectrum over the field of view, with diamonds indicating the different wavelength positions of the filter. Red and blue: profiles of RBEs. \[fig:ca-detsp\]](rouppe_fig09.pdf){width="\columnwidth"}
Figures \[fig:rbe-cuts-ca\], \[fig:xt-slice-ca\] and \[fig:ca-detsp\] show details of two RBEs in the [Ca II 854.2 nm]{} line, in identical format as the [H$\alpha$]{} figures.
Figure \[fig:rbe-cuts-ca\] shows the image sequences of ROI4 and ROI5. The RBEs are invisible at the largest blueshift column, instead they appear clearly at a blueshift of $-$30.6 km s$^{-1}$.
The left-hand panels show the appearance and disappearance of two partially overlapping RBEs. The first RBE appears at the arrow in the panel marked ‘a’. It extends, reaching maximum length in in panel b, after which it retracts and fades. The second RBE appears in panel b, on top of the first one, but at a slightly different angle. It gains opacity in the following panels, bringing out its curved lower end, indicating that it is indeed a different RBE than the first one. It moves upward and fades from view, and is gone in the last panel. The left-hand panels of Fig. \[fig:xt-slice-ca\] show timeslices of the intensity along the trajectory of the second RBE. It shows up as a diagonal streak in the $\Delta v {\ensuremath{\!=\!}}\mathrm{-}30.6$ km s$^{-1}$ panel, again without counterpart in the panels at other Doppler shifts.
The right-hand panels show a slow-moving roundish RBE appear at the arrow in panel c. It moves slowly away from the network elements at the top of the ROI and fades from view in the last two panels. The timeslices in the right-hand panels of Fig. \[fig:xt-slice-ca\] show the RBE only in the $\Delta v {\ensuremath{\!=\!}}-30.6$ km s$^{-1}$ panel.
Figure \[fig:ca-detsp\] shows typical line profiles of two RBEs. The profiles are asymmetric, with extra absorption in the blue wing, and a red wing very similar to the average profile. The largest extra absorption occurs around $-$30 km s$^{-1}$, at significantly lower velocity than in the [H$\alpha$]{} profiles.
Statistical properties of rapid blue excursions
-----------------------------------------------
![Histograms of RBE properties. Length (top panel), measured Doppler velocity at mid-point of extracted feature (middle panel) and width of the blue component (bottom panel) for [H$\alpha$]{} (solid line) and [Ca II 854.2 nm]{} (dashed line). \[fig:haca\_hist\]](rouppe_fig10.pdf){width="\columnwidth"}
{width="\textwidth"} {width="\textwidth"}
{width="\textwidth"}
![Histogram of RBE properties for the [H$\alpha$]{} subsample. Lifetime (top panel), measured Doppler velocity at mid-point of extracted feature (middle panel) and apparent velocity along the length of the RBE (bottom panel). \[fig:ha\_hist\]](rouppe_fig13.pdf){width="\columnwidth"}
![Histogram of RBE properties for the [Ca II 854.2 nm]{} subsample. Lifetime (top panel), measured Doppler velocity at mid-point of extracted feature (middle panel) and apparent velocity along the length of the RBE (bottom panel). \[fig:ca\_hist\]](rouppe_fig14.pdf){width="\columnwidth"}
![Histogram of transverse motions of RBEs. Transverse displacement (top panel) and transverse velocity (bottom panel) of 35 [H$\alpha$]{} RBEs. Most RBEs undergo significant motion perpendicular to their long axis. \[fig:haca\_alfven\]](rouppe_fig15.pdf){width="\columnwidth"}
The automatic detection procedure identified 413 features in the [Ca II 854.2 nm]{} dataset and 608 in the [H$\alpha$]{} dataset. For each of the identified features we subtract the average spectral profile (averaged over the whole field of view) from the profile at each pixel along the feature. We use the first and second moments with respect to wavelength as estimates of the Doppler velocity, $v_{\mathrm{Doppler}}$ and width of the blue-shifted component, $W$, i.e., $$v_{\mathrm{Doppler}} = \frac{c}{\lambda_0} \frac{\int^{\lambda_0}_{\lambda_\mathrm{min}} (\lambda-\lambda_0)
(I_\mathrm{avg}-I) \, \mathrm{d} \lambda}{\int^{\lambda_0}_{\lambda_\mathrm{min}}
(I_\mathrm{avg}-I) \, \mathrm{d} \lambda},$$ $$W = \frac{c}{\lambda_0}
\sqrt{\frac{\int^{\lambda_0}_{\lambda_\mathrm{min}} (\lambda - \lambda_{\mathrm{c}})^2
(I_\mathrm{avg}-I) \, \mathrm{d} \lambda}{\int^{\lambda_0}_{\lambda_\mathrm{min}}
(I_\mathrm{avg}-I) \, \mathrm{d} \lambda}},$$ where $c$ is the velocity of light, $\lambda_0$ the line center wavelength, $I$ the observed intensity, $I_{\mathrm{avg}}$ the average intensity and $\lambda_\mathrm{c} = \lambda_0+\lambda_0 v_{\mathrm{Doppler}} / c$. The integration ranges from the most blue-ward wavelength we observed ($\lambda_\mathrm{min}$) to line center, and is only executed for wavelengths for which $I_\mathrm{avg}-I > 0$, i.e., where the feature shows absorption in the blue wing. We note that these estimates are necessarily rather crude, as it is impossible to rigorously derive the atmospheric velocity structure from the line profile. Cloud modelling might in some cases have yielded a more precise estimate, but in the majority of cases the RBEs are overlapping a myriad of chromospheric structures, on which single-component cloud-model inversion is unlikely to yield reliable results. Based on the detailed line profiles we expect our estimates of $v_{\mathrm{Doppler}}$ to be on the low side, and most RBEs might well have components with a $v_{\mathrm{Doppler}}$ that is 10–20 km s$^{-1}$ higher than estimated.
The measured lengths, Doppler velocities and widths are shown in Fig. \[fig:haca\_hist\] for both datasets. The distribution of lengths has a sharp lower cutoff at 0.8 Mm because this was used as a criterion for acceptance of a feature in the automated detection algorithm. The length has been defined as the maximum distance along which a feature shows absorption in the blue wing compared to the average spectral profile. This definition removes some of the arbitrariness introduced by the fact that the features are detected at a specific velocity. The values for the Doppler velocities and widths are calculated by averaging over the whole length of each feature.
We find that the values for length, Doppler velocity and width are consistently higher for the [H$\alpha$]{} RBEs compared to the [Ca II 854.2 nm]{} RBEs. [H$\alpha$]{} RBEs are on average somewhat longer (of order 3 Mm vs. 2 Mm), have higher average Doppler shifts towards the blue (35 km s$^{-1}$ vs. 15 km s$^{-1}$) and higher widths (13 km s$^{-1}$ vs. 7 km s$^{-1}$). We should note that there is a wide range of values for Doppler shifts and widths from RBE to RBE.
In addition, the measured Doppler velocities and widths of the blue components vary systematically along the length of the RBEs. This is shown for two sample RBEs in Fig. \[fig:haca\_dop\_zoom\]. We find that many RBEs show an increase of Doppler shifts and widths from their footpoint to the top. The footpoints (blue-coded profiles and lines) typically show absorption that is at smaller Doppler shifts and narrower than the middle (green coded) and top end (red coded) of the RBEs. We see this effect for both [H$\alpha$]{} and [Ca II 854.2 nm]{} RBEs. This effect is visually quite striking when we plot the measured parameters for all automatically detected RBEs for a subset of the field of view (Fig. \[fig:haca\_vw\]). Here we also see that the [Ca II 854.2 nm]{} RBEs are typically shorter and closer to the footpoints (photospheric magnetic field concentrations) than the [H$\alpha$]{} RBEs.
We see clear outward motions in many of the detected features but measuring these apparent velocities is very difficult to do in an automated fashion. Instead, we used a manual approach in which we used the extracted positions of the features to extract $xt$-slices from the data cube (see Fig. \[fig:fov\] for outlines of these positions). We then measured the apparent velocities,[$v_{\mathrm{apparent}}$]{} from the slopes of the RBEs in these $xt$-slices manually. This was difficult in most cases because of short lifetimes and unclear paths in the $xt$-slices. We thus have [$v_{\mathrm{apparent}}$]{} for a subset of only 25 RBEs for the [H$\alpha$]{} dataset and 28 for the [Ca II 854.2 nm]{}dataset. There is likely to be a strong selection effect for these measurements favoring long lifetimes and low velocities. The lifetimes of the RBEs were determined from the first and last occurrences in the time sequence. The statistics for these subsamples are shown in Figs. \[fig:ha\_hist\] and \[fig:ca\_hist\]. The lifetimes of the RBEs range from 13 s (i.e., two time steps for [H$\alpha$]{}), up to 120 s, with peaks around 40 s.
The horizontal velocity of the [H$\alpha$]{} RBEs ranges from 0 km s$^{-1}$ to 120 km s$^{-1}$. The [Ca II 854.2 nm]{} RBEs have a lower maximum horizontal velocity, with most events having ${\ensuremath{v_{\mathrm{apparent}}}}< 40$ km s$^{-1}$.
We have also measured the transverse motions of about 35 [H$\alpha$]{} RBEs and found that most RBEs move transversely to their long axis, with transverse displacements of order 0.3 Mm (within a range of 0 to 0.8 Mm) and transverse velocities of order $8$ km s$^{-1}$ (with a range from 0 to 20 km s$^{-1}$). This is illustrated in Fig. \[fig:haca\_alfven\]. The measured lifetimes of these RBEs at one single height are between 15 and 60 s, with an average of 33 s.
Discussion & Conclusions
========================
We have analyzed rapid blueshifted events in [Ca II 854.2 nm]{} and [H$\alpha$]{}data obtained with the CRISP instrument at the Swedish 1-m Solar Telescope. In blue wing time sequences, we observe these RBEs as short-lived, narrow streaks that move at high speed away from areas with magnetic field concentrations. Close to disk center, the [Ca II 854.2 nm]{} events we observe are similar to those found by @2008ApJ...679L.167L. We measure similar lifetimes (45 s) and Doppler velocities (20 km s$^{-1}$). Our observations have a higher spatial and temporal resolution, allowing us to study their evolution over time and measure their apparent motion. In addition, our automated detection algorithm allows us to greatly expand on the statistics of RBEs. The RBEs we identified in [H$\alpha$]{} have generally higher Doppler velocities and larger Doppler width than their [Ca II 854.2 nm]{} counterparts. The RBEs are longer in [H$\alpha$]{} than in [Ca II 854.2 nm]{}, with the [Ca II 854.2 nm]{}RBEs located a bit closer to the magnetic field concentrations that are at the root of the RBEs. For those selected RBEs where we could make a reliable estimate of the apparent velocity, we see a slight trend of higher apparent motion in [H$\alpha$]{}. The lifetimes of RBEs in [Ca II 854.2 nm]{} and [H$\alpha$]{} seem to be similar.
@2008ApJ...679L.167L suggested that the [Ca II 854.2 nm]{} RBEs are linked to the Hinode [Ca II H]{} type II spicules observed at the limb [@2007PASJ...59S.655D]. This suggestion was based on the similarity in lifetimes, spatial extent, location near the network and sudden disappearance or fading. In addition, the fact that RBEs exclusively show blueshifts corresponds well with the fact that type II spicules only show upward flows. Our observations make the connection between RBEs observed on disk and type II spicules at the limb much stronger. First of all we find the same similarities in lifetimes, location and temporal evolution. In time sequences at high Doppler displacement from the line center, we also see the RBEs moving away from the network as narrow streaks – dynamical behavior that agrees well with the upward moving type II spicules. We find that most RBEs undergo a significant sideways motion during their lifetime. This is similar to what @2007Sci...318.1574D observed in type II spicules. The amplitude of the transverse velocities of RBEs ($\sim 8$ km s$^{-1}$) is slightly lower than that of type II spicules (of order 12 km s$^{-1}$). Our observations show a clearly separated blueshifted component in our line profiles which for the [H$\alpha$]{} line occurs at much higher velocities (of order 40–50 km s$^{-1}$) than in the [Ca II 854.2 nm]{}line. Such high velocities compare well with those reported for type II spicules (40–100 km s$^{-1}$), especially when taking into account the inevitable projection effects that reduce the line of sight velocity when observing at disk center.
The differences in RBE properties between [Ca II 854.2 nm]{} and [H$\alpha$]{} can be explained by the difference in chromospheric opacity of both lines, with [H$\alpha$]{} sampling higher layers than [Ca II 854.2 nm]{}. Our finding of higher velocities (both apparent and in Doppler) in [H$\alpha$]{} as compared to [Ca II 854.2 nm]{} is highly compatible with the idea of RBEs being disk counterparts of type II spicules: the higher opacity in [H$\alpha$]{}allows to sample higher layers where lower density plasma is propelled to higher velocities. The fact that on average the [Ca II 854.2 nm]{} RBEs occur closer to the footpoints and show lower velocities than [H$\alpha$]{} RBEs fits in well with the Hinode/SOT limb observations which have shown evidence of acceleration of plasma along type II spicules [@2007PASJ...59S.655D], with higher velocities at the top of the spicules. In addition, such acceleration is directly seen in our individual RBEs, with increasing Doppler shifts towards the top end of the RBEs in both [H$\alpha$]{} and [Ca II 854.2 nm]{}.
In addition to the opacity and acceleration argument, the difference in velocities in [Ca II 854.2 nm]{} and [H$\alpha$]{} could also partly be explained by the idea that @2008ApJ...679L.167L proposed. They performed numerical experiments to explore a scenario of reconnection jets driven by constant energy release at different atmospheric heights. In this scenario, reconnection at lower heights (i.e., higher density) gives rise to lower velocities – this could be another reason why disk center [Ca II 854.2 nm]{} RBEs have lower Doppler velocity than the off-limb type II spicules. In [H$\alpha$]{} however, higher atmospheric layers can be probed where reconnection at lower density would lead to higher velocities. More generally, we know that the limb observations introduce an observational bias towards those events that rocket high above the chromospheric fibrilar junk at lower heights. The large line of sight obscuration close to the limb may well, to a large extent, hide the lower velocity type II events at lower inclination with the surface. The highest velocities ($\sim 100$ km s$^{-1}$) of the type II spicules may thus be missed on the disk because of this selection effect, in addition to the low opacity of the low density tops of the spicules. The latter effect may also be the reason for the slightly lower transverse velocities we find in the [H$\alpha$]{} RBEs compared to those of the type II spicules at the limb. Generally, the RBEs likely sample slightly lower regions in the Ca II spicules where both longitudinal and transverse velocities are also a bit lower.
These and other selection effects imply that the histograms of Figs. \[fig:ha\_hist\] and \[fig:ca\_hist\] should not be interpreted as “the” properties of RBEs. After all, it is quite possible that our selection criteria might exclude events that have too little absorption or absorption at different Doppler velocities. In addition, the velocities we measure here are only rough estimates of the real mass motion. For example, the apparent velocities can sometimes be a bit more difficult to measure and interpret. This is because visual inspection of the movies at various wavelengths suggests that measurements of such projected velocities can sometimes depend considerably on the wavelength studied. However, the presence of a separate blue-shifted absorbing component certainly indicates strong mass flows of at least 30–50 km s$^{-1}$ in these features.
The events we measure thus have a wide variety of properties (lifetimes, velocities, temporal evolution, location, acceleration, etc.) that are highly similar to those of the type II spicules observed in [Ca II H]{} on the limb. We are therefore confident that we have found the on-disk counterparts of type II spicules.
This finding is further strengthened when we consider the occurrence rate of RBEs and compare it to the occurrence rate of type II spicules at the limb. The latter can be deduced from a visual analysis of Hinode/SOT data such as that studied by @2007PASJ...59S.655D. There we find between 1.5 and 3 [Ca II H]{}type II spicules per linear arcsec along the limb in a coronal hole. To compare this to our disk center observations of a coronal hole, we use our automated algorithm on the [H$\alpha$]{} images at $-$35 km s$^{-1}$ (the peak of the RBE Doppler shift distribution, Fig. \[fig:haca\_hist\]) and find on average about 40 RBEs per image. The field of view is about 70by 70. Observations at the limb show a large amount of line of sight superposition since spicules are quite tall (up to 10,000 km): we can still discern spicules (of 5,000 km height) at the limb even when they occur up to 115 arcsec in front or behind the limb. This means that a typical limb observation samples spicules over a spatial range of 230 arcseconds along the line of sight. This implies that our 40 RBEs per image would translate to $40\times230/70=131$ RBEs per 70 arcsec along the limb, i.e., 1.9 RBEs per linear arcsec. This compares very favorably with the 1.5 to 3 type II spicules we observe on average in Hinode/SOT limb data.
The discovery of the disk counterpart of type II spicules provides exciting new avenues towards resolving several of the major unresolved issues of these features. By providing a top view that is unhampered by the enormous line of sight superposition at the limb, RBEs will allow for a much improved study of the formation mechanism that drives these highly dynamic jets. Our preliminary finding that many of the RBEs show an increase in velocity and width of the blueshifted absorbing component as one travels along the jet provides a much needed constraint on theoretical models. For example, such an increase in velocity along the jet may well be compatible with (instantaneous) velocity profiles along reconnection jets in 2D MHD simulations [see, e.g. @2009arXiv0902.0977H]. The increase in width of the absorbing component along the feature, and the sudden disappearance or fading of RBEs at the end of their lifetime are highly compatible with a scenario that includes ongoing heating of the plasma along the jet. Such heating provides strict constraints on the acceleration and heating process that drives these jets. Detailed comparisons of the statistical properties of RBEs with advanced numerical models will be required to ascertain whether reconnection or other formation mechanisms (e.g., energy deposition at chromospheric heights from electron beams) can produce the details of the observed velocity and width profiles.
Now that we have established the disk counterpart of type II spicules, we will also be able to study in detail the relationship between these jets and the magnetic field concentrations with which they are associated. Such a relationship is bound to be quite complex, since we observe RBEs not only in association with mostly unipolar network in coronal holes, but also in quiet Sun, and highly unipolar plage regions. If reconnection plays a role in the formation of these jets, it is likely that component reconnection (e.g., at tangential discontinuities) plays a significant role [see e.g., @parker94current_sheets]. By studying the underlying magnetic field concentrations and its interaction with the flowfield, as well as the spectral line profiles at the footpoints of the RBEs, we should be able to constrain the various suggested driving mechanisms.
This is exemplified by the discovery of the black beads. These roundish features with strong absorption in the blue wing have many properties of RBEs with some significant differences. They appear along a neutral line between two regions of opposite polarity, and not around unipolar network like the other RBEs. Their shape and high velocities suggests that these beads may be jets in the direction of the line of sight. This can help explain the extremely high Doppler velocities in excess of $-$74 km s$^{-1}$ in the beads. It is tempting to speculate that the different magnetic topology also plays a role in the larger velocities. Perhaps the reconnection events release more energy because of opposite-polarity cancellation instead of reconnection at a tangential discontinuity? A preliminary study of the line profiles of the black beads indicates that some profiles are broadened on the red side of the spectral line as well. This could be indicative of heating at the footpoint of the jet. We will study these events and more generally the spectral line profiles at the footpoints of RBEs in a more extensive follow-up paper.
The presence of RBEs on the solar disk also provides an exciting way of establishing the connection of these jets to the recently discovered upflows that are seen as faint asymmetric profiles of EUV and UV emission lines which are formed at transition region and coronal temperatures [@2009arXiv0906.5434D]. These upflows in the transition region and corona have velocities of order 50–100 km s$^{-1}$ and have been connected to so-called straws: jet-like features observed as highly dynamic elongated brightenings on the disk in the Hinode/SOT [Ca II H]{} passband. Since the observations of @2009arXiv0906.5434D lacked chromospheric velocity information, significant uncertainties remain about the heating of type II spicules to coronal temperatures. The velocities, thermal evolution and occurrence rate of the RBEs we report on here are certainly compatible with the scenario of @2009arXiv0906.5434D. Follow-up observational studies with Hinode and CRISP will be invaluable to fully determine the thermal evolution of these spicules and the role they play in providing the corona with hot plasma.
This research was supported by the Research Council of Norway through grant 170935/V30. J.L. is supported by the European Commission funded Research Training Network SOLAIRE. B.D.P. thanks the Oslo group for excellent hospitality and is supported by NASA grants NNM07AA01C (HINODE), NNG06GG79G and NNX08AH45G. G.V. is supported by a Marie Curie Early Stage Research Training Fellowship of the European Community’s 6th Framework Programme (MEST-CT-2005-020395): The USO-SP International School for Solar Physics. We thank Ada Ortiz Carbonell, Viggo Hansteen, Sven Wedemeyer-B[ö]{}hm, Pit S[ü]{}tterlin, and Michiel van Noort for their help during the observations. We thank Viggo Hansteen for extensive discussions. The Swedish 1-m Solar Telescope is operated on the island of La Palma by the Institute for Solar Physics of the Royal Swedish Academy of Sciences in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrof[í]{}sica de Canarias.
[28]{} natexlab\#1[\#1]{}
, J. M. 1968, , 3, 367
, J., [Wang]{}, H., [Lee]{}, C.-Y., [Goode]{}, P. R., & [Schuhle]{}, U. 1998, , 504, L123+
, B., [Hansteen]{}, V. H., [Rouppe van der Voort]{}, L., [van Noort]{}, M., & [Carlsson]{}, M. 2007, ApJ, 655, 624
, B., [McIntosh]{}, S., [Hansteen]{}, V. H., [et al.]{} 2007, , 59, 655
, B., [McIntosh]{}, S. W., [Carlsson]{}, M., [et al.]{} 2007, Science, 318, 1574
, B., [McIntosh]{}, S. W., [Hansteen]{}, V. H., & [Schrijver]{}, C. J. 2009, ArXiv e-prints
, V. H., [De Pontieu]{}, B., [Rouppe van der Voort]{}, L., [van Noort]{}, M., & [Carlsson]{}, M. 2006, , 647, L73
, L., [De Pontieu]{}, B., & [Hansteen]{}, V. H. 2007, , 666, 1277
, L., [De Pontieu]{}, B., & [Hansteen]{}, V. H. 2009, ArXiv e-prints
, T., [Matsuzaki]{}, K., [Sakao]{}, T., [et al.]{} 2007, , 243, 3
, [Ø]{}., [De Pontieu]{}, B., [Carlsson]{}, M., [et al.]{} 2008, , 679, L167
, J., [Carlsson]{}, M., [Hansteen]{}, V., & [Rouppe van der Voort]{}, L. 2009, , 694, L128
, J., [Rutten]{}, R. J., [Carlsson]{}, M., & [Uitenbroek]{}, H. 2006, , 452, L15
, J., [Rutten]{}, R. J., [S[ü]{}tterlin]{}, P., [Carlsson]{}, M., & [Uitenbroek]{}, H. 2006, , 449, 1209
, J., [Hansteen]{}, V., [De Pontieu]{}, B., & [Carlsson]{}, M. 2009, ArXiv e-prints
, E. 1994, Spontaneous current sheets in magnetic fields : with applications to stellar x-rays (Oxford University Press)
, L. H. M., [De Pontieu]{}, B., [Hansteen]{}, V. H., [Carlsson]{}, M., & [van Noort]{}, M. 2007, , 660, L169
, G. B. 2006, , 447, 1111
, G. B., [Bjelksjo]{}, K., [Korhonen]{}, T. K., [Lindberg]{}, B., & [Petterson]{}, B. 2003, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4853, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, ed. S. L. [Keil]{} & S. V. [Avakyan]{}, 341–350
, G. B., [Dettori]{}, P. M., [Lofdahl]{}, M. G., & [Shand]{}, M. 2003, in Presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference, Vol. 4853, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, ed. S. L. [Keil]{} & S. V. [Avakyan]{}, 370–380
, G. B., [Narayan]{}, G., [Hillberg]{}, T., [et al.]{} 2008, , 689, L69
, A. C. 2000, , 196, 79
, S., [Ichimoto]{}, K., [Katsukawa]{}, Y., [et al.]{} 2008, , 249, 167
, M., [Rouppe van der Voort]{}, L., & [L[ö]{}fdahl]{}, M. G. 2005, , 228, 191
, M. J. & [Rouppe van der Voort]{}, L. H. M. 2006, ApJL, 648, L67
, M. J. & [Rouppe van der Voort]{}, L. H. M. 2008, , 489, 429
, A., [Cauzzi]{}, G., & [Reardon]{}, K. P. 2009, , 494, 269
, H., [Chae]{}, J., [Gurman]{}, J. B., & [Kucera]{}, T. A. 1998, , 183, 91
| {
"pile_set_name": "ArXiv"
} |
=5000 8.5in 0.25in .25in
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
=5000 8.5in 0.25in .25in
[**THE HIGHEST ENERGY NEUTRINOS**]{}\
Enrique Zas\
[*Departamento de Física de Partículas,\
Universidade de Santiago de Compostela, E-15706 Santiago, Spain*]{}
Introduction
============
High energy neutrino detection is one of the most exciting challenges in particle astrophysics because neutrinos provide an alternative view of the Universe. Efforts to build such detectors in the forthcoming years [@physrep] have granted an afternoon session on exprimental high energy neutrinos in this conference. Two of these projects, AMANDA and Baikal, are already in operation [@here]. The “km$^3$” initiative, to instrument 1 km$^3$ of water or ice with photodetectors, is the natural extention of the lower scale prototypes in view of the expected neutrino fluxes [@km3case]. There is some more motivation in the “Pierre Auger Project” to build an air shower detector of 6000 km$^2$ in search for the highest energy cosmic rays [@here]. The project is linked to neutrino astronomy in a double way. The production mechanism for the highest energy cosmic rays must make neutrinos at least in the interactions of the cosmic rays with the cosmic microwave background and, also, the array itself can be used to detect neutrinos of the highest energies [@venice].
I have been asked to review the possible sources of high energy neutrinos which is of course a pretty difficult task to do with justice in the light of all the activity in the field and the short space available. There is an excellent review provided by Ref. [@physrep] where more details and complete references can be found, and a discussion of neutrino fluxes close to this one in Ref. [@protflux]. Keeping this in mind I will restrict to some of the production mechanisms that predict the highest energy neutrinos. I will discuss their energy shape and comment on the plausibility of the mechanisms proposed, which is of course pretty subjective, stressing the developments in models with neutrinos produced in the jets of Active Galaxies.
Neutrino production by cosmic rays
==================================
In the majority of mechanisms most neutrinos arise from the decay of charged pions (or kaons), produced in different type of high energy particle interactions. The pions can be produced in proton-proton or photon-proton interactions or alternatively from direct fragmentation of quarks, in the same way they are produced routinely in electron positron colliders. For relativistic pions in flight it can be assumed that, on average, each of the four leptons produced in the reaction and subsequent muon decay carries one fourth of the parent energy.
The existence of high energy cosmic rays leaves little doubt about the actual production of neutrinos in their interactions with well understood targets. Atmospheric neutrinos fall in this category and are known to within about $10\%$ certainty at energies below 1 PeV [@physrep]. These neutrinos constitute the background for observation of other neutrinos sources. Their flux is zenith angle dependent because of the competition between interaction and decay of the parent pions. The vertical and horizontal atmospheric neutrino fluxes are shown in Fig 1A. At high enough energies the Lorentz expanded lifetime of the pion leads to more pion interactions decreasing the relative number of neutrinos to their parent pions. This causes that neutrinos from the decays of charmed particles (that have considerably shorter lifetimes), the “prompt” neutrinos, dominate the atmospheric flux above some unknown energy somewhere above 100 TeV. A typical prompt neutrino prediction [@hsvum] is illustrated in Fig. 1A. The uncertainty in the prompt component is due to the poorly known charm production cross section.
Cosmic rays must also interact with nucleons in the galaxy, such as dust, molecular clouds or compact objects like the sun. The interactions with the galactic disk matter are most relevant and do not have large uncertainties [@protflux]. The results [@domokos] are also shown in Fig. 1A evidencing that these neutrinos dominate the conventional atmospheric flux in the energy region where the prompt neutrinos are expected. This difficults their possible identification but it is hoped that the prompt neutrinos can be indirectly determined by measuring the atmospheric prompt muons which are produced by the same mechanism [@hsvum].
=6.5in
The interactions of cosmic rays with the cosmic microwave background is also an unavoidable source of neutrinos assuming the higher energy cosmic rays are of extragalactic origin and hence universal. This is supported by the non observation of cosmic ray anisotropies at high energies. Several groups that have calculated these fluxes[@physrep; @stecker91; @yoshida] and their results are within a couple of orders of magnitudes, mostly depending on the different assumptions made. I will refer to these as GZK neutrinos to stress their relation to the cosmic ray energy cutoff. For this calculation the cosmic ray spectrum has to be estimated at the production site. This implies extrapolating to energies above the maximum currently observed ($\sim3~10^{20}$ eV) and making some assumptions about the evolution of cosmic ray luminosity with cosmological time. The production mechanism has to be integrated over time (or redshift) up some earlier epoch ($z_{ult}$) which is expected to be provided by the Galaxy formation era ($z_{ult}$=2-4). These flux predictions are all fairly flat because the proton photon interaction cross section has a threshold behavior at the resonant $\Delta$ production. Most neutrinos are produced with energies about a factor 20 (see next section) below the Greisen-Zatsepin-Kuz’min cutoff energy $\sim 10^{20}~$eV. Depending on ($z_{ult}$) the interactions of the highest energy neutrinos with the cosmic neutrino background can play a more important role altering their shapes in the highest energy region. Fig. 1A includes some of these calculated fluxes indicating the levels of uncertainty.
Regardless of the uncertainties in the GZK neutrinos, all these mechanisms are certain, at least in the sense that if by some means they were found not to be there, the hypothetical implications of such non-discovery would have a larger impact in physics and/or cosmology than their actual observation.
Neutrino production in objects known to exist
=============================================
I will now consider another category of neutrino fluxes which is plausible in the sense that they can be produced in objects that we know exist. Some of them can be galactic such as accretion in binary systems, supernova remnants, but those reaching to highest energies are likely to be extragalactic. The most representative are Active Galactic Nuclei (AGN), and possibly Gamma Ray Bursts (GRB), although the origin of GRB’s is still in debate. AGN have also been dedicated a good part of a morning session in this conference. The recent detection of GeV gamma rays from over sixty AGN by the Compton Gamma Ray Observatory (GRO)[@GRO] together with de detection of TeV photons from three other nearby AGN with the imaging technique in Cherenkov telescopes [@weekes], place them at the forefront of particle astrophysics. These objects have also been proposed as sites for acceleration of the highest energy cosmic rays, as they have physical parameters which are dimensionally compatible with high energy cosmic rays. For all these reasons I will discuss them in some detail.
AGN are the most luminous objects that we observe. They display remarkable jets that stream highly collimated out of their cores to distances of several parsecs. They also show inner structure with superluminal motion, which is naturally explained by particle flows with bulk relativistic speeds. These jets observed in the radio band are very likely due to synchrotron emission from electrons that are accelerated along the jet axis. If protons are accelerated along with the electrons as some authors claim, then neutrino production is unavoidable because of photoproduction of pions in the ambient radiation field which is close to the Eddington limit. Earlier models of such fluxes considered their production in AGN cores [@stecker95] but the recent identification of all gamma ray detections with blazars [@mattox], believed to be AGN with their jets pointing towards us [@padovani], has shifted the interest to models in which protons are accelerated in the jets [@mannheim95; @protheroe96]. In these models the neutrinos are Lorentz boosted to energies higher than in AGN cores, what has important implications for their detection.
The models can be dimensionally explained with 3 simple assumptions [@apjzas]: protons are accelerated in the jets with an $E^{-2}$ spectrum as expected in shock acceleration, the maximum energy for the protons is $10^{20}$ eV and, finally, the target photon density behaves as a negative power law $E^{-\alpha}$ (for AGN in the broad infrared to X-ray band $\alpha$ is typically around 1).
It can be shown by simple energetics of the photopion production that the ratio of neutrino to photon luminosities is roughly $3/13$. The result is obtained using a cross section for $\pi^0$ production twice that for $\pi^+$, assuming each neutrino has exactly one fourth of the $\pi^+$ energy and adding a small correction for pair production [@physrep]. The neutrino energy flux can be obtained scaling the measured GeV to TeV gamma ray energy flux for Markarian 421, $J_{\gamma} \sim 5~10^{-10} $ TeV cm$^{-2}$ s$^{-1}$ with this ratio to get $J_{\nu} \sim 10^{-10} $ TeV cm$^{-2}$ s$^{-1}$. Mrk 421 is a nearby blazar which has been well established by GRO and by two Cherenkov telescopes. The shape of the neutrino spectrum here is also finally obtained by the threshold behavior of the photoproduction cross section at the $\Delta$ resonance. For a given proton energy $E_p$ the required energy of the target photon for resonance scales as $E_p^{-1}$. Combining the proton spectrum and the target photon density spectrum at resonance gives a power law $E_{\nu}^{-2+\alpha}$ with maximum neutrino energy of $2~10^{18}~eV \sim 0.25 <x_F>~E_p^{max}$ (where $<x_F>$=0.2 is the average Feynman-$x$ for photopion production at resonance). Provided $\alpha > 0$, the total energy flux, the spectral index and the maximum energy determine the flux uniquely because then the energy integral is insensitive to the lower limit. The shape of the spectrum obtained is very close to that predicting in the two alternative models for acceleration in AGN jets. It is a straightforward matter to rescale the flux with some factor of order 100 sr$^{-1}$ corresponding to the equivalent number of Mrk421 flux-like AGN per stereoradian[@apjzas], to get also the order of the normalization for the diffuse neutrino flux from all AGN. The exercise stresses the important assumptions in the models and explains the overall shift of energy to the $10^{18}$ eV region as illustrated in Fig. 1B where the two predictions for acceleration of protons in jets are compared to neutrinos from their cores.
Exotic neutrino sources
=======================
Lastly there is a third category of more exotic sources whose existence has only been postulated on theoretical grounds. Such is the case of Primordial Black Holes, decays of topological defects or WIMP annihilation. Topological defect (TD) scenarios arise in grand unified theories of particle interactions with spontaneous symmetry breakdown. They are naturally formed as some field vacuum goes through a phase transition to a new degenerate vacuum as the Universe cools down in its expansion. Different regions of space go to different vacuua and the net distribution may evolve later into a vacuum field with non-trivial topology, surrounding a point (monopole), a line (string) or a surface (domain wall). These cosmological objects accumulate energy and when they interact with themselves or with other objects of their own nature, they annihilate liberating large amounts of energy in the form of $X$ particles, the Gauge bosons of the underlying grand unified theory. TD scenarios have been recently heavily discussed as the possible origin of the highest energy cosmic rays. Several authors have normalized the defect abundances to cosmic or gamma ray measurements and bounds [@bahtta92; @sigl; @protstan]. Such a mechanism avoids the conceptual difficulties involved in accelerating protons or nuclei in our venicity to energies above the Greisen Zatsepin Kuz’min cutoff.
The models are however very uncertain because they are significantly affected by a variety of parameters besides the normalization itself. In general they extend to very high energies dictated by the mass of the $X$ particles expected to be of order $10^{14}-10^{16}~$GeV. The shape of the fluxes predicted are very flat and are somewhat different depending on the behavior of the time evolution of the effective injection rate of $X$ particles. This is usually parameterized as $t^{p-4}$ with $p=0$ for superconducting cosmic strings, $p=1$ for monopoles and cosmic strings and $p=2$ for constant injection rates in comoving volume [@bahtta92]. The main uncertainty in the neutrino spectrum shape is due to the fragmentation function assumed which is used with large extrapolations. Moreover the normalization to cosmic or gamma rays is also subject to uncertainties due to the interactions of the cosmic rays in their propagation, mostly with the poorly known extragalactig $B$ fields [@protstan]. Fig. 1B illustrates some of the produced neutrino fluxes by different authors and for different assumptions.
Experiment: present and future
==============================
There are already some experimental results in the form of upper bounds provided by three types of experiments. One is from underground muon detectors, of which Frèjus provides the most stringent limit [@frejus], the other two are from Extensive Air Shower detectors, particle detector arrays such as AKENO and EAS-TOP and a fluorescence light detector, Fly’s Eye. Their results are not straightforward to convert to bounds on differential neutrino spectra because the conversion involves an assumption on the shape of the neutrino spectrum. Moreover there are important uncertainties in the high energy neutrino cross sections, besides the usual experimental uncertainties associated with each of the experiments. Fig. 2 compares atmospheric fluxes, some TD fluxes and fluxes from AGN jets to these bounds. Some of the results are clearly in conflict with experiment. Some Superconducting Cosmic String models are ruled out by underground detectors and by the muon poor horizontal shower bound from AKENO [@blanco].
Our discussion of fluxes has stressed the importance that the highest energy neutrinos have for the future of this field, particularly in the light of recent theoretical developments. Possibly the largest challenge is provided by the low level of the GZK neutrinos which should however be there. The shift of interest to the higher energies has some important implications for detection because of the rise in the neutrino cross section. At these energies the Earth will be completely opaque to neutrinos so underground muon detectors will have to look for horizontal showers or vertical downgoing showers. Moreover the showers will be in dense media where interesting new effects such as the Landau-Pomerančuck- Migdal will markedly show up and difficult the energy measurements. This will certainly affect the optimal separation of their optical modules. A shift to neutrino energies in the $10^{18}~$eV and above adds considerable more interest to the alternative techniques such as the detection of horizontal air showers with giant array detectors like the Auger project or the yet unproven detection of the coherent radio pulses from the excess charge in the showers[@zas]. In table 1 some preliminary results of expected neutrino event rates in the Pierre Auger project for a number of the discussed fluxes are reported [@venice].
[$\nu$ source]{} Range of yearly event rates
-------------------------------- -----------------------------
[AGN-cores]{} [@stecker95] 0.2-1.5
[AGN-jets]{} [@mannheim95] 2-7
[GZK $z_{ult}=4$]{} [@yoshida] 0.1-0.4
[TD $p=1.5$]{} [@bahtta92] 2-10
: Expected neutrino event rates in the Pierre Auger Project for several fluxes.
Hopefully in the near future we will have some neutrino events. Underground muon detectors with a 1 km$^2$ surface area will have very enhanced acceptances for muon neutrinos because of the long range of the muon produced in charged current interactions, and can detect neutrinos for energies starting from roughly 100 GeV or so. The Pierre Auger Project could have an acceptance comparable to 1 km$^3$ for contained events and electron neutrinos. Lastly the radio technique if proven to be viable may open new possibilities of exploring even larger energies and lower fluxes. The complementarity between each of the detector types would no doubt constrain any hypothetically detected flux and allow the extraction of much more precise information.
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank Gonzalo Parente for reading the manuscript and J. Alvarez Muñiz, J.J. Blanco Pillado, F. Halzen, A. Letessier-Selvon, K. Mannheim, R. Protheroe, G. Sigl and R. Vázquez for discussions. This work was supported in part by CICYT (AEN96-1773) and by Xunta de Galicia (XUGA-20604A96).
[999]{} T.K. Gaisser, F. Halzen and T. Stanev, Phys. Rep. 238 (1995) 173, and references therein See these proceedings. F. Halzen, Proc. of the High Energy Neutrino Workshop, pp. 499, Venice February 1996. G. Parente and E. Zas, [*ibid*]{}, pp. 435, Venice, February 1996, astro-ph/9606091. R.J. Protheroe [*in “Towards the Millennium in Astrophysics: Problems and Prospects”*]{}, Erice 1996, eds. M.M. Shapiro and J.P. Wefel (World Scientific, Singapore), in press, and references therein G. Domokos et al., J. Phys. G: Nucl. Part. Phys. [**19**]{} (1993) 899. E. Zas, F. Halzen and R.A. Vázquez, Astropart. Phys. [**1**]{}, 297 (1993). F.W. Stecker, C. Done, M.H. Salamon and P. Sommers, Phys. Rev. Lett. [**66**]{} (1991) 2697. S. Yoshida and M. Teshima, Prog. of Theo. Phys. 89 (1993) 833. D.J. Thompson et al., [*Ap. J. Suppl.*]{} [**101**]{} (1995) 259; [*Ap. J. Suppl.*]{} in press (1996). M. Punch [*et al.*]{}, [*Nature [**358**]{} (1992) 477–478*]{}, and T.C. Weekes these proceedings. F.W. Stecker and M.H. Salamon, preprint astro-ph/9501064. J.R. Mattox et al., to appear in [*Ap. J.*]{} [**481**]{} (1997). C.M. Urry and P. Padovani, [*Pub. Astr. Soc. Pacif.*]{} [**107**]{}, 803 (1995); and these proc. K. Mannheim, Astrop. Phys. 3 (1995) 295-302. R.J. Protheroe, in Proc. IAU Colloq. 163, Accretion Phenomena and Related Outflows, ed. D. Wickramasinghe et al., in press (1996). F. Halzen and E. Zas, astro-ph/9702193 submitted to [*Ap. J.*]{} P. Battacharjee, C.T. Hill, D.N. Schramm, Phys. Rev.Lett. [**69**]{} (1992) 567. G. Sigl, S. Lee and D.N. Schramm, astro-ph/9610221 submitted to [*Phys. Lett.*]{} [**B**]{}. R.J. Protheroe and T. Stanev, [*Phys. Rev. Lett.*]{}, [**77**]{}, 3708 (1996). W. Rhode, K. Daum, [*et al.*]{} Proc. 24th ICRC, Rome 1995. vol. 1, p. 781. Also, W. Rhode, [*et al.*]{} [*Astropart. Phys.*]{} [**4**]{} (1996) 217. J.J. Blanco-Pillado, R.A. Vázquez and E. Zas, in press Phys. Rev. Lett. Zas, E., Halzen, F. and Stanev, T., [*Phys. Rev.*]{} [**D45**]{} 362 (1992).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Advanced machine learning techniques have boosted the performance of natural language processing. Nevertheless, recent studies, e.g., @jieyu2017men show that these techniques inadvertently capture the societal bias hidden in the corpus and further amplify it. However, their analysis is conducted only on models’ top predictions. In this paper, we investigate the gender bias amplification issue from the distribution perspective and demonstrate that the bias is amplified in the view of predicted probability distribution over labels. We further propose a bias mitigation approach based on posterior regularization. With little performance loss, our method can almost remove the bias amplification in the distribution. Our study sheds the light on understanding the bias amplification.'
author:
- |
Shengyu Jia$^{\clubsuit}$[^1], Tao Meng$^{\spadesuit}$, Jieyu Zhao$^\spadesuit$, Kai-Wei Chang$^\spadesuit$\
$^\clubsuit$ Tsinghua University\
$^\spadesuit$ University of California, Los Angeles\
[jiasy16@mails.tsinghua.edu.cn, ]{}\
[{mengt18, jieyuzhao, kwchang}@ucla.edu]{}\
bibliography:
- 'acl2020.bib'
title: |
Mitigating Gender Bias Amplification in Distribution by\
Posterior Regularization
---
Introduction
============
Data-driven machine learning models have achieved high performance in various applications. Despite the impressive results, recent studies (e.g., @wang2019balanced [@hendricks2018women]) demonstrate that these models may carry societal biases exhibited in the dataset they trained on. In particular, @jieyu2017men show that a model trained on a biased dataset may amplify the bias. For example, we can consider a task of labeling the activity and objects depicted in an image. The training set contains 30% more images with “woman cooking” than “man cooking”. However, when evaluating the top predictions of a trained model, the disparity between males and females is amplified to around 70%. Based on this observation, @jieyu2017men conduct a systematic study and propose to calibrate the top predictions of a learned model by injecting corpus-level constraints to ensure that the gender disparity is not amplified. However, when analyzing the top predictions, the models are forced to make one decision. Therefore, even if the model assigns high scores to both labels of “woman cooking” and “man cooking”, it has to pick one as the prediction. This process obviously has a risk to amplify the bias. However, to our surprise, we observe that gender bias is also amplified when analyzing the posterior distribution of the predictions. Since the model is trained with regularized maximal likelihood objective, the bias in distribution is a more fundamental perspective of analyzing the bias amplification issue.
In this paper, we conduct a systematic study to quantify the bias in the predicted distribution over labels. Our analysis demonstrates that when evaluating the distribution, though not as significant as when evaluating top predictions, the bias amplification exists. About half of activities show significant bias amplification in the posterior distribution, and on average, they amplify the bias by 3.2%.
We further propose a new bias mitigation technique based on posterior regularization because the approaches described in @jieyu2017men can not be straightforwardly extended to calibrate bias amplification in distribution. With the proposed technique, we successfully remove the bias amplification in the posterior distribution while maintain the performance of the model. Besides, the bias amplification in the top predictions based on the calibrated distribution is also mitigated by around 30%. These results suggest that the bias amplification in top predictions comes from both the requirement of making hard predictions and the bias amplification in the posterior distribution of the model predictions. Our study advances the understanding of the bias amplification issue in natural language processing models. The code and data are available at <https://github.com/uclanlp/reducingbias>.
Related Work
============
#### Algorithmic Bias
Machine learning models are becoming more and more prevalent in the real world, and algorithmic bias will have a great societal impact [@tonry2010social; @buolamwini2018gender]. Researchers have found societal bias in different applications such as coreference resolution [@rudinger2018gender; @zhao2018gender], machine translation [@stanovsky2019evaluating] and online advertisement [@sweeney2013discrimination]. Without appropriate adjustments, the model can amplify the bias [@jieyu2017men]. Different from the previous work, we aim at understanding the bias amplification from the posterior perspective instead of directly looking at the top predictions of the model.
#### Posterior Regularization
The posterior regularization framework [@ganchev2010posterior] is aiming to represent and enforce constraints on the posterior distribution. It has been shown effective to inject domain knowledge for NLP applications. For example, @ji2012question [@gao2014learning] design constraints based on similarity to improve question answering and machine translation, respectively. @yang2014context propose constraints based on lexical patterns in sentiment analysis. @meng2019target apply corpus-level constraints to guide a dependency parser in the cross-lingual transfer setting. In this paper we leverage corpus-level constraints to calibrate the output distribution. Our study resembles to the confidence calibration [@guo2017calibration; @pakdaman2015obtaining]. However, the temperature turning and binning methods proposed in these papers cannot straightforwardly be extended to calibrate the bias amplification.
Background {#sec:background}
==========
We follow the settings in @jieyu2017men to focus on the imSitu vSRL dataset [@yatskar2016situation], in which we are supposed to predict the activities and roles in given images and this can be regraded as a structure prediction task (see Fig. \[fig:imsitu\]).
![An instance from the imSitu dataset. Given an input image, the task it to identify the activity depicted in the image as well as the objects (noun) and their semantic role.[]{data-label="fig:imsitu"}](imsitu2.png){width="\linewidth"}
We apply the Conditional Random Field (CRF) model for the structure prediction task. We denote $\mathbf{y}$ as a joint prediction result for all instances, and $\mathbf{y}^i$ as a prediction result for instance $i$. We use $\mathbf{y}_v$ to denote the predicted activity, and $\mathbf{y}_r$ to denote the predicted role. An activity can have multiple roles and usually one of them conveys the gender information. For an instance $i$, the CRF model predicts the scores for every activity and role, and the score for a prediction is the summation of all these scores. Formally, $$f_\theta(\mathbf{y}^i,i)=s_\theta(\mathbf{y}^i_v,i)+\sum\nolimits_{e\in \mathbf{y}^i_r}s_\theta(\mathbf{y}^i_v,e,i),$$ where $s_\theta(\mathbf{y}^i_v,i)$ and $s_\theta(\mathbf{y}^i_v,e,i)$ are the scores for activity $\mathbf{y}^i_v$ of instance $i$, and the score for role $e$ of instance $i$ with activity $\mathbf{y}^i_v$, respectively. We can infer the top structure for instance $i$ by: $$\arg\max\nolimits_{\mathbf{y}^i\in\mathcal{Y}^i}f_\theta(\mathbf{y}^i,i),$$ where $\mathcal{Y}^i$ refers to all the possible assignments to the instance.
Bias Amplification Quantification and Corpus-level Constraints {#sec:corpusconstr}
==============================================================
@jieyu2017men demonstrate bias amplification in the top prediction and present a bias mitigation technique by inference with corpus-level constraints. In the following, we extend their study to analyze the bias amplification in the posterior distribution by the CRF model and define the corresponding corpus-level constraints.
Formally, the probability of prediction $\mathbf{y}^i$ for instance $i$ and the joint prediction $\mathbf{y}$ defined by CRF model with parameters $\theta$ are given by $$\label{eq:crfprob}
\begin{aligned}
&p_\theta(\mathbf{y}^i,i)\propto \exp(f_\theta(\mathbf{y}^i, i)),\\
&p_\theta(\mathbf{y})=\prod\nolimits_i p_\theta(\mathbf{y}^i,i),
\end{aligned}$$ since instances are mutually independent.
In this section, we will define how to quantify the bias and the bias amplification in the distribution, and introduce the corpus-level constraints towards restricting the bias in the distribution.
We focus on the gender bias on activities in the vSRL task. To quantify the gender bias given a particular activity $v^*$, @jieyu2017men uses the percentage that $v^*$ is predicted together with male agents among all prediction with genders. This evaluation focuses on the top prediction. In the contrast, we define bias function $B(p,v^*,D)$ w.r.t distribution $p$ and activity $v^*$, evaluating the bias toward male in dataset $D$ based on the conditional probability $P(X|Y)$, where $event\ Y:$ given an instance, its activity is predicted to be $v^*$ and its role is predicted to have a gender; $event\ X:$ this instance is predicted to have gender male. Formally, $$\label{eq:rfunc}
\begin{aligned}
& B(p,v^*,D) \\
=& \mathbb{P}_{i\sim D, \mathbf{y}\sim p}(\mathbf{y}^i_r\in M|\mathbf{y}^i_v=v^*\wedge \mathbf{y}^i_r\in M\cup W) \\
=& \frac{\sum_{i\in D} \sum_{\mathbf{y}^i:\mathbf{y}^i_v=v^*,\mathbf{y}^i_r\in M}p(\mathbf{y}^i, i)}{\sum_{i\in D} \sum_{\mathbf{y}^i:\mathbf{y}^i_v=v^*,\mathbf{y}^i_r\in M\cup W}p(\mathbf{y}^i, i)}. \\
\end{aligned}$$
This bias can come from the training set $D_{tr}$. Here we use $b^*(v^*, male)$ to denote the “dataset bias” toward male in the training set, measured by the ratio of between male and female from the labels:
$$b^*=\frac{\sum_{i\in D_{tr}} \mathbf{1}[\hat{\mathbf{y}}^i_v=v^*,\hat{\mathbf{y}}^i_r\in M]}{\sum_{i\in D_{tr}} \mathbf{1}[\hat{\mathbf{y}}^i_v=v^*,\hat{\mathbf{y}}^i_r\in M\cup W]},$$ where $\hat{\mathbf{y}}^i$ denotes the label of instance $i$.
Ideally, the bias in the distribution given by CRF model should be consistent with the bias in the training set, since CRF model is trained by maximum likelihood. However, the amplification exists in practice. Here we use the difference between the bias in the posterior distribution and in training set to quantify the bias amplification, and average it over all activities to quantify the amplification in the whole dataset: $$\begin{aligned}
A(p,v^*,D)&=sgn(b^*-0.5)[B(p,v^*,D)-b^*],\\
\bar{A}(p,D)&=\frac{1}{|V|}\sum_{v^*\in V} A(p,v^*,D).\\
\end{aligned}$$ Note that if we use the top prediction indicator function to replace $p$ in $A,\bar{A}$, it is the same as the definition of the bias amplification in top prediction in @jieyu2017men.
The corpus-level constraints aim at mitigating the bias amplification in test set $D_{ts}$ within a pre-defined margin $\gamma$, $$\label{eq:constr}
\forall v^*,\ |A(p,v^*,D_{ts})| \leq \gamma.$$
Posterior Regularization
========================
Posterior regularization [@ganchev2010posterior] is an algorithm leveraging corpus-level constraints to regularize the posterior distribution for a structure model. Specifically, given corpus-level constraints and a distribution predicted by a model, we 1) define a feasible set of the distributions with respect to the constraints; 2) find the closest distribution in the feasible set from given distribution; 3) do maximum a posteriori (MAP) inference on the optimal feasible distribution.
The feasible distribution set $Q$ is defined by the corpus-level constraints defined in Eq. : $$\label{eq:constrQ}
Q=\{q\ |\ \forall v^*,\ |B(q, v^*, D_{ts}) - b^*
|\leq \gamma\},$$ where $B(\cdot)$ is defined in Eq. .
Given the feasible set $Q$ and the model distribution $p_\theta$ defined by Eq. , we want to find the closest feasible distribution $q^*:$ $$\label{eq:probj}
q^*=\arg\min\nolimits_{q\in Q} KL(q\|p_\theta).$$
This is an optimization problem and our variable is the joint distribution $q$ with constraints, which is intractable in general. Luckily, according to the results in @ganchev2010posterior, if the feasible set $Q$ is defined in terms of constraints feature functions $\phi$ and their expectations: $$\label{eq:expQ}
Q=\{q\ |\ \mathbb{E}_{\mathbf{y}\sim q}[\phi(\mathbf{y})\leq\mathbf{c}]\},$$ Eq. will have a close form solution $$\label{eq:qstar}
q^*(\mathbf{y})=\frac{p_\theta(\mathbf{y})\exp(-\lambda^*\cdot \phi(\mathbf{y}))}{Z(\lambda^*)},$$ where $\lambda^*$ is the solution of $$\label{eq:lambda}
\begin{aligned}
\lambda^* &= \arg\max\nolimits_{\lambda\geq 0}-\mathbf{c}\cdot \lambda-\log Z(\lambda). \\
Z(\lambda) &= \sum\nolimits_{\mathbf{y}} p_\theta(\mathbf{y})\exp(-\lambda\cdot \phi(\mathbf{y})).
\end{aligned}$$
Actually, we can derive the constraints into the form we want. We set $\mathbf{c}=\mathbf{0}$ and $$\label{eq:phi}
\phi(\mathbf{y})=\sum\nolimits_i \phi^i(\mathbf{y}^i).$$ We can choose a proper $\phi^i(\mathbf{y}^i)$ to make Eq. equal to Eq. . The detailed derivation and the definition of $\phi^i(\mathbf{y}^i)$ are shown in Appendix \[app:phi\].
We can solve Eq. by gradient-based methods to get $\lambda^*$, and further compute the close form solution in Eq. . Actually, considering the relation between $\mathbf{y}$ and $\mathbf{y}^i$ in Eq. and , we can factorize the solution in Eq. on instance level: $$q^*(\mathbf{y}^i,i)=\frac{p_\theta(\mathbf{y}^i,i)\exp(-\lambda^*\cdot \phi^i(\mathbf{y}^i))}{Z^i(\lambda^*)},$$ and the derivation details are in Appendix \[app:expectation\]. With this, we can reuse original inference algorithm to conduct MAP inference based on the distribution $q^*$ for every instance seperately.
[0.45]{} {width="\linewidth"} \[fig:dist\_before\]
[0.45]{} {width="\linewidth"} \[fig:dist\_after\]
\
[0.45]{} {width="\linewidth"} \[fig:inf\_before\]
[0.45]{} {width="\linewidth"} \[fig:inf\_after\]
Experiments
===========
We conduct experiments on the vSRL task to analyze the bias amplification issue in the posterior distribution and demonstrate the effectiveness of the proposed bias mitigation technique.
#### Dataset
Our experiment settings follow @jieyu2017men. We evaluate on imSitu [@yatskar2016situation] that activities are selected from verbs, roles are from FrameNet [@baker1998framenet] and nouns from WordNet [@fellbaum1998wordnet]. We filter out the non-human oriented verbs and images with labels that do not indicate the genders.
#### Model
We analyze the model purposed together with the dataset. The score functions we describe in Sec. \[sec:background\] are modeled by VGG [@simonyan2015vgg] with a feedforward layer on the top of it. The scores are fed to CRF for inference.
Bias Amplification in Distribution
----------------------------------
Figures \[fig:dist\_before\] and \[fig:inf\_before\] demonstrate the bias amplification in both posterior distribution $p_\theta$ and the top predictions $\mathbf{y}$ defined in Sec.\[sec:corpusconstr\], respectively. For most activities with the bias toward male (i.e., higher bias score) in the training set, both the top prediction and posterior distribution are even more biased toward male, vise versa. If the bias is not amplified, the dots should be scattered around the reference line. However, most dots are on the top-right or bottom-left, showing the bias is amplified. The black regression line with $slope>1$ also indicates the amplification. Quantitatively, $109$ and $173$ constraints are violated when analyzing the bias in distribution an in top predictions. Most recent models are trained by minimizing the cross-entropy loss which aims at fitting the model’s predicted distribution with observed distribution on the training data. In the inference time, the model outputs the top predictions based on the underlying prediction distribution. Besides, in practice, the distribution has been used as an indicator of confidence in the prediction. Therefore, understanding bias amplification in distribution provides a better view about this issue.
![The curve of training and test accuracy, and bias amplification with the number of training epochs. The optimal model evaluated on the development set is found in the grey shade area.[]{data-label="fig:early"}](early.pdf){width="\linewidth"}
To analyze the cause of bias amplification, we further show the degree of amplification along with the learning curve of the model (see Fig. \[fig:early\]). We observed that when the model is overfitted, the distribution of the model prediction becomes more peaky[^2]. We suspect this is one of the key reasons causes the bias amplification.
Bias Amplification Mitigation
-----------------------------
We set the margin $\gamma=0.05$ for every constraint in evaluation. However, we employ a stricter margin ($\gamma=0.001$) in performing posterior regularization to encourage the model to achieve a better feasible solution. We use mini-batch to estimate the gradient w.r.t $\lambda$ with Adam optimizer [@kingma2014adam] when solving Eq. . We set the batchsize to be $39$ and train for $10$ epochs. The learning rate is initialized as $0.1$ and decays after every mini-batch with the decay factor $0.998.$
#### Results
We then apply the posterior regularization technique to mitigate the bias amplification in distribution. Results are demonstrated in Figures \[fig:dist\_after\] (distribution) and \[fig:inf\_after\] (top predictions). The posterior regularization effectively calibrates the bias in distribution and only $5$ constraints are violated after the calibration. The average bias amplification is close to $0$ ($\bar{A}$: $0.032$ to $-0.005$). By reducing the amplification of bias in distribution, the bias amplification in top predictions also reduced by 30.9% ($\bar{A}$: $0.097$ to $0.067$). At the same time, the model’s performance is kept (accuracy: $23.2\%$ to $23.1\%$).
Note that calibrating the bias in distribution cannot remove all bias amplification in the top predictions. We posit that the requirement of making hard predictions (i.e., maximum a posteriori estimation) also amplifies the bias when evaluating the top predictions.
Conclusion
==========
We analyzed the bias amplification from the posterior distribution perspective, which provides a better view to understanding the bias amplification issue in natural language models as these models are trained with the maximum likelihood objective. We further proposed a bias mitigation technique based on posterior regularization and show that it effectively reduces the bias amplification in the distribution. Due to the limitation of the data, we only analyze the bias over binary gender. However, our analysis and the mitigation framework is general and can be adopted to other applications and other types of bias.
One remaining open question is why the gender bias in the posterior distribution is amplified. We posit that the regularization and the over-fitting nature of deep learning models might contribute to the bias amplification. However, a comprehensive study is required to prove the conjecture and we leave this as future work.
#### Acknowledgement
This work was supported in part by National Science Foundation Grant IIS-1927554. We thank anonymous reviewers and members of the UCLA-NLP lab for their feedback.
[^1]: Both authors contributed equally to this work and are listed in alphabetical order.
[^2]: This effect, called overconfident, has been also discussed in the literature [@guo2017calibration].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper we propose a model to learn multimodal multilingual representations for matching images and sentences in different languages, with the aim of advancing multilingual versions of image search and image understanding. Our model learns a common representation for images and their descriptions in two different languages (which need not be parallel) by considering the image as a pivot between two languages. We introduce a new pairwise ranking loss function which can handle both symmetric and asymmetric similarity between the two modalities. We evaluate our models on image-description ranking for German and English, and on semantic textual similarity of image descriptions in English. In both cases we achieve state-of-the-art performance.'
author:
- |
Spandana Gella Rico Sennrich Frank Keller Mirella Lapata\
Institute for Language, Cognition and Computation\
School of Informatics, University of Edinburgh\
{spandana.gella, rico.sennrich}@ed.ac.uk\
{keller,mlap}@inf.ed.ac.uk
bibliography:
- 'references.bib'
title: Image Pivoting for Learning Multilingual Multimodal Representations
---
Introduction
============
In recent years there has been a significant amount of research in language and vision tasks which require the joint modeling of texts and images. Examples include text-based image retrieval, image description and visual question answering. An increasing number of large image description datasets has become available [@hodosh:flickr8k:2013; @young:flickr30k:2014; @mscoco:2014] and various systems have been proposed to handle the image description task as a generation problem [@Bernardi:ea:16; @mao2014deep; @google:show-tell:2014; @fang:msr:captions:2015]. There has also been a great deal of work on sentence-based image search or cross-modal retrieval where the objective is to learn a joint space for images and text [@hodosh:flickr8k:2013; @devise:2013; @karpathy:deep-fragments:2014; @kiros2014unifying; @sdt-rnn:2015; @lrcn:2015]. Previous work on image description generation or learning a joint space for images and text has mostly focused on English due to the availability of English datasets. Recently there have been attempts to create image descriptions and models for other languages [@uiuc:jp:2015; @multi30k:2016; @bridgeCorr:2016; @yjcaptions:2016; @specia:mmt:2016; @li:flickr8kcn:2016; @imagePivotACL16; @YoshikawaST17]. Most work on learning a joint space for images and their descriptions is based on Canonical Correlation Analysis (CCA) or neural variants of CCA over representations of image and its descriptions [@hodosh:flickr8k:2013; @andrew:dcca:2013; @yan:dcca:2015; @gong:tcca:2014; @chandar:corrnet:2016]. Besides CCA, a few others learn a visual-semantic or multimodal embedding space of image descriptions and representations by optimizing a ranking cost function [@kiros2014unifying; @sdt-rnn:2015; @ma2015multimodal; @vendrov2015order] or by aligning image regions (objects) and segments of the description [@karpathy:deep-fragments:2014; @plummer2015flickr30k] in a common space. Recently @lin:leveraging:vqa:embeddings:2016 have leveraged visual question answering models to encode images and descriptions into the same space.
However, all of this work is targeted at monolingual descriptions, i.e., mapping images and descriptions in a single language onto a joint embedding space. The idea of pivoting or bridging is not new and language pivoting is well explored for machine translation [@wu2007pivot; @FiratSAYC16] and to learn multilingual multimodal representations [@bridgeCorr:2016; @calixto:mlmme:2017]. @bridgeCorr:2016 propose a model to learn common representations between $M$ views and assume there is parallel data available between a pivot view and the remaining $M-1$ views. Their multimodal experiments are based on English as the pivot and use large parallel corpora available between languages to learn their representations.
Related to our work @calixto:mlmme:2017 proposed a model for creating multilingual multimodal embeddings. Our work is different from theirs in that we choose the image as the pivot and use a different similarity function. We also propose a single model for learning representations of images and multiple languages, whereas their model is language-specific.
In this paper, we learn multimodal representations in multiple languages, i.e., our model yields a joint space for images and text in multiple languages using the image as a pivot between languages. We propose a new objective function in a multitask learning setting and jointly optimize the mappings between images and text in two different languages.
Dataset {#sec:dataset}
=======
We experiment with the Multi30k dataset, a multilingual extension of Flickr30k corpus [@young:flickr30k:2014] consisting of English and German image descriptions [@multi30k:2016]. The Multi30K dataset has 29k, 1k and 1k images in the train, validation and test splits respectively, and contains two types of multilingual annotations: (i) a corpus of one English description per image and its translation into German; and (ii) a corpus of five independently collected English and German descriptions per image. We use the independently collected English and German descriptions to train our models. Note that these descriptions are not translations of each other, i.e., they are not parallel, although they describe the same image.
Problem Formulation
===================
Given an image $i$ and its descriptions $c_1$ and $c_2$ in two different languages our aim is to learn a model which maps $i$, $c_1$ and $c_2$ onto same common space $\mathbb{R}^{N}$ (where $N$ is the dimensionality of the embedding space) such that the image and its gold-standard descriptions in both languages are mapped close to each other (as shown in Figure \[fig:proposed\_model\]). Our model consists of the embedding functions $f_i$ and $f_{c}$ to encode images and descriptions and a scoring function $S$ to compute the similarity between a description–image pair.
In the following we describe two models: (i) the [<span style="font-variant:small-caps;">Pivot</span>]{}model that uses the image as pivot between the description in both the languages; (ii) the [<span style="font-variant:small-caps;">Parallel</span>]{}model that further forces the image descriptions in both languages to be closer to each other in the joint space. We build two variants of [<span style="font-variant:small-caps;">Pivot</span>]{}and [<span style="font-variant:small-caps;">Parallel</span>]{}with different similarity functions $S$ to learn the joint space.
Multilingual Multimodal Representation Models {#sec:model}
---------------------------------------------
In both [<span style="font-variant:small-caps;">Pivot</span>]{}and [<span style="font-variant:small-caps;">Parallel</span>]{}we use a deep convolutional neural network architecture (CNN) to represent the image $i$ denoted by $f_i(i)= W_{i} \cdot CNN(i)$ where $W_{i}$ is a learned weight matrix and $CNN(i)$ is the image vector representation. For each language we define a recurrent neural network encoder $f_c(c_{k}) = GRU(c_{k})$ with gated recurrent units (GRU) activations to encode the description $c_{k}$.
In [<span style="font-variant:small-caps;">Pivot</span>]{}, we use monolingual corpora from multiple languages of sentences aligned with images to learn the joint space. The intuition of this model is that an image is a universal representation across all languages, and if we constrain a sentence representation to be closer to image, sentences in different languages may also come closer. Accordingly we design a loss function as follows: $$\small
\begin{split}
loss_{pivot} = \sum_{k} \bigg[ \sum_{(c_{k}, i)} \bigg( \sum_{c'_{k}} \max\{0, \alpha - S(c_{k},i) + S(c'_{k}, i)\} \\
+ \sum_{i'} \max\{0, \alpha - S(c_{k},i) + S(c_{k}, i')\} \bigg) \bigg]
\end{split}
\label{eq:contrastive}$$ where $k$ stands for each language. This loss function encourages the similarity $S(c_{k},i)$ between gold-standard description $c_{k}$ and image $i$ to be greater than any other irrelevant description $c'_{k}$ by a margin $\alpha$. A similar loss function is useful for learning multimodal embeddings in a single language [@kiros2014unifying]. For each minibatch, we obtain invalid descriptions by selecting descriptions of other images except the current image of interest and vice-versa.
In [<span style="font-variant:small-caps;">Parallel</span>]{}, in addition to making an image similar to a description, we make multiple descriptions of the same image in different languages similar to each other, based on the assumption that these descriptions, although not parallel, share some commonalities. Accordingly we enhance the previous loss function with an additional term: $$\small
\begin{split}
loss_{para} = loss_{pivot} +
\sum_{(c_{1}, c_{2})} \bigg( \sum_{c'_{1}} \max\{0, \alpha - S(c_{1},c_{2}) \\
\noindent + S(c'_{1}, c_{2})\} + \sum_{c'_{2}}\max\{0, \alpha - S(c_{1},c_{2}) + S(c_{1}, c'_{2})\} \bigg)
\end{split}
\label{eq:contrastive-lang}$$ Note that we are iterating over all pairs of descriptions $(c_1, c_2)$, and maximizing the similarity between descriptions of the same image and at the same time minimizing the similarity between descriptions of different images.
We learn models using two similarity functions: symmetric and asymmetric. For the former we use cosine similarity and for the latter we use the metric of which is useful for learning embeddings that maintain an order, e.g., dog and cat are more closer to pet than animal while being distinct. Such ordering is shown to be useful in building effective multimodal space of images and texts. An analogy in our setting would be two descriptions of an image are closer to the image while at the same time preserving the identity of each (which is useful when sentences describe two different aspects of the image). The similarity metric is defined as: $$\small
S(a,b) = - ||max(0, b - a)||^2 \label{eq:order-score}$$ where $a$ and $b$ are embeddings of image and description.
We call the symmetric similarity variants of our models as [<span style="font-variant:small-caps;">Pivot-Sym</span>]{}and [<span style="font-variant:small-caps;">Parallel-Sym</span>]{}, and the asymmetric variants [<span style="font-variant:small-caps;">Pivot-Asym</span>]{}and [<span style="font-variant:small-caps;">Parallel-Asym</span>]{}.
Experiments and Results
=======================
We test our model on the tasks of image-description ranking and semantic textual similarity. We work with each language separately. Since we learn embeddings for images and languages in the same semantic space, our hope is that the training data for each modality or language acts complementary data for the another modality or language, and thus helps us learn better embeddings.
#### Experiment Setup
We sampled minibatches of size 64 images and their descriptions, and drew all negative samples from the minibatch. We trained using the Adam optimizer with learning rate 0.001, and early stopping on the validation set. Following @vendrov2015order we set the dimensionality of the embedding space and the GRU hidden layer $N$ to 1024 for both English and German. We set the dimensionality of the learned word embeddings to 300 for both languages, and the margin $\alpha$ to 0.05 and 0.2, respectively, to learn asymmetric and symmetric similarity-based embeddings.[^1] We keep all hyperparameters constant across all models. We used the L2 norm to mitigate over-fitting [@kiros2014unifying]. We tokenize and truecase both English and German descriptions using the Moses Decoder scripts.[^2]
To extract image features, we used a convolutional neural network model trained on 1.2M images of 1000 class ILSVRC 2012 object classification dataset, a subset of ImageNet [@imagenet:2014]. Specifically, we used VGG 19-layer CNN architecture and extracted the activations of the penultimate fully connected layer to obtain features for all images in the dataset [@vgg:2014]. We use average features from 10 crops of the re-scaled images.[^3]
#### Baselines
As baselines we use monolingual models, i.e., models trained on each language separately. Specifically, we use Visual Semantic Embeddings (VSE) of and Order Embeddings (OE) of . We use a publicly available implementation to train both VSE and OE.[^4]
Image-Description Ranking Results
---------------------------------
To evaluate the multimodal multilingual embeddings, we report results on an image-description ranking task. Given a query in the form of a description or an image, the task its to retrieve all images or descriptions sorted based on the relevance. We use the standard ranking evaluation metrics of recall at position $k$ (R@K, where higher is better) and median rank (Mr, where lower is better) to evaluate our models. We report results for both English and German descriptions. Note that we have one single model for both languages.
In Tables \[tab:english-embedding-results\] and \[tab:german-embedding-results\] we present the ranking results of the baseline models of @kiros2014unifying and @vendrov2015order and our proposed [<span style="font-variant:small-caps;">Pivot</span>]{}and [<span style="font-variant:small-caps;">Parallel</span>]{}models. We do not compare our image-description ranking results with @calixto:mlmme:2017 since they report results on half of validation set of Multi30k whereas our results are on the publicly available test set of Multi30k. For English, [<span style="font-variant:small-caps;">Pivot</span>]{}with asymmetric similarity is either competitive or better than monolingual models and symmetric similarity, especially in the R@10 category it obtains state-of-the-art. For German, both [<span style="font-variant:small-caps;">Pivot</span>]{}and [<span style="font-variant:small-caps;">Parallel</span>]{}with the asymmetric scoring function outperform monolingual models and symmetric similarity. We also observe that the German ranking experiments benefit the most from the multilingual signal. A reason for this could be that the German description corpus has many singleton words (more than $50\%$ of the vocabulary) and English description mapping might have helped in learning better semantic embeddings. These results suggest that the multilingual signal could be used to learn better multimodal embeddings, irrespective of the language. Our results also show that the asymmetric scoring function can help learn better embeddings. In [Table \[tab:model-analysis\]]{} we present a few examples where [<span style="font-variant:small-caps;">Pivot-Asym</span>]{}and [<span style="font-variant:small-caps;">Parallel-Asym</span>]{}models performed better on both the languages compared to baseline order embedding model even using descriptions of very different lengths as queries.
Semantic Textual Similarity Results
-----------------------------------
In the semantic textual similarity task (STS), we use the textual embeddings from our model to compute the similarity between a pair of sentences (image descriptions in this case). We evaluate on video task from STS-2012 and image tasks from STS-2014, STS-2015 (@sts:2012, @sts:2014, @sts:2015). The video descriptions in the STS-2012 task are from the MSR video description corpus [@chen:msrvid:2011] and the image descriptions in STS-2014 and 2015 are from UIUC PASCAL dataset [@uiuc:pascal:2010].
In [Table \[tab:sts-results\]]{}, we present the Pearson correlation coefficients of our model predicted scores with the gold-standard similarity scores provided as part of the STS image/video description tasks. We compare with the best reported scores for the STS shared tasks, achieved by MLMME [@calixto:mlmme:2017], paraphrastic sentence embeddings [@wieting2017learning], visual semantic embeddings [@kiros2014unifying], and order embeddings [@vendrov2015order]. The shared task baseline is computed based on word overlap and is high for both the 2014 and the 2015 dataset, indicating that there is substantial lexical overlap between the STS image description datasets. Our models outperform both the baseline system and the best system submitted to the shared task. For the 2012 video paraphrase corpus, our multilingual methods performed better than the monolingual methods showing that similarity across paraphrases can be learned using multilingual signals. Similarly, @wieting2017learning have reported to learn better paraphrastic sentence embeddings with multilingual signals. Overall, we observe that models learned using the asymmetric scoring function outperform the state-of-the-art on these datasets, suggesting that multilingual sharing is beneficial. Although the task has nothing to do German, because our models can make use of datasets from different languages, we were able to train on significantly larger training dataset of approximately 145k descriptions. @calixto:mlmme:2017 also train on a larger dataset like ours, but could not exploit this to their advantage. In [Table \[tab:sts-analysis\]]{} we present the example sentences with the highest and lowest difference between gold-standard and predicted semantic textual similarity scores using our best performing [<span style="font-variant:small-caps;">Parallel-Asym</span>]{}model.
Conclusions
===========
We proposed a new model that jointly learns multilingual multimodal representations using the image as a pivot between languages. We introduced new objective functions that can exploit similarities between images and descriptions across languages. We obtained state-of-the-art results on two tasks: image-description ranking and semantic textual similarity. Our results suggest that exploiting multilingual and multimodal resources can help in learning better semantic representations.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work greatly benefited from discussions with Siva Reddy and Desmond Elliot. The authors would like to thank the anonymous reviewers for their helpful comments. The authors gratefully acknowledge the support of the European Research Council (Lapata: award number 681760).
[^1]: We constrain the embeddings of descriptions and images to have non-negative entries when using asymmetric similarity by taking their absolute value.
[^2]: <https://github.com/moses-smt/mosesdecoder/tree/master/scripts>
[^3]: We rescale images so that the smallest side is 256 pixels wide, we take 224 $\times$ 224 crops from the corners, center, and their horizontal reflections to get 10 crops for the image.
[^4]: <https://github.com/ivendrov/order-embedding>
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Using high resolution adaptive mesh refinement simulations in 3D, we investigate the formation of relativistic jets from rotating magnetospheres. Here, we focus on the development of non-axisymmetric modes due to internal and external perturbations to the jet. These originate either from injection of perturbations with the flow or from a clumpy external medium. In the helical field geometry of the accelerating jet, the m=1 to m=5 modes are analyzed and found to saturate at a height of $\sim 20$ inner disk radii. We also discuss a means to control artificial amplification of $m=4$ noise in the cartesian simulation geometry. Strong perturbations due to an in-homogeneous ambient medium lead to flow configurations with increased magnetic pitch and thus indicate a self-stabilization of the jet formation mechanism.'
author:
- |
O. Porth$^{1,2,3}$[^1]\
$^{1}$Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany\
$^{2}$Center for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium\
$^{3}$School of Maths, University of Leeds, LS2 9JT, Great Britain
bibliography:
- 'astro.bib'
title: Three Dimensional Structure of Relativistic Jet Formation
---
=1
\[firstpage\]
accretion, accretion disks – ISM: jets and outflows – MHD – galaxies: active – galaxies: jets – relativity
Introduction
============
Relativistic jets from active galactic nuclei (AGN) are host to a multitude of physical processes that lead to emission of radiation with the highest energies in the universe [e.g. @bottcher2007]. The observed non-thermal emission of high-energy particles is subject to beaming in a highly collimated jet of plasma moving with Lorentz factors in the range $10-40$ [@jorstad2005; @lister2009]. The leading paradigm of jet formation suggests that the acceleration of the bulk flow can be understood in the relativistic magnetohydrodynamic (RMHD) approximation, where the jet is endowed with a large-scale current circuit. Current axisymmetric simulation models of jet formation [@2007MNRAS.380...51K; @2008MNRAS.388..551T; @porth2011] have matured to an overall agreement on the underlying process of jet formation. However, since nature does not obey axisymmetry, 2D modeling falls short in two important aspects: *1.* The axisymmetric induction equation ($\partial_{\phi}\equiv0$) lacks a mechanism to transform toroidal magnetic field back into poloidal field. Therefore the toroidal field can only be amplified, leading to a potential overproduction of $B_{\phi}$ and the collimating pinch force. This might cast a shadow of doubt on the validity of results from axisymmetric jet formation simulations. *2.* In order to also study the stability of jets, a three dimensional treatment is necessary. It is well known that instability of current carrying plasmas is connected to non-axisymmetric perturbations of the toroidal magnetic field [@Bateman:1978]. Among these current driven instabilities, the $m=1$ mode known as the “kink” is the most violent one. It leads to a helical displacement of the flow from the axis of the plasma-cylinder. In the context of young stellar objects, the helical kink can yield an explanation for the wiggly structure observed in some stellar jets (e.g. HH 46/47) [@todo1993; @lery2000]. If not brought to a halt by regulating non-linear mechanisms, the exponential instability growth must lead to complete disruption of the jet. At the presence of instability, jets can dissipate magnetic energy via reconnection and shocks (e.g. [@Blandford1979]), leading to heating, acceleration of high-energy particles and radiation. As noted by [@heinz2000] and , the dissipation into a fully tangled magnetic field could also promote efficient quasi-thermal acceleration of the bulk flow out of magnetic enthalpy. In principle, the current-driven instabilities can be accompanied by other types of instabilities such as the Kelvin-Helmholtz instability (KH) caused by shear between jet and ambient material. The KH instability can thus lead to an efficient mixing of the jet and environment. However, in the presence of strong magnetic fields, growth of the KH modes is strongly suppressed [@keppens1999]. In particular, toroidal fields appear to hinder mixing and thus exert a stabilizing influence [@appl1992; @Mignone:2010a].
As pointed out by [@mckinney2009], when the well-known Kruskal-Shafranov (KS) instability criterion $$\begin{aligned}
\left|\frac{B_{\phi}}{B_{p}}\right|>\frac{2\pi r}{z}\end{aligned}$$ for cylindrical force-free equilibria is applied to relativistic jets, we obtain the result that jets become unstable already at the Alfvén point $z_{A}\simeq 10 r_{\rm S}$ (where $B_{\phi}\gtrsim B_{p}$ and $z\gtrsim r$; $r_{\rm S}$ denotes the Schwarzschild radius) - before accelerating to highly relativistic velocities. This is in stark contrast to the finding of some AGN FR-II jets propagating unperturbed out to distances of $10^{7} r_{\rm S}$.
Linear stability analysis in non-relativistic, but otherwise complete MHD was conducted for current-free and current-carrying jets with helical magnetic fields by [@appl1992]. The stability of *relativistic* MHD jets on the other hand is still not fully addressed in linear analysis and subject of ongoing research. Considering linear analysis of *force-free* cylindrical configurations, [@istomin1994] and [@istomin1996] concluded stability of $B_{z}=const.$ jets with respect to axisymmetric and helical perturbations. [@begelman1998] on the other hand demonstrated the violent instability of the $m=1$ mode which he postulated as a possible solution to the long-standing $\sigma-$problem. Also [@Lyubarskii:1999] stressed the importance of the $m=1$ mode by considering more realistic configurations with decreasing flux. However, [@Lyubarskii:1999] and recently [@narayan2009] note the time-dilated slow growth rate of the kink. This can lead to a substantial displacement between launching and dissipation scales and thus yield a possible explanation for the extent of the blazar zone of AGN.
[@Tomimatsu:2001] could extend the classical KS criterion and demonstrated a stabilizing effect of the relativistic field line rotation. Their analysis (TKS) yields the simple criterion for instability $$\begin{aligned}
\left|\frac{B_{\phi}}{B_{p}}\right|>\frac{2\pi r}{z}\ \hspace{1cm}{\rm and}\hspace{1cm}
\left|\frac{B_{\phi}}{B_{p}}\right|>\frac{r\Omega}{c}\end{aligned}$$ where $\Omega$ denotes the angular velocity of the field line $\Omega= (v_{\phi} - v_{p} B_{\phi}/B_{p})/r$, such that the toroidal field strength also has to overcome the stabilizing electric field. The asymptotic relation for relativistic jets $B_{\phi}\simeq -r\Omega/c B_{p}$ suggests marginal stability of the unperturbed flow.[^2] Taken at face value, the TKS criterion thus suggests that relativistic jets are always on the verge of instability with the ultimate fate strongly depending on details of the modeling. In more recent studies, several authors investigated the stabilizing influence of environmental effects such as shear [e.g. @2007ApJ...662..835M], external wind with relativistic bulk motion [@2003ApJ...583..116H] and sideways expansion [@2000ApJ...542..750R], emphasizing the influence of modeling details for jet and ambient material. While there is now growing consensus that the kink instability can also operate in the relativistic regime, to answer whether it can grow indefinitely to finally disrupt the jet or rather saturate, requires a study of the non-linear evolution via numerical simulations [e.g. @2009ApJ...700..684M; @Mignone:2010a; @mizuno2011a]. Also [@oneill2012a] studied the stability of various local jet configurations in force-free, pressure confined and rotational equilibriua. These authors found pressure confined models corresponding to astrophysical jets furthest from the origin to be the most unstable ones.
On the other hand we would like to point out that jet stability can best be studied by including the initial acceleration and collimation region of the jet. Thereby, self-consistent helical magnetic fields are obtained and the number of ad-hoc assumptions for the jet base can be reduced to a minimum of physically well motivated choices. [@mckinney2009] were the first to present global 3D simulations of jet formation including a turbulent accretion disk. In their seminal paper, no significant disruption or dissipation out to scales of $10^{3}{r_{\rm S}}$ was observed, however by using comparatively low resolution GRMHD simulations with $256\times128\times32$ cells in a spherical $r\times \theta\times\phi$ grid. [^3] Although launching jets directly from the turbulent accretion flow promises high realism, a systematic study of jet instability growth can hardly be performed this way. For non-relativistic disk jets, [@2003ApJ...582..292O], [@anderson2006], and [@staff2010] followed an alternative approach by treating the rotating magnetosphere as a fixed-in time injection boundary. As a natural generalization of previous axisymmetric work, the boundary can be used to model the corona of a Keplerian accretion disk. Also we will follow this strategy for the case of relativistic jets, as it allows to systematically control both fixed in time and time variable injection conditions.
In this paper, we show our first results concerning the stability of relativistic jets near the launching region. We will discuss 3D RMHD simulations exposed to non-axisymmetric perturbations triggered by the accretion disk and due to jet-cloud interactions. To complement previous considerations of axisymmetric (2.5D) jet formation outlined in [@2010ApJ...709.1100P] (PF I) and [@porth2011] (PF II), we focus mainly on non-axisymmetric features within the jet.
Model Setup
===========
The jet is launched from a rotating inlet resembling the corona of an accretion disk similar to PF I. The initially purely poloidal magnetic field is transformed into a helical shape giving rise to a global electric current system which accelerates and collimates the flow into a jet. [Since the disk enters our description essentially as a boundary (allowing for outgoing Alfvén and fast waves) we do not investigate the back reaction from the jet to the disk.]{} Due to the increased complexity of the three dimensional treatment compared to previous 2.5D work, we neglect stratification caused by a gravitational source term, since the flow is accelerated above escape velocity very near the disk and sonic surface (see also PF I). Note that in the following, the axis of symmetry is denoted as the cartesian ’y-axis’ (not the ’z-axis’ as usual).
Initial and Boundary Conditions {#sec:3dboundary}
-------------------------------
We initialize the domain with constant values for density and pressure $\rho\equiv\rho_{0}=0.01$, $p\equiv p_{0}$, threaded by an initial poloidal magnetic field of monopolar shape with the “source” outside of the domain at $y=-4$. The initial pressure follows from the plasma $\beta$ parameter at the inner disk edge $r=1$ to $p_{0}=\beta/2 B_{1}^{2}$, where $B_{1}$ denotes the corresponding poloidal magnetic field strength at $(r,y)=(1,0)$ and we set to $B_{1}=1$ for convenience. A sketch of the computational domain is shown in figure \[fig:domain\].
![Sketch of the computational domain in the x-y plane *(left)* and the x-z plane *(right)*. Within $r_{\rm out}$, rotating injection conditions are applied. The monopolar magnetic field is indicated by dashed lines in the xy plane and has its “source” outside of the domain at $(x,y,z) = (0,-4,0)$. []{data-label="fig:domain"}](f1.pdf){width="48.00000%"}
To avoid the poorly determined states in the corners of the domain where disk-boundary and outflow boundary would meet, we truncate the disk inlet at $r_{\rm out}$. Not truncating the disk introduces additional $m=4$ noise triggered predominantly at the corners of the domain. At the injection boundary within $r<r_{\rm out}$ we assign boundary conditions for a rotating magnetosphere, injecting a sonic flow aligned with the poloidal field-lines. To provide the axisymmetric boundary constraints at the bottom plane, we first transform to cylindrical coordinates around the jet axis in the middle of the cartesian domain, compute the required quantities and subsequently transform back to the cartesian domain to assign the updated boundary conditions. We specify fixed-in time axisymmetric profiles for the five quantities $p_{\rm d}\equiv p_{0}, \rho_{\rm d}\equiv 100\rho_{0}=1, v_{\phi}=r \omega(r), v_{p}\equiv c_{\rm s}, E_{\phi}\equiv0$ to account for the downstream Alfvén and fast magnetosonic waves leaving the domain across the boundary. In addition, the magnetic flux $B_{y}$ is fixed to the initial profile. Since the evolution of $B_{y}$ is already suppressed via the choice of $E_{\phi}\equiv0$, this does not represent an additional boundary constraint. The radial field component is obtained by a field extrapolation satisfying $j_{\phi}\equiv0$ of the domain values $B_{r}$, given the fixed profile of the vertical field component $B_{y}$. The toroidal field $B_{\phi}$ follows from the domain via zero gradient extrapolation to yield also $j_{r}=0$. This way, the boundary current is under control and no spurious current sheets are allowed to arise at the bottom plane injection boundary. For the rotation profile, we adopt $$\begin{aligned}
\omega(r) = 0.5 c/r_{\rm in} \left\{\begin{array}{lll}
1 &; & r<r_{\rm in}\\
\left(\frac{r}{r_{\rm in}}\right)^{-3/2} &; & 1\le r <r_{\rm out}
\end{array}
\right.\end{aligned}$$ representing an inner solid-body rotation law and an outer Keplerian profile within $r_{\rm in}=1 $ as in [@2008MNRAS.388..551T]. Beyond $r_{\rm out}$, we simply freeze the initial conditions in the boundary.
[ We should note here that in a general relativistic treatment, frame dragging of a monopolar magnetosphere leads to an approximate solid-body field-line rotation with $\Omega \simeq 0.5\, \Omega_{\rm H}$, where $\Omega_{\rm H}$ denotes the outer horizon angular velocity]{} [@1977MNRAS.179..433B; @komissarov2001]. [In the extreme Kerr case, this would yield $\Omega \simeq 0.25\, c/r_{\rm in}$ hence we are even exaggerating the central field line rotation and thus the putative Blandford & Znajek power output. Naively, one could thus expect a fast axial jet to arise in our simulations. However, since the mass influx into the domain is in fact increasing as one approaches the origin (given constant *poloidal* disk- velocity and density) and due to the necessity of vanishing Poynting flux on the axis, our simulations will not produce a high Lorentz factor spine. This would require an accurate modeling of the mass flux in the plunging region using a GRMHD approach which we must defer to future work. ]{}
For the set of parameters shown here, $\beta=0.01, v_{\phi,\rm r=1}=0.5c$ and $\rho_{d}=1$, the injection speed follows as $c_{\rm s}\simeq0.1 c$ and hence the maximal [@1969ApJ...158..727M] magnetization parameter can be written $$\begin{aligned}
\sigma_{\rm M} \simeq \frac{v_{\phi}^{2} B_{1}^{2}}{4\pi\Gamma\rho_{\rm d} c_{\rm s}c^{3}}. \end{aligned}$$ It becomes $\sigma_{\rm M} = 2.5$ giving an upper limit of the Lorentz factor to be obtained by the flow. Higher values of $\sigma_{\rm M}$ can be achieved by reducing the disk density or the plasma-$\beta$ as the magnetization depends on these parameters according to $\sigma_{\rm M}\propto \rho^{-1/2} \beta^{-1/2}$.
Due to the accumulation of numerical errors and decreased resolution compared to previous axisymmteric studies, steep gradients of the rotation law at the inner disk edge must be avoided if no special treatment for the velocity profile is given.[^4] It is also for similar stability reasons that we specify $\omega$ as a boundary constraint instead of the field rotation law $\Omega$ as customary.
To close the set of ideal special relativistic magnetohydrodynamic equations (RMHD), we employ the equation of state proposed by @meliani2004, [approximating an ideal one-component monoatomic]{} @synge1957 [gas (see also equation 6 of ]{}@Keppens2012718).
Additional passive tracer scalars $\iota_{1}-\iota_{3}$ are advected with the flow for the purposes of refinement and post-processing.
Numerical Grid Setup
--------------------
We perform simulations in a cartesian adaptive grid using MPI-AMRVAC [@Keppens2012718]. With the adaptive mesh refinement in a cartesian domain, a uniformly high resolution for all regions of interest is obtained without the directional biases present in stretched grid simulations , especially, a high resolution can be achieved also at the jet head which will prove necessary to resolve the helically displaced tip of the jet. The cartesian discretization and quadrantal symmetry of the grid can however introduce notable noise leading to a “pumping” of the $m=4$ mode. We comment on the measures taken to analyze and circumvent such spurious effects further in section \[sec:modeanalysis\].
The largest domain size considered in the following extends over $x\in [-32,32]$, $y\in[0,128]$ and $z\in [-32,32]$ in units of the inner disk radius. A base resolution of $n_{x}\times n_{y} \times n_{z} = 48\times 96 \times 48$ cells is chosen, adaptively refined by four additional grid levels (three for the smaller domain case). Thus 12 grid cells per inner disk radius are achieved, totaling in 240 cells across the entire jet inlet radius. To our knowledge this high resolution is unprecedented in 3D jet formation simulations. The effective resolution for the large domain is $768\times1536\times768=9.06\times10^{8}$ cells and we observe a grid filling with $\simeq3\times10^{8}$ cells at the termination time.
### Refinement Strategy
Refinement to the highest level is enforced for the disk inlet $r<r_{\rm out}; y<1$ and for the region around the axis $x,z<1;y<10$ to resolve the steep gradients of the monopole field. The jet is refined based on the scheme proposed by [@lohner1987] with weighted contributions of density, Lorentz factor and magnetic field strength.
As additional criterion, the initial domain is coarsened to the lowest level until the jet comes within reach. This decision is based on the Lorentz factor ($\Gamma>1.05$) and the passive tracer scalars described earlier. The criterion in terms of the Lorentz-factor is required in order to avoid accidental coarsening of the torsional Alfvén wave launched at the switch on of disk-rotation; it does not transport jet-inlet material and is thus unaffected by the tracers injected along the jet. With this strategy, the fast jet is always resolved appropriately, while the enforced coarsening of the initial domain significantly speeds up the simulations. Especially when the domain is initialized with a non-homogeneous medium, the coarsening-strategy is crucial.
Perturbations
-------------
In order to investigate the behavior upon instability and to break the quandrantal symmetry of the grid, the initially axisymmetric setup needs to be distorted by non-axisymmetric perturbations. The physical origin of the perturbation can be found in a non-axisymmetric evolution of the accretion disk, for example due to orbit of vortices and quasi-periodic oscillations [@van-der-klis1985], or in a non-homogeneous external medium. In active galactic nuclei, the presence of such a medium within the central region is well established as the broad line region that is possibly comprised of clouds orbiting the black hole at high velocity ($\sim 0.1 c$) [e.g. @davidson1979; @araudo2010] or in manifestation of a clumpy torus surrounding the accretion disk [e.g. @dullemond2005]. The eventual collision of such clouds with a relativistic jet was already proposed by [@Blandford1979] as a mechanism to explain transient features in compact radio sources. The scenario of jet-cloud collision is thus of interest not only in respect to jet stability, but also concerning the triggering of particle acceleration at the shock surface. We take the presence of a clumpy ambient medium as a motivation for our second setup of jet perturbations, where we model how the emerging jet propagates through such a density structure.
### Mode Injection
To model non-axisymmetric features of the accretion disk, we employ perturbations to the rotation velocity $\omega+\Delta\omega$ following a mode decomposition. This method was already successfully applied by [@rossi2008] and [@Mignone:2010a]. In particular, we adopt $$\begin{aligned}
\begin{split}
\Delta \omega(r,t) = \\
\frac{\epsilon_{\omega} \omega_{0}(r)}{32}\sum_{m=0}^{3}\sum_{l=1}^{8} \cos(m\phi+\omega(l)\omega_{0}(r) t+\phi_0(l))
\end{split}\end{aligned}$$ to obtain modes with sub- and super-Keplerian frequencies $\omega(n)\in\{0.5,1,2,3,0.03,0.06,0.12,0.25\}$ featuring a random phase offset $\phi_{0}(l)$. The maximum amplitude of the perturbation is set to $\epsilon_{\omega}=2\%$.
### Clumpy medium
A static clumpy medium is modeled as density perturbation by prescribing the size spectrum in Fourier space and subsequent transformation of the random-phased Fourier coefficients to real space. To obtain a particular cloud size, the following spectrum is adopted: $$\begin{aligned}
f(k)= k_{\rm min}^{-s-1}\frac{s+1}{2s}\left\{\begin{array}{clc}
k^{s}&; & k\ge k_{\rm min} \\
k_{\rm min}^{s}&; & k<k_{\rm min} \\
\end{array}\right.\end{aligned}$$ where $k_{\rm min}$ relates to the cloud size $\lambda$ and domain size via $\lambda={\rm max}(\Delta x, \Delta y, \Delta z)/k_{\rm min}$. The highest frequency $k_{\rm max}$ is limited by the base-level resolution of the grid to $k_{\rm max}<0.5 \min(n_{x},n_{y},n_{z})$ as we do not allow refinement on the initial condition. We adopt a slope of the cloud size function $s=5/3$. In the following, the maximal cloud density is set to $100$ times the ambient density $\rho_{0}$.
Results and Discussion
======================
We now describe the simulations performed and the analysis of non-axisymmetric features. The non-linear temporal and spatial evolution of angular modes is discussed and we show evidence for jet self-stabilization in the launching region.
Overview of the Simulations
---------------------------
Table \[tab:3D\] summarizes our parameter runs and Figure \[fig:fancyplots\] shows a rendering of our reference unperturbed solution labeled as M3D.
ID $\beta$ $\epsilon_{\omega}$ $r_{\rm out}$ $\lambda$ Domain levels $T_{\rm end}$
------- --------- --------------------- --------------- ----------- ----------------------- -------- ---------------
L3D 0.01 0 20 - $64\times128\times64$ 5 168
L3Dm 0.01 0.02 20 - $64\times128\times64$ 5 168
M3D 0.01 0 10 - $32\times64\times32$ 4 103
M3Dd 0.01 0 10 16 $32\times64\times32$ 4 189
M3Dmd 0.01 0.02 10 16 $32\times64\times32$ 4 222
: Parameter summary of the 3D simulations. Simulation names indicate box size (L,M) and the nature of perturbation (m - mode injection, d - density perturbations). \[tab:3D\]
![Rendering of the reference simulation, showing cut open pressure isocontour colored according to Lorentz factor with magnetic field lines (run M3D). []{data-label="fig:fancyplots"}](f2.pdf){width="45.00000%"}
Various flow quantities in the $z=0$ plane containing the jet “axis” are shown in Figure \[fig:valuecomp1\]. [Here, we also illustrate the momentary run of the critical surfaces as in PF I. Note that these are strictly only valid under the assumption of stationarity and axisymmetry. Compared to PF I, our new simulations are not evolved long enough to adopt the near-stationary state and still reflect the initial transient phase of the sudden spin up of the disk. None the less, the flow transcends the critical Alfvén surface (blue contour) and becomes super-luminal (black contour) but stays sub magnetofast (red contour) within the domain.]{}
At this scale, the Lorentz factor of the disk wind is still moderate ($\Gamma<2$), it rises above $\Gamma\simeq2$ only in regions localized adjacent to the high pressure backbone.
![Slice along the jet in simulation M3D at $t=103$ $z=0$. Shown are (left to right, top to bottom): Thermal pressure (log-scale); co-moving density (log-scale); (toroidal) magnetic field strength across the plane and magnitude of four velocity $u=\Gamma v$. [Critical points are indicated in the plot of density, where we show the (critical) Alfén surface (blue), the light cylinder (black) and the fast magnetosonic surface (red) according to the definitions of]{} @Camenzind1986. []{data-label="fig:valuecomp1"}](f3a.pdf "fig:"){width="23.00000%"} ![Slice along the jet in simulation M3D at $t=103$ $z=0$. Shown are (left to right, top to bottom): Thermal pressure (log-scale); co-moving density (log-scale); (toroidal) magnetic field strength across the plane and magnitude of four velocity $u=\Gamma v$. [Critical points are indicated in the plot of density, where we show the (critical) Alfén surface (blue), the light cylinder (black) and the fast magnetosonic surface (red) according to the definitions of]{} @Camenzind1986. []{data-label="fig:valuecomp1"}](f3b.pdf "fig:"){width="23.00000%"} ![Slice along the jet in simulation M3D at $t=103$ $z=0$. Shown are (left to right, top to bottom): Thermal pressure (log-scale); co-moving density (log-scale); (toroidal) magnetic field strength across the plane and magnitude of four velocity $u=\Gamma v$. [Critical points are indicated in the plot of density, where we show the (critical) Alfén surface (blue), the light cylinder (black) and the fast magnetosonic surface (red) according to the definitions of]{} @Camenzind1986. []{data-label="fig:valuecomp1"}](f3c.pdf "fig:"){width="23.00000%"} ![Slice along the jet in simulation M3D at $t=103$ $z=0$. Shown are (left to right, top to bottom): Thermal pressure (log-scale); co-moving density (log-scale); (toroidal) magnetic field strength across the plane and magnitude of four velocity $u=\Gamma v$. [Critical points are indicated in the plot of density, where we show the (critical) Alfén surface (blue), the light cylinder (black) and the fast magnetosonic surface (red) according to the definitions of]{} @Camenzind1986. []{data-label="fig:valuecomp1"}](f3d.pdf "fig:"){width="23.00000%"}
Mode Analysis {#sec:modeanalysis}
-------------
We now first evaluate the impact of artificial $m=4$ pumping due to the quadrantal symmetry and quantify the growth of non-axisymmetric modes within the jet formation region. To first give an impression of the azimuthal variations, a qualitative comparison of the unperturbed simulation L3D with run L3Dm is shown in slices of selected quantities midway across the jet in Figures \[fig:nomodes\] and \[fig:modes\].
{width="95.00000%"}
{width="95.00000%"}
When no measure of perturbing the quadrantal symmetry is taken, we observe multiples of the $m=4$ mode in all flow quantities. Higher order modes are most apparent in the density $\rho$ and vertical current density $j_{y}$. In the mode injected slices on the other hand, we observe a slight dominance of the $m=3$ mode.
To quantify the growth of non-axisymmetric modes within the jet, we calculate the fast Fourier transform of the variables on the slice in cylindrical $(r,\phi)$ representation. For this purpose, we re-grid the unstructured slice data $x\in[-12,12]$, $z\in[-12,12]$ containing the jet spine to a uniform grid before transformation to the cylindrical coordinates $r\in[0,12]$, $\phi\in[0,2\pi]$. Thereafter, the Fourier-transformation of the $(r,\phi)$-plane $$\begin{aligned}
\begin{split}
\tilde{f}(n,m)&=\sum_{n_{r}=-N_{r}}^{N_{r}}\sum_{n_{\phi}=-N_{\phi}}^{N_{\phi}}f(n_{r},n_{\phi})\\
&\times \exp{\left(-2\pi i \left(m \frac{n_{\phi}}{N_{\phi}}+ n \frac{n_{r}}{N_{r}}\right)\right)}
\end{split}\end{aligned}$$ is executed which yields the radial $(n)$ and angular $(m)$ Fourier amplitudes of the input scalar via $A(n,m)\equiv|\tilde{f}(m,n)|^{2}$. To quantify fluctuations of the angular part alone, we define the normalized cumulative Fourier amplitudes $$\begin{aligned}
A(m)\equiv\frac{1}{|\tilde{f}(0,0)|^{2}}\sum_{n=-N_{n}+1}^{N_{n}-1}|\tilde{f}(n,m)|^{2}\end{aligned}$$ which measures the fluctuations with angular frequency $m$ in relation to the squared mean of the scalar $f$ expressed by ${|\tilde{f}(0,0)|^{2}}$. The Fourier amplitude planes of the density fluctuations of Figure \[fig:nomodes\] and \[fig:modes\] are shown in Figure \[fig:powerspecs\].
![*Top:* Radial (n) and angular (m) Fourier amplitudes of density across $y=32$ at $t=100$ for the unperturbed case L3D *(left)* and for the perturbed case L3Dm *(right)* . *Bottom:* Cumulative modes for the two cases. To guide the eye, we show the empirical mode-decay following the power-law $m^{-3}$. []{data-label="fig:powerspecs"}](f6a.pdf "fig:"){width="23.00000%"} ![*Top:* Radial (n) and angular (m) Fourier amplitudes of density across $y=32$ at $t=100$ for the unperturbed case L3D *(left)* and for the perturbed case L3Dm *(right)* . *Bottom:* Cumulative modes for the two cases. To guide the eye, we show the empirical mode-decay following the power-law $m^{-3}$. []{data-label="fig:powerspecs"}](f6b.pdf "fig:"){width="23.00000%"} ![*Top:* Radial (n) and angular (m) Fourier amplitudes of density across $y=32$ at $t=100$ for the unperturbed case L3D *(left)* and for the perturbed case L3Dm *(right)* . *Bottom:* Cumulative modes for the two cases. To guide the eye, we show the empirical mode-decay following the power-law $m^{-3}$. []{data-label="fig:powerspecs"}](f6c.pdf "fig:"){width="46.00000%"}
The $m=4$ pollution of the unperturbed run is clearly visible, mode injection on the other hand can be used to get rid of this effect almost entirely as shown in the lower panel of Figure \[fig:powerspecs\].
### Temporal Evolution
To quantify the temporal growth of the modes, we calculate the Fourier amplitudes at the $y=32$ slice in run L3Dm for various snapshots. At this altitude, the magnetic “backbone” $B_{y}$ becomes distorted at $t>80$ with dominating $m=1$ and $m=2$ modes, however the amplitudes grow no further. The time evolution of the $A(m)$ function is shown in the left panel of Figure \[fig:ampvstime\]. After an exponential rise where the growth time in $m=1$ is shortest, followed by the $m=4$ mode, the perturbations saturate at $t\simeq80$ and subsequently fluctuate about a mean value. Mostly, the amplitudes are ordered according to $A(m)>A(m+1)$ although the $m=2$ occasionally surpasses the $m=1$ contribution.
![*Left:* Mode growth of $B_{y}$ at $y=32$ in simulation L3Dm. After initial exponential rise, the modes tend to saturate. Due to grid-noise, the $m=4$ mode is initially comparable to the dominant $m=1$ mode. *Right:* Barycenter motion on the $y=32$ slice. []{data-label="fig:ampvstime"}](f7.pdf){width="46.00000%"}
### Spatial Evolution
The clearest indicator of the kink instability can be observed in the deflection of the jet barycenter. For this purpose we define the barycenter $\bar{r}=\sqrt{\bar{x}^{2}+\bar{z}^{2}}$ of the quantity Q $$\begin{aligned}
\bar {x}\equiv \frac{\int x\ Q\ dx\ dz}{\int Q\ dx\ dz};\hspace{1cm}
\bar {z}\equiv \frac{\int z\ Q\ dx\ dz}{\int Q\ dx\ dz} \label{eq:barycenters}\end{aligned}$$ in analogy with [@Mignone:2010a]. For the density-displacement $r_{\rm cm}$, we define $Q_{\rm cm}=\chi \Gamma \rho$, where the $\chi$ is computed from the tracer scalars to pick out the jet contribution alone. This quantity is also shown against simulation time in the right panel of figure \[fig:ampvstime\]. For the current displacement $r_{jy>0}$ we set $Q_{jy>0}=\chi j^{+}_{y}$ taking only the positive values of the current density $j^{+}_{y}$. Finally, the motion of the magnetic flux is defined via $Q_{By>0}=\chi B_{y}^{+}$, taking into account only the positive flux $B_{y}^{+}$. Instantaneous barycenter position and mode population along the jet is shown in Figure \[fig:barycenters\] for run L3D and in Figure \[fig:barycentersmodes\] for the mode-injected run L3Dm. Let’s first focus on run L3D. The barycenter displacement is small compared to the inner disk radius and even tends to decrease along the jet. Only near the jet-head, at about $y\simeq 80$ a significant displacement can be observed. The modes are dominated by ubiquitous $m=4$ noise, which seems to suppress all other fluctuations starting at height $y=20$. The $m=4$ dominance prevails all the way to the jet head.
![Instantaneous barycenter displacements in units of inner disk radius and cumulative Fourier amplitudes of density $\rho$, all along the unperturbed jet L3D at time $t=168$. The jet base is at left. []{data-label="fig:barycenters"}](f8.pdf){width="46.00000%"}
The behavior of run L3D is different, here, the kink mode surpasses the $m=4$ at $y\simeq10$. The angular fluctuations saturate around $y=20$. Until $y=60$, the displacements in current and magnetic flux stay roughly constant. This hints to a self-stabilization of the jet formation region. The kink mode starts to rise again towards the jet head, accompanied with a notable barycenter displacement. We note that the magnetic field configuration near the jet head is strongly toroidally dominated as the monopolar initial field configuration is overtaken by the jet (see e.g. Figure \[fig:fancyplots\]). This toroidal dominance could yield an explanation for the stronger growth of the kink mode at the jet head.
![As Figure \[fig:barycenters\] but for run L3Dm at time $t=160$. The mode injection at the base (y=0) avoids artificial $m=4$ dominance. []{data-label="fig:barycentersmodes"}](f9.pdf){width="46.00000%"}
To quantify this further, we introduce the co-moving magnetic pitch defined as $$\begin{aligned}
P\equiv -2\pi r \frac{B'_{p}}{B'_{\phi}}\end{aligned}$$ which plays a major role in the stability of current carrying plasmas in the laboratory [e.g. @Bateman:1978] and astrophysical jets [e.g. @appl2000; @lery2000]. Small values of $P/r<1$ and thus toroidally dominated configurations are particularly susceptible to the kink instability. The co-moving fields are obtained by applying the projection $B'^{\alpha} = u_{\beta} F^{*{\alpha \beta}}$, where $F^{*{\alpha \beta}}$ denotes the dual Faraday tensor and $u_{\beta}$ the co-variant Four-velocity as customary. The radius $r$ and the fields $B_{\phi}$, $B_{p}$ involved in the definition of the pitch are well-defined however only in axi-symmetry. For small perturbations from the cylindrical shape, we can re-orient the symmetry axis to the magnetic backbone at the position $(\bar{x},\bar{z})$ and define effective values for $\bar{r}$, $B_{\bar{\phi}}$ and $B_{\bar{r}}$ with respect to the new origin. To define the magnetic backbone position using equations (\[eq:barycenters\]), we apply the kernel $Q_{\rm BB}=\chi B_{y}^{2}$ which reliably finds the peak of magnetic flux. To investigate also effects due to the electric field, we define an effective light surface via the comparison of the field strengths $$\begin{aligned}
x=\frac{E}{B_{\bar{p}}} \label{eq:lceff}\end{aligned}$$ where $x=1$ marks the light surface and $B_{\bar{p}}=\sqrt{B_{\bar{r}}^{2}+B_{y}^{2}}\simeq B_{y}$ is the poloidal field strength with respect to the magnetic backbone. Since the jet is well collimated $B_{y}\gg B_{\bar{r}}$, the location of the backbone is in fact only of secondary importance for the definition of the light surface and we obtain similar results when considering only the vertical field $B_{y}$ in Equation (\[eq:lceff\]). This light surface location is indicated with the gray contour in the top panel of figure \[fig:kinkfit\] where the pitch profiles across the jet is shown.
![*Top:* Co-moving pitch and effective light surface with respect to the center of magnetic flux (shown as grey contours) through the surfaces $y=60$ for run L3Dm at $t=160$. Black arrows indicate the direction of the electric force $\rho_{e} \mathbf{E}$ and white arrows show the direction of the Lorentz force $(\nabla \times \mathbf{B}) \times \mathbf{B}$. Both force vectors are projected onto the image-plane. The axis is marked by a black “+” and the center of magnetic flux is shown as gray “+”. Corresponding magnetic flux $B_{y}$ with velocity (white) and magnetic field vectors (black) in the plane. []{data-label="fig:kinkfit"}](f10a.pdf "fig:"){width="49.00000%"} ![*Top:* Co-moving pitch and effective light surface with respect to the center of magnetic flux (shown as grey contours) through the surfaces $y=60$ for run L3Dm at $t=160$. Black arrows indicate the direction of the electric force $\rho_{e} \mathbf{E}$ and white arrows show the direction of the Lorentz force $(\nabla \times \mathbf{B}) \times \mathbf{B}$. Both force vectors are projected onto the image-plane. The axis is marked by a black “+” and the center of magnetic flux is shown as gray “+”. Corresponding magnetic flux $B_{y}$ with velocity (white) and magnetic field vectors (black) in the plane. []{data-label="fig:kinkfit"}](f10b.pdf "fig:"){width="49.00000%"}
To visualize the Lorentz force across the flow, we show force vectors of the electric field $\rho_{e}\mathbf{E}=(\boldsymbol{\nabla}\cdot \mathbf{E)\ E}$ and of the classical Lorentz force $\mathbf{j\times B}$ where we neglected the displacement-current for simplicity.
The axial flow is outside of the light surface, then follows a region of super-luminal field line rotation and a third region outside of the main jet where field lines rotate again sub-luminal. The magnetic backbone is markedly seen in the pitch and we note that electric forces exhibit a de-collimating contribution at the central core while they tend to collimate near the outer light cylinder. In most regions, the additional electric component is opposed to the classical Lorentz force. As a general trend, we obtain a radially decreasing pitch profile from the backbone to its minimal value $P/r\lesssim1$, from where the pitch increases again to the boundary of the jet. Due to the m=1 motion, a spiral pattern with increased pitch is shed of the magnetic backbone. This ’smearing’ leads to a diminishing of the toroidally dominated regions. These regions of low pitch coincide with the locations of super-luminal field-line rotation and the stabilizing effect of electric forces becomes apparent as they tend to counter-act the magnetic contribution.
Jet-Cloud interaction
---------------------
We now study how the jet reacts to external perturbations exerted by a clumpy ambient medium. As the jet head punches its way through the in-homogeneous environment, also the upstream flow becomes deflected. Figure \[fig:cloudplot\] shows a rendering of the simulation M3Dmd with mode injection and clumpy medium in comparison to the unperturbed case M3D. Field lines are colored according to the magnetic pitch. The clouds with a density contrast of 100:1 represent a strong perturbation and we find the jet heavily distorted. Accordingly, the motion of the jet barycenter is increased and we find a strong dominance of the $m=1$ mode along the whole jet (Figure \[fig:barycl\]).
![Field lines of the unperturbed *(top)* and perturbed *(bottom)* simulations at times $t=103$ and $t=194$, respectively. To guide the eye along the bent jet, pressure isocontours are added in the figures. The perturbed run shows a wider magnetic backbone and decreased toroidal field. []{data-label="fig:cloudplot"}](f11.pdf "fig:"){width="49.00000%"} ![Field lines of the unperturbed *(top)* and perturbed *(bottom)* simulations at times $t=103$ and $t=194$, respectively. To guide the eye along the bent jet, pressure isocontours are added in the figures. The perturbed run shows a wider magnetic backbone and decreased toroidal field. []{data-label="fig:cloudplot"}](f11b.pdf "fig:"){width="49.00000%"}
![Barycenter motion and low order angular modes in density of simulation M3Dmd at $t=194$. []{data-label="fig:barycl"}](f12a.pdf){width="46.00000%"}
Due to the increased density of the external medium, the jet propagation speed is reduced and the bow-shock is much wider (compare with Figure \[fig:fancyplots\]). We therefore compare the jet morphology between runs M3D and M3Dmd at times of roughly equal jet propagation length. The precession of the magnetic backbone against the toroidal magnetic field direction tends to “smear” out the high-pitched axial region and the tightly wound helix is effectively unwound. This represents an efficient mechanism of jet self-stabilization also noted by [@2003ApJ...582..292O]. In their study the effect was described as follows: “The appearance of the $|m|=1$ modes pumps energy into the poloidal magnetic field, causing the jet Alfvén Mach number to fall below unity and stabilize the jet”. From our simulations we come to a similar conclusion, in addition, we note that the “unwinding” of the helical field also decreases the field-line rotation $\Omega$ and thus also the influence of electric fields. We show the increase in magnetic pitch compared to the unperturbed simulation L3D in Figure \[fig:pitchclouds\]. Regions of low pitch are reduced to a thin sheet where the effective light cylinder is found. In this regime, electric stabilization can not be of importance since the typical value of the light-cylinder $x$ is only of order unity or below.
It is interesting to visualize the return current in the heavily perturbed run M3Dmd. The structure of the current density across the jet is shown in figure \[fig:jyclouds\]. Fine layered current sheets form on small scales that could in principle dissipate and give rise to particle acceleration within the jet. It is only due to the high resolution that this effect can be observed.
![As in Figure \[fig:kinkfit\]; comparison of the pitch at $y=32$ in runs M3Dmd *(left)* and L3Dm *(right)* at times $t=194$ respectively $t=160$. The motion of the backbone in the heavily perturbed simulation increases the poloidal field [on account of]{} the unwinding of the helical field. []{data-label="fig:pitchclouds"}](f13a.pdf "fig:"){width="23.00000%"} ![As in Figure \[fig:kinkfit\]; comparison of the pitch at $y=32$ in runs M3Dmd *(left)* and L3Dm *(right)* at times $t=194$ respectively $t=160$. The motion of the backbone in the heavily perturbed simulation increases the poloidal field [on account of]{} the unwinding of the helical field. []{data-label="fig:pitchclouds"}](f13b.pdf "fig:"){width="23.00000%"}
![Out of plane current density at $y=32$, $t=194$ for the run with external perturbation M3Dmd. A layered filamentary structure of alternating current directions develops. []{data-label="fig:jyclouds"}](f14a.pdf){width="45.00000%"}
Conclusions
===========
We have presented first results of high-resolution 3D simulations of relativistic jet formation from magnetospheres in Keplerian rotation. When the flow is perturbed by non-axisymmetric internal perturbations of the accretion disk corona, the modes first grow exponentially at the base, approach saturation along the jet and grow again towards the jet head. The $m=1$ kink is the dominant mode of departure from axisymmetry. At a given height above the accretion disk, the temporal evolution of the modes was considered. Also here we find a saturation of perturbations before a notable dissipation or even disruption is encountered. As an aside, we also performed simulations where the only measure of perturbation is the ubiquitous discretization noise of the grid. In result, the modes are dominated by multiples of $m=4$ which grow along the flow at [the expense of]{} other modes and we observe virtually no motion of the jet barycenter.
To further investigate the stability of the jet structure, we considered the co-moving magnetic pitch. As in the axisymmetric case, a stabilizing backbone of high pitch $P\gg1$ develops, surrounded by an intermediate, toroidally dominated region $P<1$ and an outer high-pitch region at the border of the jet. The locations of super-luminal field line rotation (the light surface) approximately coincide with the low-pitch region. Forces due to the electric field $\rho_{e}\mathbf{E}$ oppose the classical magnetic Lorentz force $\mathbf{(\boldsymbol{\nabla}\times B) \times B}$ which could thus add to jet stabilization in the relativistic case.
In order to study external perturbations, we initialized the domain with a static clumpy ambient medium following a power-law spectrum in Fourier space. While this should not be mistaken for a fully developed MHD-turbulent medium, it allows us to investigate perturbations “external” to the jet as a general scenario. The maximum amplitude in cloud density to the jet density was chosen as $100:1$ which thus represent a strong perturbation to the emerging jet.
In result, the jet funnels its way through the path of least resistance which leads to large departures of the barycenter from the axis and dominating $m=1$ modes. When compared to the unperturbed case, the magnetic pitch is largely increased which can be interpreted as a mechanism of jet self-stabilization. Due to the precession of the magnetic backbone against the toroidal magnetic field direction, the helical structure tends to be “unwound” leading to an increase of poloidal field which also reduces the amount of field-line rotation. The external perturbation and accompanying motion of the magnetic backbone also gives rise to filamentary small-scale structure visible in the return currents. The accompanying dissipation could facilitate particle acceleration within the jet. Further investigation with high-resolution 3D simulations is needed to quantify this effect. [Another point of interest is the validity of the axisymmetric assumption in the formation of jets in general. We find that due to the early saturation of non-axisymmetric instabilities, 2D results on the inner scales of jet acceleration and collimation (e.g. PF I, PF II) appear largely unaffected by non-axisymmetric effects. Detailed comparisons of the dynamics in two- and three- dimensions will be provided in a forthcoming paper.]{}
Acknowledgments {#acknowledgments .unnumbered}
===============
The author likes to thank Serguei Komissarov for comments on the early manuscript and Rony Keppens for remarks on the later version. Sincere thanks go to Christian Fendt for invaluable advice and many discussions on the formation of jets. Computing for this work was carried out under the HPC-EUROPA2 project (project number: 228398), with the support of the European Community - Research Infrastructure Action of the FP7. Post-processing of the simulations was performed on the VIZ cluster of the Max Planck Society.
\[lastpage\]
[^1]: E-mail: o.porth@leeds.ac.uk
[^2]: However, this result should be handled with care, since the TKS criterion strictly only applies in the sub-Alfvénic region of the flow; see also the discussion by [@mckinney2009].
[^3]: Although a fourth order limiter was used, we note here that an accurate study of the higher-order mode growth in the jet requires sufficient actual angular resolution ([@mckinney2009] report $20\%$ convergence in the $m=1,2,3$ power when comparing 16 and 32 angular cells).
[^4]: See also the discussion in Appendix 3 of [@2003ApJ...582..292O].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We investigate a scenario that the top quark is the only window to the dark matter particle. We use the effective Lagrangian approach to write down the interaction between the top quark and the dark matter particle. Requiring the dark matter satisfying the relic density we obtain the size of the effective interaction. We show that the scenario can be made consistent with the direct and indirect detection experiments by adjusting the size of the effective coupling. Finally, we calculate the production cross section for $t\bar t + \chi \bar \chi$ at the Large Hadron Collider (LHC), which will give rise to an interesting signature of a top-pair plus large missing energy.'
author:
- |
Kingman Cheung$^{1,2,3}$, Kentarou Mawatari$^{4,5}$,\
Eibun Senaha$^{3}$, Po-Yan Tseng$^{2}$, and Tzu-Chiang Yuan$^6$
title: The Top Window for dark matter
---
Introduction
============
The presence of cold dark matter (CDM) in our Universe is now well established by a number of observational experiments, especially the very precise measurement of the cosmic microwave background radiation in the Wilkinson Microwave Anisotropy Probe (WMAP) experiment [@wmap]. The measured value of the CDM relic density is $$\Omega_{\rm CDM}\, h^2 = 0.1099 \;\pm 0.0062 \;,$$ where $h$ is the Hubble constant in units of $100$ km/Mpc/s. Though the gravitation nature of the dark matter is established, we know almost nothing about the particle nature, except that it is, to a high extent, electrically neutral.
One of the most appealing and natural CDM particle candidates is [*weakly-interacting massive particle*]{} (WIMP). It is a coincidence that if the dark matter is produced thermally in the early Universe, the required annihilation cross section is right at the order of weak interaction. The relation between the relic density and the thermal annihilation cross section can be given by the following simple formula [@hooper] $$\label{rate}
\Omega_\chi h^2 \simeq \frac{ 0.1 \;{\rm pb} }{\langle \sigma v \rangle} \;,$$ where $\langle \sigma v \rangle$ is the annihilation rate of the dark matter around the time of freeze-out. Given the measured $\Omega_{\rm CDM} h^2$ the annihilation rate is about $1$ pb or $10^{-26}\;{\rm cm}^3 \, {\rm s}^{-1}$. This is exactly the size of the cross sections that one expects from a weak interaction process and that would give a large to moderate production rate at the LHC. In general, production of dark matter at the LHC would give rise to a large missing energy. Thus, the anticipated signature in the final state is high-$p_T$ jets or leptons plus a large missing energy. Note that there could be non-thermal sources for the dark matter, such as decay from exotic relics like moduli fields, cosmic strings, etc. In such cases, the annihilation rate in Eq. (\[rate\]) can be larger than the value quoted above.
The most studied dark matter candidate is perhaps the neutralino of the supersymmetric models with $R$-parity conservation. In this work, we study a different scenario. The dark matter is in a hidden sector and the only standard model (SM) particle that it interacts with is the top quark. The top quark, having a mass so close to the electroweak symmetry breaking scale, makes itself so unique among the fermions. It is perhaps one of the best windows to probe the electroweak symmetry breaking. The dark matter, if it is a WIMP, is also closely related to electroweak symmetry breaking. The logic is that since both the top quark and the WIMP are closely related to electroweak symmetry breaking, we argue that the top quark may be the only window to probe the dark matter. This is our motivation. [^1] We use an effective Lagrangian approach to parameterize the interactions between the top quark and the dark matter particle, without specifying the detailed communication between the top quark and the hidden sector. One simple example would be a hidden-sector gauge boson that can couple to the top quark. If it is heavy enough we can shrink the propagator to a 4-fermion vertex. One form of the 4-fermion interaction is $(\bar t \gamma_\mu t)\, (\overline{\chi} \gamma^\mu \chi)$ for a vector-type interaction or $(\bar t \gamma_\mu \gamma^5 t)\, (\overline{\chi} \gamma^\mu \gamma^5\chi)$ for an axial-vector-type interaction. We can estimate the size of the new interaction based on the fact that it is the only interaction that can thermalize the dark matter particle in the early Universe. The most interesting implication of the scenario is the collider signature. The final state consists of a top-quark pair and a pair of dark matter particles, giving rise to a top-quark pair plus a large missing energy. On the other hand, we anticipate that the spin-independent and spin-dependent cross sections in direct detection would be consistent with the current limit. This is easy to understand because the top content inside the nucleon is so small that it hardly contributes to the DM-nucleon scattering. We will explicitly show that. In addition, the annihilation of the dark matter in the Galactic Halo would give rise to positrons and antiprotons that can be observed by antimatter search experiments, e.g., PAMELA and AMSII. We use the present data on positron and antiproton spectra from PAMELA to constrain the size of the effective interactions.
The organization of the paper is as follows. In the next section, we describe the interaction between the top quark and the dark matter particle, and estimate the size of the interaction based on the relic density. In Sec. III, we calculate the spin-independent and spin-dependent cross sections for direct detection. In Sec. IV, we calculate the positron and antiproton spectra due to the DM annihilation in Galactic halo. In Sec. V, we discuss the collider signature. Finally, we conclude in Sec. VI.
There are a few recent works [@cao; @bai; @tait] that assumed some form of effective interactions between the dark matter and light quarks and studied the corresponding collider phenomenology. Fan [*et al.*]{} [@fan] also wrote down the effective nonrelativistic interactions between the dark matter and nuclei.
Effective Interactions and Relic Density
========================================
Our simple model consists of the SM and a hidden sector, in which there is a pair of Dirac/Majorana fermions and a gauge boson. For some reasons this gauge boson couples this hidden fermion only to the top quark on the SM side. If the mass of this gauge boson is heavy enough, we can integrate it out. More generally, below the heavy mass scale $\Lambda$ the interaction between the top quark and the dark matter particle, denoted by $\chi$, is given by $$\label{eff}
{\cal L} = \frac{g_\chi^2}{\Lambda^2} \,
\left ( \overline{\chi} \Gamma \chi \right )\;
\left ( \bar{t} \Gamma t \right ) \;,$$ where $\Gamma = \gamma^\mu$ for a vector gauge boson, $\Gamma =
\gamma^\mu \gamma^5$ for an axial-vector gauge boson, $\Gamma=1 \,
(\gamma^5) $ for a scalar (pseudoscalar) boson interaction, and $\Gamma = \sigma^{\mu\nu} (\gamma^5)$ with $\sigma^{\mu\nu} \equiv i (\gamma^\mu \gamma^\nu -\gamma^\nu \gamma^\mu ) /2$ for a tensor (axial-tensor) interaction, and $g_\chi$ is an effective coupling constant. For Majorana fermions the $\Gamma= \gamma^\mu$ or $\sigma^{\mu\nu}$ type interaction is identically zero, and so for vector or tensor type interaction the fermion $\chi$ in Eq.(\[eff\]) must be Dirac. Explicitly, we assume the dark matter candidate to be Dirac, but the results are also applicable to Majorana dark matter. With this interaction we can calculate the thermal averaged cross sections and thus the relic density, the direct and indirect detection rates, and also the production cross section of $pp \to t \bar t + \chi \overline{\chi}$ at the LHC.
![\[fig1\] The calculated $\sigma \, v$ versus $g_\chi^2$ for the effective interaction $\frac{g_\chi^2}{\Lambda^2} \,
\left ( \overline{\chi} \Gamma \chi \right )\;
\left ( \bar{t} \Gamma t \right ) $ of various Dirac structures $\Gamma$ with $\Lambda = 1$ TeV, $m_\chi = 200 $ GeV, and $v \approx 0.3$. ](xsecv_gchi2.eps){width="4.5in"}
![\[fig2\] Contours of $\sigma v = 0.91$ pb in the plane of $(g_\chi^2,\;m_\chi)$ for vector, axial-vector, pseudoscalar, scalar, tensor and axial-tensor interactions. $\Lambda$ is set at 1 TeV.](mchi_gchi2.eps){width="4.5in"}
We start with a vector gauge boson type interaction: $\Gamma = \gamma^\mu$. The differential cross section for $\chi (p_1) \; \overline\chi (p_2) \to t (k_1) \;\bar t (k_2)$, with the 4-momenta listed in the parentheses, is $$\label{dsigma}
\frac{d\sigma}{dz} = \frac{g_\chi^4}{\Lambda^4}\frac{N_C}{16\pi s} \,
\frac{\beta_t}{\beta_\chi}\, \left[
u_m^2 + t_m^2 + 2 s ( m_\chi^2 + m_t^2) \right ]$$ where $N_C=3$ for the top quark color, $\beta_{t,\chi} = ( 1- 4 m^2_{t,\chi}/s)^{1/2}$, $t_m = t - m_\chi^2 - m_t^2 = - s( 1- \beta_t \beta_\chi z)/2$, $u_m = u - m_\chi^2 - m_t^2 = - s( 1 + \beta_t \beta_\chi z)/2$, $s=(p_1+p_2)^2$ is the square of the center-of-mass energy, $t=(p_1 - k_1)^2$, $u=(p_1- k_2)^2$, and $z \equiv \cos\Theta$ with $\Theta$ the scattering angle. The quantity $\sigma v$, where $v\approx
2 \beta_\chi$ in the non-relativistic limit, can be obtained by integrating over the variable $z$ in Eq. (\[dsigma\]). Instead of solving the Boltzmann equation, we can naively estimate the size of the interaction by the following equation $$\Omega_\chi h^2 \simeq \frac{ 0.1\; {\rm pb}}{\langle \sigma v \rangle }\;.$$ With the most recent WMAP result on dark matter density $\Omega_{\rm CDM} h^2 = 0.1099 \pm 0.0062$ we obtain the size of $\sigma v$ $$\langle \sigma v \rangle \simeq 0.91 \; {\rm pb} \;.$$ We show in Fig. \[fig1\] $\sigma v$ versus the coefficient $g_\chi^2$ for a dark matter mass of 200 GeV. The result shown is for $v\approx 0.3$ to approximate the velocity of the dark matter particle at around the freeze-out time. We can repeat the calculation with $\Gamma =
\sigma^{\mu\nu}(\gamma^5), \; \gamma^\mu \gamma^5, \; \gamma^5,1$ for tensor (axial-tensor), axial-vector, pseudoscalar, and scalar type interactions. The results are $$\begin{aligned}
\frac{d\sigma}{dz} &=&
\label{tensor-axialtensor}
\frac{g_\chi^4}{\Lambda^4}\frac{N_C}{4\pi s} \,
\frac{\beta_t}{\beta_\chi}\, \left[
2 \, (t_m^2 + u_m^2 )+ 2 s ( m_t^2 +m_\chi^2 ) +
8 m_t^2 m_\chi^2 - s^2 \right ] \\
\frac{d\sigma}{dz} &=& \frac{g_\chi^4}{\Lambda^4}\frac{N_C}{16\pi s} \,
\frac{\beta_t}{\beta_\chi}\, \left[
t_m^2 + u_m^2 - 2 s ( m_t^2 + m_\chi^2 ) + 16 m_t^2 m_\chi^2 \right ] \\
\frac{d\sigma}{dz} &=& \frac{g_\chi^4}{\Lambda^4}\frac{N_C}{32\pi} s
\frac{\beta_t}{\beta_\chi} \\
\frac{d\sigma}{dz} &=& \frac{g_\chi^4}{\Lambda^4}\frac{N_C}{32\pi} s
\beta_\chi \beta_t^3 \end{aligned}$$ for $\Gamma = \sigma^{\mu\nu}(\gamma^5), \; \gamma^\mu \gamma^5,\;
\gamma^5,\; 1$, respectively. We note that the axial-tensor case has the expression as in the tensor one given by Eq.(\[tensor-axialtensor\]). The results are shown in Fig. \[fig1\] as well. We can see that the tensor-type interaction gives the largest cross section, followed by vector, pseudoscalar, and axial-vector. These four types of interactions require $g_\chi^2$ falling into the range of $0.2 -0.6$ which is about the size of weak-scale interaction. On the other hand, the scalar-type interaction always gives a very small annihilation cross section for a similar range of $g_\chi^2$, which is in danger of over-closing the Universe.
In Fig. \[fig2\], we show the contour of the cross section for the various types of interactions as a function of $g^2_\chi$ and $m_\chi$ as allowed by the WMAP result.
Direct Detection
================
Recently, the CDMSII finalized their search in Ref. [@cdms]. When they opened the black box in their blind analysis, they found two candidate events, which are consistent with background fluctuation at a probability level of about 23%. Nevertheless, the signal is not conclusive. The CDMS then improves upon the upper limit on the spin-independent (SI) cross section $\sigma^{\rm SI}_{\chi N}$ to $3.8 \times 10^{-44} \; {\rm cm}^2$ for $m_\chi \approx 70 $ GeV. The XENON100 Collaboration [@xenon100] also recently announced their newest result. Although XENON100 has the best sensitivity in the lower mass range, the CDMSII limit is currently still the best in the world for dark matter mass larger than about 100 GeV. We will adopt a limit of order $4-10 \times 10^{-44}\; {\rm cm}^2$ for dark matter mass of $200-500$ GeV. In the following, we will check if the spin-independent cross section generated by the 4-fermion interactions is consistent with the new limit.
Spin-independent cross sections can arise from the scalar-type and vector-type interactions between the DM and quarks. If the effective interactions between the dark matter particle and the quarks are given by $${\cal L } = \sum_{q=u,d,s,c,b,t} \{ \alpha_q^S \,\overline{\chi} \chi \,
\bar q q +
\alpha_q^V \,\overline{\chi} \gamma^\mu \chi \, \bar q \gamma_\mu q \} \;\; ,$$ then the spin-independent cross section between $\chi$ and each of the nucleon (taking the average between proton and neutron) is given by $$\label{111}
\sigma^{\rm SI}_{\chi N} = \frac{4 \mu_{\chi N}^2}{\pi}\; \left (
\left|G^N_s \right |^2 + \frac{|b_N|^2}{256} \right )\;,$$ where $\mu_{\chi N} = m_\chi m_N / ( m_\chi + m_N)$ is the reduced mass between the dark matter particle and the nucleon $N$, and $$G^N_s = \sum_{q=u,d,s,c,b,t} \langle N | \bar q q | N \rangle \;
\alpha^S_q \;,$$ where $\langle N | \bar q q | N \rangle$ denotes the various nucleon matrix elements. The expression for $b_N$ of a [*whole nucleus*]{} $(A,Z)$ is $b_N \equiv \alpha_u^V (A+Z) + \alpha^V_d (2A-Z)$, we take the average between proton and neutron (assume the number of protons is about the same as that of neutrons in the nuclei) and thus obtain the expression for a single nucleon $$b_N = \frac{3}{2} \left( \alpha_u^V + \alpha_d^V \right ) \;\; .$$ Nevertheless, the contributions to $b_N$ come from valence quarks only. Therefore, in our scenario the only contribution to the SI cross section comes from the top quark, thus only $\alpha_t^S $ is nonzero in the expression of $G^N_s$, which is then given by $$G^N_s = \langle N | \bar t t | N \rangle \;
\left ( \frac{g^2_{\chi}}{\Lambda^2} \right ) \;\;,$$ where $\langle N | \bar t t | N \rangle = f^N_{Tt} (m_N/m_t)$. [^2] The default value of the parameters $f^N_{Tt}$ used, e.g. in DarkSUSY [@darksusy], is $$f^p_{Tt} = 0.0595\;, \qquad f^n_{Tt} = 0.0592 \;.$$ Taking the average between proton and neutron the value of $G^N_s$ is $$G^N_s \simeq \frac{f^N_{Tt} m_N}{m_t} \left( \frac{g^2_{\chi}}{\Lambda^2}
\right ) \;.$$ For $m_\chi \sim O(100)$ GeV, $\mu_{\chi N} \approx m_N$. The spin-independent cross section is $$\sigma^{\rm SI}_{\chi N} \approx \frac{ 4 \mu^2_{\chi N}}{\pi} \left(
\frac{f^N_{Tt} m_N}{m_t} \right )^2 \,
\left( \frac{g^2_{\chi}}{\Lambda^2} \right )^2 \; .$$
We show in Fig. \[si\] the spin-independent cross section versus $g_{\chi}^2$. Note that the axial-vector interactions contributes to spin-dependent cross sections. Since the constraint from spin-dependent cross sections is a few orders of magnitude weaker than that from spin-independent cross sections, we simply focus the spin-independent one to obtain the meaningful range of $g_\chi^2$ and $\Lambda$. We found that the limit on spin-independent cross section of the order of $10^{-44}\; {\rm cm}^{2}$ allows $g_\chi^2$ as large as $30$ for $\Lambda = 1$ TeV. Note that for a strongly coupled theory, one can have $g^2 = (4 \pi)^2$. Such a large $g_\chi^2$ is allowed by spin-independent cross section constraint as well as by the WMAP relic density constraint. [^3] However, one must be cautious that for such a large effective coupling constant, perturbative calculation becomes less reliable.
We next turn to the indirect detection of the dark matter, which then gives the strongest constraint on the present scenario.
![\[si\] Spin-independent cross sections for the vector type interaction versus $g_\chi^2$.](SIxsec_gchi2.eps){width="4in"}
Indirect Detection
==================
Another important method to detect the dark matter is by measuring its annihilation products in Galactic halo. Current experiments can detect the positron, antiproton, gamma ray, and deuterium from dark matter annihilation. The Milky Way Halo may contain clumps of dark matter, from where the annihilation of dark matter particles may give rise to large enough signals, such as positron and antiproton, that can be identified by a number of antimatter search experiments. The most recent ones come from PAMELA [@pamela-e; @pamela-p], which showed a spectacular rise in the positron spectrum but an expected spectrum for antiproton. It may be due to nearby pulsars or dark matter annihilation or decays. If it is really due to dark matter annihilation, the dark matter would have very strange properties, because it only gives positrons in the final products but not antiproton. Here we adopt a conservative approach. We use the observed antiproton and positron spectra as constraints on the annihilation products in $\chi\bar \chi$ annihilation.
We first consider the positron coming from the process $$\chi \bar \chi \to t \bar t \to ( b W^+) (\bar b W^-) \to (b e^+ \nu_e) + X$$ in which the most energetic $e^+$ comes from the $W^+$ decay. There are also positrons coming off in the subsequent decays of $b,\bar b$, $\tau^+$, or $\mu^+$, but these positrons are in general softer than those coming directly from the $W^+$ decay. For a first order estimate of the size of the coupling $g_\chi^2$ in Eq. (\[eff\]) we only include the positrons coming directly from the $W^+$ decay.
The expression for annihilation has already been given in Eq. (\[dsigma\]), but now with a present time velocity $v\approx 10^{-3}$. The positron flux observed at the Earth is given by $$\Phi_{e^+} (E) = \frac{ v_{e^+} } { 4 \pi} \, f_{e^+} (E) \;,
\label{semiflux}$$ with $v_{e^+}$ close to the velocity of light $c$. The function $f_{e^+} (E)$ satisfies the diffusion equation of $$\frac{\partial f}{\partial t} - K(E) \nabla^2 f
- \frac{\partial}{\partial E} \left( b(E) f \right ) = Q \;,$$ where the diffusion coefficient is $K(E) = K_0(E/{\rm GeV})^\delta$ and the energy loss coefficient is $b(E) = E^2/ ({\rm GeV} \times \tau_E)$ with $\tau_E = 10^{16}$ sec. The source term $Q$ due to the annihilation is $$Q_{\rm ann} = \eta \left( \frac{\rho_{\rm CDM} }{M_{\rm CDM}} \right )^2
\, \sum \langle \sigma v \rangle_{e^+} \, \frac{d N_{e^+}}{ d E_{e^+} } \;,$$ where $\eta = 1/2 \; (1/4)$ for (non-)identical DM particle in the initial state. The summation is over all possible channels that can produce positrons in the final state, and $dN_{e^+}/dE_{e^+}$ denotes the spectrum of the positron energy per annihilation in that particular channel. In our analysis, we employ the vector-type interaction for Dirac fermions and thus the source term is given by $$Q_{\rm ann} = \frac{1}{4} \left(
\frac{\rho_{\rm CDM} }{M_{\rm CDM}} \right )^2
\, \langle \sigma v \rangle_{\chi\bar \chi\to t\bar t} \, \frac{d N_{e^+}}
{ d E_{e^+} } \;,$$ where the normalization of $N_{e^+}$ is $$\int \frac{d N_{e^+} }{d x} dx = B(t \to b W^+ \to b e^+ \nu_e) \;.$$ We then put the source term into GALPROP[@GALPROP] to solve the diffusion equation. In Fig. \[e+\] we show the predicted energy spectrum for the positron fraction for various values of $g_\chi^2$. With a visual inspection the $g_\chi^2 \alt 8$ is allowed by the spectrum.
![\[e+\] Spectrum for the positron fraction predicted for the vector type interactions for various $g_\chi^2$. PAMELA data are shown.](positron.eps){width="6in"}
![\[pbar\] Spectrum for the antiproton fraction predicted for the vector type interactions for various $g_\chi^2$. PAMELA data are shown.](antiproton.eps){width="6in"}
Next we turn to the antiproton fraction as it was also measured by PAMELA. Similarly, the antiproton flux can be obtained by solving the diffusion equation with the corresponding terms and the appropriate source term for the input antiproton spectrum: $$Q_{\rm ann} = \eta \left( \frac{\rho_{\rm CDM} }{M_{\rm CDM}} \right )^2
\, \sum \langle \sigma v \rangle_{\bar p} \, \frac{d N_{\bar p}}{ d T_{\bar p} }
\;,$$ where $\eta =1/2\;(1/4) $ for (non-)identical initial state, and $T_{\bar p}$ is the kinetic energy of the antiproton which is conventionally used instead of the total energy. We again solve the diffusion equation using GALPROP[@GALPROP].
In our case, the dominant contribution comes from $$\chi \bar \chi \to t \bar t \to (b W^+) (\bar b W^-) \to ( b q \bar q')
(\bar b q \bar q') \to \bar p + X \;.$$ In the last step, all the $b\,\bar b, q, \bar q'$ have probabilities fragmenting into $\bar p$. We adopt a publicly available code[@kniehl] to calculate the fragmentation function $D_{q\to h}(z)$ for any quark $q$ into hadrons $h$, e.g., $p,\bar p, \pi$. The fragmentation function is then convoluted with energy spectrum $d N/ dE$ of the light quark to obtain the energy spectrum of the antiproton $d N /d E_{\bar p}$. The source term $dN /d T_{\bar p}$ is then implemented into GALPROP to calculate the propagation from the halo to the Earth. We display the energy spectrum for the antiproton fraction in Fig. \[pbar\]. It is easy to see that $g_\chi^2$ is constrained to be $$g^2_\chi \alt 4-5 \;.$$ We will use this allowed range to estimate what we would expect from the LHC.
Collider Signature
==================
Collider signatures are perhaps the most interesting part of the scenario – $t\bar t$ pair plus large missing energy. We first calculate using the effective 4-fermion interaction with $\Gamma = \gamma^\mu$ the production cross section for $pp \to t \bar t + \chi \bar \chi$. There are two contributing subprocesses for $t\bar t$ production at the LHC: $$q\bar q \to t \bar t\;, \qquad \qquad gg \to t \bar t \;,$$ on which we can attach one 4-fermion interaction vertex to each fermion leg including internal fermion line to further produce a $\chi \overline{\chi}$ pair. A typical Feynman diagram is shown in Fig. \[feyn\]. We employ MADGRAPH [@madgraph] to calculate the signal and background cross sections.
![\[feyn\] A contributing Feynman diagram for the subprocess $gg \to t \bar t + \chi
\overline{\chi}$. The other two diagrams can be obtained by attaching the black dot to the $t$ and $\bar t$ leg, respectively.](ttxx1.eps){width="3in"}
![\[fig7\] Missing transverse energy $\not\!{E}_T$ distributions for the signal $ pp \to t\bar t + \chi \bar \chi$ and the background $pp\to t\bar t Z$ for $g_\chi^2=3-30$ with $m_\chi=200$ GeV.](etmiss.eps){width="4in"}
![\[fig8\] Event numbers for the invariant mass $t\bar t$ distributions for the signal $ pp \to t\bar t + \chi \bar \chi$ and the background $pp\to t\bar t Z$ (a) before and (b) after applying the missing transverse momentum cut of 400 GeV. The assumed luminosity is 100 fb$^{-1}$, which corresponds to 240 signal events and 420 background events after the cut.](mtt.eps){width="6in"}
\(a) (b)
The irreducible background is $t\bar t + Z \to t\bar t \nu\bar \nu$. Before applying any cuts we calculate the signal cross section versus background cross section: 8.2 fb (for $g_\chi^2=5$) to 140 fb, in which we have chosen scale $Q = (2m_t + 2m_\chi)/2$ in the running coupling constant and the parton distribution functions for the signal, while $Q= (2m_t + m_Z)/2$ for the background. We first compare the missing $E_T$ distribution between the dark matter signal and the $t\bar t Z$ background, shown in Fig. \[fig7\]. It is clear that the signal has a harder missing $E_T$ spectrum than the background. This plot suggests a cut as large as 400 GeV in the missing transverse energy can substantially reduces the background to a level similar to the signal. The cross sections in fb for the signal and the background using various cuts on the missing energy are shown in Table \[table\]. The background can indeed be cut down to the level as the signal with a missing energy cut of 400 GeV. Note that the signal cross section scales as $g_\chi^4$. The significance of the signal $S/\sqrt{B}$ for an integrated luminosity of 100 fb$^{-1}$ stays around $11$ with a cut of $300-500$ GeV. Since the significance scales as $\sqrt{\cal L}$, with a reduced luminosity of 30 fb$^{-1}$ the significance is still as large as $6$.
We then compare the $t\bar t$ invariant mass distribution between the dark matter signal and the $t\bar t Z$ background before and after applying the missing $E_T$ cuts, shown in Fig. \[fig8\]. In Table \[table2\], we also show the signal cross sections and the significance for axial-vector, pseudoscalar, and scalar interactions [^4] in decreasing order. Note that the cross section at the LHC for scalar interaction is not severely suppressed, in sharp contrast to the annihilation cross section calculated in Sec. II.
$\not\!{E}_T > $ $pp\to t\bar t + \chi\bar \chi$ $p\to t\bar t Z \to t\bar t \nu \bar \nu$ $S/B$ $S/\sqrt{B}$ (100 (30) fb$^{-1}$)
------------------ --------------------------------- -------------------------------------------- -------- -----------------------------------
$0$ GeV $8.2$ $140.3$ $0.06$ $6.9\;(3.8) $
$300$ GeV $3.6$ $10.7$ $0.34$ $11.0\;(6.0)$
$400$ GeV $2.4$ $4.2$ $0.57$ $11.8\;(6.4) $
$500$ GeV $1.5$ $1.9$ $0.78$ $10.6\;(5.9) $
: \[table\] Cross sections in fb for the signal $pp \to t\bar t + \chi\bar\chi$ and the background $pp\to t\bar t Z \to t\bar t + \nu\bar\nu$ at the LHC. We used $g_\chi^2 = 5$ for illustration. The signal cross section scales as $g_\chi^4$. The significance $S/\sqrt{B}$ is calculated with an integrated luminosity of 100 (30) fb$^{-1}$.
Signal cross section (fb) $S/B$ $S/\sqrt{B}$
-------------- --------------------------- --------- ---------------
Vector $2.4$ $ 0.57$ $11.8\;(6.4)$
Axial-vector $1.9$ $0.45$ $9.3\;(5.1)$
Pseudoscalar $0.82$ $0.20$ $4.0\;(2.2)$
Scalar $0.55$ $0.13$ $2.7\;(1.5)$
: \[table2\] Cross sections in fb for the signal $pp \to t\bar t + \chi\bar\chi$ for vector, axial-vector, pseudoscalar, and scalar interactions at the LHC. We have imposed the $\not\!\!{E}_T > 400$ GeV cut. The $S/B$ and $S/\sqrt{B}$ are shown. The background is from Table I. The significance $S/\sqrt{B}$ is calculated with an integrated luminosity of 100 (30) fb$^{-1}$.
Conclusions
===========
In this paper we have studied an interesting scenario where the dark matter couples exclusively to the top quark using an effective field theory approach, with the intuition that both the top quark and the dark matter may be closely related to electroweak symmetry breaking. We did not specify any particular connector linking the SM sector and the invisible dark matter sector, except that this connector sector was taken to be heavy, probably at the TeV scale. Integrating out the heavy connector sector may give rise to effective 4-fermion interactions of tensor, axial-tensor, vector, axial-vector, pseudoscalar, or scalar types between the dark matter and the top quark. We studied the constraints of these effective couplings from WMAP as well as from the direct and indirect detection of dark matter at CDMSII and PAMELA, respectively.
If we require all the dark matter in the Universe comes from the thermal equilibrium, the coupling $g_\chi^2 \approx 0.3 - 0.6$. However, if we just require that the dark matter does not overclose the Universe the $g_\chi^2$ can be much larger. Since only the top quark inside the nucleon contributes to direct detection cross section, the coupling $g_\chi^2$ can be as large as $40$. On the other hand, the strongest constraint comes from the positron and antiproton fraction spectra. The PAMELA antiproton spectrum constrains the coupling to be $g_\chi^2 \alt 4-5$.
This model can be tested at colliders with a very distinct signature, namely, $t\bar t$ plus missing energies. The top quark and antiquark would mostly have high $p_T$ and boosted. The detection of such boosted top quarks has attracted some recent studies that it can be sufficiently distinguished from the background [@boost]. Our results suggested that this interesting scenario can be testable at the LHC.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work was supported in parts by the National Science Council of Taiwan under Grant Nos. 96-2628-M-007-002-MY3, 99-2112-M-007-005-MY3, and 98-2112-M-001-014-MY3, the NCTS, and the WCU program through the KOSEF funded by the MEST (R31-2008-000-10057-0).
[99]{} J. Dunkley [*et al.*]{} \[WMAP Collaboration\], Astrophys. J. Suppl. [**180**]{}, 306 (2009). G. Bertone, D. Hooper, and J. Silk, Phys. Rept. [**405**]{}, 279 (2005) \[arXiv:hep-ph/0404175\]. C. B. Jackson, G. Servant, G. Shaughnessy, T. M. P. Tait and M. Taoso, JCAP [**1004**]{}, 004 (2010) \[arXiv:0912.0004 \[hep-ph\]\]; M. Battaglia and G. Servant, arXiv:1005.4632 \[hep-ex\]. Q. H. Cao, C. R. Chen, C. S. Li and H. Zhang, arXiv:0912.4511 \[hep-ph\]; Y. Bai, P. J. Fox and R. Harnik, arXiv:1005.3797 \[hep-ph\]; J. Goodman, M. Ibe, A. Rajaraman, W. Shepherd, T. M. P. Tait and H. B. P. Yu, arXiv:1005.1286 \[hep-ph\]; J. Goodman, M. Ibe, A. Rajaraman, W. Shepherd, T. M. P. Tait and H. B. P. Yu, arXiv:1008.1783 \[hep-ph\]. J. Fan, M. Reece and L. T. Wang, arXiv:1008.1591 \[hep-ph\]. Z. Ahmed [*et al.*]{} \[The CDMS-II Collaboration and CDMS-II Collaboration\], arXiv:0912.3592 \[astro-ph.CO\]. E. Aprile [*et al.*]{} \[XENON100 Collaboration\], arXiv:1005.0380 \[astro-ph.CO\]. P. Gondolo, J. Edsjo, P. Ullio, L. Bergstrom, M. Schelke and E. A. Baltz, JCAP [**0407**]{}, 008 (2004) \[arXiv:astro-ph/0406204\]. O. Adriani [*et al.*]{} \[PAMELA Collaboration\], Nature [**458**]{}, 607 (2009). O. Adriani [*et al.*]{}, Phys. Rev. Lett. [**102**]{}, 051101 (2009). A. W. Strong, I. V. Moskalenko, T. A. Porter, G. Johannesson, E. Orlando and S. W. Digel, arXiv:0907.0559 \[astro-ph.HE\]. S. Albino, B. A. Kniehl and G. Kramer, Nucl. Phys. B [**725**]{}, 181 (2005). F. Maltoni and T. Stelzer, JHEP [**0302**]{}, 027 (2003) \[arXiv:hep-ph/0208156\]; J. Alwall [*et al.*]{}, JHEP [**0709**]{}, 028 (2007) \[arXiv:0706.2334 \[hep-ph\]\]. D. E. Kaplan, K. Rehermann, M. D. Schwartz and B. Tweedie, Phys. Rev. Lett. [**101**]{}, 142001 (2008) \[arXiv:0806.0848 \[hep-ph\]\]; S. Rappoccio \[CMS Collaboration\], PoS E [**PS-HEP2009**]{}, 360 (2009); T. Plehn, M. Spannowsky, M. Takeuchi and D. Zerwas, arXiv:1006.2833 \[hep-ph\].
[^1]: A possibility of realizing such a scenario can be found in Ref. [@servant]
[^2]: Equivalently, the top content inside the nucleon can be replaced by the gluon content with $f^N_{Tq}$ replaced by $\frac{2}{27} f^N_{Tg}$ [@hooper]. Numerically, they are very close to each other.
[^3]: When the annihilation cross section is larger than that required by thermal production, the resulting relic density from thermal production is just too low. However, there could be some other non-thermal sources, such as decay from heavier fields.
[^4]: Note that the tensor interaction is not present in current version of MADGRAPH [@madgraph].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'One-stage detector basically formulates object detection as dense classification and localization (i.e., bounding box regression). The classification is usually optimized by Focal Loss and the box location is commonly learned under Dirac delta distribution. A recent trend for one-stage detectors is to introduce an *individual* prediction branch to estimate the quality of localization, where the predicted quality facilitates the classification to improve detection performance. This paper delves into the *representations* of the above three fundamental elements: quality estimation, classification and localization. Two problems are discovered in existing practices, including (1) the inconsistent usage of the quality estimation and classification between training and inference (i.e., separately trained but compositely used in test) and (2) the inflexible Dirac delta distribution for localization when there is ambiguity and uncertainty which is often the case in complex scenes. To address the problems, we design new representations for these elements. Specifically, we merge the quality estimation into the class prediction vector to form a joint representation of localization quality and classification, and use a vector to represent arbitrary distribution of box locations. The improved representations eliminate the inconsistency risk and accurately depict the flexible distribution in real data, but contain *continuous* labels, which is beyond the scope of Focal Loss. We then propose Generalized Focal Loss (GFL) that generalizes Focal Loss from its discrete form to the *continuous* version for successful optimization. On COCO [test-dev]{}, GFL achieves 45.0% AP using ResNet-101 backbone, surpassing state-of-the-art SAPD (43.5%) and ATSS (43.6%) with higher or comparable inference speed, under the same backbone and training settings. Notably, our best model can achieve a single-model single-scale AP of 48.2%, at 10 FPS on a single 2080Ti GPU. Code and pretrained models are available at https://github.com/implus/GFocal.'
author:
- |
Xiang Li$^{1,2}$, Wenhai Wang$^{3,2}$, Lijun Wu$^{4}$, Shuo Chen$^{1}$, Xiaolin Hu$^{5}$, Jun Li$^{1}$, Jinhui Tang$^{1}$,\
and Jian Yang$^{1}$[^1]
bibliography:
- 'neurips\_2020.bib'
title: |
Generalized Focal Loss: Learning Qualified and Distributed Bounding Boxes for\
Dense Object Detection
---
Introduction {#intro}
============
Recently, dense detectors have gradually led the trend of object detection, whilst the attention on the *representation* of bounding boxes and their localization quality estimation leads to the encouraging advancement. Specifically, bounding box *representation* is modeled as a simple Dirac delta distribution [@he2019bounding; @lin2017focal; @zhang2019freeanchor; @tian2019fcos; @zhang2019bridging], which is widely used over past years. As popularized in FCOS [@tian2019fcos], predicting an additional localization quality (e.g., IoU score [@wu2020iou] or centerness score [@tian2019fcos]) brings consistent improvements of detection accuracy, when the quality estimation is combined (usually multiplied) with classification confidence as final scores [@jiang2018acquisition; @huang2019mask; @tian2019fcos; @wu2020iou; @zhu2019iou] for the rank process of Non-Maximum Suppression (NMS) during inference. Despite their success, we observe the following problems in existing practices: **Inconsistent usage of localization quality estimation and classification score between training and inference:** (1) In recent dense detectors, the localization quality estimation and classification score are usually trained independently but compositely utilized (e.g., multiplication) during inference [@tian2019fcos; @wu2020iou] (Fig. \[fig\_QFL\_drawback\_1\_cropped\](a)); (2) The supervision of the localization quality estimation is currently assigned for positive samples only [@jiang2018acquisition; @huang2019mask; @tian2019fcos; @wu2020iou; @zhu2019iou], which is unreliable as negatives may get chances to have uncontrollably higher quality predictions (Fig. \[fig\_QFL\_drawback\_2\_cropped\](a)). These two factors result in a gap between training and test, and would potentially degrade the detection performance, e.g., negative instances with randomly high-quality scores could rank in front of positive examples with lower quality prediction during NMS.
[l]{}[0.6]{}
**Inflexible representation of bounding boxes:** The widely used bounding box representation can be viewed as Dirac delta distribution [@girshick2015fast; @ren2015faster; @he2017mask; @cai2018cascade; @lin2017focal; @tian2019fcos; @kong2019foveabox; @zhang2019bridging] of the target box coordinates. However, it fails to consider the ambiguity and uncertainty in datasets (see the unclear boundaries of the figures in Fig. \[fig\_DFL\_drawback\_22\_cropped\]). Although some recent works [@he2019bounding; @choi2019gaussian] model boxes as Gaussian distributions, it is too simple to capture the real distribution of the locations of bounding boxes. In fact, the real distribution can be more arbitrary and flexible [@he2019bounding], without the necessity of being symmetric like the Gaussian function.
To address the above problems, we design new representations for the bounding boxes and their localization quality. **For localization quality representation**, we propose to merge it with the classification score into a single and unified representation: a classification vector where its value at the ground-truth category index refers to its corresponding localization quality (typically the IoU score between the predicted box and the corresponding ground-truth box in this paper). In this way, we unify classification score and IoU score into a joint and single variable (denoted as “classification-IoU joint representation”), which can be trained in an end-to-end fashion, whilst directly utilized during inference (Fig. \[fig\_QFL\_drawback\_1\_cropped\](b)). As a result, it eliminates the training-test inconsistency (Fig. \[fig\_QFL\_drawback\_1\_cropped\](b)) and enables the strongest correlation (Fig. \[fig\_QFL\_drawback\_2\_cropped\] (b)) between localization quality and classification. Further, the negatives will be supervised with 0 quality scores, thereby the overall quality predictions become more confidential and reliable. It is especially beneficial for dense object detectors as they rank all candidates regularly sampled across an entire image. **For bounding box representation**, we propose to represent the arbitrary distribution (denoted as “General distribution” in this paper) of box locations by directly learning the discretized probability distribution over its continuous space, without introducing any other stronger priors (e.g., Gaussian [@he2019bounding; @choi2019gaussian]). Consequently, we can obtain more reliable and accurate bounding box estimations, whilst being aware of a variety of their underlying distributions (see the predicted distributions in Fig. \[fig\_DFL\_drawback\_22\_cropped\] and Supplementary Materials).
The improved representations then pose challenges for optimization. Traditionally for dense detectors, the classification branch is optimized with Focal Loss [@lin2017focal] (FL). FL can successfully handles the class imbalance problem via reshaping the standard cross entropy loss. However, for the case of the proposed classification-IoU joint representation, in addition to the imbalance risk that still exists, we face a new problem with continuous IoU label (0$\sim$1) as supervisions, as the original FL only supports discrete $\{1, 0\}$ category label currently. We successfully solve the problem by extending FL from $\{1, 0\}$ discrete version to its continuous variant, termed Generalized Focal Loss (GFL). Different from FL, GFL considers a much general case in which the globally optimized solution is able to target at any desired continuous value, rather than the discrete ones. More specifically in this paper, GFL can be specialized into Quality Focal Loss (QFL) and Distribution Focal Loss (DFL), for optimizing the improved two representations respectively: QFL focuses on a sparse set of hard examples and simultaneously produces their *continuous* 0$\sim$1 quality estimations on the corresponding category; DFL makes the network to rapidly focus on learning the probabilities of values around the *continuous* locations of target bounding boxes, under an arbitrary and flexible distribution.
We demonstrate three advantages of GFL: (1) It bridges the gap between training and test when one-stage detectors are facilitated with additional quality estimation, leading to a simpler, joint and effective representation of both classification and localization quality; (2) It well models the flexible underlying distribution for bounding boxes, which provides more informative and accurate box locations; (3) The performance of one-stage detectors can be consistently boosted without introducing additional overhead. On COCO [test-dev]{}, GFL achieves 45.0% AP with ResNet-101 backbone, surpassing state-of-the-art SAPD (43.5%) and ATSS (43.6%). Our best model can achieve a single-model single-scale AP of 48.2% whilst running at 10 FPS on a single 2080Ti GPU.
Related Work
============
**Representation of localization quality.** Existing practices like Fitness NMS [@tychsen2018improving], IoU-Net [@jiang2018acquisition], MS R-CNN [@huang2019mask], FCOS [@tian2019fcos] and IoU-aware [@wu2020iou] utilize a separate branch to perform localization quality estimation in a form of IoU or centerness score. As mentioned in Sec. \[intro\], this separate formulation causes the inconsistency between training and test as well as unreliable quality predictions. Instead of introducing an additional branch, PISA [@cao2019prime] and IoU-balance [@wu2019iou] assign different weights in the classification loss based on their localization qualities, aiming at enhancing the correlation between the classification score and localization accuracy. However, the weight strategy is of implicit and limited benefits since it does not change the optimum of the loss objectives for classification.
**Representation of bounding boxes.** Dirac delta distribution [@girshick2015fast; @ren2015faster; @he2017mask; @cai2018cascade; @lin2017focal; @tian2019fcos; @kong2019foveabox; @zhang2019bridging] governs the representation of bounding boxes over past years. Recently, Gaussian assumption [@he2019bounding; @choi2019gaussian] is adopted to learn the uncertainty by introducing a predicted variance. Unfortunately, existing representations are either too rigid or too simplified, which can not reflect the complex underlying distribution in real data. In this paper, we further relax the assumption and directly learn the more arbitrary, flexible General distribution of bounding boxes, whilst being more informative and accurate.
Method
======
In this section, we first review the original Focal Loss [@lin2017focal] (FL) for learning dense classification scores of one-stage detectors. Next, we present the details for the improved representations of localization quality estimation and bounding boxes, which are successfully optimized via the proposed Quality Focal Loss (QFL) and Distribution Focal Loss (DFL), respectively. Finally, we summarize the formulations of QFL and DFL into a unified perspective termed Generalized Focal Loss (GFL), as a flexible extension of FL, to facilitate further promotion and general understanding in the future.
**Focal Loss (FL)**. The original FL [@lin2017focal] is proposed to address the one-stage object detection scenario where an extreme imbalance between foreground and background classes often exists during training. A typical form of FL is as follows (we ignore $\alpha_t$ in original paper [@lin2017focal] for simplicity): $$\setlength{\abovedisplayskip}{0pt}
\setlength{\belowdisplayskip}{0pt}
\textbf{FL}(p) = - (1 - p_t)^\gamma\log(p_t), p_t = \left\{\begin{array}{rc}p, & \text{when}\ \ y = 1 \\1 - p, & \text{when}\ \ y = 0\end{array} \right.
\label{eq_fl}$$ where $y \in \{1, 0\}$ specifies the ground-truth class and $p \in [0, 1]$ denotes the estimated probability for the class with label $y = 1$. $\gamma$ is the tunable focusing parameter. Specifically, FL consists of a standard cross entropy part $-\log(p_t)$ and a dynamically scaling factor part $(1 - p_t)^\gamma$, where the scaling factor $(1 - p_t)^\gamma$ automatically down-weights the contribution of easy examples during training and rapidly focuses the model on hard examples.
**Quality Focal Loss (QFL)**. To solve the aforementioned inconsistency problem between training and test phases, we present a joint representation of localization quality (i.e., IoU score) and classification score (“classification-IoU” for short), where its supervision softens the standard one-hot category label and leads to a possible float target $y \in [0, 1]$ on the corresponding category (see the classification branch in Fig. \[fig\_gfocal\_cropped\]). Specifically, $y = 0$ denotes the negative samples with 0 quality score, and $0 < y \le 1$ stands for the positive samples with target IoU score $y$. Note that the localization quality label $y$ follows the conventional definition as in [@wu2020iou; @jiang2018acquisition]: IoU score between the predicted bounding box and its corresponding ground-truth bounding box during training, with a dynamic value being 0$\sim$1. Following [@lin2017focal; @tian2019fcos], we adopt the multiple binary classification with sigmoid operators $\sigma(\cdot)$ for multi-class implementation. For simplicity, the output of sigmoid is marked as $\sigma$.
Since the proposed classification-IoU joint representation requires dense supervisions over an entire image and the class imbalance problem still occurs, the idea of FL must be inherited. However, the current form of FL only supports $\{1, 0\}$ discrete labels, but our new labels contain decimals. Therefore, we propose to extend the two parts of FL for enabling the successful training under the case of joint representation: (1) The cross entropy part $-\log(p_t)$ is expanded into its complete version $-\big((1 - y)\log(1 - \sigma) + y\log(\sigma)\big)$; (2) The scaling factor part $(1 - p_t)^\gamma$ is generalized into the absolute distance between the estimation $\sigma$ and its continuous label $y$, i.e., $|y - \sigma|^\beta$ ($\beta \ge 0$), here $|\cdot|$ guarantees the non-negativity. Subsequently, we combine the above two extended parts to formulate the complete loss objective, which is termed as Quality Focal Loss (QFL): $$\setlength{\abovedisplayskip}{0pt}
\setlength{\belowdisplayskip}{0pt}
\textbf{QFL}(\sigma) = -\big|y - \sigma\big|^{\beta}\big((1 - y)\log(1 - \sigma) + y\log(\sigma)\big).$$Note that $\sigma = y$ is the global minimum solution of QFL. QFL is visualized for several values of $\beta$ in Fig. \[fig\_fcos\_atss\_reg\_cropped\](a) under quality label $y = 0.5$. Similar to FL, the term $\big|y - \sigma\big|^{\beta}$ of QFL behaves as a modulating factor: when the quality estimation of an example is inaccurate and deviated away from label $y$, the modulating factor is relatively large, thus it pays more attention to learning this hard example. As the quality estimation becomes accurate, i.e., $\sigma \to y$, the factor goes to 0 and the loss for well-estimated examples is down-weighted, in which the parameter $\beta$ controls the down-weighting rate smoothly ($\beta = 2$ works best for QFL in our experiments).
**Distribution Focal Loss (DFL).** Following [@tian2019fcos; @zhang2019bridging], we adopt the relative offsets from the location to the four sides of a bounding box as the regression targets (see the regression branch in Fig. \[fig\_gfocal\_cropped\]). Conventional operations of bounding box regression model the regressed label $y$ as Dirac delta distribution $\delta(x - y)$, where it satisfies $\int_{-\infty}^{+\infty} \delta(x - y){\mathop{}\!\mathrm{d}}x = 1$ and is usually implemented through fully connected layers. More formally, the integral form to recover $y$ is as follows: $$\setlength{\abovedisplayskip}{0pt}
\setlength{\belowdisplayskip}{0pt}
y = \int_{-\infty}^{+\infty} \delta(x - y) x {\mathop{}\!\mathrm{d}}x.$$ According to the analysis in Sec. \[intro\], instead of the Dirac delta [@ren2015faster; @he2017mask; @cai2018cascade; @tian2019fcos; @zhang2019bridging] or Gaussian [@choi2019gaussian; @he2019bounding] assumptions, we propose to directly learn the underlying General distribution $P(x)$ without introducing any other priors. Given the range of label $y$ with minimum $y_0$ and maximum $y_n$ ($y_0 \le y \le y_n, n \in \mathbb{N}^+$), we can have the estimated value $\hat{y}$ from the model ($\hat{y}$ also meets $y_0 \le \hat{y} \le y_n$): $$\setlength{\abovedisplayskip}{0pt}
\setlength{\belowdisplayskip}{0pt}
\hat{y} = \int_{-\infty}^{+\infty} P(x) x {\mathop{}\!\mathrm{d}}x = \int_{y_0}^{y_n} P(x) x {\mathop{}\!\mathrm{d}}x.
\label{eq_integral}$$ To be consistent with convolutional neural networks, we convert the integral over the continuous domain into a discrete representation, via discretizing the range $[y_0, y_n]$ into a set $\{y_0, y_1, ..., y_i, y_{i+1}, ..., y_{n-1}, y_n\}$ with even intervals $\varDelta$ (we use $\varDelta = 1$ for simplicity). Consequently, given the discrete distribution property $\sum_{i = 0}^{n} P(y_i) = 1$, the estimated regression value $\hat{y}$ can be presented as: $$\setlength{\abovedisplayskip}{0pt}
\setlength{\belowdisplayskip}{0pt}
\hat{y} = \sum_{i = 0}^{n} P(y_i) y_i.$$ As a result, $P(x)$ can be easily implemented through a softmax $\mathcal{S(\cdot)}$ layer consisting of $n + 1$ units, with $P(y_i)$ being denoted as $\mathcal{S}_i$ for simplicity. Note that $\hat{y}$ can be trained in an end-to-end fashion with traditional loss objectives like SmoothL1 [@girshick2015fast], IoU Loss [@tychsen2018improving] or GIoU Loss [@rezatofighi2019generalized]. However, there are infinite combinations of values for $P(x)$ that can make the final integral result being $y$, as shown in Fig. \[fig\_fcos\_atss\_reg\_cropped\](b), which may reduce the learning efficiency. Intuitively compared against (1) and (2), distribution (3) is compact and tends to be more confident and precise on the bounding box estimation, which motivates us to optimize the shape of $P(x)$ via explicitly encouraging the high probabilities of values that are close to the target $y$. Furthermore, it is often the case that the most appropriate underlying location, if exists, would not be far away from the coarse label. Therefore, we introduce the Distribution Focal Loss (DFL) which forces the network to rapidly focus on the values near label $y$, by explicitly enlarging the probabilities of $y_i$ and $y_{i+1}$ (nearest two to $y$, $y_i \le y \le y_{i+1}$). As the learning of bounding boxes are only for positive samples without the risk of class imbalance problem, we simply apply the complete cross entropy part in QFL for the definition of DFL: $$\setlength{\abovedisplayskip}{4pt}
\setlength{\belowdisplayskip}{4pt}
\textbf{DFL}(\mathcal{S}_i, \mathcal{S}_{i+1})=-\big((y_{i+1} - y)\log(\mathcal{S}_i)+(y - y_{i})\log(\mathcal{S}_{i+1})\big).$$Intuitively, DFL aims to focus on enlarging the probabilities of the values around target $y$ (i.e., $y_i$ and $y_{i+1}$). The global minimum solution of DFL, i.e, $\mathcal{S}_i = \frac{y_{i+1} - y}{y_{i+1} - y_i}, \mathcal{S}_{i+1} = \frac{y - y_i}{y_{i+1} - y_i}$ (see Supplementary Materials), can guarantee the estimated regression target $\hat{y}$ infinitely close to the corresponding label $y$, i.e., $\hat{y} = \sum_{j = 0}^{n} P(y_j) y_j = \mathcal{S}_iy_i + \mathcal{S}_{i+1}y_{i+1} = \frac{y_{i+1} - y}{y_{i+1} - y_i} y_i + \frac{y - y_i}{y_{i+1} - y_i} y_{i+1} = y$, which also ensures its correctness as a loss function.
**Generalized Focal Loss (GFL).** Note that QFL and DFL can be unified into a general form, which is called the Generalized Focal Loss (GFL) in the paper. Assume that a model estimates probabilities for two variables $y_l, y_r (y_l < y_r)$ as $p_{y_l}, p_{y_r}$ ($p_{y_l} \ge 0, p_{y_r} \ge 0, p_{y_l} + p_{y_r}=1$), with a final prediction of their linear combination being $\hat{y} = y_l p_{y_l} + y_r p_{y_r} (y_l \le \hat{y} \le y_r)$. The corresponding continuous label $y$ for the prediction $\hat{y}$ also satisfies $y_l \le y \le y_r$. Taking the absolute distance $|y - \hat{y}|^\beta$ ($\beta \ge 0$) as modulating factor, the specific formulation of GFL can be written as: $$\setlength{\abovedisplayskip}{0pt}
\setlength{\belowdisplayskip}{0pt}
\textbf{GFL}(p_{y_l}, p_{y_r}) = - \big|y - (y_l p_{y_l} + y_r p_{y_r})\big|^{\beta} \big((y_r - y)\log(p_{y_l}) + (y - y_l)\log(p_{y_r})\big).
\label{eq_gfl}$$ **Properties of GFL.** $\textbf{GFL}(p_{y_l}, p_{y_r})$ reaches its global minimum with $p_{y_l}^* = \frac{y_r - y}{y_r - y_l}, p_{y_r}^* = \frac{y - y_l}{y_r - y_l}$, which also means that the estimation $\hat{y}$ perfectly matches the continuous label $y$, i.e., $\hat{y} = y_l p_{y_l}^* + y_r p_{y_r}^* = y$ (see the proof in ). Obviously, the original FL [@lin2017focal] and the proposed QFL and DFL are all *special cases* of GFL (see Supplementary Materials for details). Note that GFL can be applied to any one-stage detectors. The modified detectors differ from the original detectors in two aspects. First, during inference, we directly feed the classification score (joint representation with quality estimation) as NMS scores without the need of multiplying any *individual* quality prediction if there exists (e.g., centerness as in FCOS [@tian2019fcos] and ATSS [@zhang2019bridging]). Second, the last layer of the regression branch for predicting each location of bounding boxes now has $n+1$ outputs instead of $1$ output, which brings *negligible* extra computing cost as later shown in Table \[tab:qfl\_dfl\_atss\]. $$\setlength{\abovedisplayskip}{0pt}
\setlength{\belowdisplayskip}{0pt}
\textbf{GFL}(1-p, p)=- \big|y\!-\! p\big|^{\gamma} \big((1\!-\!y)\log(1\!-\!p)+y \log(p)\big)=\left\{
\begin{array}{rc}
- (p)^\gamma \log(1\!-\!p), & \text{when}\ \ y = 0 \\
- (1\!-\!p)^\gamma \log(p), & \text{when}\ \ y = 1
\end{array} \right.
\label{eq_gfl_fl}$$
**Training Dense Detectors with GFL.** We define training loss $\mathcal{L}$ with GFL: $$\setlength{\abovedisplayskip}{0pt}
\setlength{\belowdisplayskip}{0pt}
\mathcal{L} = \frac{1}{N_{pos}}\sum_{z}{\mathcal{L_{Q}}} + \frac{1}{N_{pos}}\sum_{z} \textbf{1}_{\{c^*_{z} > 0\}} \big( \lambda_{0}\mathcal{L_{B}} + \lambda_{1}\mathcal{L_{D}} \big),$$ where $\mathcal{L_{Q}}$ is QFL and $\mathcal{L_{D}}$ is DFL. Typically, $\mathcal{L_{B}}$ denotes the GIoU Loss as in [@tian2019fcos; @zhang2019bridging]. $N_{pos}$ stands for the number of positive samples. $\lambda_0$ (typically 2 as default, similarly in [@chen2019mmdetection]) and $\lambda_1$ (practically $\frac{1}{4}$, averaged over four directions) are the balance weights for $\mathcal{L_{Q}}$ and $\mathcal{L_{D}}$, respectively. The summation is calculated over all locations $z$ on the pyramid feature maps [@lin2017feature]. $\textbf{1}_{\{c^*_{z} > 0\}}$ is the indicator function, being 1 if $c^*_{z} > 0$ and 0 otherwise. Following the common practices in the official codes [@chen2019mmdetection; @tian2019fcos; @zhang2019bridging; @li2019learning], we also utilize the quality scores to weight $\mathcal{L_{B}}$ and $\mathcal{L_{D}}$ during training.
\[tab\_1\]
Experiment
==========
Our experiments are conducted on COCO benchmark [@lin2014microsoft], where [trainval35k]{} (115K images) is utilized for training and we use [minival]{} (5K images) as validation for our ablation study. The main results are reported on [test-dev]{} (20K images) which can be obtained from the evaluation server. For fair comparisons, all results are produced under mmdetection [@chen2019mmdetection], where the default hyper-parameters are adopted. Unless otherwise stated, we adopt 1x learning schedule (12 epochs) without multi-scale training for the following studies, based on ResNet-50 [@he2016deep] backbone. More training/test details can be found in .
[r]{}[0.44]{}
We first investigate the effectiveness of the QFL (Table \[tab\_1\]). In Table \[tab\_1\](a), we compare the proposed joint representation with its separate or implicit counterparts. Two alternatives for representing localization quality: IoU [@wu2020iou; @jiang2018acquisition] and centerness [@tian2019fcos; @zhang2019bridging] are also adopted in the experiments. In general, we construct 4 variants that use separate or implicit representation, as illustrated in Fig. \[fig\_QFL\_compare\_cropped\]. According to the results, we observe that the joint representations optimized by QFL consistently achieve better performance than all the counterparts, whilst IoU always performs better than centerness as a measurement of localization quality ([Supplementary Materials]{}). Table \[tab\_1\](b) shows that QFL can also boost the performance of other popular one-stage detectors, and Table \[tab\_1\](c) shows that $\beta = 2$ is the best setting for QFL. We illustrate the effectiveness of joint representation by sampling instances with its predicted classification and IoU scores of both IoU-branch model and ours, as shown in Fig. \[fig\_QFL\_drawback\_2\_cropped\](b). It demonstrates that the proposed joint representation trained with QFL can benefit the detection due to its more reliable quality estimation, and yields the strongest correlation between classification and quality scores according to its definition. In fact, in our joint representation, the predicted classification score is equal to the estimated quality score exactly.
[l]{}[0.4]{}
\[tab:qfl\_dfl\_atss\]
Second, we investigate the effectiveness of the DFL (Table \[tab\_2\]). To quickly select a reasonable value of $n$, we first illustrate the distribution of the regression targets in Fig. \[fig\_fcos\_atss\_reg\_cropped\](c). We will show in later experiments, the recommended choice of $n$ for ATSS is 14 or 16. In Table \[tab\_2\](a), we compare the effectiveness of different data representations for bounding box regression. We find that the General distribution achieves superior or at least comparable results, whilst DFL can further boost its performance. Qualitative comparisons are depicted in Fig. \[fig\_unimodal\]. It is observed that the proposed General distribution can provide more accurate bounding box locations than Gaussian and Dirac delta distribution, especially under the case with considerable occlusions (More discussions in ). Based on the improved ATSS trained by GFL, we report the effect of $n$ and $\varDelta$ in DFL by fixing one and varying another in Table \[tab\_2\](b) and (c). The results demonstrate that the selection of $n$ is not sensitive and $\varDelta$ is suggested to be small (e.g., 1) in practice. To illustrate the effect of General distribution, we plot several representative instances with its distributed bounding box over four directions in Fig. \[fig\_DFL\_drawback\_22\_cropped\], where the proposed distributed representation can effectively reflect the uncertainty of bounding boxes by its shape (see more examples in Supplementary Materials).
\[tab\_3\]
Third, we perform the ablation study on ATSS with ResNet-50 backbone to show the relative contributions of QFL and DFL (Table \[tab:qfl\_dfl\_atss\]). FPS (Frames-per-Second) is measured on the same machine with a single GeForce RTX 2080Ti GPU using a batch size of 1 under the same mmdetection [@chen2019mmdetection] framework. We observe that the improvement of DFL is orthogonal to QFL, and joint usage of both (i.e., GFL) improves the strong ATSS baseline by absolute 1% AP score. Furthermore, according to the inference speeds, GFL brings negligible additional overhead and is considered very practical.
Finally, we compare GFL (based on ATSS) with state-of-the-art approaches on COCO [test-dev]{} in Table \[tab\_3\]. Following previous works [@lin2017focal; @tian2019fcos], the multi-scale training strategy and 2x learning schedule (24 epochs) are adopted during training. For a fair comparison, we report the results of single-model single-scale testing for all methods, as well as their corresponding inference speeds (FPS). GFL with ResNet-101 [@he2016deep] achieves 45.0% AP at 14.6 FPS, which is superior than all the existing detectors with the same backbone, including SAPD [@zhu2019soft] (43.5%) and ATSS [@zhang2019bridging] (43.6%). Further, Deformable Convolutional Networks (DCN) [@zhu2019deformable] consistently boost the performances over ResNe(X)t backbones, where GFL with ResNeXt-101-32x4d-DCN obtains state-of-the-art 48.2% AP at 10 FPS. Fig. \[fig\_sota\] demonstrates the visualization of the accuracy-speed trade-off, where it can be observed that our proposed GFL pushes the envelope of accuracy-speed boundary to a high level.
Conclusion
==========
To effectively learn qualified and distributed bounding boxes for dense object detectors, we propose Generalized Focal Loss (GFL) that generalizes the original Focal Loss from $\{1, 0\}$ discrete formulation to the continuous version. GFL can be specialized into Quality Focal loss (QFL) and Distribution Focal Loss (DFL), where QFL encourages to learn a better joint representation of classification and localization quality, and DFL provides more informative and precise bounding box estimations by modeling their locations as General distributions. Extensive experiments validate the effectiveness of GFL. We hope GFL can serve as a simple yet effective baseline for the community.
More Discussions about the Distributions
========================================
Fig. \[fig\_distribution\_three\] depicts the ideas of Dirac delta, Gaussian, and the proposed General distributions, where the assumption goes from rigid (Dirac delta) to flexible (General). We also list several key comparisons about these distributions in Table \[tab\_distribution\]. It can be observed that the loss objective of the Gaussian assumption is actually a dynamically weighted L2 Loss, where its training weight is related to the predicted variance $\sigma$. It is somehow similar to that of Dirac delta (standard L2 Loss) when optimized at the edge level. Moreover, it is not clear how to integrate the Gaussian assumption into the IoU-based Loss formulations, since it heavily couples the expression of the target representation with its optimization objective. Therefore, it can not enjoy the benefits of the IoU-based optimization [@rezatofighi2019generalized], as it is proved to be very effective in practice. In contrast, our proposed General distribution decouples the representation and loss objective, making it feasible for any type of optimizations, including both edge level and box level.
\[tab\_distribution\]
We also find that the bounding box regression of Dirac delta distribution (including Gaussian distribution based on the analysis from Table \[tab\_distribution\]) behaves more sensitive to feature perturbations, making it less robust and susceptible to noise, as shown in the simulation experiment (Fig. \[fig\_DFL\_drawback\_1\_cropped\]). It proves that General distribution enjoys more benefits than the other counterparts.
Global Minimum of $\textbf{GFL}(p_{y_l}, p_{y_r})$
==================================================
Let’s review the definition of $\textbf{GFL}$: $$\textbf{GFL}(p_{y_l}, p_{y_r}) = - \big|y - (y_l p_{y_l} + y_r p_{y_r})\big|^{\beta} \big((y_r - y)\log(p_{y_l}) + (y - y_l)\log(p_{y_r})\big), \ \ \ \text{given}\ \ p_{y_l} + p_{y_r} = 1. \nonumber$$
For simplicity, $\textbf{GFL}(p_{y_l}, p_{y_r})$ can then be expanded as: $$\begin{aligned}
\textbf{GFL}(p_{y_l}, p_{y_r}) & = - \big|y - (y_l p_{y_l} + y_r p_{y_r})\big|^{\beta} \big((y_r - y)\log(p_{y_l}) + (y - y_l)\log(p_{y_r})\big)\\
& = \underbrace{\left\{ \big|y - (y_l p_{y_l} + y_r p_{y_r})\big|^{\beta}\right\}}_{\textbf{L}(\cdot,\cdot)} \underbrace{\left\{ -\big((y_r - y)\log(p_{y_l}) + (y - y_l)\log(p_{y_r})\big)\right\}}_{\textbf{R}(\cdot,\cdot)}\\
& = \textbf{L}(p_{y_l}, p_{y_r}) \textbf{R}(p_{y_l}, p_{y_r}), \nonumber
\end{aligned}$$
$$\begin{aligned}
\textbf{R}(p_{y_l}, p_{y_r}) &= -\big((y_r - y)\log(p_{y_l}) + (y - y_l)\log(p_{y_r})\big) \\
& = -\big((y_r - y)\log(p_{y_l}) + (y - y_l)\log(1 - p_{y_l})\big) \\
& \ge -\big( (y_r - y)\log(\frac{y_r - y}{y_r - y_l}) + (y - y_l)\log(\frac{y - y_l}{y_r - y_l}) \big) \\
& = \textbf{R}(p_{y_l}^*, p_{y_r}^*) > 0, \ \ \ \text{where}\ \ \ p_{y_l}^* = \frac{y_r - y}{y_r - y_l}, p_{y_r}^* = \frac{y - y_l}{y_r - y_l}.\\
\textbf{L}(p_{y_l}, p_{y_r}) &= \big|y - (y_l p_{y_l} + y_r p_{y_r})\big|^{\beta} \\
& \ge \textbf{L}(p_{y_l}^*, p_{y_r}^*) = 0, \ \ \ \text{where}\ \ \ p_{y_l}^* = \frac{y_r - y}{y_r - y_l}, p_{y_r}^* = \frac{y - y_l}{y_r - y_l}. \nonumber
\end{aligned}
$$
Furthermore, given $\epsilon \neq 0$, for arbitrary variable $(p_{y_l}, p_{y_r}) = (p_{y_l}^* + \epsilon, p_{y_r}^* - \epsilon)$ in the domain of definition, we can have: $$\begin{aligned}
\textbf{R}(p_{y_l}^* + \epsilon, p_{y_r}^* - \epsilon) > \textbf{R}(p_{y_l}, p_{y_r}) > 0, \\
\textbf{L}(p_{y_l}^* + \epsilon, p_{y_r}^* - \epsilon) = \big|\epsilon(y_r - y_l)\big|^{\beta} > 0 = \textbf{L}(p_{y_l}^*, p_{y_r}^*). \nonumber
\end{aligned}$$ Therefore, it is easy to deduce: $$\begin{aligned}
\textbf{GFL}(p_{y_l}, p_{y_r}) & = \textbf{L}(p_{y_l}, p_{y_r}) \textbf{R}(p_{y_l}, p_{y_r}) \ge \textbf{L}(p_{y_l}^*, p_{y_r}^*)\textbf{R}(p_{y_l}^*, p_{y_r}^*) = 0, \nonumber
\end{aligned}$$ where “$=$” holds only when $p_{y_l} = p_{y_l}^*, p_{y_r} = p_{y_r}^*$.
The global minimum property of GFL somehow explains why the IoU or centerness guided variants in Fig. \[fig\_QFL\_compare\_cropped\] would not have obvious advantages. In fact, the weighted guidance does not essentially change the global minimum of the original classification loss (e.g., Focal Loss), whilst their optimal classification targets are still one-hot labels. In contrast, the proposed GFL indeed modifies the global minimum and force the predictions to approach the accurate IoU between the estimated boxes and ground-truth boxes, which is obviously beneficial for the rank process of NMS.
FL, QFL and DFL are special cases of GFL
========================================
In this section, we show how GFL can be specialized into the form of FL, QFL and DFL, respectively.
**FL**: Letting $\beta = \gamma, y_l = 0, y_r = 1, p_{y_r} = p, p_{y_l} = 1 - p$ and $y \in \{1, 0\}$ in GFL, we can obtain FL: $$\begin{aligned}
\textbf{FL}(p) &= \textbf{GFL}(1-p, p)= -\big|y - p\big|^{\gamma} \big((1 - y)\log(1 - p) + y\log(p)\big), y \in \{1, 0\} \\
& = -(1 - p_t)^\gamma\log(p_t), p_t = \left\{\begin{array}{rc}p, & \text{when}\ \ y = 1 \\1 - p, & \text{when}\ \ y = 0\end{array} \right.
\end{aligned}
$$
**QFL**: Having $y_l = 0, y_r = 1, p_{y_r} = \sigma$ and $p_{y_l} = 1 - \sigma$ in GFL, the form of QFL can be written as: $$\setlength{\abovedisplayskip}{0pt}
\setlength{\belowdisplayskip}{0pt}
\textbf{QFL}(\sigma) = \textbf{GFL}(1-\sigma, \sigma) = -\big|y - \sigma\big|^{\beta}\big((1 - y)\log(1 - \sigma) + y\log(\sigma)\big).$$
**DFL**: By substituting $\beta = 0, y_l = y_i, y_r = y_{i+1}, p_{y_l} = P(y_l) = P(y_i) = \mathcal{S}_i, p_{y_r} = P(y_r) = P(y_{i+1}) = \mathcal{S}_{i+1}$ in GFL, we can have DFL: $$\setlength{\abovedisplayskip}{4pt}
\setlength{\belowdisplayskip}{4pt}
\textbf{DFL}(\mathcal{S}_i, \mathcal{S}_{i+1})=\textbf{GFL}(\mathcal{S}_i, \mathcal{S}_{i+1})=-\big((y_{i+1} - y)\log(\mathcal{S}_i)+(y - y_{i})\log(\mathcal{S}_{i+1})\big).$$
Details of Experimental Settings
================================
**Training Details:** The ImageNet pretrained models [@he2016deep] with FPN [@lin2017feature] are utilized as the backbones. During training, the input images are resized to keep their shorter side being 800 and their longer side less or equal to 1333. In ablation study, the networks are trained using the Stochastic Gradient Descent (SGD) algorithm for 90K iterations (denoted as 1x schedule) with 0.9 momentum, 0.0001 weight decay and 16 batch size. The initial learning rate is set as 0.01 and decayed by 0.1 at iteration 60K and 80K, respectively. **Inference Details:** During inference, the input image is resized in the same way as in the training phase, and then passed through the whole network to output the predicted bounding boxes with a predicted class. Then we use the threshold 0.05 to filter out a variety of backgrounds, and output top 1000 candidate detections per feature pyramid. Finally, NMS is applied under the IoU threshold 0.6 per class to produce the final top 100 detections per image as results.
Why is IoU-branch always superior than centerness-branch?
=========================================================
The ablation study in original paper also demonstrates that for FCOS/ATSS, IoU performs consistently better than centerness, as a measurement of localization quality. Here we give a convincing reason why this is the case. We discover the major problem of centerness is that its definition leads to unexpected small ground-truth label, which makes a possible set of ground-truth bounding boxes extremely hard to be recalled (as shown in Fig. \[fig\_centerness\_vs\_iou\_cropped\]). From the label distributions demonstrated in Fig. \[fig\_distribution\_atss\_centerness\_iou\_label\], we observe that most of IoU labels is larger than 0.4 yet centerness labels tend to be much smaller (even approaching 0). The small values of centerness labels prevent a set of ground-truth bounding boxes from being recalled, as their final scores for NMS would be potentially small since their predicted centerness scores are already supervised by these extremely small signals.
More Examples of Distributed Bounding Boxes
===========================================
We demonstrate more examples with General distributed bounding boxes predicted by GFL (ResNet-50 backbone). As demonstrated in Fig. \[fig\_more\_bad\_cropped\], we show several cases with boundary ambiguities: does the slim and almost invisible backpack strap belong to the box of the bag (left top)? does the partially occluded umbrella handle belong to the entire umbrella (left down)? In these cases, our models even produce more reasonable coordinates of bounding boxes than the ground-truth ones. In Fig. \[fig\_more\_good\_cropped\], more examples with clear boundaries and sharp General distributions are shown, where GFL is very confident to generate accurate bounding boxes, e.g., the bottom parts of the orange and skiing woman.
[^1]: Corresponding author.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
The goal of two-sample tests is to assess whether two samples, $S_P \sim
P^n$ and $S_Q \sim Q^m$, are drawn from the same distribution. Perhaps intriguingly, one relatively unexplored method to build two-sample tests is the use of binary classifiers. In particular, construct a dataset by pairing the $n$ examples in $S_P$ with a positive label, and by pairing the $m$ examples in $S_Q$ with a negative label. If the null hypothesis “$P =
Q$” is true, then the classification accuracy of a binary classifier on a held-out subset of this dataset should remain near chance-level. As we will show, such *Classifier Two-Sample Tests* (C2ST) learn a suitable representation of the data on the fly, return test statistics in interpretable units, have a simple null distribution, and their predictive uncertainty allow to interpret where $P$ and $Q$ differ.
The goal of this paper is to establish the properties, performance, and uses of C2ST. First, we analyze their main theoretical properties. Second, we compare their performance against a variety of state-of-the-art alternatives. Third, we propose their use to evaluate the sample quality of generative models with intractable likelihoods, such as Generative Adversarial Networks (GANs). Fourth, we showcase the novel application of GANs together with C2ST for causal discovery.
author:
- |
David Lopez-Paz$^1$, Maxime Oquab$^{1,2}$\
$^1$Facebook AI Research, $^2$WILLOW project team, Inria / ENS / CNRS\
`dlp@fb.com`, `maxime.oquab@inria.fr`
bibliography:
- 'classifier\_tests.bib'
title: 'Revisiting Classifier Two-Sample Tests'
---
Introduction
============
One of the most fundamental problems in statistics is to assess whether two samples, $S_P \sim P^n$ and $S_Q \sim Q^m$, are drawn from the same probability distribution. To this end, *two-sample tests* [@lehmann2006testing] summarize the differences between the two samples into a real-valued test *statistic*, and then use the value of such statistic to accept[^1] or reject the null hypothesis “$P=Q$”. The development of powerful two-sample tests is instrumental in a myriad of applications, including the evaluation and comparison of generative models. Over the last century, statisticians have nurtured a wide variety of two-sample tests. However, most of these tests are only applicable to one-dimensional examples, require the prescription of a fixed representation of the data, return test statistics in units that are difficult to interpret, or do not explain *how* the two samples under comparison differ.
Intriguingly, there exists a relatively unexplored strategy to build two-sample tests that overcome the aforementioned issues: training a binary classifier to distinguish between the examples in $S_P$ and the examples in $S_Q$. Intuitively, if $P=Q$, the test accuracy of such binary classifier should remain near chance-level. Otherwise, if $P\neq Q$ and the binary classifier is able to unveil some of the distributional differences between $S_P$ and $S_Q$, its test accuracy should depart from chance-level. As we will show, such *Classifier Two-Sample Tests* (C2ST) learn a suitable representation of the data on the fly, return test statistics in interpretable units, have simple asymptotic distributions, and their learned features and predictive uncertainty provide interpretation on *how* $P$ and $Q$ differ. In such a way, this work brings together the communities of statistical testing and representation learning.
The goal of this paper is to establish the theoretical properties and evaluate the practical uses of C2ST. To this end, our **contributions** are:
- We review the basics of two-sample tests in Section \[sec:two\], as well as their common applications to measure statistical dependence and evaluate generative models.
- We analyze the attractive properties of C2ST (Section \[sec:neural\]) including an analysis of their exact asymptotic distributions, testing power, and interpretability.
- We evaluate C2ST on a wide variety of synthetic and real data (Section \[sec:experiments\]), and compare their performance against multiple state-of-the-art alternatives. Furthermore, we provide examples to illustrate how C2ST can interpret the differences between pairs of samples.
- In Section \[sec:gmodels\], we propose the use of classifier two-sample tests to evaluate the sample quality of generative models with intractable likelihoods, such as Generative Adversarial Networks [@goodfellow2014generative], also known as GANs.
- As a novel application of the synergy between C2ST and GANs, Section \[sec:causality\] proposes the use of these methods for causal discovery.
Two-Sample Testing {#sec:two}
==================
The goal of two-sample tests is to assess whether two samples, denoted by $S_P \sim P^n$ and $S_Q \sim Q^m$, are drawn from the same distribution [@lehmann2006testing]. More specifically, two-sample tests either accept or reject the *null hypothesis*, often denoted by $H_0$, which stands for “$P= Q$”. When rejecting $H_0$, we say that the two-sample test favors the *alternative hypothesis*, often denoted by $H_1$, which stands for “$P \neq Q$”. To accept or reject $H_0$, two-sample tests summarize the differences between the two *samples* (sets of identically and independently distributed *examples*): $$\begin{aligned}
S_P := \{ x_1, \ldots, x_n \} \sim P^n(X) \text{ and } S_Q := \{ y_1,
\ldots, y_m \} \sim Q^m(Y)\label{eq:samples}
\end{aligned}$$ into a statistic $\hat{t} \in \mathbb{R}$. Without loss of generality, we assume that the two-sample test returns a small statistic when the [null hypothesis]{} “$P=Q$” is true, and a large statistic otherwise. Then, for a sufficiently small statistic, the two-sample test will accept $H_0$. Conversely, for a sufficiently large statistic, the two-sample test will reject $H_0$ in favour of $H_1$.
More formally, the statistician performs a two-sample test in four steps. First, decide a *significance level* $\alpha \in [0,1]$, which is an input to the two-sample test. Second, compute the two-sample test statistic $\hat{t}$. Third, compute the *p-value* $\hat{p} = P(T \geq \hat{t} |
H_0)$, the probability of the two-sample test returning a statistic as large as $\hat{t}$ when $H_0$ is true. Fourth, reject $H_0$ if $\hat{p} < \alpha$, and accept it otherwise.
Inevitably, two-sample tests can fail in two different ways. First, to make a *type-I error* is to reject the null hypothesis when it is true (a “false positive”). By the definition of $p$-value, the probability of making a type-I error is upper-bounded by the significance level $\alpha$. Second, to make a *type-II error* is to accept the null hypothesis when it is false (a “false negative”). We denote the probability of making a type-II error by $\beta$, and refer to the quantity $\pi = 1-\beta$ as the *power* of a test. Usually, the statistician uses domain-specific knowledge to evaluate the consequences of a type-I error, and thus prescribe an appropriate significance level $\alpha$. Within the prescribed significance level $\alpha$, the statistician prefers the two-sample test with maximum power $\pi$.
Among others, two-sample tests serve two other uses. First, two-sample tests can *measure statistical dependence* [@gretton2012kernel]. In particular, testing the independence null hypothesis “the random variables $X$ and $Y$ are independent” is testing the two-sample null hypothesis “$P(X,Y)=P(X)P(Y)$”. In practice, the two-sample test would compare the sample $S =
\{(x_i,y_i)\}_{i=1}^n \sim P(X,Y)^n$ to a sample $S_\sigma = \{(x_i,
y_{\sigma(i)})\}_{i=1}^n \sim (P(X)P(Y))^n$, where $\sigma$ is a random permutation of the set of indices $\{1, \ldots, n \}$. This approach is consistent when considering all possible random permutations. However, since independence testing is a subset of two-sample testing, specialized independence tests may exhibit higher power for this task [@gretton2005measuring].
Second, two-sample tests can *evaluate the sample quality of generative models* with intractable likelihoods, but tractable sampling procedures. Intuitively, a generative model produces good samples $\hat{S} =
\{\hat{x}_i\}_{i=1}^n$ if these are indistinguishable from the real data $S=\{x_i\}_{i=1}^n$ that they model. Thus, the two-sample test statistic between $\hat{S}$ and $S$ measures the fidelity of the samples $\hat{S}$ produced by the generative model. The use of two-sample tests to evaluate the sample quality of generative models include the pioneering work of @box1980sampling, the use of Maximum Mean Discrepancy (MMD) criterion [@bengio2013bounding; @karolina; @lloyd2015statistical; @bounliphone2015test; @gretton2016], and the connections to density-ratio estimation [@kanamori2010; @Wornowizki2016; @Menon2016; @Mohamed2016].
Over the last century, statisticians have nurtured a wide variety of two-sample tests. Classical two-sample tests include the $t$-test [@student1908probable], which tests for the difference in means of two samples; the Wilcoxon-Mann-Whitney test [@wilcoxon1945individual; @mann1947test], which tests for the difference in rank means of two samples; and the Kolmogorov-Smirnov tests [@kolmogorov1933sulla; @smirnov1939estimation] and their variants [@kuiper], which test for the difference in the empirical cumulative distributions of two samples. However, these classical tests are only efficient when applied to one-dimensional data. Recently, the use of kernel methods [@smola98] enabled the development of two-sample tests applicable to multidimensional data. Examples of these tests include the MMD test [@gretton2012kernel], which looks for differences in the empirical kernel mean embeddings of two samples, and the Mean Embedding test or ME [@jit2; @metests], which looks for differences in the empirical kernel mean embeddings of two samples at optimized locations. However, kernel two-sample tests require the prescription of a manually-engineered representation of the data under study, and return values in units that are difficult to interpret. Finally, only the ME test provides a mechanism to interpret how $P$ and $Q$ differ.
Next, we discuss a simple but relatively unexplored strategy to build two-sample tests that overcome these issues: the use of binary classifiers.
Classifier Two-Sample Tests (C2ST) {#sec:neural}
==================================
Without loss of generality, we assume access to the two samples $S_P$ and $S_Q$ defined in , where $x_i, y_j \in \mathcal{X}$, for all $i = 1, \ldots, n$ and $j = 1, \ldots, m$, and $m=n$. To test whether the null hypothesis $H_0 : P=Q$ is true, we proceed in five steps. First, construct the dataset $$\mathcal{D} = \{(x_i, 0)\}_{i=1}^n \cup \{(y_i, 1)\}_{i=1}^n =: \{(z_i, l_i)\}_{i=1}^{2n}.$$ Second, shuffle $\mathcal{D}$ at random, and split it into the disjoint *training* and [testing]{} subsets $\mathcal{D}_\text{tr}$ and $\mathcal{D}_\text{te}$, where $\mathcal{D} = \mathcal{D}_\text{tr} \cup \mathcal{D}_\text{te}$ and $n_\text{te} := |\mathcal{D}_\text{te}|$. Third, train a binary classifier $f
: \mathcal{X} \to [0,1]$ on $\mathcal{D}_\text{tr}$; in the following, we assume that $f(z_i)$ is an estimate of the conditional probability distribution $p(l_i = 1 | z_i)$. Fourth, return the classification accuracy on $\mathcal{D}_\text{te}$: $$\label{eq:stat}
\hat{t} = \frac{1}{n_\text{te}} \sum_{(z_i,l_i) \in \mathcal{D}_\text{te}}
\mathbb{I}\left[ \mathbb{I}\left(f(z_i) > \frac{1}{2}\right) = l_i \right]$$ as our *C2ST statistic*, where $\mathbb{I}$ is the indicator function. The intuition here is that if $P=Q$, the test accuracy should remain near chance-level. In opposition, if $P \neq Q$ and the binary classifier unveils distributional differences between the two samples, the test classification accuracy should be *greater* than chance-level. Fifth, to accept or reject the null hypothesis, compute a p-value using the null distribution of the C2ST, as discussed next.
Null and Alternative Distributions {#sec:null}
----------------------------------
Each term $\mathbb{I}\left[ \mathbb{I}(f(z_i) > 1/2) = l_i\right]$ appearing in is an independent $\text{Bernoulli}(p_i)$ random variable, where $p_i$ is the probability of classifying correctly the example $z_i$ in $\mathcal{D}_\text{te}$.
First, under the null hypothesis $H_0 : P = Q$, the samples $S_P \sim P^n$ and $S_Q \sim Q^m$ follow the same distribution, leading to an impossible binary classification problem. In that case, $n_\text{te} \hat{t}$ follows a $\text{Binomial}(n_\text{te},p=\frac{1}{2})$ distribution. Therefore, for large $n_{\text{te}}$, we can use the central limit theorem to approximate the null distribution of by $\mathcal{N}(\frac{1}{2}, \frac{1}{4 n_\text{te}})$.
Second, under the alternative hypothesis $H_1 : P \neq Q$, the statistic $n_{\text{te}} \hat{t}$ follows a Poisson Binomial distribution, since the constituent Bernoulli random variables may not be identically distributed. In the following, we will approximate such Poisson Binomial distribution by the $\text{Binomial}(n,\bar{p})$ distribution, where $\bar{p} = \frac{1}{n}
\sum_{i=1}^n p_i$ [@Ehm91]. Therefore, we can use the central limit theorem to approximate the alternative distribution of by $\mathcal{N}(\bar{p},
\frac{\bar{p}(1-\bar{p})}{n_\text{te}})$.
Testing power
-------------
To analyze the power (probability of correctly rejecting false null hypothesis) of C2ST, we assume that the our classifier has an expected (unknown) accuracy of $H_0: t = \frac{1}{2}$ under the null hypothesis “$P=Q$”, and an expected accuracy of $H_1: t = \frac{1}{2} +
\epsilon$ under the alternative hypothesis “$P\neq Q$”, where $\epsilon \in
(0, \frac{1}{2})$ is the *effect size* distinguishing $P$ from $Q$. Let $\Phi$ be the Normal cdf, $n_\text{te}$ the number of samples available for testing, and $\alpha$ the significance level. Then,
\[thm:thm1\] Given the conditions described in the previous paragraph, the approximate power of the statistic is $\Phi\left(
\frac{\epsilon\sqrt{n_\text{te}}-\Phi^{-1}(1-\alpha)/2}{\sqrt{\frac{1}{4}-\epsilon^2}}\right)$.
See Appendix \[app:proof\] for a proof. The power bound in Theorem \[thm:thm1\] has an optimal order of magnitude for multi-dimensional problems [@bai1996effect; @gretton2012kernel; @reddi2015high]. These are problems with fixed $d$ and $n \to \infty$, where the power bounds do not depend on $d$.
We leave for future work the study of quadratic-time C2ST with optimal power in high-dimensional problems [@Ramdas15]. These are problems where the ratio $n/d \to c \in [0,1]$, and the power bounds depend on $d$. One possible line of research in this direction is to investigate the power and asymptotic distributions of quadratic-time C2ST statistics $
\frac{1}{n_\text{te}(n_\text{te}-1)}\sum_{i\neq j} \mathbb{I}[
\mathbb{I}(f(z_i,z_j)>\frac{1}{2}) = l_i]$, where the classifier $f(z,z')$ predicts if the examples $(z,z')$ come from the same sample.
Theorem \[thm:thm1\] also illustrates that maximizing the power of a C2ST is a trade-off between two competing objectives: choosing a classifier that *maximizes the test accuracy* $\epsilon$ and *maximizing the size of the test set* $n_{\text{te}}$. This relates to the well known bias-variance trade-off in machine learning. Indeed, simple classifiers will miss more nonlinear patterns in the data (leading to smaller test accuracy), but call for less training data (leading to larger test set sizes). On the other hand, flexible classifiers will miss less nonlinear patterns in the data (leading to higher test accuracy), but call for more training data (leading to smaller test sizes). Formally, the relationship between the test accuracy, sample size, and the flexibility of a classifier depends on capacity measures such as the VC-Dimension [@vapnik98]. Note that there is no restriction to perform model selection (such as cross-validation) on $\mathcal{D}_\text{tr}$.
\[remark:loss\] We have focused on test statistics built on top of the zero-one loss $\ell_{0-1}(y,y') = \mathbb{I}[y=y'] \in \{0,1\}$. These statistics give rise to Bernoulli random variables, which can exhibit high variance. However, our arguments are readily extended to real-valued binary classification losses. Then, the variance of such real-valued losses would describe the norm of the decision function of the classifier two-sample test, appear in the power expression from Theorem \[thm:thm1\], and serve as a hyper-parameter to maximize power as in [@gretton2012optimal Section 3].[^2]
Interpretability
----------------
There are three ways to interpret the result of a C2ST. First, recall that the classifier predictions $f(z_i)$ are estimates of the conditional probabilities $p(l_i = 1 |z_i)$ for each of the samples $z_i$ in the test set. Inspecting these probabilities together with the true labels $l_i$ determines which examples were correctly or wrongly labeled by the classifier, with the least or the most confidence. Therefore, the values $f(z_i)$ explain *where* the two distributions differ. Second, C2ST inherit the interpretability of their classifiers to explain which *features* are most important to distinguish distributions, in the same way as the ME test [@metests]. Examples of interpretable features include the filters of the first layer of a neural network, the feature importance of random forests, the weights of a generalized linear model, and so on. Third, C2ST return statistics $\hat{t}$ in interpretable units: these relate to the percentage of samples correctly distinguishable between the two distributions. These interpretable numbers can complement the use of $p$-values.
Prior Uses
----------
The reduction of two-sample testing to binary classification was introduced in [@friedman2003multivariate], studied within the context of information theory in [@perez2009estimation; @reid2011information], discussed in [@Fukumizu2009; @gretton2012kernel], and analyzed (for the case of linear discriminant analysis) in [@ramdas16]. The use of binary classifiers for two-sample testing is increasingly common in neuroscience: see [@pereira2009machine; @olivetti2012induction] and the references therein. Implicitly, binary classifiers also perform two-sample tests in algorithms that discriminate data from noise, such as unsupervised-as-supervised learning [@friedman2001elements], noise contrastive estimation [@gutmann2012noise], negative sampling [@mikolov2013distributed], and GANs [@goodfellow2014generative].
Experiments on Two-Sample Testing {#sec:experiments}
=================================
We study two variants of classifier-based two-sample tests (C2ST): one based on neural networks (C2ST-NN), and one based on $k$-nearest neighbours (C2ST-KNN). C2ST-NN has one hidden layer of 20 ReLU neurons, and trains for $100$ epochs using the Adam optimizer [@adam]. C2ST-KNN uses $k = \lfloor n_{\text{tr}}^{1/2}\rfloor$ nearest neighbours for classification. Throughout our experiments, we did not observe a significant improvement in performance when increasing the flexibility of these classifiers (e.g., increasing the number of hidden neurons or decreasing the number of nearest neighbors). When analyzing one-dimensional data, we compare the performance of C2ST-NN and C2ST-KNN against the Wilcoxon-Mann-Whitney test [@wilcoxon1945individual; @mann1947test], the Kolmogorov-Smirnov test [@kolmogorov1933sulla; @smirnov1939estimation], and the Kuiper test [@kuiper]. In all cases, we also compare the performance of C2ST-NN and C2ST-KNN against the linear-time estimate of the Maximum Mean Discrepancy (MMD) criterion [@gretton2012kernel], the ME test [@metests], and the SCF test [@metests]. We use a significance level $\alpha =0.05$ across all experiments and tests, unless stated otherwise. We use Gaussian approximations to compute the null distributions of C2ST-NN and C2ST-KNN. We use the implementations of the MMD, ME, and SCF tests gracefully provided by @metests, the scikit-learn implementation of the Kolmogorov-Smirnov and Wilcoxon tests, and the implementation from [<https://github.com/aarchiba/kuiper>]{} of the Kuiper test. The implementation of our experiments is available at [<https://github.com/lopezpaz/classifier_tests>]{}.
{width="\textwidth"}
\[fig:synthetic\]
Experiments on Two-Sample Testing {#sec:smallexps}
---------------------------------
#### Control of type-I errors
We start by evaluating the correctness of all the considered two-sample tests by examining if the prescribed significance level $\alpha = 0.05$ upper-bounds their type-I error. To do so, we draw $x_1, \ldots, x_n, y_1, \ldots, y_n \sim \mathcal{N}(0,1)$, and run each two-sample test on the two samples $\{x_i\}_{i=1}^n$ and $\{y_i\}_{i=1}^n$. In this setup, a type-I error would be to reject the true null hypothesis. Figure \[fig:synthetic\](a) shows that the type-I error of all tests is upper-bounded by the prescribed significance level, for all $n\in \{25,50,100,500,1000,5000,10000\}$ and $100$ random repetitions. Thus, all tests control their type-I error as expected, up to random variations due to finite experiments.
#### Gaussian versus Student
We consider distinguishing between samples drawn from a Normal distribution and samples drawn from a Student’s t-distribution with $\nu$ degrees of freedom. We shift and scale both samples to exhibit zero-mean and unit-variance. Since the Student’s t distribution approaches the Normal distribution as $\nu$ increases, a two-sample test must focus on the peaks of the distributions to distinguish one from another. Figure \[fig:synthetic\](b,c) shows the percentage of type-II errors made by all tests as we vary separately $n$ and $\nu$, over $100$ trials (random samples). We set $n = 2000$ when $\nu$ varies, and let $\nu =
3$ when $n$ varies. The Wilcoxon-Mann-Whitney exhibits the worst performance, as expected (since the ranks mean of the Gaussian and Student’s t distributions coincide) in this experiment. The best performing method is the the one-dimensional Kuiper test, followed closely by the multi-dimensional tests C2ST-NN and ME.
#### Independence testing on sinusoids
For completeness, we showcase the use two-sample tests to measure statistical dependence. This can be done, as described in Section \[sec:two\], by performing a two-sample test between the observed data $\{(x_i,y_i)\}_{i=1}^n$ and $\{(x_i,y_{\sigma(i)})\}_{i=1}^n$, where $\sigma$ is a random permutation. Since the distributions $P(X)P(Y)$ and $P(X,Y)$ are bivariate, only the C2ST-NN, C2ST-KNN, MMD, and ME tests compete in this task. We draw $(x_i,
y_i)$ according to the generative model $x_i \sim \mathcal{N}(0,1)$, $\epsilon_i \sim \mathcal{N}(0,\gamma^2)$, and $y_i \sim \cos(\delta x_i) +
\epsilon_i$. Here, $x_i$ are iid examples from the random variable $X$, and $y_i$ are iid examples from the random variable $Y$. Thus, the statistical dependence between $X$ and $Y$ weakens as we increase the frequency $\delta$ of the sinusoid, or increase the variance $\gamma^2$ of the additive noise. Figure \[fig:synthetic\](d,e,f) shows the percentage of type-II errors made by C2ST-NN, C2ST-KNN, MMD, and ME as we vary separately $n$, $\delta$, and $\gamma$ over $100$ trials. We let $n = 2000$, $\delta = 1$, $\gamma = 0.25$ when fixed. Figure \[fig:synthetic\](d,e,f) reveals that among all tests, C2ST-NN is the most efficient in terms of sample size, C2ST-KNN is the most robust with respect to high-frequency variations, and that C2ST-NN and ME are the most robust with respect to additive noise.
#### Distinguishing between NIPS articles
We consider the problem of distinguishing between some of the categories of the 5903 articles published in the Neural Information Processing Systems (NIPS) conference from 1988 to 2015, as discussed in @metests. We consider articles on Bayesian inference (Bayes), neuroscience (Neuro), deep learning (Deep), and statistical learning theory (Learn). Table \[table:nips\] shows the type-I errors (Bayes-Bayes row) and powers (rest of rows) for the tests reported in [@metests], together with C2ST-NN, at a significance level $\alpha =
0.01$, when averaged over $500$ trials. In these experiments, C2ST-NN achieves maximum power, while upper-bounding its type-I error by $\alpha$.
#### Distinguishing between facial expressions
Finally, we apply C2ST-NN to the problem of distinguishing between positive (happy, neutral, surprised) and negative (afraid, angry, disgusted) facial expressions from the Karolinska Directed Emotional Faces dataset, as discussed in [@metests]. See the fourth plot of Figure \[fig:interpretability\], first two-rows, for one example of each of these six emotions. Table \[table:expressions\] shows the type-I errors ($\pm$ vs $\pm$ row) and the powers ($+$ vs $-$ row) for the tests reported in [@metests], together with C2ST-NN, at $\alpha = 0.01$, averaged over $500$ trials. C2ST-NN achieves a near-optimal power, only marginally behind the perfect results of SCF-full and MMD-quad.
\[table:nips\]
\[table:expressions\]
Experiments on Generative Adversarial Network Evaluation {#sec:gmodels}
========================================================
Since effective generative models will produce examples barely distinguishable from real data, two-sample tests arise as a natural alternative to evaluate generative models. Particularly, our interest is to evaluate the sample quality of generative models with intractable likelihoods, such as GANs [@goodfellow2014generative]. GANs implement the adversarial game $$\label{eq:ganobj}
\min_{g} \max_{d} \operatorname*{\mathbb{E}}_{x \sim P(X)}\, \left[\log(d(x))\right] + \operatorname*{\mathbb{E}}_{z \sim P(Z)} \left[\log(1-d(g(z)))\right],$$ where $d(x)$ depicts the probability of the example $x$ following the data distribution $P(X)$ versus being synthesized by the generator. This is according to a trainable *discriminator* function $d$. In the adversarial game, the generator $g$ plays to fool the discriminator $d$ by transforming noise vectors $z \sim P(Z)$ into real-looking examples $g(z)$. On the opposite side, the discriminator plays to distinguish between real examples $x$ and synthesized examples $g(z)$. To approximate the solution to , alternate the optimization of the two losses [@goodfellow2014generative] given by $$\begin{aligned}
L_d(d) &= \mathbb{E}_{x}\, \left[\ell(d(x), 1)\right] + \mathbb{E}_z\, \left[ \ell(d(g(z)),0) \right],\nonumber\\
L_g(g) &= \mathbb{E}_{x}\, \left[\ell(d(x), 0)\right] + \mathbb{E}_{z}\,\left[ \ell(d(g(z)),1) \right].\label{eq:ganobj2}
\end{aligned}$$ Under the formalization , the adversarial game reduces to the sequential minimization of $L_d(d)$ and $L_g(g)$, and reveals the true goal of the discriminator: to be the C2ST that best distinguishes data examples $x \sim P$ and synthesized examples $\hat{x}
\sim \hat{P}$, where $\hat{P}$ is the probability distribution induced by sampling $z \sim P(Z)$ and computing $\hat{x} = g(z)$. The formalization unveils the existence of an arbitrary binary classification loss function $\ell$ (See Remark \[remark:loss\]), which in turn decides the divergence minimized between the real and fake data distributions [@nowozin2016f].
Unfortunately, the evaluation of the log-likelihood of a GANs is intractable. Therefore, we will employ a two-sample test to evaluate the quality of the fake examples $\hat{x} = g(z)$. In simple terms, evaluating a GAN in this manner amounts to withhold some real data from the training process, and use it later in a two-sample test against the same amount of synthesized data. When the two-sample test is a binary classifier (as discussed in Section \[sec:neural\]), this procedure is simply *training a fresh discriminator on a fresh set of data*. Since we train and test this *fresh* discriminator on held-out examples, it may differ from the discriminator trained along the GAN. In particular, the discriminator trained along with the GAN may have over-fitted to particular artifacts produced by the generator, thus becoming a poor C2ST.
We evaluate the use of two-sample tests for model selection in GANs. To this end, we train a number of DCGANs [@radford2015unsupervised] on the bedroom class of LSUN [@lsun] and the Labeled Faces in the Wild (LFW) dataset [@lfw]. We reused the Torch7 code of @radford2015unsupervised to train a set of DCGANs for $\{1,10,50,100,200\}$ epochs, where the generator and discriminator networks are convolutional neural networks [@lecun1998gradient] with $\{1,2,4,8\}\times\text{gf}$ and $\{1,2,4,8\}\times\text{df}$ filters per layer, respectively. We evaluate each DCGAN on $10,000$ held-out examples using the fastest multi-dimensional two-sample tests: MMD, C2ST-NN, and C2ST-KNN.
Our first experiments revealed an interesting result. When performing two-sample tests directly on pixels, all tests obtain near-perfect test accuracy when distinguishing between real and synthesized (fake) examples. Such near-perfect accuracy happens consistently across DCGANs, regardless of the visual quality of their examples. This is because, albeit visually appealing, the fake examples contain checkerboard-like artifacts that are sufficient for the tests to consistently differentiate between real and fake examples. @odena2016deconvolution discovered this phenomenon concurrently with us.
On a second series of experiments, we featurize all images (both real and fake) using a deep convolutional ResNet [@he2015deep] pre-trained on ImageNet, a large dataset of natural images [@imagenet]. In particular, we use the `resnet-34` model from @resnet. Reusing a model pre-trained on natural images ensures that the test will distinguish between real and fake examples based only on natural image statistics, such as Gabor filters, edge detectors, and so on. Such a strategy is similar to perceptual losses [@perceptual] and inception scores [@salimans16]. In short, in order to evaluate how natural the images synthesized by a DCGAN look, one must employ a “natural discriminator”. Table \[table:littlegans\] shows three GANs producing poor samples and three GANs producing good samples for the LSUN and LFW datasets, according to the MMD, C2ST-KNN, C2ST-NN tests on top of ResNet features. See Appendix \[sec:app\] for the full list of results. Although it is challenging to provide with an objective evaluation of our results, we believe that the rankings provided by two-sample tests could serve for efficient early stopping and model selection.
Evaluating generative models is a delicate issue [@theis2015note], but two-sample tests may offer some guidance. In particular, good (non-overfitting) generative models should produce similar two-sample test statistics when comparing their generated samples to both the train-set and the test-set samples. [^3] As a general recipe, prefer *the smallest* (in number of parameters) generative model that achieves the *same and small* two-sample test statistic when comparing their generated samples to both the train-set and test-set samples.
We have seen that GANs of different quality may lead to the same (perfect) C2ST statistic. To allow a finer comparison between generative models, we recommend implementing C2ST using a margin classifier with finite norm, or using as statistic the whole area under the C2ST training curve (on train-set or test-set samples).
Experiments on Interpretability
-------------------------------
[0.26]{} {width="\textwidth"}
[0.26]{} {width="\textwidth"}
[0.26]{}
[0.18]{} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"}\
{width="32.00000%"} {width="32.00000%"} {width="32.00000%"}\
{width="32.00000%"} {width="32.00000%"} {width="32.00000%"}
\[fig:interpretability\]
We illustrate the interpretability power of C2ST. First, the predictive uncertainty of C2ST sheds light on where the two samples under consideration agree or differ. In the context of GANs, this interpretability is useful to locate captured or dropped modes. In the first plot of Figure \[fig:interpretability\], a C2ST-NN separates two bivariate Gaussian distributions with different means. When performing this separation, the C2ST-NN provides an explicit decision boundary that illustrates *where* the two distributions separate from each other. In the second plot of Figure \[fig:interpretability\], a C2ST-NN separates a Gaussian distribution from a Student’s t distribution with $\nu = 3$, after scaling both to zero-mean and unit-variance. The plot reveals that the peaks of the distributions are their most differentiating feature. Finally, the third plot of Figure \[fig:interpretability\] displays, for the LFW and LSUN datasets, five examples classified as real with high uncertainty (first row, better looking examples), and five examples classified as fake with high certainty (second row, worse looking examples).
Second, the features learnt by the classifier of a C2ST are also a mechanism to understand the differences between the two samples under study. The third plot of Figure \[fig:interpretability\] shows six examples from the Karolinska Directed Emotional Faces dataset, analyzed in Section \[sec:smallexps\]. In that same figure, we arrange the weights of the first linear layer of C2ST-NN into the feature most activated at positive examples (bottom left, positive facial expressions), the feature most activated at negative examples (bottom middle, negative facial expressions), and the “discriminative feature”, obtained by substracting these two features (bottom right). The discriminative feature of C2ST-NN agrees with the one found by [@metests]: positive and negative facial expressions are best distinguished at the eyebrows, smile lines, and lips. A similar analysis [@metests] on the C2ST-NN features in the NIPS article classification problem (Section \[sec:smallexps\]) reveals that the features most activated for the “statistical learning theory” category are those associated to the words *inequ*, *tight*, *power*, *sign*, *hypothesi*, *norm*, *hilbert*. The features most activated for the “Bayesian inference” category are those associated to the words *infer*, *markov*, *graphic*, *conjug*, *carlo*, *automat*, *laplac*.
Experiments on Conditional GANs for Causal Discovery {#sec:causality}
====================================================
In causal discovery, we study the causal structure underlying a set of $d$ random variables $X_1, \ldots, X_d$. In particular, we assume that the random variables $X_1, \ldots, X_d$ share a causal structure described by a collection of Structural Equations, or SEs [@pearl2009causality]. More specifically, we assume that the random variable $X_i$ takes values as described by the SE $X_i = g_i(\text{Pa}(X_i, \mathcal{G}), N_i)$, for all $i=1,\ldots,d$. In the previous, $\mathcal{G}$ is a Directed Acyclic Graph (DAG) with vertices associated to each of the random variables $X_1,
\ldots, X_d$. Also in the same equation, $\text{Pa}(X_i, \mathcal{G})$ denotes the set of random variables which are parents of $X_i$ in the graph $\mathcal{G}$, and $N_i$ is an independent noise random variable that follows the probability distribution $P(N_i)$. Then, we say that $X_i \to X_j$ if $X_i \in \text{Pa}(X_j)$, since a change in $X_i$ will *cause* a change in $X_j$, as described by the $i$-th SE.
The goal of causal discovery is to infer the causal graph $\mathcal{G}$ given a sample from $P(X_1, \ldots, X_d)$. For the sake of simplicity, we focus on the discovery of causal relations between two random variables, denoted by $X$ and $Y$. That is, given the sample $\mathcal{D} = \{(x_i,y_i)\}_{i=1}^n \sim P^n(X,Y)$, our goal is to conclude whether “$X$ causes $Y$”, or “$Y$ causes $X$”. We call this problem *cause-effect discovery* [@mooij2014distinguishing]. In the case where $X \to Y$, we can write the cause-effect relationship as: $$\begin{aligned}
x \sim P(X),\quad
n \sim P(N),\quad
y \leftarrow g(x,n).\label{eq:mechanism}
\end{aligned}$$ The current state-of-the-art in the cause-effect discovery is the family of Additive Noise Models, or ANM [@mooij2014distinguishing]. These methods assume that the SE allow the expression $y \leftarrow g(x)+n$, and exploit the independence assumption between the cause random variable $X$ and the noise random variable $N$ to analyze the distribution of nonlinear regression residuals, in both causal directions.
Unfortunately, assuming independent additive noise is often too simplistic (for instance, the noise could be heteroskedastic or multiplicative). Because of this reason, we propose to use Conditional Generative Adversarial Networks, or CGANs [@conditionalGans] to address the problem of cause-effect discovery. Our motivation is the shocking resemblance between the generator of a CGAN and the SE : the random variable $X$ is the conditioning variable input to the generator, the random variable $N$ is the noise variable input to the generator, and the random variable $Y$ is the variable synthesized by the generator. Furthermore, CGANs respect the independence between the cause $X$ and the noise $N$ by construction, since $n \sim P(N)$ is independent from all other variables. This way, CGANs bypass the additive noise assumption naturally, and allow arbitrary interactions $g(X,N)$ between the cause variable $X$ and the noise variable $N$.
To implement our cause-effect inference algorithm in practice, recall that training a CGAN from $X$ to $Y$ minimizes the two following objectives in alternation: $$\begin{aligned}
L_d(d) &= \mathbb{E}_{x,y}\, \left[\ell(d(x,y), 1)\right] + \mathbb{E}_{x,z}\, \left[\ell(d(x,g(x,z)), 0)\right],\nonumber\\
L_g(g) &= \mathbb{E}_{x,y}\, \left[\ell(d(x,y), 0)\right] + \mathbb{E}_{x,z}\, \left[\ell(d(x,g(x,z)),1)\right].\nonumber
\end{aligned}$$ Our recipe for cause-effect is to learn two CGANs: one with a generator $g_y$ from $X$ to $Y$ to synthesize the dataset $\mathcal{D}_{X\to
Y} = \{(x_i, g_y(x_i,z_i))\}_{i=1}^n$, and one with a generator $g_x$ from $Y$ to $X$ to synthesize the dataset $\mathcal{D}_{X\leftarrow Y} =
\{(g_x(y_i,z_i),y_i)\}_{i=1}^n$. Then, we prefer the causal direction $X \to Y$ if the two-sample test statistic between the real sample $\mathcal{D}$ and $\mathcal{D}_{X\to Y}$ is smaller than the one between $\mathcal{D}$ and $\mathcal{D}_{Y \to X}$. Thus, our method is Occam’s razor at play: declare the simplest direction (in terms of conditional generative modeling) as the true causal direction.
\[table:tuebingen\]
Table \[table:tuebingen\] summarizes the performance of this procedure when applied to the $99$ Tübingen cause-effect pairs dataset, version August 2016 [@mooij2014distinguishing]. RCC is the Randomized Causation Coefficient of [@dlp-clt]. The Ensemble-CGAN-C2ST trains 100 CGANs, and decides the causal direction by comparing the top generator obtained in each causal direction, as told by C2ST-KNN. The need to ensemble is a remainder of the unstable behaviour of generative adversarial training, but also highlights the promise of such models for causal discovery.
Conclusion
==========
Our *take-home message* is that modern binary classifiers can be easily turned into powerful two-sample tests. We have shown that these *classifier two-sample tests* set a new state-of-the-art in performance, and enjoy unique attractive properties: they are easy to implement, learn a representation of the data on the fly, have simple asymptotic distributions, and allow different ways to interpret how the two samples under study differ. Looking into the future, the use of binary classifiers as two-sample tests provides a flexible and scalable approach for the evaluation and comparison of generative models (such as GANs), and opens the door to novel applications of these methods, such as causal discovery.
Results on Evaluation of Generative Adversarial Networks {#sec:app}
========================================================
\[table:bedrooms\]
\[table:faces\]
Proof of Theorem \[thm:thm1\] {#app:proof}
=============================
Our statistic is a random variable $T \sim
\mathcal{N}\left(\frac{1}{2}, \frac{1}{4 n_\text{te}}\right)$ under the null hypothesis, and $T \sim \mathcal{N}\left(\frac{1}{2} + \epsilon,
n_\text{te}^{-1}\left(\frac{1}{4}-\epsilon^2\right)\right)$ under the alternative hypothesis. Furthermore, at a significance level $\alpha$, the threshold of our statistic is $z_\alpha = \frac{1}{2} +
\frac{\Phi^{-1}(1-\alpha)}{\sqrt{4n_\text{te}}}$; under this threshold we would accept the null hypothesis. Then, the probability of making a type-II error is $$\begin{aligned}
\mathbb{P}_{T\sim \mathcal{N}\left(\frac{1}{2}+\epsilon,
\frac{\frac{1}{4}-\epsilon^2}{n_\text{te}}\right)}\left(T < z_\alpha \right) &=
\mathbb{P}_{T'\sim \mathcal{N}\left(0,
\frac{\frac{1}{4}-\epsilon^2}{n_\text{te}}\right)}\left(T' <
\frac{\Phi^{-1}(1-\alpha)}{\sqrt{4n_\text{te}}}-\epsilon\right)\\
&=\Phi\left(\sqrt{\frac{n_\text{te}}{\frac{1}{4}-\epsilon^2}}
\left(
\frac{\Phi^{-1}(1-\alpha)}{\sqrt{4n_\text{te}}}-\epsilon\right)\right)\\
&=\Phi
\left(
\frac{\Phi^{-1}(1-\alpha)/2 - \epsilon\sqrt{n_\text{te}}}{\sqrt{\frac{1}{4}-\epsilon^2}}
\right).
\end{aligned}$$ Therefore, the power of the test is $$\pi(\alpha, n_\text{te}, \epsilon) = 1-\Phi
\left(
\frac{\Phi^{-1}(1-\alpha)/2 - \epsilon\sqrt{n_\text{te}}}{\sqrt{\frac{1}{4}-\epsilon^2}}
\right) =
\Phi
\left(
\frac{\epsilon\sqrt{n_\text{te}}-\Phi^{-1}(1-\alpha)/2}{\sqrt{\frac{1}{4}-\epsilon^2}}
\right),$$ which concludes the proof.
Acknowledgements
================
We are thankful to L. Bottou, B. Graham, D. Kiela, M. Rojas-Carulla, I. Tolstikhin, and M. Tygert for their help in improving the quality of this manuscript. This work was partly supported by ERC grant LEAP (no. 336845) and CIFAR Learning in Machines & Brains program.
[^1]: For clarity, we abuse statistical language and write “accept” to mean “fail to reject”.
[^2]: For a related discussion on this issue, we recommend the insightful comment by Arthur Gretton and Wittawat Jitkrittum, available at <https://openreview.net/forum?id=SJkXfE5xx>.
[^3]: As discussed with Arthur Gretton, if the generative model memorizes the train-set samples, a sufficiently large set of generated samples would reveal such memorization to the two-sample test. This is because some unique samples would appear multiple times in the set of generated samples, but not in the test-set of samples.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We report the observation of plasma oscillations in an ultracold neutral plasma. With this collective mode we probe the electron density distribution and study the expansion of the plasma as a function of time. For classical plasma conditions, [*i.e.*]{} weak Coulomb coupling, the expansion is dominated by the pressure of the electron gas and is described by a hydrodynamic model. Discrepancies between the model and observations at low temperature and high density may be due to strong coupling of the electrons.'
address: |
National Institute of Standards and Technology, Gaithersburg, MD 20899-8424\
(Accepted *[Phys. Rev. Lett.]{})*
author:
- 'S. Kulin, T. C. Killian, S. D. Bergeson[@scott], and S. L. Rolston'
title: Plasma Oscillations and Expansion of an Ultracold Neutral Plasma
---
One of the most interesting features of neutral plasmas is the rich assortment of collective modes that they support. The most common of these is the plasma oscillation [@tla29], in which electrons oscillate around their equilibrium positions and ions are essentially stationary. This mode is a valuable probe of ionized gases because the oscillation frequency depends solely on the electron density.
In an ultracold neutral plasma as reported in [@kkb99], the density is nonuniform and changing in time. A diagnostic of the density is thus necessary for a variety of experiments, such as determination of the three-body recombination rate at ultralow temperature [@hah97], and observation of the effects of strong Coulomb coupling [@ichimaru82] in a two-component system. A density probe would also aid in the study of the evolution of a dense gas of cold Rydberg atoms to a plasma [@nnp99], which may be an analog of the Mott insulator-conductor phase transition [@mott].
In this work we excite plasma oscillations in an ultracold neutral plasma by applying a radio frequency (rf) electric field. The oscillations are used to map the plasma density distribution and reveal the particle dynamics and energy flow during the expansion of the ionized gas.
The creation of an ultracold plasma has been described in [@kkb99]. A few million metastable xenon atoms are laser cooled to approximately $10\,\mu$K. The peak density is about $2\times
10^{10}$cm$^{-3}$ and the spatial distribution of the cloud is Gaussian with an rms radius $\sigma \approx 220\,\mu$m. These parameters are determined with resonant laser absorption imaging [@matt]. To produce the plasma, up to $25$% of the atoms are photoionized in a two-photon excitation. Light for this process is provided by a Ti:sapphire laser at $882$nm and a pulsed dye laser at $514$nm ($ 10\,$ns pulse length). Because of the small electron-ion mass ratio, the resulting electrons have an initial kinetic energy ($E_{e}$) approximately equal to the difference between the photon energy and the ionization potential. In this study we vary $E_{e}/k_{B}$ between 1 and 1000K. The initial kinetic energy of the ions varies between 10$\mu$K and 4mK.
For detection of charged particles, a small DC field (about $1$mV/cm) directs electrons to a single channel electron multiplier and ions to a multichannel plate detector. The amplitude of the rf field that excites plasma oscillations, $F$, varies between $0.2-20\,$mV/cm rms. All electric fields are applied to the plasma with grids located above and below the laser-atom interaction region.
In the absence of a magnetic field, the frequency of plasma oscillations is given by $f_{e}=(1/2\pi)\sqrt{e^{2}n_{e}/\epsilon_{0}m_{e}}$ [@tla29]. Here, $e$ is the elementary charge, $n_{e}$ is the electron density, $\epsilon_{0}$ is the permittivity of vacuum, and $m_{e}$ is the electron mass. This relation is most often derived for an infinite homogeneous plasma, but it is also valid in our inhomogeneous system for modes which are localized in regions of near resonant density. Corrections to $f_{e}$ due to finite temperature [@bgr49] depend on the wavelength of the collective oscillation, which is difficult to accurately estimate. Such corrections are not expected to be large and will be neglected. We observe plasma oscillations with frequencies from $1$ to $250$MHz. This corresponds to resonant electron densities, $n_{r}$, between $1\times10^{4}$cm$^{-3}$ and $8 \times10^{8}$cm$^{-3}$. The oscillation frequency is sensitive only to $n_{e}$, but, as explained in [@kkb99], the core of the plasma is neutral. This implies that plasma oscillations measure electron and ion densities in this region ($n_{e}=n_{i}\equiv n$).
Figure \[compositepaper\]a shows electron signals from an ultracold neutral plasma created by photoionization at time $t=0$. Some electrons leave the sample and arrive at the detector at about $1\,\mu$s, producing the first peak in the signal. The resulting excess positive charge in the plasma creates a Coulomb potential well that traps the remaining electrons [@kkb99]. In the work reported here, typically $90-99$% of the electrons are trapped. Debye shielding maintains local neutrality inside a radius $r_{e}$ beyond which the electron density drops to zero on a length scale equal to $\lambda_D$. The value of $r_{e}$ depends on the fraction of electrons that has escaped, and $\lambda_D$ is the Debye screening length, $\lambda_D =\sqrt{{\epsilon_0 k_B T_{e}}/{e^2 n_{e}}}$, where $T_{e}$ is the electron temperature. For our conditions $r_{e}\,$$\,2 \sigma$, and $\lambda_D \ll \sigma$. As the plasma expands, the depth of the Coulomb well decreases, allowing the remaining electrons to leave the trap. This produces the broad peak at $\approx 25\,\mu$s.
In the presence of an rf field an additional peak appears in the electron signal (Fig. \[compositepaper\]a). We understand the generation of this peak as follows: The applied rf field excites plasma oscillations only where the frequency is resonant. Energy is thus pumped into the plasma in the shell with the appropriate electron density ($n=n_{r}$). The amplitude of the collective electron motion is much less than $\sigma$, but the acquired energy is collisionally redistributed among all the electrons within $10-1000$ns [@spitzer], raising the electron temperature. This increases the evaporation rate of electrons out of the Coulomb well, which produces the plasma oscillation response on the electron signal.
The resonant response at a given time, $S(t)$, is proportional to the number of electrons in the region where the density equals $n_{r}$. If we make a simple local density approximation and neglect decoherence of the oscillations, $S(t) \propto F^{2}\int d^{3}r\, n({\bf r},t)\,\delta[n({\bf r},t)-n_{r}]$. The width in time of the observed signal (Fig. \[compositepaper\]a) reflects the density distribution of the sample [@densnote]. At early times when the density is higher than $n_{r}$ almost everywhere, $S(t)$ is negligibly small. As the cloud expands and the density decreases, the response grows because the fraction of the plasma which is in resonance increases. The peak of the response appears approximately when the average density, $\bar n$, becomes resonant with the rf field. $S(t)$ vanishes when the peak density is less than $n_{r}$.
The resonant response arrives later for lower frequency (Fig. \[compositepaper\]b) as expected because $\bar n$ decreases in time. Assuming that the plasma density profile remains Gaussian during the expansion, $S(t)$ can be evaluated and its amplitude scales as $F^{2}/n_{r}$. In Fig. \[compositepaper\]b the data have been normalized by this factor and the resulting amplitudes are similar for all conditions.
By equating $\bar n$ to $n_{r}$ when the response peak arrives, we can plot the average plasma density as a function of time (Fig. \[paperalltemp\]). The data are well described by a self similar expansion of a Gaussian cloud, $\bar n= N/[4\pi (\sigma_0^{2}+v_{0}^{2}t^{2})]^{3/2}$, where $\sigma_{0}$ is the initial rms radius and $v_{0}$ is the rms radial velocity at long times. $N$ is determined independently by counting the number of neutral atoms with and without photoionization. The extracted values of $\sigma_0$ are equal to the size of the initial atom cloud. In such an expansion, the average kinetic energy per particle is $3mv_{0}^{2}/2$.
Figure \[expansion\] shows the dependence of $v_{0}$ on density and initial electron energy. We first discuss data with $E_{e}\ge 70\,$K, for which the expansion velocities approximately follow $v_{0}=\sqrt{E_{e}/\alpha m_{i}}$, where $m_{i}$ is the ion mass and $\alpha=1.7$ is a fit parameter. For the plasma to expand at this rate, the ions must acquire, on average, a velocity characteristic of the electron energy. This is much greater than the initial ion thermal velocity. Electron-ion equipartition of energy would yield $v_{0}=\sqrt{E_{e}/3m_{i}}$, close to the observed value. However, due to the large electron-ion mass difference, this thermalization requires milliseconds [@spitzer]. The observed expansion, in contrast, occurs on a time scale of tens of microseconds. One might expect the expansion to be dominated by the Coulomb energy arising from the slight charge imbalance of the plasma, but this energy is about an order of magnitude less than the observed expansion energy. Also, by Gauss’ law, it would only be important in the expansion of the non-neutral outer shell of the plasma. The oscillation probe provides information only on the neutral core because it relies on the presence of electrons.
A hydrodynamic model [@gru95], which describes the plasma on length scales larger than $\lambda_D$, shows that the expansion is driven by the pressure of the electron gas. The pressure is exerted on the ions by outward-moving electrons that are stopped and accelerated inward in the trap. For the hydrodynamic calculation, ions and electrons are treated as fluids with local densities $n_{a}({\bf r})$ and average velocities ${\bf u}_{a}({\bf r})=\langle{\bf v}_{a}({\bf r})\rangle$. Here, $a$ refers to either electrons or ions, and $\langle \cdots\rangle$ denotes a local ensemble average. Particle and momentum conservation lead to the momentum balance equations $$m_{a}n_{a}\left[ {\partial{\bf u}_{a} \over \partial t}
+({\bf u}_{a}\cdot \nabla){\bf
u}_{a}\right] =-{\bf \nabla }(n_{a}k_{B}T_{a})+ {\bf R}_{ab}.\nonumber$$ Here $n_{a}k_{B}T_{a}$ represents a scalar pressure [@gru95]. The ion and electron equations are coupled by ${\bf R}_{ab}$, which is the rate of momentum exchange between species $a$ and $b$. The exact form of this term is unimportant for this study, but ${\bf R}_{ab}=-{\bf R}_{ba}$. Plasma hydrodynamic equations typically have electric and magnetic field terms, but applied and internally generated fields are negligible when describing the expansion.
We can make a few simplifying approximations that are valid before the system has significantly expanded. The directed motion is negligible, so we set ${\bf u}_{a}\approx 0$ everywhere. Because $n_{e}\approx n_{i}= n$, $\partial{\bf u}_{e}/ \partial t\approx
\partial{\bf u}_{i}/ \partial t$. Due to the small electron mass, the rate of increase of average electron momentum is negligible compared to that of the ions. The electron momentum balance equation then yields ${\bf \nabla }(nk_{B}T_{e})\approx {\bf R}_{ei}$, which describes a balance between the pressure of the electron gas and collisional interactions. This is the hydrodynamic depiction of the trapping of electrons by the ions.
In the ion momentum balance equation, we eliminate ${\bf
R}_{ie}$ using the electron equation, and we drop the pressure term because the ion thermal motion is negligible. Thus $m_{i}n\partial{\bf u}_{i} / \partial t \approx -{\bf \nabla }
(nk_{B}T_{e})$, which shows that the pressure of the electron gas drives the expansion[@expnote]. This result implies that the ions acquire a velocity of order $\sqrt{k_{B}T_{e}/m_{i}}$, which is in qualitative agreement with the high $E_{e}$ data of Fig. \[expansion\]. To calculate the expansion velocity more quantitatively, one must consider that as electrons move in the expanding trap, they perform work on the ions and cool adiabatically. The thermodynamics of this process [@mmf93] is beyond the scope of this study.
The data in Fig. \[expansion\] indicate that about $90$% of the initial kinetic energy of the electrons is transferred to the ions’ kinetic energy, $3m_{i}
v^{2}_{0}/2=3E_{e}/2\alpha$. This does not imply that the [*temperature*]{} of the ions becomes comparable to $E_{e}/ k_{B}$ in this process. For the ions, ${\bf u}_{i}$ increases, but $m_{i}\langle|{\bf v}_{i}-{\bf u}_{i}|^{2}\rangle$, which measures random thermal motion and thus temperature, is expected to remain small. This follows from slow ion-electron thermalization [@spitzer] and correlation between position and velocity during the expansion [@ows99].
We now turn our attention to systems with $E_{e}<70\,$K (Fig. \[expansion\]). They expand faster than expected from an extrapolation of $v_{0}=\sqrt{E_{e}/\alpha m_{i}}$, and thus do not even qualitatively follow the hydrodynamic model. A relative measure of the deviation is $(m_{i}v_{0}^{2}-E_{e}/\alpha)/(E_{e}/\alpha)$. Figure \[expheatstudy\] shows that the relative deviation increases with increasing electron Coulomb coupling parameter [@ichimaru82], $
\Gamma_{e} = (e^{2} / 4\pi \varepsilon_{0}\,a) / k_{B}T_{e}.
$ Here, $a=(4\pi n/3)^{-1/3}$ is the Wigner-Seitz radius, $n$ is the peak density at $t=0$, and the temperature is calculated by $3k_{B}T_{e}/2=E_{e}$.
The fact that the relative deviation depends only on $\Gamma_{e}$, and that it becomes significant as $\Gamma_{e}$ approaches 1, suggests that we are observing the effects of strong coupling of the electrons [@ioncoupling]. The hydrodynamic model of the plasma is only valid when $\Gamma _{e} \ll 1$. When $\Gamma _{e}\, $$\, 1$, electron and ion spatial distributions show short range correlated fluctuations that are not accounted for in a smooth fluid description [@ich92]. Correlations between the ion and electron positions would provide the excess kinetic energy observed in the expansion by lowering the potential energy of the plasma. This satisfies overall energy conservation and it may also explain the systematically poor fits of the data for high $\Gamma_{e}$ (See Fig.\[paperalltemp\]).
Strong coupling is also predicted to alter the relation for the frequency $f_{e}$ [@kgm93], with which we extract the plasma density, size, and expansion velocity. The trend of this effect agrees qualitatively with the observed deviation, but knowledge of the wavelength of the collective oscillation is needed for a quantitative comparison.
Other possible explanations for the deviation are related to how the ultracold plasma is created. The $10\,$ns duration of the photoionization pulse is long compared to the time required for electrons to move an interparticle spacing. Photoionization late in the pulse thus occurs in the presence of free charges, which will depress the atomic ionization threshold by $\Delta E_{IP}\approx \frac{1}{2} k_{B}T_{e}
\{[(3\Gamma_{e})^{3/2}+1]^{2/3}-1\}$ [@spy66]. This effect might increase the electron kinetic energy by $\Delta
E_{IP}$ above what has been assumed. However, as shown in Fig. \[expheatstudy\], the calculated $\Delta E_{IP}$ is about an order of magnitude smaller than the observed effect. The random potential energy of charged particles when they are created may also yield a greater electron energy than $E_{e}$ [@gey].
High $\Gamma_{e}$ (high density and low temperature) conditions are desirable for studying the three-body recombination rate in an ultracold plasma. The theory [@mke69] for this process was developed for high temperature, and is expected to break down in the ultracold regime [@hah97]. Measuring or setting an upper limit for the recombination rate is not possible until the dynamics of high $\Gamma_{e}$ systems is understood. We are currently studying this problem with molecular dynamics calculations.
We have shown that plasma oscillations are a valuable probe of the ion and electron density in an ultracold neutral plasma. This tool will facilitate future experimental studies of this novel system, such as the search for other collective modes in the plasma and further investigation of the effects of correlations due to strong coupling.
We thank Lee Collins for helpful discussions and Michael Lim for assistance with data analysis. S. Kulin acknowledges funding from the Alexander-von-Humboldt foundation. This work was funded by the ONR.
Present address: Department of Physics and Astronomy, Brigham Young University, Provo UT 84602-4640.
L. Tonks and I. Langmuir, Phys. Rev. [**33**]{}, 195 (1929).
T. C. Killian, S. Kulin, S. D. Bergeson, L. A. Orozco, C. Orzel, and S. L. Rolston, Phys. Rev. Lett. [**83**]{}, 4776 (1999).
Y. Hahn, Phys. Lett. A [**231**]{}, 82 (1997).
S. Ichimaru, Rev. Mod. Phys. [**54**]{}, 1017 (1982).
S. Kulin, T. C. Killian, S. D. Bergeson, L. A. Orozco, C. Orzel, and S. L. Rolston, in [*Non-Neutral Plasma Physics III*]{}, edited by J. J. Bollinger, R. L . Spencer, and R. C. Davidson, (AIP, New York, 1999), p. 367.
, edited by P. P. Edwards and C. N. R. Rao, (Taylor & Francis Ltd., London, 1995); G. Vitrant, J. M. Raimond, M. Gross, and S. Haroche, J. Phys. B: At. Mol. Phys. [**15**]{}, L49 (1982).
M. Walhout, H. J. L. Megens, A. Witte, and S. L. Rolston, Phys. Rev. A [**48**]{}, R879 (1993).
D. Bohm and E. P. Gross, Phys. Rev. [**75**]{}, 1851 (1949).
L. Spitzer, Jr., [*Physics of Fully Ionized Gases*]{} (John Wiley & Sons, Inc., New York, 1962), chap. 5.
By applying an rf pulse, we found the response time for excitation and detection of plasma oscillations to be about $1\,\mu$s. This is short compared to the width of $S(t)$ and is probably set by the electron thermalization rate and time-of-flight to the detector.
R. J. Goldston and P. H. Rutherford, [*Introduction to Plasma Physics*]{} (Institute of Physics, Philadelphia, 1995), chap. 6.
This equation would also describe the ballistic expansion of a single-component cloud of noninteracting particles at temperature $T_{e}$. It also preserves a Gaussian density distribution, which supports our assumption of such a distribution in the data analysis.
G. Manfredi, S. Mola, and M. R. Feix, Phys. Fluids B [**5**]{}, 388 (1993).
C. Orzel, M. Walhout, U. Sterr, P. S. Julienne, and S. L. Rolston, Phys. Rev. A [**59**]{}, 1926 (1999).
We expect that the ions are also strongly coupled, although we have no direct evidence for this. The ion-ion thermalization time, equal to that of the electrons, is short, and the low initial ion kinetic energy yields $\Gamma_{i}>>1$. Strong coupling of ions is not expected to affect the experiments discussed here.
S. Ichimaru, [*Statistical Plasma Physics, Volume I*]{} (Addison-Wesley Publishing Co., Reading, MA, 1992), chap. 7.
G. Kalman, K. I. Golden, and M. Minella, in [*Strongly Coupled Plasma Physics*]{}, edited by H. M. Van Horn, and S. Ichimaru, (University of Rochester Press, Rochester, 1993), p. 323.
J. C. Stewart and K. D. Pyatt, Jr., Astrophys. J. [**144**]{}, 1203 (1966); M. Nantel, G. Ma, S. Gu, C. Y. Côté, J. Itatani, and D. Umstadter, Phys. Rev. Lett. [**80**]{}, 4442 (1998).
E. Eyler and P. Gould, private communication.
P. Mansbach, and J. Keck, Phys. Rev. [**181**]{}, 275 (1969).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We study the nonresonant oscillations between left-handed electron neutrinos $\nu_e$ and nonthermalized sterile neutrinos $\nu_s$ in the early Universe plasma. The case when $\nu_s$ do not thermalize till 2 MeV and the oscillations become effective after $\nu_e$ decoupling is discussed. As far as for this model the rates of expansion of the Universe, neutrino oscillations and neutrino interactions with the medium may be comparable, we have analysed the kinetic equations for neutrino density matrix, accounting [*simultaneously*]{} for all processes. The evolution of neutrino ensembles was described numerically by integrating the kinetic equations for the neutrino density matrix in [*momentum*]{} space for small mass differences $\delta m^2 \le 10^{-7}$ eV$^2$. This approach allowed us to study precisely the evolution of the neutrino number densities, energy spectrum distortion and the asymmetry between neutrinos and antineutrinos due to oscillations for each momentum mode.
We have provided a detail numerical study of the influence of the nonequilibrium $\nu_e \leftrightarrow \nu_s$ oscillations on the primordial production of $^4\! He$. The exact kinetic approach enabled us to calculate the effects of neutrino population depletion, the distortion of the neutrino spectrum and the generation of neutrino-antineutrino asymmetry on the kinetics of neutron-to-proton transitions during the primordial nucleosynthesis epoch and correspondingly on the cosmological $^4\! He$ production.
It was shown that the neutrino population depletion and spectrum distortion play an important role. The asymmetry effect, in case the lepton asymmetry is accepted initially equal to the baryon one, is proved to be negligible for the discussed range of $\delta m^2$. Constant helium contours in $\delta m^2$ - $\vartheta$ plane were calculated. Thanks to the exact kinetic approach more precise cosmological constraints on the mixing parameters were obtained.
address:
- |
Teoretisk Astrofysik Center\
Juliane Maries Vej 30, DK-2100, Copenhagen\
and\
Copenhagen University, Niels Bohr Institute, Blegdamsvej 17\
DK-2100, Copenhagen, Denmark [^1]
- |
Centre for Space Research and Technologies, Faculty of Physics,\
University of Sofia, 1164 Sofia, Bulgaria\
E-mail: $mih@phys.uni$-$sofia.bg$
author:
- 'D. P. Kirilova'
- 'M. V. Chizhov'
title: |
Cosmological Nucleosynthesis and Active-Sterile Neutrino Oscillations\
with Small Mass Differences: The Nonresonant Case
---
\#1\#2\#3\#4[[\#1]{}[**\#2**]{}, \#3 (\#4)]{}
\
PACS number(s): 05.70.Ln, 14.60.Pq, 26.35.+c
Introduction
============
The idea of Gamow, proposed in the 1930s and 1940s [@Gamow] about the production of elements through thermonuclear reactions in the hot ylem during the early stages of the Universe expansion, has been developed during the last 60 years into an elegant famous theory of cosmological nucleosynthesis, explaining quantitatively the inferred from observational data primordial abundances of the light elements [@CN]. Thanks to that good accordance between theory predictions and the observational facts, we nowadays believe to have understood well the physical conditions of the nucleosynthesis epoch. Still, the uncertainties of the primordial abundances values extracted from observations yet leave a room for physics beyond the standard model.
In this article we present a modification of the standard model of cosmological nucleosynthesis (CN) - [*CN with neutrino oscillations*]{}. Our aim is twofold: (1) to construct a modification of CN using a more precise kinetic approach to the problem of nonequilibrium neutrino oscillations and to illustrate the importance of such an exact approach, and (2) to determine the cosmologically allowed range for oscillation parameters from an accurate study of the oscillations effect on the primordial production of helium-4, thus helping clarify the mixing patterns of neutrinos.
The theme of neutrino oscillations is with us almost forty years, since the hypothesis for them was proposed by Pontecorvo [@pontec]. They were studied experimentally and theoretically and their cosmological and astrophysical effects have been considered in numerous publications [@REV] as far as their study helps to go deeper into the secrets of neutrino physics and neutrino mass pattern. Nowadays there are three main [*experimental indications*]{} that neutrinos oscillate, namely: the solar neutrino deficit [@SUN] (an indirect indication), the atmospheric neutrino anomaly [@ATM] (an indirect indication) and the LSND experiment results [@LSND] (a direct indication).
(a)[*Solar neutrino deficit:*]{} Already four experiments using different techniques have detected electron neutrinos from the Sun, at a level significantly lower than the predicted on the basis of the Standard Solar Model and the Standard Electroweak Theory. Moreover, there exists incompatibility between Chlorine and Kamiokande experiments data, as well as problems for predicted berilium and borum neutrinos in the gallium experiments [@SUN; @Bah]. Recently, it was realized that by changing the solar model it is hardly possible to solve these problems [@SUNMOD]. Therefore, it is interesting to find a solution beyond the Standard Electroweak Model. The only known natural solutions of that kind today are the energy dependent MSW neutrino transitions in the Sun interior [@MSW] and the “just-so” vacuum oscillations solutions, as well as the recently developed hybrid solutions of MSW transitions +vacuum oscillations type [@SUNTH].
(b)[*Atmospheric neutrino anomaly:*]{} Three of the five underground experiments on atmospheric neutrinos have observed disappearance of muon neutrinos [@ATM]. This is in contradiction with the theoretically expected flux of muon neutrinos from primary cosmic rays interacting in the atmosphere. A successful oscillatory solution of that problem requires large mixing and $\delta m^2$ of the order of $10^{-2}$ eV$^2$.
(c)[*Los Alamos LSND experiment*]{} claimed evidence for the oscillation of $\tilde{\nu}_\mu$ into $\tilde{\nu}_{e}$, with a maximal probability of the order of $0.45\times10^{-2}$. A complementary $\nu_{\mu}$ into $\nu_e$ oscillation search, with completely different systematics and backgrounds, also shows a signal, which indicates the same favoured region of oscillation parameters [@LSND].
There exists yet another observational suggestion for massive neutrinos and oscillations - the [*dark matter problem*]{}. Present models of structure formation in the Universe indicate that the observed hierarchy of structures is reproduced best by an admixture of about 20% hot dark matter to the cold one [@DM]. Light neutrinos with mass in eV range are the only particle dark matter candidates, that are actually known to exist and are the most plausible candidates provided by particle physics [@drees]. Actually, recent most popular hot plus cold dark matter models assume that two nearly degenerate massive neutrinos each with mass 2.4 eV play the role of the hot dark matter. This small mass value is now accessible only by oscillations.
However, in case we take seriously each of these experiments pointing to a neutrino anomaly and the neutrino oscillation solution to them, a fourth neutrino seems inevitable. In the case of only three species of light neutrinos with normal interactions and a see-saw hierarchy between the three masses, it is hardly possible to accommodate all the present data simultaneously. The successful attempts to reconcile the LSND results with neutrino oscillation solutions to the solar and atmospheric neutrino problems usually contain some “unnatural” features, like forth ultra-light sterile neutrino species, or inverted neutrino mass hierarchy [@ALL]. However, an additional light (with mass less than 1 MeV) flavour neutrino is forbidden both from cosmological considerations and the experiments on $Z$ decays at LEP [@NLEP]. Hence, it is reasonable to explore in more detail the possibility for an additional light [*sterile*]{} neutrino. Besides, GUT theories ($SO(10), E_6$, etc.) [@hr; @E6] and SUSY theories [@ps; @SUSY] predict the existence of a sterile neutrino. Moreover, recently models of singlet fermions, which explain the smallness of sterile neutrino mass and its mixing with the usual neutrino were proposed [@SUSY]. Therefore, it may be very useful to obtain more precise information about the cosmologically allowed range for the neutrino mixing parameters and thus present an additional independent test for the already discussed neutrino puzzles. Moreover, the very small values of mass differences, which can be explored by the oscillations cosmological effects (like the ones discussed in our model) are beyond the reach of present and near future experiments.
The present work is a step towards this: we suppose the existence of a sterile neutrino ($SU(2)$-singlet) $\nu_s$, and explore the cosmological effect of [*nonresonant neutrino oscillations*]{} $\nu_e \leftrightarrow \nu_s$ on the primordial nucleosynthesis, obtaining thus cosmological constraints on the neutrino mixing parameters. The nonresonant case in the Early Universe medium corresponds to the resonant case in the Sun, therefore, the obtained information is also of interest for the MSW solution to the solar neutrino problem. We discuss the special case of [*nonequilibrium oscillations*]{} between weak interacting and sterile neutrinos for [*small mass differences*]{} $\delta m^2$, as far as the case of large $\delta m^2$ is already sufficiently well studied [@bd1]-[@s]. Oscillations between active and sterile neutrinos, effective before neutrino freezing at 2 MeV, leading to $\nu_s$ thermalization before 2 MeV have been studied there. T.e. mainly the equilibrium oscillations were considered with rates of oscillations and neutrino weak interactions greater than the expansion rate. Here we discuss nonequilibrium oscillations between electron neutrinos $\nu_e$ and sterile neutrinos $\nu_s$ for the case when $\nu_s$ do not thermalize till $\nu_e$ decoupling at 2 MeV and oscillations become effective after $\nu_e$ decoupling. Such kind of active-sterile neutrino oscillations in vacuum was first precisely studied in [@dpk] using the accurate kinetic approach for the description of oscillating neutrinos, proposed in the pioneer work of Dolgov [@do]. However, the thermal background in the prenucleosynthesis epoch may strongly affect the propagation of neutrino [@medium; @the] and the account of the neutrino interactions with the primeval plasma is obligatory [@nr; @bd1; @ekm]. The precise kinetic consideration of oscillations in a medium was provided in [@our]. It was proved that in case when the Universe expansion, the oscillations and the neutrino interactions with the medium have comparable rates, their effects should be accounted for simultaneously, using the exact kinetic equations for the neutrino density matrix. Moreover, for the nonequilibrium oscillations energy distortion and asymmetry between neutrinos and antineutrinos may play a considerable role. As far as both neutrino collisions and active-sterile neutrino oscillations distort the initially equilibrium active neutrino momentum distribution, the momentum degree of freedom in the description of neutrino must be accounted for. Therefore, for the case of nonequilibrium oscillations the evolution of neutrino ensembles should be studied using the exact kinetic equations for the [*density matrix of neutrinos in momentum space*]{}. This approach allows an exact investigation of the different effects of neutrino oscillations [@dpk; @our; @sr]: depletion of the neutrino number densities, the energy distortion and the generation of asymmetry, for each separate momentum of the neutrino ensembles.
In the present work we expand the original investigation [@our] for the full parameter space of the nonequilibrium oscillations model for the nonresonant case. (The resonant case will be discussed in a following publication.) We have provided an exact kinetic analysis of the neutrino evolution by a numerical integration of the [*kinetic equations for the neutrino density matrix for each momentum mode*]{}. The kinetic equations are coupled nonlinear and, therefore, an analytic solution is hardly possible in the general case of oscillations in a medium. We have numerically described the evolution of the neutrino ensembles from the $\nu_e$ freezing at 2 MeV till the formation of helium-4.
We have calculated the production of helium-4 in a detail model of primordial nucleosynthesis, accounting for the direct kinetic effects of oscillations on the neutron-to-proton transitions. The oscillations effect on CN has been considered by many authors [@bd1]-[@s],[@lan], [@sv]-[@dhs]. However, mainly the excitation of an additional degree of freedom due to oscillations (i.e. an increase of the effective degrees of freedom $g$) and the corresponding increase of the Universe expansion rate $H \sim \sqrt g$, leading to an overproduction of helium-4 was discussed. The excluded regions for the neutrino mixing parameters were obtained from the requirement (based on the accordance between the theoretically predicted and the extracted from observations light elements abundances) that the neutrino types should be less than 3.4: $N_{\nu}<3.4$ [@bd1]-[@ekm]. A successful account for the electron neutrino depletion due to oscillations was first made in [@bd1] and [@ekm]. In the present work we have precisely calculated the influence of oscillations on the primordially produced helium-4 using the exact kinetic equations in momentum space for the neutron number density and the density matrix of neutrino, instead of their particle densities. The accurate numerical analysis of oscillations effect on helium production within a model of nucleosynthesis with oscillations, allowed us to account precisely for the following important effects of neutrino oscillations: neutrino population depletion, distortion of the neutrino spectrum and the generation of neutrino-antineutrino asymmetry. This enabled us to investigate the zone of very small neutrino mass differences up to $10^{-11}$ eV$^2$, which has not been reached before. As a result, we have obtained constant helium contours in the mass difference – mixing angle plane for the full range of the parameter values of our model. No matter what will be the preferred primordial helium value, favoured by future observations, it will be possible to obtain the excluded region of the mixing parameters using the results of this survey.
The paper is organized as follows. In Section II we present the model of nonequilibrium neutrino oscillations. In Section III an exact analysis of the neutrino evolution using kinetic equations for the neutrino density matrix for each momentum mode is provided. The main effects of nonequilibrium oscillations are revealed. In Section IV we investigate $\nu_e$ into $\nu_s$ oscillations effect on the primordial production of helium using a numerical nucleosynthesis code. We discuss the influence of nonequilibrium neutrino oscillations, namely electron neutrino depletion, neutrino spectrum distortion and the generation of neutrino-antineutrino asymmetry on the primordial yield of helium-4. The results and conclusions are presented in Section V.
Nonequilibrium neutrino oscillations - the model
================================================
The model of nonequilibrium oscillations between weak interacting electron neutrinos $\nu_e$ and sterile neutrinos $\nu_s$ for the case when $\nu_s$ do not thermalize till $\nu_e$ decoupling at 2 MeV and oscillations become effective after $\nu_e$ decoupling is described in detail in [@our]. The main assumptions are the following:
- Singlet neutrinos decouple much earlier, i.e. at a considerably higher temperature than the active neutrinos do: $T_{\nu_s}^F > T_{\nu_e}^F$.
This is quite a natural assumption, as far as sterile neutrinos do not participate into the ordinary weak interactions. In the models predicting singlet neutrinos, the interactions of $\nu_s$ are mediated by gauge bosons with masses $M={\cal O}$(1 TeV) [@hr; @ps; @gm]. Therefore, in later epochs after their decoupling, their temperature and number densities are considerably less than those of the active neutrinos due to the subsequent annihilations and decays of particles that have additionally heated the nondecoupled $\nu_e$ in comparison with the already decoupled $\nu_s$.
- We consider oscillations between $\nu_s$ ($\nu_s \equiv \tilde{\nu}_L$) and the active neutrinos, according to the Majorana&Dirac ($M\&D$) mixing scheme [@b] with mixing present just in the electron sector $\nu_i={\cal U}_{il}~\nu_l$, $l=e,s$: [^2]
$$\begin{array}{ccc}
\nu_1 & = & c\nu_e+s\nu_s\\
\nu_2 & = & -s\nu_e+c\nu_s,
\end{array}$$ where $\nu_s$ denotes the sterile electron antineutrino, $c=\cos(\vartheta)$, $s=\sin(\vartheta)$ and $\vartheta$ is the mixing angle in the electron sector, the mass eigenstates $\nu_1$ and $\nu_2$ are Majorana particles with masses correspondingly $m_1$ and $m_2$. We consider the nonresonant case $\delta m^2=m_2^2-m_1^2>0$, which corresponds in the small mixing angle limit to a sterile neutrino heavier than the active one.
In this model the element of nonequilibrium is introduced by the presence of a small singlet neutrino density at 2 MeV $n_{\nu_s} \ll n_{\nu_e}$, when the oscillations between $\nu_s$ and $\nu_e$ become effective. In order to provide such a small singlet neutrino density the sterile neutrinos should have decoupled from the plasma sufficiently early in comparison to the active ones and should have not regained their thermal equilibrium till 2 MeV [@ma; @dpk; @our]. Therefore, as far as the oscillations into $\nu_e$ and the following noncoherent scattering off the background may lead to the thermalization of $\nu_s$, two more assumptions are necessary for the nonequilibrium case to have place:
- Neutrino oscillations should become effective after the decoupling of the active neutrinos, $\Gamma_{osc}\ge H$ for $T\le 2$ MeV, which is realizable for $\delta m^2 \le 1.3 \times 10^{-7}$ eV$^2$ [@our].
<!-- -->
- Sterile neutrinos should not thermalize till 2 MeV when oscillations become effective, i.e. the production rate of $\nu_s$ must be smaller than the expansion rate.
The problem of sterile neutrino thermalization was discussed in the pioneer work of Manohar [@ma] and in more recent publications [@bd1]-[@ekm]. This assumption limits the allowed range of oscillation parameters for our model: $\sin^2(2\vartheta) \delta m^2 \le 10^{-7}$ eV$^2$ [@our].
We have assumed here that electron neutrinos decouple at 2 MeV. However, the neutrino decoupling process is more complicated. It has been discussed in literature in detail [@decoupl]. Decoupling occurs when the neutrino weak interaction rate $\Gamma_w \sim E^2 n_\nu(E)$ becomes less than the expansion rate $H \sim \sqrt g T^2$. Really, for electron neutrinos this happens at about 2 MeV. Nevertheless, due to the fact that weak interaction rate is greater at a higher energy, some thermal contact between neutrinos and high energy plasma remains after 2 MeV, especially for the high energy tail of the neutrino spectrum. In case these high energy neutrinos begin to oscillate before their decoupling, the account of this dependence of decoupling time on the neutrino momentum will be essential for our model. Otherwise, in case these neutrinos do not start oscillating before decoupling, there will be no harm considering them decoupled earlier, as far as they preserve their equilibrium distribution anyway due to their extremely small mass. In [@our] we have checked that neutrinos from high-energy tail start to oscillate much later than they decouple for the range of oscillation parameters considered in our model. It can easily be understood from the fact that the oscillation rate decreases with energy $\Gamma_{osc} \sim \delta m^2/E_\nu$ and, therefore, neutrinos with higher energies begin to oscillate later, namely when $\Gamma_{osc}$ exceeds the expansion rate $H \sim \sqrt g T^2$. Hence, the precise account for the momentum dependence of the decoupling does not change the results of our model but unnecessarily complicates the analysis and leads to an enormous increase of the calculation time. Therefore, in what follows we have assumed a fixed decoupling time instead of considering the real decoupling period - i.e. we have accepted that the electron neutrinos have completely decoupled at 2 MeV.
The kinetics of nonequilibrium neutrino oscillations
====================================================
The exact kinetic analysis of the neutrino evolution, discussed in this Section, though much more complicated, reveals some important features of nonequilibrium oscillations, that cannot be caught otherwise. As far as for the nonequilibrium model discussed the rates of expansion of the Universe, neutrino oscillations and neutrino interactions with the medium may be comparable, we have used kinetic equations for neutrinos accounting [*simultaneously*]{} for the participation of neutrinos into expansion, oscillations and interactions with the medium. All possible reactions of neutrinos with the plasma were considered, namely: reactions of neutrinos with the electrons, neutrons and protons, neutrinos of other flavours, and the corresponding antiparticles, as well as self interactions of electron neutrinos. These equations contain all effects due to first order on $G_F$ medium-induced energy shifts, second order effects due to non-forward collisions, and the effects non-linear on the neutrino density matrices like neutrino refraction effects in a medium of neutrinos. In the case of nonequilibrium oscillations the density matrix of neutrinos may considerably differ from its equilibrium form. [^3] Then, for the correct analysis of nonequilibrium oscillations, it is important to work in terms of density matrix of neutrinos in momentum space [@do; @dpk; @our; @sr]. Therefore, we have provided a proper kinetic analysis of the neutrino evolution using kinetic equations for the [*neutrino density matrix for each momentum mode*]{}.
Hence, the kinetic equations for the density matrix of the nonequilibrium oscillating neutrinos in the primeval plasma of the Universe in the epoch previous to nucleosynthesis, i.e. consisting of photons, neutrinos, electrons, nucleons, and the corresponding antiparticles, have the form: = H p [(t) p]{} + i +i + [O]{}([H]{}\^2\_[int]{} ), \[kin\] where $p$ is the momentum of electron neutrino and $\rho$ is the density matrix of the massive Majorana neutrinos in momentum space.
The first term in the equation describes the effect of expansion, the second is responsible for oscillations, the third accounts for forward neutrino scattering off the medium and the last one accounts for second order interaction effects of neutrinos with the medium. ${\cal H}_o$ is the free neutrino Hamiltonian: $${\cal H}_o = \left( \begin{array}{cc}
\sqrt{p^2+m_1^2} & 0 \\ 0 & \sqrt{p^2+m_2^2}
\end{array} \right),$$ while ${\cal H}_{int} = \alpha~V$ is the interaction Hamiltonian, where $\alpha_{ij}=U^*_{ie} U_{je}$, $V=G_F \left(+L - Q/M_W^2 \right)$, and in the interaction basis plays the role of an induced squared mass for electron neutrinos:
$${\cal H}_{int}^{LR} = \left( \begin{array}{cc}
V & 0 \\ 0 & 0 \end{array} \right).$$
Hence, $V$ is the time varying (due to the Universe cooling) effective potential, induced by the interactions of neutrino with the medium through which it propagates. Since $\nu_s$ does not interact with the medium it has no self-energy correction, i.e. $V_s=0$.
The first ‘local’ term in $V$ accounts for charged- and neutral-current tree-level interactions of $\nu_e$ with medium protons, neutrons, electrons and positrons, neutrinos and antineutrinos. It is proportional to the fermion asymmetry of the plasma $L=\sum_f L_f$, which is usually taken to be of the order of the baryon one i.e. $10^{-10}$ (i.e. $B-L$ conservation is assumed). $$L_f \sim {N_f-N_{\bar{f}} \over N_\gamma}~T^3 \sim
{N_B-N_{\bar{B}} \over N_\gamma}~T^3 = \beta T^3.$$ The second ‘nonlocal’ term in $V$ arises as an $W/Z$ propagator effect, $Q \sim E_\nu~T^4$ [@nr; @bd1]. For the early Universe conditions both terms must be accounted for because although the second term is of the second power of $G_F$ , the first term is proportional besides to the first power of $G_F$, also to the small value of the fermion asymmetry. Moreover, the two terms have different temperature dependence and an interesting interplay between them during the cooling of the Universe is observed. At high temperature the nonlocal term dominates, while with cooling of the Universe in the process of expansion the local one becomes more important.
The last term in the Eq. (\[kin\]) describes the weak interactions of neutrinos with the medium. For example, for the weak reactions of neutrinos with electrons and positrons $e^+ e^- \leftrightarrow
\nu_i \tilde{\nu}_j$, $e^\pm \nu_j \to e'^\pm \nu'_i$ it has the form $$\begin{array}{cl}
& \int {\rm d}\Omega(\tilde{\nu},e^+,e^-)\left[
n_{e^-} n_{e^+} {\cal A} {\cal A}^\dagger - \frac{1}{2} \left\{
\rho,~ {\cal A}^\dagger \bar{\rho} {\cal A} \right\}_+ \right] \\
+ & \int {\rm d}\Omega(e^-,\nu',e'^-)\left[
n'_{e^-} {\cal B} \rho' {\cal B}^\dagger - \frac{1}{2} \left\{
{\cal B}^\dagger {\cal B}, ~\rho \right\}_+ n_{e^-} \right] \\
+ & \int {\rm d}\Omega(e^+,\nu',e'^+)\left[
n'_{e^+} {\cal C} \rho' {\cal C}^\dagger - \frac{1}{2} \left\{
{\cal C}^\dagger {\cal C}, ~\rho \right\}_+ n_{e^+} \right],
\end{array}$$ where $n$ stands for the number density of the interacting particles, $$\begin{array}{cl}
{\rm d}\Omega(i,j,k)&={(2\pi)^4 \over 2E_\nu}
\int {{\rm d}^3 p_i \over (2\pi)^3~2E_i}{{\rm d}^3 p_j \over (2\pi)^3~2E_j}
{{\rm d}^3 p_k \over (2\pi)^3~2E_k}\\
&\times \delta^4(p_\nu+p_i-p_j-p_k)
\end{array}$$ is a phase space factor, ${\cal A}$ is the amplitude of the process $e^+ e^- \to \nu_i \tilde{\nu}_j$, ${\cal B}$ is the amplitude of the process $e^- \nu_j \to e'^- \nu'_i$ and ${\cal C}$ is the amplitude of the process $e^+ \nu_j \to e'^+ \nu'_i$. They are expressed through the known amplitudes ${\cal A}_e(e^+ e^- \to \nu_e \tilde{\nu}_e)$, ${\cal B}_e(e^- \nu_e \to e^- \nu_e)$ and ${\cal C}_e(e^+ \nu_e \to e^+ \nu_e)$: $${\cal A} = \alpha~{\cal A}_e,~~~~~
{\cal B} = \alpha~{\cal B}_e,~~~~~
{\cal C} = \alpha~{\cal C}_e.$$
An analogues equations hold for the antineutrino density matrix, the only difference being in the sign of the lepton asymmetry: $L_f$ is replaced by $-L_f$. Medium terms depend on neutrino density, thus introducing a nonlinear feedback mechanism. Neutrino and antineutrino ensembles evolve differently as far as the background is not CP symmetric. Oscillations may change neutrino-antineutrino asymmetry and it in turn affects oscillations. The evolution of neutrino and antineutrino ensembles is coupled and hence, it must be considered simultaneously.
We have analysed the evolution of the neutrino density matrix for the case when oscillations become noticeable after electron neutrinos decoupling, i.e. after 2 MeV. Then the last term in the kinetic equation can be neglected. So, the equation (\[kin\]) results into a set of coupled nonlinear integro-differential equations with time dependent coefficients for the components of the density matrix of neutrino. It is convenient instead of $\partial / \partial t$ to use $\partial / \partial \mu$, where $\mu^2=\sqrt{16\pi^3 g/45}~(\delta^2/M_{Pl})~t$, and $\delta=m_n-m_p$.
Then from eq. (\[kin\]) we obtain:
(
[c]{} ’\_[11]{}\
’\_[22]{}\
’\_[12]{}\
’\_[21]{}\
) = (
[cccc]{} 0 & 0 & +iscV & -iscV\
0 & 0 & -iscV & +iscV\
+iscV & -iscV & -iM & 0\
-iscV & +iscV & 0 & +iM\
) (
[c]{} \_[11]{}\
\_[22]{}\
\_[12]{}\
\_[21]{}\
), \[neutrino\]
where prime denotes $\partial / \partial \mu$ and $M=\delta m^2/(2E_\nu)+(s^2-c^2)V$.
Analytical solution is not possible without drastic assumptions and, therefore, we have numerically explored the problem [^4] using the Simpson method for integration and the fourth order Runge-Kutta algorithm for the solution of the differential equations.
The neutrino kinetics down to 2 MeV does not differ from the standard case, i.e. electron neutrinos maintain their equilibrium distribution, while sterile neutrinos are absent. So, the initial condition for the neutrino ensembles in the interaction basis can be assumed of the form:
$${\cal \varrho} = n_{\nu}^{eq}
\left( \begin{array}{cc}
1 & 0 \\
0 & 0
\end{array} \right)$$ where $n_{\nu}^{eq}=\exp(-E_{\nu}/T)/(1+\exp(-E_{\nu}/T))$.
We have analyzed the evolution of nonequilibrium oscillating neutrinos by numerically integrating the kinetic equations (\[neutrino\]) for the period after the electron neutrino decoupling till the freeze out of the neutron-proton ratio ($n/p$-ratio), i.e. for the temperature interval $[0.3,2.0]$ MeV. The oscillation parameters range studied is $\delta m^2 \in [10^{-11}, 10^{-7}]$ eV$^2$ and $\vartheta \in[0,\pi/4]$.
The distributions of electrons and positrons were taken the equilibrium ones. Really, due to the enormous rates of the electromagnetic reactions of these particles the deviations from equilibrium are negligible. We have also neglected the distortion of the neutrino spectra due to residual interactions between the electromagnetic and neutrino components of the plasma after 2 MeV. This distortion was accurately studied in [@dhs], where it was shown that the relative corrections to $\nu_e$ density is less than 1 % and the effect on the primordial helium abundance is negligible.
The neutron and proton number densities, used in the kinetic equations for neutrinos, were substituted from the numnerical calculations in CN code accounting for neutrino oscillations. I.e. we have simultaneously solved the equations governing the evolution of neutrino ensembles and those describing the evolution of the nucleons (see the next section). The baryon asymmetry $\beta$, parametrized as the ratio of the baryon number density to the photon number density, was taken to be $3\times 10^{-10}$.
Three [*main effects of neutrino nonequilibrium oscillations*]{} were revealed and precisely studied, namely electron neutrino depletion, neutrino energy spectrum distortion and the generation of asymmetry between neutrinos and their antiparticles:
\(a) Depletion of $\nu_e$ population due to oscillations: As far as oscillations become effective when the number densities of $\nu_e$ are much greater than those of $\nu_s$, $N_{\nu_e} \gg N_{\nu_s}$, the oscillations tend to reestablish the statistical equilibrium between different oscillating species. As a result $N_{\nu_e}$ decreases in comparison to its standard equilibrium value due to oscillations in favour of sterile neutrinos. [^5] The effect of depletion may be very strong (up to $50\%$) for relatively great $\delta m^2$ and maximal mixing. This result of our study is in accordance with other publications concerning depletion of electron neutrino population due to oscillations, like [@ekt], however, we have provided more precise account for this effect due to the accurate kinetic approach used.
In Figs. 1 the evolution of neutrino number densities is plotted. In Fig. 1a the curves represent the evolution of the electron neutrino number density in the discussed model with a fixed mass difference $\delta m^2=10^{-8}$ eV$^2$ and for different mixings. The numerical analysis showed that for small mixing, $\sin^2(2\vartheta)< 0.01$, the results do not differ from the standard case, i.e. then oscillations may be neglected. In Fig. 1b the evolution of the electron neutrino number density is shown for a nearly maximum mixing, $\sin^2(2\vartheta)=0.98$, and different squared mass differences. Our analysis has proved, that for mass differences $\delta m^2<10^{-11}$ eV$^2$, the effect of oscillations is negligible for any $\vartheta$.
In case of oscillations effective after the neutrino freeze out, electron neutrinos are not in thermal contact with the plasma and, therefore, the electron neutrino state, depleted due to oscillations into steriles, cannot be refilled by electron-positron annihilations. That irreversible depletion of $\nu_e$ population exactly equals the increase of $\nu_s$ one (see Fig. 1c). The number of the effective degrees of freedom do not change due to oscillations in that case, as far as the electron neutrino together with the corresponding sterile one contribute to the energy density of the Universe as one neutrino unit, even in case when the steriles are brought into chemical equilibrium with $\nu_e$. This fact was first noted in [@ekt].[^6]
\(b) Distortion of the energy distribution of neutrinos: The effect was first discussed in [@do] for the case of flavour neutrino oscillations. However, as far as the energy distortion for that case was shown to be negligible [@do; @dhs], it was not paid the necessary attention it deserved. The distortion of the neutrino spectrum was not discussed in publications concerning active-sterile neutrino oscillations, and was thought to be negligible. In [@dpk] it was first shown that for the case of $\nu_e \leftrightarrow \nu_s$ vacuum oscillations this effect is considerable and may even exceed that of an additional neutrino species. In [@our] we have discussed this effect for the general case of neutrino oscillations in a medium. The evolution of the distortion is the following: Different momentum neutrinos begin to oscillate at different temperatures and with different amplitudes. First the low energy part of the spectrum is distorted, and later on this distortion concerns neutrinos with higher and higher energies. This behaviour is natural, as far as neutrino oscillations affect first low energy neutrinos, $\Gamma_{osc} \sim \delta m^2/E_{\nu}$. The Figs. 2a, 2b, 2c and 2d snapshot the evolution of the energy spectrum distortion of active neutrinos $x^2 \rho_{LL}(x)$, where $x=E_\nu/T$, for maximal mixing and $\delta m^2 =
10^{-8.5}$ eV$^2$, at different temperatures: $T=1$ MeV (a), $T=0.7$ MeV (b), $T=0.5$ MeV (c), $T=0.3$ MeV (d). As can be seen from the figures, the distortion down to temperatures of 1 MeV is not significant as far as oscillations are not very effective and/or the weak residual interactions with the background still can compensate for the difference. However, for lower temperatures the distortion increases and at 0.5 MeV is strongly expressed. Its proper account is important for the correct determination of oscillations role in the kinetics of $n$-$p$ transitions during the freeze out of nucleons at about $0.3$ MeV.
Our analysis has shown that the account for the nonequilibrium distribution by shifting the effective temperature and assuming the neutrino spectrum of equilibrium form, often used in literature (see for example [@ssf]), may give misleading results for the case $\delta m^2 < 10^{-7}$ eV$^2$. The effect cannot be absorbed merely in shifting the effective temperature and assuming equilibrium distributions. For larger neutrino mass differences oscillations are fast enough and the naive account is more acceptable, provided that $\nu_e$ have not decoupled.
\(c) The generation of asymmetry between $\nu_e$ and their antiparticles: The problem of asymmetry generation in different contexts was discussed by several authors. The possibility of an asymmetry generation due to CP-violating flavour oscillations was first proposed in Ref. [@hp]. Later estimations of an asymmetry due to CP-violating MSW resonant oscillations were provided [@la]. The problem of asymmetry was considered in connection with the exploration of the neutrino propagation in the early Universe CP-odd plasma also in [@bd1]-[@ekm] and this type of asymmetry was shown to be negligible. Recently it was realized in [@ftv; @s], that asymmetry can grow to a considerable values for the case of great mass differences, $\delta m^2 \ge 10^{-5}$ eV$^2$. The effect of asymmetry for small mass differences $\delta m^2 \le 10^{-7}$ eV$^2$ on primordial production of helium was also proved to be important for the case of resonant neutrino oscillations [@our]. Our approach allows precise description of the asymmetry evolution, as far as working with the [*self consistent kinetic equations for neutrinos in momentum space*]{} enables us to calculate the behaviour of the asymmetry at each momentum. This is important particularly when the distortion of the neutrino spectrum is considerable.
In the present work we have explored accurately the effect of the asymmetry in the nonresonant case for all mixing angles and for small mass differences $\delta m \le 10^{-7}$ eV$^2$. Our analysis showed that when the lepton asymmetry is accepted initially equal to the baryon one, (as is usually assumed for the popular $L-B$ conserving models), the effect of the asymmetry is small for all the discussed parameters range. And although the asymmetry is not wiped out by the coupled oscillations, as stated by some authors [@ka; @ekm], nonresonant neutrino oscillations really cannot generate large neutrino-antineutrino asymmetry in the early Universe. This result is in accordance with the conclusions concerning asymmetry evolution in [@bd2; @ekm; @our]. We have also checked that the neutrino asymmetry even in the case of initial neutrino asymmetry by two orders of magnitude higher does not have significant effect on the cosmologically produced $^4\! He$. Therefore, for such small initial values of the lepton asymmetry, the neutrino asymmetry should be better neglected when calculating primordial element production for the sake of computational time. Mind, however, that for higher values of the initial asymmetry the effect could be significant, and should be studied in detail. The asymmetry evolution and its effect on He-4 production for unusual high initial values of the lepton asymmetry will be studied elsewhere [@asym].
In conclusion, our numerical analysis showed that the nonequilibrium oscillations can considerably deplete the number densities of electron neutrinos (antineutrinos) and distort their energy spectrum.
Nucleosynthesis with nonequilibrium oscillating neutrinos
=========================================================
As an illustration of the importance of these effects, and hence of the proposed approach to the analysis of nonequilibrium neutrino oscillations, we discuss their influence on the primordial production of $^4\! He$. The effect of oscillations on nucleosynthesis has been discussed in numerous publications [@bd1]-[@dpk], [@sv; @hp; @la]. A detail kinetic calculation of primordial yield of helium for the case of the nonequilibrium oscillations in vacuum was made in [@dpk] and the proper consideration accounting for the neutrino forward scattering processes off the background particles was done in [@our] for some neutrino mixing parameters. [^7] In the present work we calculate precisely the influence of oscillations on the production of He-4 within a detail numerical CN model with nonresonant nonequilibrium neutrino oscillations. The analysis of [@our] is expanded for the full space of the mixing parameters values.
Working with exact kinetic equations for the nucleon number densities and neutrino density matrix in momentum space, enables us to analyze the direct influence of oscillations onto the kinetics of the neutron-to-proton transfers and to account precisely for the neutrino depletion, neutrino energy distortion and the generation of asymmetry due to oscillations.
Primordial element abundances depend primarily on the neutron-to-proton ratio at the weak freeze out ($(n/p)_f$-ratio) of the reactions interconverting neutrons and protons : $n+\nu_e \leftrightarrow p+e$ and $n+e^+ \leftrightarrow p+\tilde{\nu_e}$. The freeze out occurs when due to the decrease of temperature with Universe expansion these weak interaction rates $\Gamma_w \sim E_\nu^2 n_{\nu}$ become comparable and less than the expansion rate $H \sim \sqrt{g}~ T^2$. Hence, the $(n/p)_f$-ratio depends on the effective relativistic degrees of freedom $g$ (through the expansion rate) and the neutrino number densities and neutrino energy distribution (through the weak rates). Therefore, we calculate accurately the evolution of neutron number density till its freeze-out. Further evolution is due to the neutron decays $n \rightarrow p+e+\tilde{\nu_e}$ that proceed till the effective synthesis of deuterium begins. As far as the expansion rate exceeds considerably the decay rate for the characteristic period before the freeze out, decays are not essential. Therefore, we have accounted for them adiabatically.
The master equation, describing the evolution of the neutron number density in momentum space $n_n$ for the case of oscillating neutrinos $\nu_e \leftrightarrow \nu_s$, reads:
[l]{} (n\_n / t ) = H p\_n (n\_n / p\_n ) +\
+ (e\^-,p,) |[A]{}(e\^- pn)|\^2\
\
- (e\^+,p,) |[A]{}(e\^+np)|\^2\
.
The first term on the right-hand side describes the effect of expansion while the next ones – the processes $e^- + p \leftrightarrow n + \nu_e$ and $p + \tilde{\nu}_e \leftrightarrow e^+ + n$, directly influencing the nucleon density. It differs from the standard scenario one only by the substitution of $\rho_{LL}$ and $\bar{\rho}_{LL}$ instead of $n_{\nu}^{eq} = [1-\exp(E_\nu/T)]^{-1}$. The neutrino and antineutrino density matrices differ $\bar{\rho}_{LL} \ne \rho_{LL}$, contrary to the standard model, as a result of the different reactions with the CP-odd plasma of the prenucleosynthesis epoch. We have accounted for the final state Pauli blocking for neutrinos and electrons.
Particle number densities per unit volume are expressed as $N = (2\pi)^{-3}\int{\rm d}^3 p~n(p)$. Performing the integration on the right-hand side of the equation also one gets the final equations for the time evolution of the neutron number density:
$$\begin{aligned}
&&\left(\partial N_n / \partial t \right)
= -3H N_n + G_F^2~ \frac{g^2_V+3 g^2_A}{\pi^3}~ T^5~ \times
\nonumber \\
&\times& \Big\{ N_p \int_0^{\infty} [1-\rho_{LL}(x)]~ \frac{{\rm e}^{-x-y}}
{1+{\rm e}^{-x-y}}~
f(x,y){\rm d}x \nonumber \\
&-& N_n\int_0^{\infty} \rho_{LL}(x)~ \frac{1}{1+{\rm e}^{-x-y}}~ f(x,y)
{\rm d}x \nonumber \\
&+& N_p\int_{(1+\zeta)y}^{\infty} \bar{\rho}_{LL}(x)~
\frac{1}{1+{\rm e}^{-x+y}}
f(x,-y){\rm d}x \nonumber \\
&-& N_n\int_{(1+\zeta)y}^{\infty} [1-\bar{\rho}_{LL}(x)]~
\frac{{\rm e}^{-x+y}}{1+{\rm e}^{-x+y}}~ f(x,-y){\rm d}x \Big\}\end{aligned}$$
where $f(x,y)=x^2(x+y)\sqrt{(x+y)^2+\zeta^2y^2}$ and $y=(\delta + m_e)/T$, $\zeta=m_e/\delta$, $\delta = m_n - m_p$.
The first term on the right-hand side describes the dilution effect of expansion, the next describe the weak processes, as pointed above. We have numerically integrated this equation for the temperature range of interest $T \in [0.3,2.0]$ MeV for the full range of oscillation parameters of our model. The value of $\rho_{LL}(x)$ at each integration step was taken from the simultaneously performed integration of the set of equations (\[neutrino\]), i.e. the evolution of neutrino and the nucleons was followed self consistently. The initial values at $T=2$ MeV for the neutron, proton and electron number densities are their equilibrium values. Although the electron mass is comparable with the temperature in the discussed temperature range, the deviation of the electron density from its equilibrium value is negligible due to the enormous rate of the reactions with the plasma photons [@do]. The parameters values of the CN model, adopted in our calculations, are the following: the mean neutron lifetime is $\tau= 887$ sec, which corresponds to the present weighted average value [@partdata], the effective number of relativistic flavour types of neutrinos during the nucleosynthesis epoch $N_{\nu}$ is assumed equal to the standard value $3$. This is a natural choice as far as it is in good agreement both with the CN arguments [@N] [^8] and with the precision measurements of the $Z$ decay width at LEP [@NLEP].
Results and conclusions
=======================
The results of the numerical integration are illustrated in Fig. 3. As it can be seen from the figure the kinetic effects (neutrino population depletion and distortion of neutrino spectrum) due to oscillations play an important role and lead to a considerable overproduction of helium.
Qualitatively the effect of oscillations on helium production can be described as follows:
The depletion of the electron neutrino number densities due to oscillations into sterile ones strongly affects the $n \leftrightarrow p$ reactions rates. It leads to an effective decrease in the processes rates, and hence to an increase of the freezing temperature of the $n/p$-ratio and the corresponding overproduction of the primordially produced $^4\! He$.
The effect of the distortion of the energy distribution of neutrinos has two aspects. On one hand an average decrease of the energy of active neutrinos leads to a decrease of the weak reactions rate, $\Gamma_w \sim E_\nu^2$ and subsequently to an increase in the freezing temperature and the produced helium. On the other hand, there exists an energy threshold for the reaction $\tilde{\nu}_e+p \to n+e^+$. And in case when, due to oscillations, the energy of the relatively greater part of neutrinos becomes smaller than that threshold the $n/p$- freezing ratio decreases leading to a corresponding decrease of the primordially produced helium-4 [@ki]. The numerical analysis showed that the latter effect is less noticeable compared with the former ones.
The asymmetry calculations showed a slight predominance of neutrinos over antineutrinos, not leading to a noticeable effect on the production of helium in case the lepton asymmetry is accepted initially equal to the baryon one. So, the effect of asymmetry is proved to be negligible for all the discussed parameter range, i.e. for any $\vartheta$ and for $\delta m^2 \le 10^{-7}$ eV$^2$. We have partially (not for the full range of model parameters) investigated the problem for higher than the baryon one initial lepton asymmetry. The preliminary results point that even lepton asymmetry initially by two orders of magnitude higher does not have noticeable effect on the cosmologically produced $^4\! He$. Higher than those lepton asymmetries, however, should be accounted for properly even in the nonresonant case.
Thus, the total result of nonequilibrium neutrino oscillations is an overproduction of helium in comparison to the standard value.
In Fig. 4 the dependence of the frozen neutron number density relative to nucleons $X_n=N_n/(N_p+N_n)$ on the mixing angle for different fixed $\delta m^2$ is illustrated. The dependence of the frozen neutron number density relative to nucleons $X_n=N_n/(N_p+N_n)$ on the $\delta m^2$ for fixed different mixing angles, is presented in Fig. 5. The effect of oscillations is maximal at maximal mixing for the nonresonant case of neutrino oscillations. As it can be seen from the figures, it becomes almost negligible (less than $1\%$) for mixings as small as 0.1 for any $\delta m^2$ of the discussed range of our model. The value of the frozen $n/p$-ratio is a smoothly increasing function of the mass difference. Our analysis shows that the effect of oscillation for $\delta m^2$ smaller than $10^{-10}$ eV$^2$ even for maximal mixing is smaller than $1\%$. The nonresonant oscillations with $\delta m^2 \le 10^{-11}$ eV$^2$ do not have any observable effect on the primordial production of elements, i.e. the results coincide with the standard model values with great accuracy.
From the numerical integration for different oscillation parameters we have obtained the primordial helium yield $Y_p(\delta m^2,\vartheta)$, which is illustrated by the surface in Fig. 6. Some of the constant helium contours calculated in the discussed model of cosmological nucleosynthesis with nonresonant neutrino oscillations on the $\delta m^2-\vartheta$ plane are presented in Fig. 7.
On the basis of these results, requiring an agreement between the theoretically predicted and the observational values of helium, it is possible to obtain cosmological constraints on the neutrino mixing parameters. At present the primordial helium values extracted from observations differ considerably: for example some authors believe that the systematic errors have already been reduced to about the same level as the statistical one and obtain the bounds for the primordial helium: $Y_p(^4\! He)=0.232\pm 0.003$ [@Ystand], while others argue that underestimation of the systematic errors, such as errors in helium emissivities, inadequatisies in the radiative transfer model used, corrections for underlying stellar absorption and fluorescent enhancement in the He I lines, corrections for neutral helium, may be significant and their account may raise the upper bound on $Y_p$ as high as 0.26 [@Ynstand]. Thus besides the widely adopted “classical” bound $Y_p<0.24$ [@Yclas] it is reasonable to have in mind the more “reliable” upper bound to the primordial helium abundance $Y_p<0.25$ and even the extreme value as high as 0.26 [@Ynclas]. Therefore, we considered it useful to provide the precise calculations for helium contours up to 0.26. So, whatever the primordial abundance of $^4\! He$ will be found to be in future (within this extreme range) the results of our calculations may provide the corresponding bound on mixing parameters of neutrino for the case of nonresonant active-sterile oscillations with small mass differences. Assuming the conventional observational bound on primordial $^4\! He$ $0.24$ the cosmologically excluded region for the oscillation parameters is shown on the plane $\sin^2(2\vartheta)$ - $\delta m^2$ in Fig. 7. It is situated to the right of the $Y_p=0.245$ curve, which gives $5\%$ overproduction of helium in comparison with the accepted 0.24 observational value.
The curves, corresponding to helium abundance $Y_p=0.24$, obtained in the present work, and in previous works, analyzing the nonresonant active-sterile neutrino oscillations, are plotted in Fig. 8. In [@bd1] and [@ekt] the authors estimated the effect of excitement of an additional degree of freedom due to oscillations, and the corresponding increase of the Universe expansion rate, leading to an overproduction of helium-4. The excluded regions for the neutrino mixing parameters were obtained from the requirement that the neutrino types should be less than 3.4: $N_{\nu}<3.4$. In these works the depletion effect was considered. The asymmetry was neglected and the distortion of the neutrino spectrum was not studied as far as the kinetic equations for neutrino mean number densities were considered. Our results are in good accordance with the estimations in [@bd1] and the numerical analysis in [@ekt], who have made very successful account for one of the discussed effects of nonequilibrium oscillations - the neutrino population depletion. The results of [@ssf], as can be seen from the Fig. 8, differ more both from the ones of the previously cited works and from our results. Probably the account for nonequilibrium oscillations merely by shifting the effective neutrino temperature, as assumed there is not acceptable for a large range of model parameters.
As can be seen from the curves, for large mixing angles, we exclude $\delta m^2\ge 10^{-9}$ eV$^2$, which is almost an order of magnitude stronger constraint than the previously existing. This more stringent constraints obtained in our work for the region of great mixing angles and small mass differences is due to the more accurate kinetic approach we have used and to the precise account of neutrino depletion, energy distortion and asymmetry due to oscillations.
As far as we already have at our disposal some impressive indications for neutrino oscillations, it is interesting to compare our results also with the range of parameters which could eventually explain the observed neutrino anomalies:
The vacuum oscillation interpretation of the solar neutrino problem requires extremely small mass differences squared, less and of the order of $10^{-10}$ eV$^2$. It is safely lower than the excluded region, obtained in our work, and is, therefore, allowed from CN considerations. The MWS small mixing angle nonadiabatic solution (see for example Krastev, Liu and Petcov in [@SUNTH]) is out of the reach of our model. However, as we are in a good accordance with the results of active-sterile neutrino oscillation models with higher mass differences, it is obvious that a natural extrapolation of our excluded zone towards higher mass differences will rule out partially the possible solution range for large mixing angles.
Our pattern of neutrino mixing is compatible with models of degenerate neutrino masses of the order of 2.4 eV, necessary for the successful modelling of the structure formation of the Universe in Hot plus Cold Dark Matter Models [@DM].
As a conclusion, we would like to outline the main achievements of this work: In a model of nonequilibrium nonresonant active-sterile oscillation, we had studied the effect of oscillations on the evolution of the neutrino number densities, neutrino spectrum distortion and neutrino-antineutrino asymmetry. We have used kinetic equations for the density matrix of neutrinos in [*momentum*]{} space, accounting [*simultaneously*]{} for expansion, oscillations and interactions with the medium. This approach enabled us to describe precisely the behaviour of neutrino ensembles in the Early Universe in the period of interest for CN. The analysis was provided for small mass differences. We have shown that the energy distortion may be significant, while the asymmetry in case it is initially (i.e. before oscillations become effective) of the order of the baryon one, may be neglected.
Next, we have made a precise survey of the influence of the discussed type of oscillations on the cosmological production of helium-4. We have calculated the evolution of the corresponding neutron-to-proton ratio from the time of freeze out of neutrinos at 2 MeV till the effective freeze out of nucleons at 0.3 MeV for the full range of model parameters. As a result we have obtained the dependence $Y_p(\delta m^2, \vartheta)$ and constant helium contours on the $\delta m^2 - \vartheta$ plane. Requiring an agreement between the observational and the theoretically predicted primordial helium abundances, we have calculated accurately the excluded regions for the neutrino mixing parameters, for different assumptions about the preferred primordial value of helium.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank prof. A. Dolgov for useful discussions and encouragement. D.K. is grateful to prof. I. Novikov and prof. P. Christensen for the opportunity to work at the Theoretical Astrophysics Center. She is glad to thank the Theoretical Astrophysical Center for the warm hospitality and financial support. She acknowledges the hospitality and support of the Niels Bohr Institute. This work was supported also by 1996/1997 Danish Governmental Scholarship grant. M.C. thanks NORDITA for the hospitality.
This work was supported in part by the Danish National Research Foundation through its establishment of the Theoretical Astrophysics Center.
[99]{}
G. Gamow, Ohio Journal of Science 35, 406 (1935);\
G. Gamow, Journal of the Washington Academy of Sciences 32, 353 (1942);\
G. Gamow, [*Phys. Rev.*]{} 70, 572 (1946).
A. M. Boesgaard and G. Steigman, Ann. Rev. Astr. Astophys. 23, 319 (1985);\
R. A. Malaney and G. J. Mathews, ;\
M. S. Smith, L. H. Kawano and R. A. Malaney, ;\
B. Pagel, in Nucleosynthesis and Chemical Evolution of Galaxies, Cambridge Univ. Press, 1997, chapter 4;\
K. A. Olive, E. Skillman and G. Steigman , astro-ph/9611166;\
Hogan C.J., astro-ph/9609138, astro-ph/9702044 ;\
G. Steigman , astro-ph/9608084; to be published in Proceedings of the Conf. Critical Dialogues in Cosmology, Princeton, NJ, June 1996;\
S. Sarkar, [*Rep. Prog. Phys.*]{} 59, 1493 (1996) and the references there in.
B. Pontecorvo, , ;\
B. Pontecorvo, .
J. Ellis, Inv. talk at NEUTRINO 96 Conf., Helsinki, June 1996; hep-ph/9612209;\
A.D. Dolgov, ;\
J. W. F. Valle, ;\
A. Yu. Smirnov, Proc. 28 Int. Conf.on High Energy Physics (ICHEP 96), Warsaw, Poland, 1996; hep-ph/9611465;\
P. Elmfors et al., hep-ph/9703214 and the references there in;\
K. Kainulainen, hep-ph/9608215 and the references there in; to be published in Proc. of the 17 Int. Conf. of Neutrino Physics and Astrophysics NEUTRINO96, Helsinki, 1996;\
N. Hata and P. Langacker, hep-ph/9705339;\
K. Enqvist, hep-ph/9310226, published in New Physics with New Experiments ed. Z. Adjuk, S.Pokorski and A. Wroblewski, 1994, 380.
K. Lande, (Homestake Collaboration), Talk given at the NEUTRINO 96 Conf., 1996, Helsinki (to be published in the Proc.);\
R. Davis, Prog. Part. Nucl. Phys. 32, 13 (1994);\
B. T. Cleveland et al.,(Homestake Coll.), ;\
S. T. Petcov, ;\
J. N. Bahcall, Neutrino Astrophysics, Cambridge Univ.Press, Cambridge, 1989;\
J. N. Bahcall and M. H. Pinsonneault, ;\
W. Hampel et al., (GALLEX Coll.), ;\
T. Kirsten, (GALLEX Coll.), Talk given at the NEUTRINO 96 Conf., 1996, Helsinki (to be published in the Proc.);\
P. Anselmann et al., ; ;\
V. N. Gavrin, (SAGE Coll.), Talk given at the NEUTRINO 96 Conf., 1996, Helsinki (to be published in the Proc.);\
J. N. Abdurashitov et al., (SAGE Coll.), ;\
J. N. Abdurashitov et al., (SAGE Coll.), ;\
Y. Fukuda et al., (Kamiokande Coll.), ;\
Y. Suzuki, (Kamiokande Coll.), .
Y. Fukuda et al., (Kamiokande Coll.), ;\
K. S. Hirata et al., (Kamiokande Coll.), ; ;\
D. Casper et al., (IBM Coll.), ;\
R. Becker-Szendy et al., (IBM Coll.), ;\
R. Becker-Szendy et al., (IBM Coll.), ;\
W. W. M. Allison et al., (Soudan-2 Coll.), .
C. Athanassopoulos et al. (LSND Coll.), ; ; nucl-ex/9706006 submitted to [*Phys. Rev. C*]{};\
J. E. Hill, .
J. N. Bahcall, astro-ph/9702057, Proc. 18 Texas Symp. on Relativistic Astrophysics, Chicago, 1996, eds. A. Olinto, J. Frieman, D. Schramm, World Scientific Press.
V. S. Berezinsky et al., ;\
B. Ricci et al., astro-ph/9705164;\
J. N. Bahcall and M.H. Pinsonneault, ;\
N. Hata and P. Langacker, ;\
J. N. Bahcall, .
S. P. Mikheyev and A. Yu. Smirnov, ; ;\
L. Wolfenstein, .
P. I. Krastev and S. T. Petcov, Proc. Neutrino’96 Conf., 1996, Helsinki;\
P. I. Krastev, Q. Y. Liu and S. T. Petcov, ;\
P. I. Krastev and S. T. Petcov, ;\
Q. Y. Liu and S. T. Petcov, SISSA 174/96/EP and hep-ph/9702400;\
P. I. Krastev and S. T. Petcov, ;\
E. Calabresu et al., Astropart. Phys. 4, 159 (12995);\
Z. G. Berezhiani and A. Rossi, ;\
A. Dar and G. Shaviv, ;\
N. Hata and P. Langacker, ; ;\
P. I. Krastev and A. Yu. Smirnov, .
J. R. Primack et al., ;\
J. R. Primack, astro-ph/9610078, to appear in Critical Dialogues in Cosmology, ed. N. Turok, World Sci., 1996;\
D. G. Lee and R. N. Mohapatra, ;\
D. O. Caldwell and R. N. Mohapatra, ;\
D. O. Caldwell and R. N. Mohapatra, ;\
J.A. Holtzman and J.R. Primack, ;\
A.N. Taylor and M. Rowan-Robinson, ;\
M. Davis et al., ;\
E.L. Wright et al., .
M. Drees, hep-ph/9703260.
R. N. Mohapatra, hep-ph/9702229;\
P. Binetruy et al., hep-ph/9610481, ;\
N. Okada and O. Yasuda, hep-ph/9606411, ; S. M. Bilenky, ;\
S. Bilenky et al.,hep-ph/9604364, SISSA 35/96/EP and the references there in;\
H. Minakata, hep-ph/9612259;\
H. Minakata, ;\
H. Minakata, ;\
G. M. Fuller et al, ;\
E. Ma and J. Pantaleone, ;\
R. Foot and R. R. Volkas, ;\
Z. G. Berezhiani and R. N. Mohapatra, ;\
E. J. Chun et al., ;\
S. T. Petcov and A.Yu. Smirnov , ;\
A. S. Joshipura, ;
Note, however, that in case the zenith angle dependence reported by Kamioka is not a real effect, as proposed by IBM collaboration report (R. Clark et al., submitted to Phys. Rev. Lett.) and the SuperKamiokande preliminary results, the atmospheric $\delta m^2$ solution may be compatible with the LSND, i.e. oscillations between the three flavour neutrinos could explain all the existing anomalies:
G. Fogli, E. Lisi, G. Scioscia, hep-ph/ 9702298;\
C. Cardall and G. Fuller, .
V. F. Shvartsman, ;\
B. Adeva et al., ;\
The LEP Collaborations et al., CERN-PPE/96-183, 1996.
J. Hewet and T. Rizzo, Phys. Rep. 183 , 195 (1989).
E. Ma and P. Roy, ;\
E. Ma, .
J. Pati and A. Salam, ; R. Mohapatra and S. Pati, .
E. Chun, A. Joshipura, and A. Smirnov, ; .
R. Barbieri and A. Dolgov, .
K. Kainulainen, .
R. Barbieri and A. Dolgov, .
K. Enqvist, K. Kainulainen and M. Thomson, .
K. Enqvist, K. Kainulainen and J. Maalampi, , .
J. M. Cline, .
X. Shi, D. N. Schramm and B. D. Fields, .
X. Shi, .
D. P. Kirilova, preprint JINR E2-88-301, 1988.
A. D. Dolgov, .
P.G. Langacker et al., ;\
M.G.Savage et al.,.
H. Weldon, ;\
P. B. Pal, T. N. Pham, ;\
J. F. Nieves, ;\
N. P. Lausman, Ch. van Weert, . P. B. Pal and T.N. Pham, ;\
see also Langacker in [@la];\
L. Wolfenstein, ;\
P. Langacker, J. Leveille and J. Sheiman, ;\
V. Barger et al., .
D. Nötzold and G. Raffelt, ;\
R.A.Harris and L.Stodolsky, ; ;\
L.Stodolsky, .
D. P. Kirilova and M. V. Chizhov, ;\
D. P. Kirilova and M. V. Chizhov, Talk at NEUTRINO 96 Int. Conf., to be published in the Proc. of the Conf., Helsinki, 1996; hep-ph/9704269.
G. Sigl and G. Raffelt, .
V. B. Semikoz and J. W. F. Valle, , erratum - ; preprint FTUV-94-05-ERR, July 1996 (hep-ph/9607208).
D. Fargion and M. Shepkin, ;\
L. Stodolsky in [@nr];\
P. Langacker, Pennsylvania preprint UPR 0401T (89).
M. Yu. Khlopov and S. T. Petcov, , Erratum: .
P. Langacker, S. Petcov, G. Steigman and S. Toshev, ;\
L. Wolfenstein, in [@MSW];\
S. P. Mikheyev and A. Yu. Smirnov, in [@MSW];\
T. K. Kuo and J. Panteleone, .
R. Foot, M. J. Thomson, R. R. Volkas, ;\
R. Foot and R. R. Volkas, .
A.D. Dolgov, S. H. Hanested and D. V. Semikoz, preprint TAC-1997-10, hep-ph/9703315;\
See also previous works on that theme and references there in:\
S. Hannestad and J. Madsen, ;\
A. D. Dolgov and M. Fukugita, ;\
A. D. Dolgov and M. Fukugita, .
J.A.Grifols and T.G. Masso, .
S. M. Bilenky, preprint JINR P2-83-441, 1983.
A. Manohar, ;\
see also G. Fuller and R. Malaney, and E. K. Semikoz, .
M. A. Herrera and S. Hacyan, ;\
N. C. Rana and B. Mitra, ;\
T. P. Walker et al., ;\
S. Dodelson and M. S. Turner, ;\
D. A. Dicus et al., ;\
A.D.Dolgov and Y.B.Zeldovich, ;\
A. D. Dolgov and M. Fukugita, in [@dhs];\
S. Hannestad and J. Madsen in [@dhs];\
B. Fields, S. Dodelson and M. S. Turner, ;\
V. A. Kostelecký and S. Samuel, .
D. P. Kirilova and M. V. Chizhov , in preparation.
Particle Data Group, .
V. F. Shvartsman, in [@NLEP];\
G. Steigman et al., ;\
G. Steigman et al., ;\
A. Dolgov and Ya. Zeldovich, Usp. Fiz. Nauk 130, 559 (1980);\
J. Yang et al., ;\
T. Walker et al., .
N. Terasawa and K. Sato, ;\
S. Dodelson, G. Gyuk and M. S. Turner, .
A. Dolgov and D. Kirilova, .
K. A. Olive and G. Steigman, ;\
E. Skillman et al., Ann. N. Y. Acad. Sci. 688, 739 (1993);\
B. Pagel et al., MNRAS 255, 325, 1992.
D. Sasselov and D. Goldwirth, ;\
K. Davidson and T. D. Kinman, ;\
N. Hata et al., ;\
N. Hata and G. Steigman, ;\
G. M. Fuller and C. Y. Cardall, ;\
Yu. I. Izotov et al., .
T. P. Walker in [@decoupl];\
M. S. Smith, L. H. Kawano and R. A. Malaney, ;\
P. J. Kernan and L. M. Krauss, .
P. J. Kernan and S. Sarkar, ;\
L. M. Krauss and P. J. Kernan, ;\
C. J. Copi et al., .
[Figure Captions]{}
\
[**Figure 1a**]{}: The curves represent the calculated evolution of the electron neutrino number density in the discussed model of active-sterile neutrino oscillations with a mass difference $\delta m^2=10^{-8}$ eV$^2$ and different mixing, parametrized by $\sin^2(2\vartheta)$, namely: $1$, $10^{-0.01}$, $10^{-0.1}$ and 0.1.
\
[**Figure 1b**]{}: The curves show the evolution of the electron neutrino number density in the discussed model of nonresonant active-sterile neutrino oscillations for a nearly maximum mixing, $\sin^2(2\vartheta)=0.98$, and different squared mass differences $\delta m^2$, namely $10^{-7}$, $10^{-8}$, $10^{-9}$ and $10^{-10}$ in eV$^2$.
\
[**Figure 1c**]{}: The curves show the evolution of the electron neutrino number density (the solid curve) and the sterile neutrino number density (the dashed curve) in the case of the nonresonant active-sterile neutrino oscillations for a maximal mixing and $\delta m^2=10^{-8}$ eV$^2$. The reduction of the active neutrino population is exactly counterbalanced by a corresponding increase in the sterile neutrino population.
\
[**Figure 2**]{}: The figures illustrate the evolution of the energy spectrum distortion of active neutrinos $x^2 \rho_{LL}(x)$, where $x=E_\nu/T$, for the case of nonresonant $\nu_e$-$\nu_s$ oscillations with a maximal mixing and $\delta m^2 = 10^{-8.5}$ eV$^2$, at different temperatures: $T=1$ MeV (a), $T=0.7$ MeV (b), $T=0.5$ MeV (c), $T=0.3$ MeV (d).
\
[**Figure 3**]{}: The evolution of the neutron number density relative to nucleons $X_n(t)=N_n(t)/(N_p+N_n)$ for the case of nonresonant oscillations with maximal mixing and different $\delta m^2$ is shown. For comparison the standard model curve is plotted also.
\
[**Figure 4**]{}: The figure illustrates the dependence of the frozen neutron number density relative to nucleons $X_n=N_n/(N_p+N_n)$ on the mixing angle for different $\delta m^2$.
\
[**Figure 5**]{}: The figure illustrates the dependence of the frozen neutron number density relative to nucleons $X_n=N_n/(N_p+N_n)$ on the mass difference for different mixing angles.
\
[**Figure 6**]{}: The dependence of the primordially produced helium on the oscillation parameters is represented by the surface $Y_p(\delta m^2,\vartheta)$.
\
[**Figure 7**]{}: On the $\delta m^2-\vartheta$ plane some of the constant helium contours calculated in the discussed model of cosmological nucleosynthesis with nonresonant neutrino oscillations are shown.
\
[**Figure 8**]{}: The curves, corresponding to helium abundance $Y_p=0.24$, obtained in the present work and in previous works, analyzing the nonresonant active-sterile neutrino oscillations, are plotted on the $\delta m^2-\vartheta$ plane.
[**Figure 1a**]{}: The curves represent the calculated evolution of the electron neutrino number density in the discussed model of active-sterile neutrino oscillations with a mass difference $\delta m^2=10^{-8}$ eV$^2$ and different mixing, parametrized by $\sin^2(2\vartheta)$, namely: $1$, $10^{-0.01}$, $10^{-0.1}$ and 0.1.
[**Figure 1b**]{}: The curves show the evolution of the electron neutrino number density in the discussed model of nonresonant active-sterile neutrino oscillations for a nearly maximum mixing, $\sin^2(2\vartheta)=0.98$, and different squared mass differences $\delta m^2$, namely $10^{-7}$, $10^{-8}$, $10^{-9}$ and $10^{-10}$ in eV$^2$.
[**Figure 1c**]{}: The curves show the evolution of the electron neutrino number density (the solid curve) and the sterile neutrino number density (the dashed curve) in the case of the nonresonant active-sterile neutrino oscillations for a maximal mixing and $\delta m^2=10^{-8}$ eV$^2$. The reduction of the active neutrino population is exactly counterbalanced by a corresponding increase in the sterile neutrino population.
[**Figure 2**]{}: The figures illustrate the evolution of the energy spectrum distortion of active neutrinos $x^2 \rho_{LL}(x)$, where $x=E_\nu/T$, for the case of nonresonant $\nu_e$-$\nu_s$ oscillations with a maximal mixing and $\delta m^2 = 10^{-8.5}$ eV$^2$, at different temperatures: $T=1$ MeV (a), $T=0.7$ MeV (b), $T=0.5$ MeV (c), $T=0.3$ MeV (d).
[**Figure 3**]{}: The evolution of the neutron number density relative to nucleons $X_n(t)=N_n(t)/(N_p+N_n)$ for the case of nonresonant oscillations with maximal mixing and different $\delta m^2$ is shown. For comparison the standard model curve is plotted also.
[**Figure 4**]{}: The figure illustrates the dependence of the frozen neutron number density relative to nucleons $X_n=N_n/(N_p+N_n)$ on the mixing angle for different $\delta m^2$.
[**Figure 5**]{}: The figure illustrates the dependence of the frozen neutron number density relative to nucleons $X_n=N_n/(N_p+N_n)$ on the mass difference for different mixing angles.
[**Figure 6**]{}: The dependence of the primordially produced helium on the oscillation parameters is represented by the surface $Y_p(\delta m^2,\vartheta)$.
[**Figure 7**]{}: On the $\delta m^2-\vartheta$ plane some of the constant helium contours calculated in the discussed model of cosmological nucleosynthesis with nonresonant neutrino oscillations are shown.
[**Figure 8**]{}: The curves, corresponding to helium abundance $Y_p=0.24$, obtained in the present work and in previous works, analyzing the nonresonant active-sterile neutrino oscillations, are plotted on the $\delta m^2-\vartheta$ plane.
[^1]: Permanent address: Institute of Astronomy, Bulgarian Academy of Sciences,\
blvd. Tsarigradsko Shosse 72, Sofia, Bulgaria\
E-mail:$dani@libra.astro.acad.bg$
[^2]: The transitions between different neutrino flavours were proved to have negligible effect on the neutrino number densities and on primordial nucleosynthesis because of the very slight deviation from equilibrium in that case $T_f \sim T_f'$ ($f$ is the flavour index) [@do; @lan; @dhs].
[^3]: When neutrinos are in equilibrium their density matrix has its equilibrium form, namely $\rho_{ij}=\delta_{ij} \exp(\mu/T-E/T)$, so that one can work with particle densities instead of $\rho$. In an equilibrium background, the introduction of oscillations slightly shifts $\rho$ from its diagonal form, due to the extreme smallness of the neutrino mass in comparison with the characteristic temperatures and to the fact that equilibrium distribution of massless particles is not changed by the expansion [@do].
[^4]: For the case of vacuum neutrino oscillations this equation was analytically solved and the evolution of density matrix was given explicitly in Ref. [@dpk].
[^5]: Note, that while neutrinos are in thermal equilibrium with the plasma no dilution of their number density is expected as far as it is kept the equilibrium one due to the annihilations of the medium electrons and positrons.
[^6]: Note the essential difference from the case of electron neutrinos in thermal equilibrium, when the oscillations into sterile neutrinos bring an additional degree of freedom into thermal contact.
[^7]: Calculations of helium production within the full Big Bang Nucleosynthesis code with oscillations were provided also in [@ssf], however, there the momentum degree of freedom of neutrino was not considered and a simplifying account of the nonequilibrium was used - by merely shifting the neutrino effective temperature and working in terms of equilibrium particle densities.
[^8]: However mind also the possibilities for somewhat relaxation of that kind of bound in modifications of the CN model with decaying particles as in [@Nvar; @ki].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We consider the uplink of a cellular massive MIMO network. Since the spectral efficiency of these networks is limited by pilot contamination, the pilot allocation across cells is of paramount importance. However, finding efficient pilot reuse patterns is non-trivial especially in practical asymmetric base station deployments. In this paper, we approach this problem using coalitional game theory. Each cell has its own unique pilots and can form coalitions with other cells to gain access to more pilots. We develop a low-complexity distributed algorithm and prove convergence to an individually stable coalition structure. Simulations reveal fast algorithmic convergence and substantial performance gains over one-cell coalitions and full pilot reuse.'
author:
-
bibliography:
- 'IEEEabrv.bib'
- 'refs.bib'
title: 'Pilot Clustering in Asymmetric Massive MIMO Networks [^1] '
---
[^1]: © 2015 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'This paper is devoted to the evaluation of the pionic scalar density at finite temperature and baryonic density. We express the latter effect in terms of the nuclear response evaluated in the random phase approximation. We discuss the density and temperature evolution of the pionic density which governs the quark condensate evolution. Numerical evaluations are performed.'
address:
- |
$^a$Institut de Physique Nucléaire de Lyon, IN2P3-CNRS et Université Cl. Bernard,\
43, Bvd. du 11 novembre 1918, F-69622 Villeurbanne Cedex, France
- '$^b$ CERN, Theory Division, CH-1211 Geneva 23, Switzerland'
author:
- 'G. Chanfray$^a$, D. Davesne$^a$, J. Delorme$^a$, M. Ericson$^{a,b}$ and J. Marteau$^a$'
title: Pion scalar density and chiral symmetry restoration at finite temperature and density
---
1 true cm
LYCEN 99128\
CERN-TH/99-362
Introduction {#introduction .unnumbered}
============
Pions play a crucial role in chiral symmetry restoration, due to their Goldstone boson character. The amount of restoration is measured by the modification of the order parameter, i.e. the quark condensate density, with respect to the vacuum value. In this context the evaluation of the expectation value of the squared pion field, linked to the scalar density of pions $\rho ^\pi_S$ by $\rho ^\pi _S =m_\pi \langle \Phi^2
\rangle$, is of a great interest. For a single nucleon this quantity governs the total amount of chiral symmetry restoration of pionic origin, according to [@CE; @CEW]: $$\label{eq1}
\frac{m_\pi^2}{2} \int d^3\vec x {\langle N|} \Phi^2(x)
{|N \rangle} = 2 m_q \int d^3\vec x {\langle N|} \Delta ^\pi \overline q q(x) {|N \rangle}
= \Sigma^\pi_N = {m_\pi\over 2} N_\pi,$$ where $N_\pi$ is the scalar number of pions in the nucleon cloud, $\Sigma^\pi_N$ is the part of the nucleon $\Sigma$ commutator of pionic origin and ${\langle N|} \Delta ^\pi \overline q q(x) {|N \rangle}$ represents the corresponding modification of the quark condensate with respect to the vacuum value. Similarly, in a uniform nuclear medium of density $\rho$, or in a heat bath, the evolution of the quark condensate originating from the pions is linked, to one-pion loop order, to the average value $\langle \Phi^2 \rangle$ by : $$\label{eq2}
\frac{\Delta ^\pi \langle \bar{q} q (\rho ,T) \rangle}{\langle \bar{q} q (0,0)
\rangle} = - \frac{\langle \Phi^2 \rangle}{2f_\pi^2}\,. \label{qqbar}$$
The same quantity $\langle \Phi^2 \rangle$ governs also the quenching factor, $1-\langle \Phi^2 \rangle/3f_\pi ^2$, of coupling constants such as the nucleonic axial coupling constant $g_A$ or the pion decay one $f_\pi$, originating from pion loops, which is the counterpart of the mixing of axial and vector currents [@IOF; @CDE1]. This type of quenching has to be seen as a manifestation of chiral symmetry restoration and should also apply to the case of $\rho$ meson excitation by virtual photons as enters in relativistic heavy ion collisions. It is therefore interesting to evaluate the quantity $\langle \Phi^2\rangle$ in the conditions of such experiments. Now, the fireball which is the source of the dileptons contains, besides thermal pions, a significant residual baryonic background We have therefore to understand how the pion density evolves at finite values of both temperature and baryonic chemical potential. The first order approximation for the quantity $\langle \Phi^2 \rangle $ adds the values for a pure heat bath and for a cold baryonic medium, writing with obvious notations : $$\langle \Phi^2 \rangle(\rho , T) = \langle \Phi^2\rangle_T (\rho=0)
+ \langle \Phi^2\rangle_\rho (T=0) \,.
\label{addit}$$ However this approximation is likely to be crude since the temperature has an effect on the pion density of the nuclear medium and on the other hand the presence of the baryonic background modifies the number of pions thermally excited. As an illustration of the second point, the pion density in the baryonic vacuum is fixed by the Bose-Einstein factor which is $\left ( e^{\omega _k/T} -1 \right ) ^{-1}$ for pions of momentum $\vec k$, with $\omega _k= \sqrt{\vec k ^2 + m_\pi ^2}$. In the nuclear medium the pion becomes a quasi-particle with a broad width. It can decay for instance into a particle-hole pair which has a smaller energy than the free pion. Its excitation is then favored by the thermal factor. There is therefore a mutual influence between temperature and density that we will investigate.
The article is organized as follows. In the first section we evaluate the pion density of a nuclear medium at zero temperature. We relate this quantity to the nuclear response to a pion-like excitation. We evaluate this response in the RPA scheme, taking also into account the two particle-two hole ($2p$-$2h$) excitations. Within this framework, we study the deviation with respect to the independent nucleon approximation. In the second section we introduce the effect of the temperature through the modification of the nuclear responses. In the third section we incorporate the influence of the finite baryonic chemical potential in the heat bath case.
Pion scalar density in the cold nuclear medium
==============================================
This quantity was discussed in relation to the quark condensate modification by Chanfray and Ericson [@CE]. They discussed its deviations with respect to free nucleons, introducing the nuclear response to pion-like excitation, treated in the static case $M_N \rightarrow
\infty$. The extension to the non-static situation can be performed through the time-ordered graphs of Fig. \[fig1\] where the cross represents the point at which the pions are created or annihilated. The pion momentum is denoted $q$ and $\omega$ is the excitation energy of the nuclear system in the intermediate state. The sum of the four graphs leads to the following expression which has also been derived with a different method in Ref. [@CD] : $$\begin{aligned}
Eq.
\label{eq6}
\langle \Phi^2 \rangle = \frac{\rho}{A} \, 3 \, \frac{g_{\pi NN}^2}{4M_N^2} \,
\int \frac{d^3q}{(2\pi)^3 \, 2 \omega_q^2} \int_0^\infty d\omega \left[
\frac{1}{(\omega+\omega_q)^2} + \frac{1}{\omega_q}\frac{1}{(\omega+\omega_q)}
\right] \, v^2(Q^2) \,R_L(\omega,{\hbox{\boldmath$q$\unboldmath}}) \end{aligned}$$ where $Q^2 =\omega^2-{\hbox{\boldmath$q$\unboldmath}}^2$, $v(Q^2)$ is the form factor of the pion vertex for which we use a monopole form : $$\label{formfactor}
v(Q^2) = \frac{\Lambda^2 - m_\pi^2}{\Lambda^2 - Q^2}.$$ Finally, $R_L$ represents the spin-isospin longitudinal response function : $$R_L(\omega,{\hbox{\boldmath$q$\unboldmath}})=
\sum_n\,\vert\langle n|\sum_{i=1}^A\,\sigma_i.{\hbox{\boldmath$q$\unboldmath}}\, \tau_i^a
\,e^{i {\hbox{\boldmath$q$\unboldmath}}.{\hbox{\boldmath$r$\unboldmath}}_i}|0\rangle\vert ^2\,\delta(\omega-E_n).$$
![\[fig1\] *Time-ordered graphs for pion exchange.*](diag.eps){width="8" height="2"}
In the actual calculation, we add in the response $R_L$ the excitation of the Delta resonance with the standard replacement : $${\hbox{\boldmath$\sigma$\unboldmath}}_i.{\hbox{\boldmath$q$\unboldmath}} \,\, \tau^a_i \longrightarrow
\frac{g_{\scriptscriptstyle \pi N\Delta}}{g_{\scriptscriptstyle \pi NN}} \,
{\hbox{\boldmath$S$\unboldmath}}^\dagger_i.{\hbox{\boldmath$q$\unboldmath}} \,\, (T^a_i)^\dagger$$ where ${\hbox{\boldmath$S$\unboldmath}}^\dagger$ (${\hbox{\boldmath$T$\unboldmath}}^\dagger$) is the spin (isospin) transition operator connecting the spin (isospin) $\frac{1}{2}$ and $\frac{3}{2}$ states [@EW]. Note that for a matter of convenience we have incorporated in this operator the ratio of the $\pi N\Delta$ and $\pi NN$ coupling constants. We have also assumed the same form factors, $v(Q^2)$, at the $\pi NN$ vertex and at the $\pi N\Delta$ vertices. Finally we recall the link between the response function and the polarization propagator $\Pi(\omega,{\hbox{\boldmath$q$\unboldmath}},{\hbox{\boldmath$q'$\unboldmath}})$ which we will use in the following : $$\label{resp}
R(\omega,{\hbox{\boldmath$q$\unboldmath}}) = -\frac{V}{\pi} \, \mathrm{Im} \Pi(\omega,{\hbox{\boldmath$q$\unboldmath}},
{\hbox{\boldmath$q$\unboldmath}})$$ We will first discuss the result for the free nucleon.
Free nucleon.
-------------
In the nucleon case, where the response $R_L$ reduces to a simple expression, Eq. (\[eq6\]) provides for the (scalar) pion number $N_\pi$ in the nucleon cloud : $$\begin{aligned}
N_\pi & = & m_\pi\int d^3 \vec x {\langle N|} \Phi^2 ({\bf x}{|N \rangle} \nonumber \\
& = & m_\pi \, 3 \, \frac{g_{\pi NN}^2}{4M^2} \, \int
\frac{d^3q}{(2\pi)^3} {\hbox{\boldmath$q$\unboldmath}}^2 \left\{\frac{1}{\omega_q} \left[
\frac{1}{2\omega_q} \frac{1}{(\varepsilon_q+\omega_q)^2} +
\frac{1}{2\omega_q^2}\frac{1}{(\varepsilon_q+\omega_q)} \right]
\, v^2(Q^2) \right. \nonumber \\
& & \left. + \frac{4}{9} \, (\frac{g_{\scriptscriptstyle \pi N\Delta}}
{g_{\scriptscriptstyle \pi NN}})^2 \int_0^\infty \frac{d\omega}{\omega_q}
\left[ \frac{1}{2\omega_q} \frac{1}{(\omega+\omega_q)^2} +
\frac{1}{2\omega_q^2}\frac{1}{(\omega+\omega_q)} \right] \,
v^2(Q^2)\left(- {1\over \pi}\mathrm{Im}
\frac{1}{\omega-\omega_\Delta
+ i\frac{\Gamma_\Delta}{2}}\right)\right\}\,.\end{aligned}$$ In the above equation we have used the following notations : $\varepsilon_q =
{\hbox{\boldmath$q$\unboldmath}}^2/2M_N$ and $\omega_\Delta = M_\Delta - M_N + {\hbox{\boldmath$q$\unboldmath}}^2/2M_\Delta$. The energy dependence of the Delta width $\Gamma_\Delta$ is taken from the analysis of the pion-nucleon scattering [@EW].
Our numerical inputs for the evaluation of $N_\pi$ are defined as follows. Without form-factor the integrals diverge linearly. The resulting value is then quite sensitive to the cut-off function. We stress, however, that the specific case of the single nucleon is not the purpose of this paper. It is for us an element of comparison to introduce the nuclear effects. Since we want to restrict the calculation to a region where nuclear effects are reasonably under control, we have limited the integrals to : $q = 1$ GeV and $\omega = 1$ GeV. The parameters are chosen so as to obtain a value $\Sigma^\pi_N =
\frac{1}{2} m_\pi N_\pi = 30$ MeV which is well in the accepted range [@BMG; @JAM; @CDE2]. This is achieved with $\Lambda = 1$GeV and $(g_{\scriptscriptstyle \pi N\Delta}/g_{\scriptscriptstyle \pi NN})^2 = 3.8$.
Infinite nuclear matter.
------------------------
We now turn to the case of infinite nuclear matter. In order to evaluate the response functions, we use the method of Delorme and Guichon [@DG] who calculated the zero order response in the local density approximation and then solved exactly the RPA equations. Their zero order response function also includes the $2p$-$2h$ excitations. The corresponding Feynman diagrams of such processes are displayed in Fig. \[fig2\].
![\[fig2\] *Feynman diagrams for the $2p$-$2h$ processes. The double line represents the Delta resonance.*](fig4.eps){width="16.5" height="4cm"}
They are calculated in two steps. First we single out the contributions which reduce to a medium modification of the $\Delta$ self-energy, for which the parametrization of Ref. [@OST] is used. This parametrization includes some $3p$-$3h$ excitation states as well. For the rest we use the results of Shimizu-Faessler [@SF] who evaluated the two nucleons p-wave pion absorption at threshold $(\omega = m_\pi)$, from which the $\Delta$ self-energy part, already taken into account, is separated out. As for each the remaining contributions, an energy extrapolation suggested by the many-body diagrammatic interpretation is performed.
We solve the RPA equations in the ring approximation : $\Pi = \Pi^0 +
\Pi^0 \, {\mathcal {V}} \, \Pi$. Here ${\mathcal {V}}$ is the particle-hole ($p$-$h$) interaction with the standard formulation : ${\mathcal {V}} = V_\pi + V_{g'}$ where the second piece is the short-range Landau-Migdal part. More explicitely with our definition of the response, ${\mathcal {V}}$ reads : $${\mathcal {V}} = {v^2(Q^2)\over {\hbox{\boldmath$q$\unboldmath}}^2}\left ({{\hbox{\boldmath$q^2$\unboldmath}}\over
Q^2-m^2_\pi}+g'\right)\,.$$ The corresponding Landau-Migdal parameters $g'$ are different in the various channels : ${g'}_{NN}$ for the $NN$ sector, ${g'}_{\Delta\Delta}$ for the $\Delta$ one, ${g'}_{N\Delta}$ for the mixing of $N N$ and $\Delta N$ excitations. We adopt the following values: $${g'}_{NN} = 0.7,\,\,\, {g'}_{N\Delta} = {g'}_{\Delta\Delta} = 0.5\,.$$ The results are illustrated on Fig. \[fig3\] which shows the energy dependence of the zero order and RPA responses, for a fixed value of the momentum $q = 300 $MeV. The bare response presents a low-energy peak corresponding to the $NN^{-1}$ excitation and a high-energy one ($\Delta N ^{-1}$ excitations).
![\[fig3\] *Response function per nucleon at normal nuclear density as a function of the energy for a fixed momentum $q$ = 300 MeV. The dot-dashed and continuous lines represent respectively the zero order and RPA responses.*](qfrpa3.eps){width="9"}
The figure displays the RPA enhancement of the low energy peak, introduced by Alberico et [*al.*]{} [@AEM], which arises from the attractive nature of the $p$-$h$ interaction. It also displays the collective behaviour of the $\Delta$ excitation with a splitting into two branches.
Once the energy and momentum integrations are performed, we find, as in Ref. [@CE], a moderate increase as compared to the free nucleon value, with the values per nucleon at normal nuclear density : $$\tilde\Sigma^\pi_N \equiv\Sigma^\pi_A/A = 38.5 \,\,\, \mathrm{MeV} \,\,\,
\mathrm{and} \,\,\, N^\pi_{A}/A = 0.55\,$$ versus $30$MeV and 0.40 respectively for the free nucleon. The quantity $\tilde\Sigma^\pi_N$ is the (medium modified) effective sigma commutator.
More precisely, we decompose the response function $\Pi$ (accordingly $\Sigma ^\pi$) into four types, depending on the kind of states which are excited at each external vertices : $NN$, $N\Delta$, $\Delta N$ and $\Delta\Delta$ (Fig. \[fig4\]).
![\[fig4\] *Symbolic representation of the $NN$, $\Delta N$, $N\Delta$and $\Delta\Delta$ response functions. The Delta resonance is represented by the double line.*](fig5.eps){width="16.5" height="4cm"}
The density evolution of the different components of the sigma commutator are represented on Fig. \[fig5\] both without and with RPA. Notice that, in absence of the RPA, there is already a contribution of the $N\Delta$ channel at finite density due to the $2p$-$2h$ excitations. The overall RPA increase of the sigma commutator mainly comes from that of the $\Sigma_{N\Delta} + \Sigma_{\Delta N}$ parts. These last quantities embody the mixing of the $\Delta N^{-1}$ configurations into the $NN^{-1}$ ones. This is well known to be responsible for the enhancement of the low energy response ([*i.e.*]{} the $NN^{-1}$ excitations) [@ME].
Inclusion of temperature
========================
Influence on the virtual pion cloud
-----------------------------------
We now introduce the temperature, [*via*]{} the Matsubara formalism : all the integrals over energy are replaced by an infinite sum over Matsubara frequencies. For the $NN^{-1}$ sector, the generalized Lindhart function (the imaginary part of which is proportional to the response function) at finite temperature is discussed in textbooks (see e.g. Ref. [@FW]). The result is the replacement of the Heaviside functions that characterize the occupation of fermion states at zero temperature by the Fermi-Dirac distributions. We have generalized this procedure to particle with width and applied it to $\Delta$-$N^{-1}$ , $N$-$\Delta^{-1}$ and $\Delta$-$\Delta ^{-1}$ rings. The generalized Lindhart function $L_{ab^{-1}}$ for a process involving a particle of type (a) and a hole of type (b) is found to be : $$L_{ab^{-1}}(\omega, \vec q) = - \frac{N_{S,I}}{4\pi^2}\int k^2 dk d(\cos
\theta) \frac{f(\omega _k ^b)(1-f(\omega _{k+q} ^a))}{\omega -\omega _{k+q}
^a + \omega _k ^b +\frac{i}{2}\Gamma ^a (\omega _k ^b + \omega)
+ \frac{i} {2}\Gamma ^b
(\omega _{k+q} ^a- \omega)}$$ $$- \frac{f(\omega _k ^b)(1-f(\omega _{k+q} ^a))}
{\omega +\omega _{k+q} ^a
- \omega _k ^b +\frac{i}{2}\Gamma ^a (\omega _k ^b - \omega) +
\frac{i} {2}\Gamma ^b
(\omega _{k+q} ^a + \omega)}$$ where $N_{S,I}$ is a constant arising from the summation of spin and isospin and $\Gamma ^{a,b} (\omega)$ represents the width of the particle of type (a,b) for an energy $\omega$. The occupation number of hadron species $a$ is : $$f(\omega^a_k)={1\over exp({\omega^a_k-\mu\over T})+1}$$ where $\mu$ is the (common) chemical potential for baryons, the value of which fixes the baryonic density $\rho$ at a given temperature. As implicitely stated before, we limit ourselves to nucleons and deltas [*i.e.*]{} $\rho=\rho_N+\rho_\Delta$.
Concerning the $\Delta$ width, things are somewhat more complicated. In the medium, the pionic decay channel $\Delta \rightarrow \pi \, N$ is partly suppressed due to Pauli blocking. At the same time, other channels open, the pion being replaced by $1p$-$1h$, $2p$-$2h$, etc... At normal density and zero temperature the pionic channel remains dominant according to Ref. [@OST]. It represents approximatively 75% of the total width ($\simeq 90$ MeV to be compared with 120 MeV at the resonance energy) and the non-pionic decay channels only the remaining 25%. In view of the difficulties of a full calculation of the temperature effects, we have adopted the simplified following strategy : we have kept the parametrization of Ref. [@OST], derived at $T = 0$, for the non-pionic decay channel. We have introduced the temperature effects [*via*]{} the Matsubara formalism only for the main pionic part of the width.
The Figure \[tfinienu\] displays the temperature evolutiom of the zero order response function. The dashed line, which represents the $T=0$ case, exhibits the $N$-$N^{-1}$ and $\Delta$-$N^{-1}$ structures. Increasing temperature tends to wash out more and more these peaks. At the same time an overall suppression effect occurs.
![*Zero order response at a baryonic density equal half nuclear matter density ($\rho=0.25 m^3_\pi$) as a function of energy for a fixed momentum $q=300$ MeV for three temperatures. Dashed lines : $T=0$. Thin full line : $T=50$ MeV. Thick full line : $T=150$ MeV.*[]{data-label="tfinienu"}](tfinienu.eps){width="10"}
Note that the $\Delta$ branch is less affected by the temperature because of the higher energies involved. In the RPA case (see Fig. \[tfinierpa\]) , one observes a similar behavior : an important general decrease and the loss of the lower energy structures.
![*Same as before but for the RPA response.*[]{data-label="tfinierpa"}](tfinierpa.eps){width="10"}
We now turn to the question of the thermal pions present in the heat bath.
Inclusion of thermal pions.
---------------------------
The effects previously discussed concerned only the modification, due to the temperature, of the virtual pion density present in the nuclear medium. At finite temperature, thermally excited pionic modes (quasi-pions) are also present and they give an additional contribution to the pion scalar density. Here for a better illustration we give the result for $\langle \Phi^2\rangle/2 f^2_\pi$ which according to Eq.(\[qqbar\]) governs the amount of chiral symmetry restoration of pionic origin: $$\begin{aligned}
{\langle \Phi^2\rangle\over 2 f^2_\pi}
& = & {\rho\over A}\frac{3}{2f_{\pi}^2} {g^2_{\pi NN}\over 4 M_N^2}
\int{d^3 q\over (2\pi)^3}\, \int_0^\infty d\omega\,
\left({1\over 2\omega^2_q (\omega+\omega_q)^2}\,+ \,
{1\over 2\omega^3_q (\omega+\omega_q)}\right)\,v^2(Q^2)\,R_L(\omega,
{\hbox{\boldmath$q$\unboldmath}}) \\
& & + \frac{3}{2f_{\pi}^2} \int{d^3 q\over (2 \pi)^3}\,\int_0^\infty d\omega\,
n(\omega)\,\left(-{2\over \pi}\right)
\mathrm{Im}\,D(\omega,{\hbox{\boldmath$q$\unboldmath}}) \\
& \equiv & \frac{\rho \tilde \Sigma_B^{\pi}}{f^2_\pi m^2_\pi}+
\frac{<\Phi^2>_T}{2f_{\pi}^2}\end{aligned}$$ where $n(\omega)=1/\left(exp(\omega/T) -1\right)$ is the Bose occupation factor and $D(\omega,{\hbox{\boldmath$q$\unboldmath}})$ the quasi-pion propagator. The second identity defines the quantities $\tilde\Sigma_B^{\pi}$ and $<\Phi^2>_T$, associated respectively with the first and second pieces of the r.h.s. of the above equation: $\tilde\Sigma_B^{\pi}$ represents an effective, temperature dependent, sigma commmutator per baryon, whereas $<\Phi^2>_T$ is the scalar density of quasi-pions thermally excited.
Results
=======
In order to display the condensate evolution, all the forthcoming figures show its relative decrease, [*i.e.*]{} the quantity $ \Phi^2/2f_\pi^2$ according to Eq. (\[eq2\]). We stress again that the points of interest are the influence of temperature on the nuclear pionic cloud contribution, the influence of the baryonic density on the thermal pions one and finally how large is the deviation from the additive approximation of Eq. (\[addit\]).
We first present on Fig. \[sigrpa\] the contribution of the virtual pion cloud alone alone (term in $\tilde\Sigma_B^{\pi}$). Each box shows the temperature evolution at fixed baryonic density. It illustrates the suppression effect of the temperature which originates in the quenching of the nuclear response previously mentioned. The increase with baryonic density observed in this figure reflects the obvious fact that the pionic density follows the baryon one.
![*Relative decrease of the condensate coming from the virtual pion cloud part as a function of the density and the temperature. Each box corresponds to a fixed density as indicated. The density increases by steps of $0.2 \rho_0$ from left to right between $0.4$ and $1.6 \rho_0$. In each box the points correspond to temperature increases by steps of $0.10 m_\pi$ between $0.05$ and $1.05 m_\pi$.* []{data-label="sigrpa"}](sigrpa.eps){width="8"}
In Figure \[sigphi\] we present in the same fashion the condensate decrease due to thermal pions alone (term in $\langle \Phi^2\rangle_T/2f_{\pi}^2$). The iso-temperature curves show the influence of the baryonic density which pushes down the quasi-pion excitation energy, thus increasing their thermal excitation.
![*Same as the preceding figure but for the thermal pions. As a guidance in order to display the influence of the density on the thermal pions, points of equal temperature are joined by a dashed line.*[]{data-label="sigphi"}](sigtot2.eps){width="8"}
Finally in the last figure (Fig. \[sigtot1\]) we present the sum of both contributions. The competition between the variations of both terms with respect to the temperature is the source of the observed parabolic type shape. For comparison, we have plotted in open circles the approximation of Eq. (\[addit\]) where the effect of the thermal pions at zero density is simply added to that of the pion cloud at zero temperature. This approximation does not display the hollow shape of the exact calculation: it overestimates the latter by at most 15% in the medium part of the temperature range we have considered, the deviation becoming quite small beyond $T \approx$ 90-100 MeV. In this region the decrease of the pionic cloud contribution with temperature (Fig. \[sigrpa\]) practically compensates the enhancement of the thermal excitations by density effects. The most important conclusion which can be drawn from Fig. \[sigtot1\] is that, due to the nuclear pions, the pion scalar density is much larger than in the absence of nuclear effects, already for densities of the order of 0.6 $\rho_0$.
![*Sum of both contributions of the virtual pion cloud and the thermal pions (black circles). For comparison the open circles represent the additive approximation of Eq. (\[addit\])*[]{data-label="sigtot1"}](sigtot1.eps){width="8"}
Conclusion. {#conclusion. .unnumbered}
===========
In conclusion we have studied the evolution of the quark condenaste of pionic origin under the simultaneous influence of the baryonic density and temperature. It is related to the scalar pionic density which comes on the one hand from the virtual nuclear pions and on the other hand from the thermally excited ones. We have expressed the first contribution in terms of the nuclear response to a pion-like excitation and evaluated it for the case of nuclear matter in the RPA scheme, first at zero and then at finite temperature. We have shown that the RPA produces a sizeable enhancement ($\approx 30\%$), while instead the temperature washes out the peaks and suppresses the nuclear response, hence decreasing the virtual pion density.
As for the thermally excited pions we have shown that the presence of the baryonic background appreciably enhances their number. The cause has to be found in the lowering of the quasi-pion excitation energies, which favours their thermal excitation. When the densities of both types of pions are added, the mutual influences which go in opposite directions cancel their effects to a large extent. In the density and temperature domain that we have explored, the additive assumption of Eq. (\[addit\]) which neglects the mutual influence is a good approximation. It deviates from the exact result by no more than 15%, the deviation being maximum around $T \approx 50$ MeV. At this T value the additive approximation slightly overestimates the pionic density.
Our study has shown that, even at moderate baryonic density, the virtual nuclear pions are a major component of the overall scalar pion density. As an example, at nuclear matter density, they dominate in the temperature range we have considered, [*i.e.*]{} up to at least $T \approx 150$ MeV. Since the pion is the agent for the mixing of the vector and axial correlators, the consequence of our study is that the existence of a baryonic background, if any, should not be ignored in this mixing.
[abc]{}
G. Chanfray and M. Ericson, Nucl. Phys. **A556** (1993) 427.
G. Chanfray, M. Ericson and J. Wambach, Phys. Lett. **B388** (1996) 673.
M. Dey, V.L. Eletsky and B.L. Ioffe, Phys. Lett. **B252** (1990) 620.
G. Chanfray, J. Delorme and M. Ericson, Nucl. Phys. **A637** (1998) 421.
G. Chanfray and D. Davesne, Nucl. Phys. **A646** (1999) 125.
T. Ericson and W. Weise, *Pions and nuclei* (Oxford Science Publications, Clarendon Press, Oxford 1988).
M. Birse and J. Mc Govern, Phys. Lett. **B292** (1992) 242.
I. Jameson, G. Chanfray and A.W. Thomas, Journal of Physics **G18** (1992) L159.
G. Chanfray, J. Delorme and M. Ericson, Phys. Lett. **B455** (1999) 39.
J. Delorme and P.A.M. Guichon, Phys. Lett. **B264** (1991) 157; more details and earlier references can be found in the unpublished report LYCEN 8906 (1989).
E. Oset, L.L. Salcedo and D. Strottman, Phys. Lett. **B165** (1985) 13.
K. Shimizu and A. Faessler, Nucl. Ph. **A306** (1978) 311; [*ibid.*]{} **A333** (1980) 495.
W.M. Alberico, M. Ericson and A. Molinari, Phys. Lett. **92B** (1980) 153.
M. Ericson, in *Proc. International School “Mesons, isobars, quarks and nuclear excitations, Erice” 1983, edited by D. Wilkinson* (Pergamon Press 1983).
A.L. Fetter and J.D. Walecka, *Quantum theory of many-particle systems* (McGraw-Hill, New York 1971).
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Mark Olchanski[^1]'
- 'Jenny G. Sorce[^2]'
bibliography:
- 'biblicomplete.bib'
title: Merger types forming the Virgo cluster in recent gigayears
---
Introduction
============
The Virgo cluster of galaxies is our closest cluster neighbor. As such it receives much-in-depth attention from observers aiming to understand galaxy formation and evolution within clusters . However, from the numerical side, it is a real challenge to obtain a good simulacrum of the Virgo cluster to precisely compare the simulated and observed galaxy populations to test and calibrate galaxy formation and evolution models. The parameters that the numerical cluster should reproduce to be considered as an efficient simulacrum of the Virgo cluster are simply difficult to completely determine. The formation history of the cluster is of particular importance since optimal comparisons between observed and simulated galaxy populations imply that observed and numerical clusters should have formed from similar mass subhalos at the time of merging [e.g. @2015ApJ...807...88G for the stellar-to-halo mass ratio].\
Simulations that resemble the local Universe are an interesting approach to determine the detailed formation history of the Virgo cluster and even to obtain high quality Virgo simulacra [e.g. @1987ApJ...323L.103B; @2001ApJS..137....1B; @2010arXiv1005.2687G; @2010MNRAS.406.1007L; @2013MNRAS.429L..84K]. These simulations stem from initial conditions that have been constrained with observational data that are either radial peculiar velocities [e.g. @2008ApJ...676..184T; @2013AJ....146...86T; @2016AJ....152...50T] or redshift surveys [e.g. @2011MNRAS.416.2840L; @2012ApJS..199...26H]. Different techniques permit reconstructing the constrained initial conditions either forward [e.g. @2013MNRAS.432..894J; @2013MNRAS.435.2065H; @2014ApJ...794...94W] or backward [e.g. @1989ApJ...336L...5B; @1990ApJ...364..349D; @1991ApJ...380L...5H; @1992ApJ...384..448H; @1996MNRAS.281...84V; @2008MNRAS.389..497K; @2008PhyD..237.2139L; @2016MNRAS.457..172L]. Resulting simulations reproduce the local Large Scale Structure as well as smaller structures down to the cluster scale [e.g. @2016MNRAS.455.2078S]. These simulations have then the merit of reproducing the environment of the Virgo cluster and the cluster itself in its entirety nowadays. In addition, @2016MNRAS.460.2015S showed recently that the resulting Virgo halos not only share a similar quiet merging history within the last seven gigayears but also that they form along a preferential direction in agreement with the theoretical formation history of the Virgo cluster established from observations [@2000ApJ...543L..27W].\
In this paper, the merger trees of the Virgo-like halos are studied in more details. Namely, while in the previous study the main focus was onto the Virgo cluster in its entirety (i.e. all the particles that constitute the cluster at z=0), in this paper we have extended the work to the merger trees and their branches to determine how deeply the simulations are indeed constrained. In other words, we seek to quantify up to what level the merger tree scatter expected from random halos within the same mass range as the Virgo cluster is reduced for the Virgo halos, but also how efficient simulacra they can be for further studies of substructures and galaxy populations. The paper opens with a short description of the 15 constrained simulations used for the study proposed in this paper and of the unique Virgo candidate identified in each one. In a third section, the merger trees and in particular the main and second (second after the main) branches are studied in detail. Finally, we conclude that the constrained scheme proves to be efficient to some extent also at the merger tree level and provides some further indications regarding the mergers that formed the Virgo cluster within the last few gigayears.
Virgo halos
===========
Constrained simulations
-----------------------
@2016MNRAS.455.2078S described in detail the scheme used to build the constrained initial conditions and to run the simulations. Furthermore the introduction of this paper summarises the various existing techniques. Thus, we summarise in this section only the main steps required to produce the simulations constrained with observational radial peculiar velocity catalogs we use here. We also give a brief description of their purpose with the latest references in the literature of the algorithms used:
- grouping [e.g. @2015AJ....149..171T; @2015AJ....149...54T] of the radial peculiar velocity catalog to remove non-linear virial motions that would affect the linear reconstruction obtained with the linear method [e.g. @2017MNRAS.468.1812S; @2017arXiv170503020S].
- minimizing the biases [@2015MNRAS.450.2644S] inherent to any observational radial peculiar velocity catalog.
- reconstructing the cosmic displacement field with the Wiener-Filter technique [linear minimum variance estimator, in abridged form WF, @1995ApJ...449..446Z; @1999ApJ...520..413Z] applied to the peculiar velocity constraints.
- relocating constraints to the positions of their progenitors using the Reverse Zel’dovich Approximation and the reconstructed cosmic displacement field [@2013MNRAS.430..888D] and replacing noisy radial peculiar velocities by their WF 3D reconstructions [@2014MNRAS.437.3586S]. This ensures that, after evolving the structures with an N-body code from an early redshift until today, structures are at the same position (within the 2 [[ h$^{-1}$ Mpc]{}]{} limit of the linear-threshold) to that observed. Keeping constraints at their current position to build initial conditions would indeed result in a shift of some $\ge $10 [[ h$^{-1}$ Mpc]{}]{} between observed and simulated structures after a complete evolution of the initial conditions until today.
- producing density fields constrained by the modified observational peculiar velocities combined with a random realization to restore statistically the missing structures using the Constrained Realization technique [CR, @1991ApJ...380L...5H; @1992ApJ...384..448H]
- rescaling the density fields to build constrained initial conditions and increasing the resolution by adding small scale features (e.g. [[Ginnungagap]{}]{} code[^3]).
@2013AJ....146...86T supplied the observational catalog used as constraints. Since the Virgo cluster is the object of study here, Figure \[fig:cumuldistance\] permits grasping the distribution of the constraints from the grouped catalog around the Virgo cluster. It presents the cumulative distribution function of the number of constraints as a function of the distance to the Virgo cluster. Initial conditions are evolved within the Planck cosmology framework in 500 [[ h$^{-1}$ Mpc]{}]{} boxes with 512$^3$ particles (particle mass: 8$\times$10$^{10}$ ${{\,\rm h^{-1}M}_\odot}$) with the N-body code [[Gadget]{}]{} [@2005MNRAS.364.1105S].
. \[fig:cumuldistance\]
Virgo and random halos
----------------------
\
. Masses are given in ${{\,\rm h^{-1}M}_\odot}$ for each halo. \[fig:appetizer\]
Subsequently, following @2016MNRAS.460.2015S, the Virgo dark matter halo in each constrained simulation is identified using the Amiga’s Halo Finder [AHF, @2009ApJS..182..608K]. By definition of a constrained simulation, the resulting “Virgos” are at the proper location with respect to the observer (assumed to be at the center of the box, the box being oriented in the same direction as the local Universe in supergalactic coordinates) and more importantly in the proper local (large scale) environment.
To quantify the efficiency of the constraining scheme used to produce the local Universe-like simulations, we select a set of random halos that are within the same mass range as the Virgo halos. These random halos are extracted from the constrained simulations. To ensure that these halos are not constrained halos or at least that the constrained nature of the simulation does not impact the results, halos are selected successively randomly in the entire simulations and then outside and in the constrained zone of the simulations. Conclusions are identical in the three cases. The observational catalog extends indeed up to about 150 [[ h$^{-1}$ Mpc]{}]{} with 50% of the data within 60 [[ h$^{-1}$ Mpc]{}]{}. It is thus completely reasonable to assume that beyond 200 [[ h$^{-1}$ Mpc]{}]{}, halos are not affected by the constraints while within they are [@2016MNRAS.455.2078S].
For further studies, 15 Virgo halos and several sets of 15 random halos are thus at our disposal. There are two ways to determine whether a property is constrained by the constraining scheme: 1) the mean value of the property differs between random and Virgo halos, 2) the range of possible values for the property is smaller for Virgos than for random halos. In other words, the standard deviation of the property is smaller for the former than for the latter. A property can fulfill both conditions, that is, not only the mean but also the standard deviation differ significantly between Virgo and random halos with a smaller standard deviation in the former case that in the latter case.
Figure \[fig:appetizer\] acts as a summary of previous studies of the Virgo simulacra [@2016MNRAS.460.2015S] and as an illustration of the study in the rest of this paper. Its upper panel shows the gathering of particles that belong to five Virgo halos from redshift 5 to redshift 0 through redshifts 2, 0.5 and 0.25 (from dark blue to red). This same gathering is represented in the bottom panel for five random halos with similar masses as the Virgo halos. Since the position of the random halos in the box is arbitrary, the latter are relocated at (0,0,0) while Virgo halos are left at their positions in the box. Virgo halos are extremely similar in terms of positions. Moreover, particles that constitute them at redshift zero gather in the same way and come from the same location in the box. On the contrary, random halos present various formation history at every redshift presented here. Inspecting the YZ and XZ planes reveals the same behavior. These observations to be cumulated with those made by @2016MNRAS.460.2015S clearly state that choosing random halos within the same mass range as the Virgo cluster are a necessary but not sufficient condition to obtain proper simulacra of the Virgo cluster for further study of its substructure and galaxy population. In the next section, these qualitative observations are quantified using merger trees.
Progenitors
===========
\
In this paper, the merger trees of the Virgo halos are under scrutiny. Firstly, we saught to understand whether their merger trees are constrained and differ from those of random halos. Secondly, assuming that the random and constrained merger trees differ, we studied the properties of their different branches separately to draw additional and specific information regarding the formation of the Virgo cluster of galaxies at the sub-halo level. This knowledge is of extreme importance for future studies that will include baryons to compare observed and simulated galaxy populations to finally test and calibrate galaxy formation and evolution models. Galaxy populations are indeed not only sensitive to the large scale environment of the cluster but also to its formation history in particular its past mergers [@2017arXiv170703208D]. A similar formation history at the subhalo level is a requisite to legitimate comparisons between observed and simulated galaxy populations down to the details .
Merger Trees
------------
The halo finder detects halos constituted of 20 particles or more. However, approximately 100 particles ensures a better stability of the halos under study. Netherless, since 1) varying the number of particles required in a progenitor halo to be considered as such or 2) changing the resolution of the simulation obviously affects the number of branches in a merger tree, the key point is that whatever selection criterion is applied, the conclusions must be identical. Figure \[fig:nobranches\] shows that this is the case: whether all the halos detected by AHF (top panel, first row) or only those with more than 100 particles (bottom panel, first row) are considered, Virgo candidates (blue) have on average fewer branches than random candidates (red) within the last few gigayears (z$<$3). More precisely, they have about 10% fewer branches in the first case and up to 40% less branches in the second case for redshifts between 0.4 and 1.6: removing the halos with less than 100 particles strengthens the signal by smoothing out the noise. It suggests that Virgo halos have mostly tiny members in their secondary branches while random halos have more massive progenitors. The smallest members can be studied using higher resolution simulations. In a first step, we are interested in the most massive members. Indeed if the latter do not exhibit signs of being constrained, there is a priori no reason for the smaller members to be constrained. Thus, no further information will be available regarding the Virgo cluster. The resolution of the simulations used here allows us to study the most massive progenitors of the Virgo halos.
Interestingly, considering only the 100 or more particles halos, not only do Virgo halos have on average fewer branches than the random halos but also the standard deviation of this number of branches is smaller for the former case than for the latter. The second row of the second panel of Figure \[fig:nobranches\] shows indeed that the constrained to random standard deviation ratio of the number of branches is smaller than one for redshifts higher than 0.4. Specifically, the range of possible number of branches is narrower by up to 50% for Virgo halos than for random halos. The constraining scheme used to build the look-alike of Virgo affects both the mean and the standard deviation of the number of branches in the merger trees.
Given the fact that only 15 random candidates are used in the two first rows of each panel in Figure \[fig:nobranches\], one might wonder whether these conclusions are due to this specific set of 15 random halos. The last row of each panel in Figure \[fig:nobranches\] ensures that this is not the case: the standard deviation of the means and standard deviations obtained for different sets of 15 random halos have low values. Specifically, the mean (standard deviation) of the number of branches changes by less than about 0.6 (0.25) from one set of 15 random halos to another at the 1-$\sigma$ level. In other words, the difference between the constrained and random mean numbers of branches stays significant whatever random set of 15 random halos is used.
In the rest of this paper, only halos with more than 100 particles are considered and values (mean and standard deviation of the parameter under study) are given up to the redshift where a minimum of three halos out of the fifteen (constrained or random) are still available to derive statistics. 35 random sets of 15 random halos are used to study the impact of the 15 random halos used for comparisons. This number has been selected after checking that increasing it further does not affect anymore the values obtained for the standard deviation of the means and standard deviations.
Main & Second Branches
----------------------
An additional individual study of the Virgo halos’ merger trees shows that they are overall constituted of one prominent main branch and only one smaller - but larger than about a tenth of the average mass of the Virgo halos at redshift zero - second branch within the last four gigayears. In addition, all these second branches merge with the main progenitor within the last gigayear. On the contrary, at the current resolution within the same recent gigayears, random halos can not only have relevant second branches but also third or more branches with more than a few hundred particles. This indicates a first suggestion that the constraining scheme does not only constrain the merging history of the Virgo halos in general [@2016MNRAS.460.2015S] but also their merger tree. This low rate of mergers within the last few gigayears is in agreement with observations of the Virgo cluster. More precisely, observations shows that the substructures of the cluster are mostly dominated by massive early-type galaxies. The most massive of these galaxies can only have been formed through major merging events that occurred far in the past. The other galaxies are clearly located in the core of the cluster and their properties indicate that they are completely virialized and thus members since early epochs. As for blue star-forming systems, they are mainly dispersed at the periphery of the cluster suggesting that they started falling recently but independently. They are not part of a major merging event .\
Within the last few gigayears, the merger trees of Virgo halos seem to be left with only two prominent branches, the main one and the smaller second one, while they can have many more prominent branches for random halos. Hereafter, we study in more detail the two branches of the Virgo halos. For the sake of comparisons, the main branch and the most prominent second branch of random halos are also studied. The case of the random halos is a bit more difficult to deal with than that of Virgo halos. While the selection of the second branch is clear for Virgo halos - the only branch within the last 4 gigayears with a mass about a tenth of the average mass of the Virgo halos at redshift zero - in the case of the random halos the second branch to be compared with that of the Virgo halos is harder to define. We select the random second branch as the most massive progenitor after the main one that merged within the last gigayear with the latter. This selection is justified by the fact that it makes more sense to compare structures at the same age, namely structures that had the same time to form and grow.
@2016MNRAS.460.2015S showed that the Virgo cluster has had a quiet merging history within the last seven gigayears. The main progenitors of the Virgo halos accrete mass at a similar quiet pace and in that respect the main branch of their merger tree is constrained. It is not immediately obvious that once the main branch of the merger tree is somewhat constrained, the second branch is. It is solely because, in addition, the second branch is the only other prominent branch left or in other words, because the whole merger tree (in particular the number of branches) is somewhat constrained. Subsequently, Figure \[fig:mass\] shows that the mass of the second progenitors is also constrained: the constrained second progenitor (green solid line) is on average smaller than the random one (violet dashed line) by about 2$\sigma$ (2$\times$10$^{13}~{{\,\rm h^{-1}M}_\odot}$). The last row of the same figure confirms again that this result is independent of the 15 random halos used for the comparison. We note that the second branch does not go down to redshift zero by definition but stops at the latest redshift recorded before z=0 where it still exists. Since the merging happens within the last gigayear, this redshift is very close to zero (about 0.06). Hence the impression that the second branch goes down to redshift zero.
The second row of Figure \[fig:mass\] shows that the ratios of the constrained to random standard deviations for both the main (solid black line) and second (red dashed line) progenitors are for most, if not all, redshifts under study here below one (down to 0.5). While for the main progenitor, this result was part of the study of @2016MNRAS.460.2015S, for the second progenitor, this is a new result: the constraining scheme constrained both the mass of the second progenitor and its range of possible masses ((2$\pm$1)$\times$10$^{13}$ ${{\,\rm h^{-1}M}_\odot}$). We note that this result is only partially due to the quiet merging history of the Virgo cluster within recent gigayears [@2016MNRAS.460.2015S]. Indeed, instead of one prominent second progenitor, there could have been lots of small ones merging with the main progenitor in the last gigayear. This is not observed overall for the 15 Virgo halos studied here meaning that the constrained scheme efficiently regulates also the second progenitor of Virgo.\
@2000ApJ...543L..27W predicted with observations that the Virgo cluster must have formed along a preferential direction. @2016MNRAS.460.2015S reinforced this claim since the simulacra of the Virgo cluster formed along a given direction: an auto-correlation function, defined as the distribution of angles formed by two particles infalling onto the cluster and its center of mass divided by the distribution of angles formed by one random point (from an isotropical distribution), one infalling particle and the center of mass, shows clearly a preferential direction of infall. However, this does not necessarily imply that the main and second branches considered separately also ‘travel’ according to their own respective constrained scheme within this region of the box. Consequently, it is interesting to enquire whether within this region of the box, sub-regions can be defined for the different progenitors. To this end, we looked at other parameters such as the velocity components and the displacement or traveled distance of the center of the mass of the progenitor under study (main or second).
Before this study, Figure \[fig:12part\] gives a first visual impression of the formation of five Virgo halos within the last seven gigayears. Blue dots represent particles belonging to the most massive progenitor at z=0.06 and the red ones those of the second progenitor that still exists at z=0.06 (it has merged at z=0). Clearly for all the Virgo halos, some mergers greater than a tenth of the mass at redshift zero happen between four and seven gigayears ago (z=0.95 to 0.4) as clumps of blue dots merge onto the main progenitor. However after z=0.4, i.e. within the last four gigayears, the last merger with a mass about a tenth of the mass of the Virgo halos at z=0 that still needs to happen is that with the second progenitor (that is still forming). In addition, for each one of the Virgo halo, blue, red and black dots have a similar motional behavior.
Figure \[fig:dispvel\] allows us to investigate these motions more quantitatively for the x component of the velocity and displacement of the main (blue particles) and second (red particles) progenitors. The same color code as in Figure \[fig:mass\] is used. Appendix A gives similar results for the y and z components. The simulations are oriented in the same way as the observed local Universe and the same supergalactic coordinate system is used. The left (right) panel of Figure \[fig:dispvel\] shows the x component of the displacement (velocity) of the progenitors with respect to their last recorded redshift of existence (z=0 for the main progenitor, earlier redshift - about 0.06 - for the second progenitor or equivalently the main progenitor at redshift zero in both cases since results are insignificantly different). Results are indisputable: at late times (low redshifts, z$<$0.8) main and second progenitors gather according to their own given way (velocity, displacement) for all the Virgos under study to finally merge and form the Virgo cluster at redshift zero. The effect is the clearest in the x direction as shown on Figure \[fig:dispvel\] when compared to the y and z directions in Appendix A: the second progenitor travels on average along a given x direction (decrease of the x coordinate in the box oriented to correspond to the x supergalactic coordinate) faster than the main progenitor, probably following the latter and thus being accelerated by it before they both merge at redshift zero. Although the trend is less obvious along the y and z directions as shown in Appendix A, it is not completely inexistent. We note that replacing the reference “main progenitor at redshift zero” by “main progenitor at each redshift” to compute the displacement does not change the conclusions.
By comparison, Figure \[fig:dispvel\] shows that the random main and second progenitors display as expected no typical features on average in the x direction (the mean values are close to zero). This assertion is true also for the y and z directions as shown in Appendix A. Again this result does not depend on the set of 15 random halos selected for the comparison: the last row of the panels of Figure \[fig:dispvel\] shows that the mean and standard deviation vary little from one random set to the other. Additionally, the second row of the panels shows that the possible range of values for the x component of the velocity and displacement of the main progenitor is considerably narrower for the Virgo halos than for the random halos. This is in agreement with the fact that the Virgo halos all accrete matter along a similar direction. The trend is similar when comparing the velocity components of the constrained and random second progenitors. Appendix A shows again that although the signal exists in both the y and z directions, it is clearer in the x direction. The negative x direction being the leading one for the formation of the Virgo cluster and its progenitors considered individually is in complete agreement with the observations that show that the very local Universe (about 15 [[ h$^{-1}$ Mpc]{}]{}) goes toward the “Great Attractor” region located at “lower x values” than Virgo [e.g. see @1987Sci...237.1296W; @1989Natur.338..538K for the earliest references].
Figure \[fig:vel\] pushes further the comparisons with the relative velocity of the second progenitors with respect to the main branch at each redshift. Here again, the relative velocity is clearly constrained: the ratio of the constrained to random standard deviations is smaller than one. Additionally, the relative velocity of the constrained second progenitors is on average larger than that of the random second progenitors at redshifts smaller than 0.2.
Conclusion
==========
This paper aims to investigate the constraining power of the scheme - used here to build simulations that look like the local Universe - at the sub-cluster level. It focuses on our closest cluster-neighbor, the Virgo cluster of galaxies. Such a study is important in two linked aspects: 1) it permits gathering information regarding the formation of the Virgo cluster, an essential pre-requisite for later selection of numerical clusters to be able to compare legitimately observed and simulated galaxy populations down to the details ; or alternatively 2) it permits determining the scale down to which the constrained simulated clusters can be used to test and calibrate galaxy formation and evolution models without biases due to variation in the formation history of the simulated clusters with respect to the observed one.
The constrained simulations have already been proven to be excellent reproduction of the local Large Scale Structure [@2016MNRAS.455.2078S] as well as of the Virgo cluster as a whole, including its formation history in general [@2016MNRAS.460.2015S]. In this paper, we thus further studied the formation histories of the Virgo halos and we focused on their merger trees and in particular on their main and second branches defined hereafter. Comparing the latter to those of random halos within the same mass range, we show that the constraining scheme is still very efficient at these scales, namely at the largest sub-halo scales:
- For redshifts between 0.4 and 1.6, the merger trees of Virgo halos have on average 40% less branches than random ones considering branches with halos more massive than 8$\times$10$^{12}$ ${{\,\rm h^{-1}M}_\odot}$ . This is in agreement with their quiet merging history within recent gigayears.
- Within the last four gigayears, Virgo halos had overall, apart from their main progenitor, only one other prominent progenitor in contrast with the random halos that could have several other prominent progenitors. On average, this other prominent progenitor is about a tenth of the mean mass of the Virgo halos at redshift zero and it merged within the last gigayear with the main progenitor. In addition the random second progenitor, defined as the most massive progenitor after the main one that merged within the last gigayear, is 2$\sigma$ more massive than the constrained one. This is in agreement with the quiet history of the Virgo cluster within the last gigayears with the additional information that there was one merger more prominent (although moderately) than the others that happened within the last gigayear. This numerical statement is in agreement with observations of the galaxy population of the Virgo cluster that imply early major mergers .
- At late times, main and second progenitors of Virgo candidates follow their own accretion scheme that appears to be constrained with respect to that of main and second progenitors of random candidates. In particular, Virgo’s progenitors exhibit a clear peculiar trend of motion in the negative x supergalactic direction in agreement with our knowledge regarding the motion of the Virgo cluster. The second progenitor moves faster than the main one probably because it is in the wake of the latter. This highlights that the main and second progenitors follow their own accretion scheme in similars way in the different constrained simulations.
This paper extends the efficiency of the constraining scheme down to the largest sub-halo level. Besides providing more properties of the Virgo cluster to select optimal simulacra in typical cosmological simulations, it opens great perspectives regarding 1) detailed comparisons with observations, including substructures and markers of the past history visible in the galaxy population, to be conducted in a near future with a large sample of high resolution “Virgos”, 2) simulations including baryons to test and calibrate galaxy formation and evolution models down to the details.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank the referee for very useful comments that helped clarify the paper. JS thanks her collaborators in particular Gustavo Yepes, Stefan Gottlöber, Noam Libeskind and Yehuda Hoffman for interesting discussions. The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for providing computing time on the GCS Supercomputers SuperMUC at LRZ Munich and Jureca at JSC Juelich. JS acknowledges support from the Astronomy ESFRI and Research Infrastructure Cluster ASTERICS project, funded by the European Commission under the Horizon 2020 Programme (GA 653477), from the “Centre National d’études spatiales (CNES)” postdoctoral fellowship program as well as from the “l’Oréal-UNESCO Pour les femmes et la Science” fellowship program.
y and z components of the velocity and the displacement
=======================================================
Figure \[fig:appendix\] shows the average y and z components of the displacement and velocity of the main and second progenitors of Virgo and random candidates. The same conclusions as for the x component can be drawn. There exist both a privileged direction of displacement and a privileged velocity value for the main and second branches of Virgo halos that are not found for the branches of random halos as expected.
\
\[lastpage\]
[^1]: marko@physik.hu-berlin.de
[^2]: E-mail:
[^3]: https://github.com/ginnungagapgroup/ginnungagap
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we formalize Sprague-Grundy theory for combinatorial games in bounded arithmetic. We show that in the presence of Sprague-Grundy numbers, a fairly weak axioms capture PSPACE.'
author:
- |
Satoru Kuroda\
Gunma Prefectural Women’s University
title: |
Sprague-Grundy theory in bounded arithmetic\
(Preliminary Draft)
---
Introduction
============
Since the seminal paper by Bouton [@bouton], combinatorial games have been paid much attention in various branches of mathematics. The observation in [@bouton] is later generalized by Grundy [@grundy] and Sprague [@sprague] to form a powerful tool for finding winning strategies which is called Grundy number or Sprague-Grundy number.
Deciding the complexity of perfect information games is also a major problem in computational complexity theory. Many combinatorial games are related to space complexity such as PSPACE. For instance, Schaefer [@schaefer] proved that the game Node Kayles played on undirected graphs is complete for PSPACE. while some games have much weaker complexity such as P or LOGSPACE.
In this paper we show that with the aid of Sprague-Grundy number, a fairly weak theory of two-sort bounded arithmetic can capture PSPACE. More precisely, we introduce a function computing Sprague-Grundy number for Node Kayles together with strategy functions for both players using Sprague-Grundy number to the system $V^0$ and show that any alternating polynomial time machine can be simulated by a game of Node Kayles.
Specifically, for an alternating Turing machine $M$ and an input $X$, we construct in $V^0$ an undirected graph $G(M,X)$ such that Alice has an winning strategy if and only if $M$ accepts $X$. Since the strategy functions are polynomial time computable in Sprague-Grundy function, this result suggests that Sprague-Grundy number has such a strong computational power that manages search through polynomial space.
There are a number of literature concerning bounded arithmetic for PSPACE. Buss [@buss] in his seminal paper defined a second order theory $U^1_2$ whose provably total functions coincide with PSPACE. Later, Skelley [@skelley] defined a three sort system $W^1_1$ for PSPACE. While these theories require higher order objects compared to theories for classes inside the polynomial hierarchy, Eguchi [@eguchi] defined a PSPACE theory ${\Sigma^B_{0}}$-ID by extending the two sort language by predicates which represent inductive definition for ${\Sigma^B_{0}}$ definable relations. Our theory ${V_{NK}}$ presented in this paper is considered as a minimal theory for PSPACE as it is contained in any of the above theory. We also remark that an application of bounded arithmetic to combinatorial game theory is also given by Soltys and Wilson [@sw] who showed that strategy stealing argument can be formalized in $W^1_1$ and in turn proved that the game Chomp is in PSPACE.
We can alternatively formalize our theory with a stronger base theory such as $PV$ while introducing Sprague-Grundy function only. However we do not follow such an approach since formalizing in weak theory such as $V^0$ enables us to construct theories for combinatorial games having weaker computational power. Among such games we are particularly interested in the game NIM whose computational complexity is around LOGSPACE but no completeness result is known so far. We remark that this choice of base theory forces us to give a slightly more complicated construction of the graph $G(M,X)$.
There is a rich theory of combinatorial games with a number of games and so we hope that our result gives a neat framework for logical analysis of combinatorial games.
This paper is organized as follows: in section 2 we define our theory ${V_{NK}}$ by extending $V^0$ by functions computing winning strategies. In section 3, we show that ${V_{NK}}$ actually computes winning strategies for Node Kayles. Section 4 is devoted to the proof of our main theorem. In particular, we construct a graph so that players winning strategies witness accepting or rejecting computations.
Formalizing combinatorial games
===============================
We will formalize the argument for combinatorial games in the language of two-sort bounded arithmetic.
We will assume familiarity with basic notions and properties of two-sort bounded arithmetic. For a detail, readers should consult with textbooks such as [@cn].
Let ${\mathcal{L}_A^2}$ be the two sort language of Cook-Nguyen [@cn]. Basically, upper case letters denote binary strings and lower case letters denote natural numbers. We also adopt an unusual notation that vector presentation of lower case letters such as $\bar{z}$ also denote strings. For a language $L$ we denote the ${\Sigma^B_{0}}$ formulas in $L$ by ${\Sigma^B_{0}}(L)$.
The theory $V^0$ has defining axioms for symbols in ${\mathcal{L}_A^2}$ together with the bit-comprehension axiom for ${\Sigma^B_{0}}$ formulas. We use many properties of $V^0$ in this paper whose details can be found in [@cn].
For a string $X$ and a number $i<|X|$, $X(i)$ denotes both the predicate that the $i$th bit of $i$ is $1$ and the $i$th bit of $X$ itself. The sequence of numbers are coded by a string and we define the $i$th entry of a sequence $X$ by $X[k]$ and the length of $X$ by $Len(X)$. For two sequences $X$ and $Y$, we denote the concatenation by $X*Y$. Strings are sometimes identified with a binary sequence as $P={\langle p_0,\ldots,p_n\rangle}$. Coding such sequences and proving basic properties of sequences can be done in $V^0$.
The game we consider is known as Node Kayles which is played over undirected graphs. We code graphs by a two-dimensional array where we assume that any node has an edge to itself. Two-dimensional arrays represent directed graphs in general and undirected graphs are given as a symmetric relation which is coded by symmetric matrices. So we define $$\begin{array}{l}
DGraph(G)\Leftrightarrow\\
{\forall}x\in G{\exists}u,v<|G|(x={\langle u,v\rangle})\land\\
({\forall}u<|G|({\exists}v<|G|({\langle u,v\rangle}\in G\lor{\langle v,u\rangle}\in G))\rightarrow{\langle u,u\rangle}\in G).\\
UGraph(G)\Leftrightarrow DGraph(G)\land{\forall}u,v<|G|({\langle u,v\rangle}\in G\rightarrow{\langle v,u\rangle}\in G).\\
Node(G)=V_G=\{u<|G| \:\ {\langle u,u\rangle}\in G\}.
\end{array}$$
We define the game Node-Kayles over undirected graphs to be an impartial game played by two players Alice and Bob (Alice always moves first) starting from a graph $G$ and in the move with the option $G'$ which is a subgraph of $G$, the player chooses a node $x\in Node(G)$ and returns the subgraph $G_x$ which is defined by $$Node(G'_x)=\{y\in Node(G') : {\langle x,y\rangle}\not\in E_{G'}\}$$ and $${\langle y,z\rangle}\in G'_x\Leftrightarrow y,z\in Node(G'_x)\land{\langle y,z\rangle}\in G'.$$ For a sequence $\bar{w}={\langle w_1,\ldots,w_l\rangle}$ we define $G_{\bar{w}}$ inductively as $$G_\emptyset=G,\
G_{\bar{w}*v}=\left\{
\begin{array}{ll}
(G_{\bar{w}})_v&\mbox{ if }v\in Node(G_{\bar{w}})\\
G_{\bar{w}}&\mbox{ otherwise.}
\end{array}
\right.$$
The first player unable to move loses. So a game over $G$ is coded by a sequence $\bar{w}={\langle w_1,\ldots,w_l\rangle}$ such that $w_1\in G$, $w_{i+1}\in G_{{\langle w_1,\ldots,w_i\rangle}}$ for any $i<l$, $G_{w_{l-1}}\neq\emptyset$ and $G_{w_l}=\emptyset$. Alice wins in the game $W$ over $G$ if $Len(w)\bmod 2= 1$ and otherwise Bob wins.
The function computing $G_{\bar{w}}$ from $G$ and $\bar{w}$ is ${\Sigma^B_{0}}$-definable in $V^0$.
(Proof). It is easy to see that $G_{\bar{w}}$ is definable by the formula $$\begin{array}{l}
\varphi(G,G',\bar{w})\Leftrightarrow\\
{\forall}u,v<|V_G|({\langle u,v\rangle}\in G'\leftrightarrow
({\langle u,v\rangle}\in G\land\neg{\exists}w_i\in\bar{w}(w=u)).
\end{array}$$ so that ${\forall}G,\bar{w}{\exists}! G'\varphi(G,G',\bar{w})$ is provable in $V^0$. $\Box$.
Now we will define our base theory for combinatorial games. First we introduce functions $sg(G)$, $\tau(G)$, $\tau_A({\langle b_0,\ldots,b_l\rangle},G)$ and $\tau_B({\langle a_0,\ldots,a_l\rangle},G)$ with the following defining axioms: $$\begin{array}{l}
G=\emptyset\rightarrow sg(G)=0,\\
\neg UGraph(G)\rightarrow sg(G)=\max\{x\in V_G\}+1,\\
UGraph(G)\land G\neq\emptyset\rightarrow
sg(G)=\min\{k<|V_G|:{\forall}x\in V_Gk\neq sg(G_x)\}.
\end{array}$$ $$\tau(G)=
\left\{
\begin{array}{ll}
\min\{v\in V_G\ :\ sg(G_v)=0\}&\mbox{ if such }v\mbox{ exists.}\\
\max\{v\in V_G\}+1&\mbox{ otherwise.}
\end{array}
\right.$$ $$\begin{array}{l}
\tau_A(\emptyset,G)=\tau(G),\\
\tau_A({\langle b_0,\ldots,b_{l+1}\rangle},G)=
\tau_A({\langle b_0,\ldots,b_l\rangle},G)*{\langle b_{l+1},\tau(G_{\tau_A({\langle b_0,\ldots,b_l\rangle},G)*b_{l+1}})\rangle}
\end{array}$$ $$\begin{array}{l}
\tau_B(\emptyset,G)=\emptyset,\\
\tau_B({\langle a_0,\ldots,a_{l+1}\rangle},G)=
\tau_B({\langle a_0,\ldots,a_l\rangle},G)*{\langle a_{l+1},\tau(G_{\tau_B({\langle a_0,\ldots,a_l\rangle},G)*a_{l+1}})\rangle}.
\end{array}$$
Let ${\mathcal{L}_{NK}}$ be the language $L_A^2$ extended by function symbols $sg(G)$ $\tau(G)$, $\tau_A({\langle b_0,\ldots,b_l\rangle},G)$ and $\tau_B({\langle a_0,\ldots,a_l\rangle},G)$ . The ${\mathcal{L}_{NK}}$ theory ${V_{NK}}$ comprises the following axioms:
- defining axioms for symbols in ${\mathcal{L}_{NK}}$
- ${\Sigma^B_{0}}({\mathcal{L}_{NK}})$-COMP: ${\exists}X<a{\forall}y<a(X(a)\leftrightarrow\varphi(a))$,\
where $\varphi(a)\in{\Sigma^B_{0}}({\mathcal{L}_{NK}})$ which does not contain free occurrences of $X$.
Thus ${V_{NK}}$ is $V^0$ in the extended language ${\mathcal{L}_{NK}}$.
[**Remark.**]{} We need only functions $sg$ and $\tau_A$ in order to axiomatize the theory ${V_{NK}}$ since other two functions are definable from these functions. For instance, $\tau_G$ can be defined from $sg$ and $\tau_B$ can be defined by $\tau_A$. However we add these two functions to the language to make argument simple.
The following fact is well-known.
${V_{NK}}$ proves ${\Sigma^B_{0}}({\mathcal{L}_{NK}})$-IND: $$\varphi(0)\land
{\forall}x(\varphi(x)\rightarrow\varphi(x+1))\rightarrow{\forall}x\varphi(x).$$
Winning strategies in Sprague-Grundy system
===========================================
We show that strategy functions $\tau_A$ and $\tau_B$ actually computes winning game instances for Alice and Bob respectively.
Define formulas $AWS_{\tau_A}(G,l)$ and $BWS_{\tau_A}(G,l)$ as follows: $$\begin{array}{ll}
AWS_{\tau_A}(G)\Leftrightarrow&
{\forall}l{\forall}{\langle b_0,\ldots,b_l\rangle}[(l=\lfloor|V_G|/2\rfloor\land{\forall}i<l b_i\leq|V_G|+1)\\
&\phantom{{\forall}l{\forall}{\langle b_0,\ldots,b_l\rangle}[}
\rightarrow{\exists}l_0\leq l({\forall}i<l_0(b_i\in Node(G_{\tau_A({\langle b_0,\ldots,b_{i-1}\rangle},G)})\land\\
&\phantom{{\forall}l{\forall}{\langle b_0,\ldots,b_l\rangle}[\rightarrow}
\tau(G_{\tau_A({\langle b_0,\ldots,b_{i-1}\rangle},G)})\in Node(G_{\tau_A({\langle b_0,\ldots,b_{i-1}\rangle},G)}))\\
&\phantom{{\forall}l{\forall}{\langle b_0,\ldots,b_l\rangle}[\rightarrow}
\land b_{l_0}\not\in Node(G_{\tau_A({\langle b_0,\ldots,b_{l_0-1}\rangle},G)}))]
\end{array}$$ $$\begin{array}{ll}BWS_{\tau_B}(G)\Leftrightarrow&
{\forall}l{\forall}{\langle a_0,\ldots,a_l\rangle}[(l=\lceil|V_G|/2\rceil\land{\forall}i<l a_i\leq|V_G|+1)\\
&\phantom{{\forall}l{\forall}{\langle a_0,\ldots,a_l\rangle}[}
\rightarrow{\exists}l_0\leq l({\forall}i<l_0(a_i\in Node(G_{\tau_A({\langle a_0,\ldots,a_{i-1}\rangle},G)})\land\\
&\phantom{{\forall}l{\forall}{\langle b_0,\ldots,b_l\rangle}[\rightarrow}
\tau(G_{\tau_A({\langle a_0,\ldots,a_{i-1}\rangle},G)})\in Node(G_{\tau_A({\langle a_0,\ldots,a_{i-1}\rangle},G)}))\\
&\phantom{{\forall}l{\forall}{\langle a_0,\ldots,a_l\rangle}[\rightarrow}
\land a_{l_0}\not\in Node(G_{\tau_A({\langle a_0,\ldots,a_{l_0-1}\rangle},G)}))]
\end{array}$$
${V_{NK}}$ proves that $${\forall}G\left\{
Ugraph(G)\rightarrow((sg(G)\neq 0\rightarrow AWS_{\tau_A}(G)\land
(sg(G)=0\rightarrow BWS_{\tau_B}(G))))\right\}.$$
(Proof). We argue inside ${V_{NK}}$.
Suppose that $sg(G)\neq 0$ and let ${\langle b_0,\ldots,b_l\rangle}$ be a list of nodes in $G$ where $l=\lfloor|V_G|/2\rfloor$. We show that $${\forall}i<l({\forall}j\leq i b_j\in Node(G_{\tau_A({\langle b_0,\ldots,b_{j-1}\rangle},G)})
\rightarrow sg(G_{\tau_A({\langle b_0,\ldots,b_i\rangle},G)}=0)\mbox{ (*)}$$ The proof proceeds by induction on $i$.
If $i=0$ then (\*) trivially follows by the assumption.
Suppose by the inductive hypothesis that (\*) holds for $i\geq 0$ and assume that $$b_{i+1}\in Node(G_{\tau_A({\langle b_0,\ldots,b_i\rangle},G)}).$$ Since $sg(G_{\tau_A({\langle b_0,\ldots,b_i\rangle},G)})=0$, it must be that $$sg(G_{\tau_A({\langle b_0,\ldots,b_i\rangle},G)}*b_{i+1})\neq 0$$ and by the definition of $\tau$, we have $$sg(G_{\tau_A({\langle b_0,\ldots,b_{i+1}\rangle},G)})=0.$$ So we have (\*) for $i+1$.
We argue similarly for the case of $sg(G)=0$ and by noting that (\*) is a ${\Sigma^B_{0}}$ formula, the claim is obtained by ${\Sigma^B_{0}}$-IND in ${V_{NK}}$. $\Box$
Sprague-Grundy system captures PSPACE {#sec:sprag-grundy-syst}
=====================================
Now we are ready to show our main result; the theory ${V_{NK}}$ captures $PSPACE$.
A function is ${\Sigma^B_{1}}$ definable in ${V_{NK}}$ if and only if it is in PSPACE.
(Proof). It is easy to show that functions $sg$, $\tau$, $\tau_A$ and $\tau_B$ can be computed in PSPACE. So the only if part can be proved using the standard witnessing argument. Actually the provably total functions of the universal conservative extension of ${V_{NK}}$ is the $AC^0$ closure of functions $sg$ and $\tau_A$. So Herbrand theorem implies the witnessing. Thus the proof of if part is given is the rest of this section. $\Box$
We will show that any polynomial time alternating Turing machine can be simulated by a game in ${V_{NK}}$. First recall that PSPACE is equal to APTIME (cf. Papadimitriou [@papa]). So we actually show that any polynomial-time alternating Turing machine can be simulated by a game of Node Kayles.
We assume some harmless simplifications on alternating Turing machines. Let $M$ be an alternating Turing machine with time bound $p(|X|)$ on input $X$. where we assume that $p(n)$ is even for all $n$. We assume that all computation of $M$ on input $X$ terminates exactly at time $p(|X|)$. We also assume that the space bound of $M$ is $p(|X|)$. Furthermore, we assume that $M$ is binary branching. So we formalize the transition function as $$\delta_M(k,q,a)={\langle q_k,a_k,m_k\rangle}$$ where $k=0,1$, $q$ and $q_k$ are states of $Q$ and $a,a_k\leq 2$, $m_k\in\{-1,0,1\}$. We abuse the notation and write $$\delta_M(k,C,C')\Leftrightarrow
C'\mbox{ is the next configuration of }C\mbox{ along the path }k.$$ The final assumption is that $M$ computes in normal form in the sense that it first guesses the path $P={\langle p_1,\ldots,pi_{p(n)}\rangle}$ in the computation tree and then start computing using $P$.
We show that polynomial time bounded alternating Turing machines can be simulated by Node Kayles provably in ${V_{NK}}$.
Let $C_{INIT}(M,X)$ denote the initial configuration of $M$ on input $X$. For a binary string $P$, we denote by $C(P,M,X)$ the configuration of $M$ reachable from $C_{INIT}(M,X)$ along the path $P$. The predicate $Accept(C,M)$ denotes that $C$ is an accepting configuration of $M$. Note that all these functions and predicates are definable in $V^0$. We also define $$\begin{array}{l}
Comp({\langle C_0,\ldots,C_{p(|X|)}\rangle},P,M,X)\Leftrightarrow\\
\phantom{Comp}C_0=C_{INIT}(M,X)\land{\forall}i<p(|X|)\delta_M((P)_i,C_i,C_{i+1}), \\
Acomp({\langle C_0,\ldots,C_{p(|X|)}\rangle},P,M,X)\Leftrightarrow\\
\phantom{Comp}Comp({\langle C_0,\ldots,C_{p(|X|)}\rangle},P,M,X)\land Accept(C_{p(|X|)},M,X),\\
Rcomp({\langle C_0,\ldots,C_{p(|X|)}\rangle},P,M,X)\Leftrightarrow\\
\phantom{Comp}Comp({\langle C_0,\ldots,C_{p(|X|)}\rangle},P,M,X)\land\neg Accept(C_{p(|X|)},M,X),\\
\end{array}$$
\[theorem:main\] There exist functions $G(M,X)$, $Comp_A(M,X,P)$, $Comp_R(M,X,P)$, $Path_A(M,X,P)$ and $Path_R(M,X,P)$ which are ${\Sigma^B_{1}}$ definable in ${V_{NK}}$ such that the following formulas are provable in ${V_{NK}}$. $$\begin{array}{l}
\mbox{(1).}\ {\forall}M,X UGraph(G(M,X)),\\
\mbox{(2).}\ {\forall}M,X,P(|P|=p(|X|)/2\rightarrow(Len(Path_A(M,X,P))=2Len(P)\land\\
\phantom{\mbox{(2).}\ }{\forall}k<Len(Path_A(M,X,P))(Path_A(M,X,P)[2k+1]=P[k])),\\
\mbox{(3).}\ {\forall}M,X,P(|P|=p(|X|)/2\rightarrow(Len(Path_R(M,X,P))=2Len(P)\land\\
\phantom{\mbox{(2).}\ }{\forall}k<Len(Path_R(M,X,P))(Path_R(M,X,P)[2k]=P[k])),\\
\mbox{(4).}\
{\forall}M,X\\
\phantom{\mbox{(2).}\ }\{(sg(G(M,X))\neq 0\rightarrow{\forall}P(|P|=p(|X|)/2\rightarrow \\
\phantom{\mbox{(2).}\ }AComp(Comp_A(M,X,Path_A(M,X,P)),Path_A(G(M,X),P),M,X)))\\
\phantom{\mbox{(2).}\ }\land(sg(G(M,X))=0\rightarrow{\forall}P(|P|=p(|X|)/2\rightarrow \\
\phantom{\mbox{(2).}\ }RComp(Comp_R(M,X,Path_R(M,X,P)),Path_R(G(M,X),P),M,X)))\}\\
\end{array}$$
First we sketch the outline of the proof.
Let $M$ be an alternating Turing machine and $X$ be an input. We construct two graphs $G_A(M,X)$ and $G_B(M,X)$ so that each legitimate game instance of either games corresponds to a computation of $M$ on input $X$. Specifically, the first $p(|X|)$ moves of the game constitute a path $P$ with $|P|=p(|X|)$ followed by a list of moves which establishes a computation of $M$ along the path $P$, if players move correctly. We require that $G_A(M,X)$ and $G_R(M,X)$ satisfy that a game instance $I$ is A-winning if and only if $I$ corresponds to an accepting and rejecting computation of $M$ on $X$ along $P$ respectively.
Once the graph is constructed, we can extract functions $Comp_A(G,P)$, $Comp_B(G,P)$, $Path_A(G,P)$ and $Path_B(G,P)$ using strategy functions $\tau_A$ and $\tau_B$.
Now we present details of the proof.
The construction of $G_A(M,X)$ and $G_R(M,X)$ is similar to that for the graph simulating QBF games in [@schaefer]. Let $M=(Q,\Sigma,\delta,q_0,q_A)$ be an alternating Turing machine with $Q=\{q_0,\ldots,q_m\}$, $\Sigma=\{0,1,2\}$ where $2$ denotes the blank symbol and $q_A=q_1$. The transition function is given as $\delta(p,q,a)={\langle q_p,a_p,m_p\rangle}$ where $p\in\{0,1\}$, $q,q_p\in Q$ and $m_p\in\{-1,0,1\}$ whose intended meaning is that if the current state is $q$, the head reads the symbol $a$ and the path $p$ is chosen then the state changes to $q_P$, the tape content of the current head position is overwritten by $a_P$ and the head moves by $m_P$.
Let $s=p(|X|)$ be the number of alternations of $M$ on $X$, $l_0=p(|X|)+2$ be the length of the sequence coding configurations and $n_0=2(s+1)l_0$. It turns out that $s+n_0$ is equal to the number of total moves in A-winning legitimate game instances. We construct the graph of $G_A(M,X)$ and $G_R(M,X)$ with layers $P_i,A_{i,j},B_{i,j},$ of legitimate nodes, $Y_i$ of illegitimate nodes and $C_A,C_B$ of constraints nodes so that in the $i$th round, the player must choose her or his move from $i$th legitimate layer. Nodes in each layers are given as follows:
- $P$-layers $P_i=\{p_{i,0},p_{i,1}\}$ for $0\leq i<s$ represent the choice of $i$th path in the computation.
- $A$-layers $A_{i,j}$ corresponds to computation by Alice after the path is decided by choices from $P_0,\ldots,P_{s-1}$ and consists of nodes as follows: $$\begin{array}{l}
A_{i,j}=\{a_{i,j,0}^T,a_{i,j,1}^T,a_{i,j,2}^T\},\ 0\leq j<s\\
\vspace{3pt}
A_{i,s}=\{a_{i,k}^H\ :\ 0\leq k<s\}\\
\vspace{3pt}
A_{i,s+1}=\{a_{i,r}^Q\ :\ 0\leq r<|Q|\}\\
\end{array}$$ The intended meaning is that if Alice chooses nodes $a_{i,0,i_0}^T,\ldots,a_{i,s-1,i_{s-1}}^T$, $a_{i,k}^H$ and $a_{i,r}^Q$ then Alice’s computation of the $i$th configuration is $C_i={\langle q_r,k,i_0,\dots,i_{s-1}\rangle}$.
- $B$-layers $B_{i,j}=\{b_{i,j}\}$ which are intended for Bob’s moves for $0\leq i<s$ and $0\leq j\leq s+1$ or $i=s$ and $0\leq j\leq s$. Note that Bob’s have no choice of moves for these rounds. Also note that the number of $B$-layers is one less than that of $A$-layers.
We list these layers in the order that players choose their moves as $$P_0,\ldots,P_{s-1},A_{0,0},B_{0,0},\ldots,A_{s,s+1},B_{s,s}.$$ So we sometimes denote layers by ignoring their types as $$L_k=\left\{
\begin{array}{ll}
P_k&\mbox{ if }0\leq k<s,\\
A_{i,j}&\mbox{ if }k=s+2(i\cdot l_0+j),\ 0\leq i\leq s,\ 0\leq j\leq s+1,\\
B_{i,j}&\mbox{ if }k=s+2(i\cdot l_0+j)+1,\ 0\leq i\leq s,\ 0\leq j\leq s.\\
\end{array}
\right.$$
We define constraint layers $C_A$ and $C_R$ for $G_A(M,X)$ and $G_R(M,X)$ respectively which expresses constraints for the computation of $M$. Nodes of these layers are labelled by propositional formulas and we identify nodes with their labels. The layer $C_A$ and $C_R$ contain the following nodes:
(A)
: Nodes of the first sort are called initial nodes and express the initial configuration of $M$ on $X$ which consists of $\rightarrow a_{0,0}^Q$, $\rightarrow a_{0,0}^H$, for $j<|X|$, $\rightarrow a_{0,j,k}^T$ where $k=X(j)$ and for $|X|\leq j<s$, $\rightarrow a_{0,j,2}^T$.
(B)
: The second sort are called transition nodes of $M$ which consists of rules expressing the transition function of $M$. Specifically, let $c\in\{0,1\}$, $0\leq j\leq m$, $z\in\{0,1,2\}$ and $\delta(c,q_j,z)={\langle q_{j'},z',d\rangle}$ for some $0\leq j\leq|Q|$, $z'\in\{0,1,2\}$ and $d\in\{-1,0,1\}$. Then for $0\leq i<s$ $0\leq j\leq|Q|$ and $0\leq k<s$ , we introduce the following rules: $$\begin{array}{l}
p_{i,c}\land a_{i,j}^Q\land a_{i,k}^H\land a_{i,k,z}^T\rightarrow a_{i+1,k,z'}^T\\
p_{i,c}\land a_{i,j}^Q\land a_{i,k}^H\land a_{i,k',a}^T\rightarrow a_{i+1,k',a}^T,\ k'\neq k\\
p_{i,c}\land z_{i,j}^Q\land z_{i,k}^H\land z_{i,k,a}^T\rightarrow a_{i+1,k+d}^H\\
p_{i,c}\land a_{i,j}^Q\land a_{i,k}^H\land a_{i,k,a}^T\rightarrow a_{i+1,j'}^Q\\
\end{array}$$ Note that these rules compute the $i+1$st configuration from the $i$th configuration which is specified by choosing the path $c$. We call a rule containing $p_{i,c}$ for $c=0,1$ as $i$-rule.
Moreover, $C_A$ contains a single accepting node denoted by $Acc$ while $C_R$ contains a single rejecting node denoted by $Rej$.
Finally, the non-legitimate nodes are defined as $$Y_{n_0-k}=\{y_{n_0-k,n_0-k+j}\ :\ 0\leq j<k+1\}.$$ for $1\leq k<n_0$.
Next we define edges among the nodes. In the following, let $C$ denote either $C_A$ or $C_R$.
1. For $0\leq i<s$ and $c\in\{0,1\}$, $p_{i,c}\in P_i$ is connected to all nodes in $C$ which contains $p_{i,1-c}$.
2. For $0\leq i\leq s$ and $0\leq j\leq s+1$, $a\in A_{i,j}$ is connected to all nodes in $C$ which either contain $a$ in the succedent or $b\in A_{i,j}$ with $b\neq a$ in the antecedent.
3. The node $a_{s,1}^Q$ in $G_A(M,X)$ is connected to the node $Acc$.
4. The node $a_{s,j}^Q$ for $j\neq 1$ in $G_A(M,X)$ is connected to the node $Rej$.
5. all nodes in $C$ are mutually connected.
6. All nodes in $L_k\cup Y_k$ for $1\leq k\leq t_0$ are mutually connected.
7. The node $y_{t_0-k,t_0-k+j}\in Y_{t_0,k}$ is connected to all nodes in $$\bigcup\{L_i\cup Y_i\ :\ t_0-k<i\leq t_0+1,\ i\neq t_0-k+j\}.$$
The function computing $G_A(M,X)$ and $G_R(M,X)$ from $M$ and $X$ is ${\Sigma^B_{1}}$ definable in $V^0$.
(Proof). We code $G(M,X)$ in such a way that indices of nodes represent their labels. For instance, the node $p_{i,c}$ in $P_i$ for $0\leq i<s$ and $c\in\{0,1\}$ is indexed by the tuple ${\langle 0,i,c\rangle}$ where the first entry $0$ represents that it belongs to a $P$-layer.
Similarly, the node $a_{i,j}^Q$ in $A_{i,s+1}$ for $0\leq i\leq s$ and $0\leq j\leq |Q|$ is indexed by the tuple ${\langle 0,s+i\cdot n_0+1,j\rangle}$ and nodes in other $A$-layers and $B$-layers are indexed as well.
The node $y_{n_0-k,n_0-k+j}$ in the layer $Y_{n_0-k}$ is indexed by the tuple ${\langle 1,n_0-k,j\rangle}$ for $0\leq j<k+1$.
Finally nodes in $C_A\cup C_R$ are indexed by tuples of the form ${\langle 0,n_0,t\rangle}$ where $t$ is a tuple coding its label. For instance the node $$p_{i,c}\land a_{i,j}^Q\land a_{i,k}^H\land a_{i,h,a}^T\rightarrow a_{i+1,j_{c,a}}^Q$$ is denoted by the tuple ${\langle 0,i,c,j,k,a,0,j_{c,a}\rangle}$.
Then it is easy to see that the edge relation of $G(M,X)$ is definable by a ${\Sigma^B_{0}}$ formula so it is defined by ${\Sigma^B_{0}}$-COMP.$\Box$
We say that a subgraph $G'$ of $G=G_A(M,X)$ or $G_R(M,X)$ is $k$-legitimate for $0\leq k\leq s+n_0$ if $$Leg(G',G,k)\Leftrightarrow
{\forall}x\in V_G
\left(
\left(
x\in\bigcup_{k'<k}L_k\rightarrow x\not\in V_{G'}
\right)
\land
\left(
x\in\bigcup_{k<k'\leq s+n_0}L_k\rightarrow x\in V_{G'}
\right)
\right).$$
In the following, we denote $G=G_A$ or $G_R$ if there is no fear of confusion.
The following lemma states that the graph $G(M,X)$ is constructed so that players are forced to choose their moves from legitimate nodes for otherwise they lead to an immediate loose.
\[lemma:legitimate1\] $V_{NK}$ proves that from any legitimate graph $G'$ of $G$, the first non-legitimate move leads to an immediate lose for either player: $${\forall}G'{\forall}<n_0+s{\forall}x((Leg(G',G,k)\land x\not\in L_{k+1})\rightarrow sg(G'_x)\neq 0).$$
(Proof). We argue in ${V_{NK}}$ to show that if $G'$ is a $k$-legitimate subgraph of $G(M,X)$ and $v\not\in L_k$ then $sg(G'_v)\neq 0$.
Let $v\not\in L_k$. Then either $v\in Y_j$ for $j\geq k$ or $v\in L_j$ for $j>k$. In the first case, we have $v=y_{j,l}$ for some $l$ and taking it from $G'$ removes all nodes except $L_l\cup Y_l$. Since $L_l\cup Y_l$ forms a complete subgraph, it must be that $sg(G_{j,l})\neq 0$.
In the second case, $G_v$ consists of all nodes in $L_l\cup Y_l$ with $l\neq j$ and nodes in $L_{N_0+1}$ which are not connected to $v$. By the construction of $G(M,X)$, $y_{k,j}$ remains in $G'_v$ and is connected to all nodes in $G'_v$. So we have $G'_{{\langle v,y_{k,j}\rangle}}=\emptyset$. This implies that $sg(G'_v)\neq 0$ as required. $\Box$
We say that a sequence $\bar{w}={\langle v_1,\cdots,v_m\rangle}$ of nodes in $G_A(M,X)$ or $G_R(M,X)$ is legitimate, denoted by $SLeg(w,G)$, if $v_i\in L_i$ for all $i\leq m$. Then the following is an immediate consequence of Lemma \[lemma:legitimate1\].
\[cor:legitimate2\] ${V_{NK}}$ proves that $${\forall}k<t_0{\forall}{\langle v_1,\ldots,v_k\rangle}\mbox{ : legitimate }{\forall}v_{k+1}
(v_{k+1}\not\in L_{k+1}\rightarrow sg(G_{v_1\cdots v_{k+1}})\neq 0).$$
(Proof). It remains to show that if ${\langle v_1,\ldots,v_m\rangle}$ is legitimate then for any $k\leq m$, $G_{v_1\cdots v_k}$ is a $k$-legitimate subgraph of $G(M,X)$ which can be proved by ${\Sigma^B_{0}}$-IND on $k\leq m$. $\Box$
If both players move legitimately, The first $s$ moves will be $p_{0,c_0},\ldots,p_{s-1,c_{s-1}}$ which decides the path $P={\langle c_0,\ldots,c_{s-1}\rangle}$ in the computation tree of $M$ on $X$.
We require that if $sg(G_A(M,X)_P)=0$ then Bob can win the game for $G_A(M,X)_P$ only if he moves consistently with the computation of $M$ on $X$ along the path $P$. Otherwise if $sg(G_A(M,X)_P)\neq 0$ then Alice can win the game for $G_R(M,X)_P$ only if she moves consistently with the computation along $P$.
In order to prove the above property of $G(M,X)$ in ${V_{NK}}$, we next show that each list of legitimate moves forms a list of configurations.
Note that we can divide A-layers and B-layers into consecutive lists $A_{i,0},\dots,A_{i,s+1}$ and $B_{i,0},\dots,B_{i,s+1}$. We call these two lists as the $i$-round. We assert that each set of legitimate move by both Alice and Bob for the $i$-round forms a configuration of $M$ on input $X$. Specifically, let Alice’s moves for the $i$-round be given as $$\bar{a}_i=a_{i,j}^Q, a_{i,k}^H, a_{i,0,a_0}^T,\ldots,a_{i,s-1,a_{s-1}}^T.$$ Then we define $conf(\bar{a}_i)={\langle j,k,a_0,\ldots,a_{s-1}\rangle}$. Thus a legitimate sequence ${\langle \bar{a}_0,\ldots,\bar{a}_s\rangle}$ of moves by Alice forms a sequence of configurations ${\langle conf(\bar{a}_0),\ldots,conf(\bar{a}_s)\rangle}$.
We define legitimate moves by Alice and Bob after $s+2$ rounds as $$\begin{array}{l}
Leg({\langle v_1,\ldots,v_k\rangle},M,X)\Leftrightarrow{\forall}j<k(v_{j+1}\in L_{s+1+i}),\\
A\mbox{-}Leg({\langle a_0,\ldots,a_k\rangle},M,X)\Leftrightarrow{\forall}j<k(a_{j+1}\in L_{s+2j}),\\
B\mbox{-}Leg({\langle b_0,\ldots,b_k\rangle},M,X)\Leftrightarrow{\forall}j<k(b_{j+1}\in L_{s+2j+1}).
\end{array}$$ We omit parameters $M$ and $X$ if it is clear from the context. We also denote legitimate sequences of Alice and Bob as ${\langle a_{0,0},a_{0,1},\ldots,a_{i,j}\rangle}\mbox{ and }{\langle b_{0,0},b_{0,1},\ldots,b_{i,j}\rangle}$ respectively for $i\leq s$ and $j\leq s+1$.
Finally we define predicates which states that a given legitimate move form a computation of $M$. $$\begin{array}{l}
Comp({\langle a_{0,0},\ldots,a_{s,s+1}\rangle},M,X,P)
\Leftrightarrow\\
\phantom{Comp}Leg(\bar{a},M,X)\land conf(\bar{a}_0)=C_{INIT}(M,X)\land
{\forall}i<s\delta_M(P(i),conf(\bar{a}_i),conf(\bar{a}_{i+1})),\\
AComp({\langle a_{0,0},\ldots,a_{s,s+1}\rangle},M,X,P)
\Leftrightarrow\\
\phantom{Comp}Comp({\langle a_{0,0},\ldots,a_{s,s+1}\rangle},M,X,P)\land Accept(\bar{a}_s,M,X),\\
RComp({\langle a_{0,0},\ldots,a_{s,s+1}\rangle},M,X,P)
\Leftrightarrow\\
\phantom{Comp}Comp({\langle a_{0,0},\ldots,a_{s,s+1}\rangle},M,X,P)\land\neg Accept(\bar{a}_s,M,X).
\end{array}$$ Note that Bob’s moves after $s$ rounds are unique if he moves legitimately. So we denote $\bar{b}={\langle b_{0,0},\ldots,b_{s,s}\rangle}$.
In the followings, $M$ and $X$ always denote a code of an alternating TM and its input respectively and we refrain from stating it explicitly.
For a sequence $X={\langle x_0,\ldots,x_l\rangle}$, We define the function $ASeq(X)=\{x_i:i\bmod 2=0\}$. Note that if $X$ codes a game instance then $ASeq(X)$ gives a list of Alice’s moves.
The next lemma states that the value of $sg(G_A(M,X)_P)$ for $|P|=s$ decides whether $M$ accepts $X$ along the path $P$.
\[lemma:comp2\] ${V_{NK}}$ proves that $$\begin{array}{l}
{\forall}M,X,P\biggl\{|P|=s\rightarrow\\
(sg(G_A(M,X)_P)\neq 0\rightarrow
AComp(ASeq(\tau_A({\langle b_{0,0},\ldots,b_{s,s}\rangle},G_A(M,X)_P),M,X,P)))\land\\
(sg(G_A(M,X)_P)=0\rightarrow
RComp(ASeq(\tau_A({\langle b_{0,0},\ldots,b_{s,s}\rangle},G_R(M,X)_P),M,X,P)))\biggl\}.
\end{array}$$
In order to prove Lemma \[lemma:comp2\], we first prepare some notations. As stated above, legitimate moves $\bar{a}_i$ by Alice in $a_i$ rounds is presented as $$\bar{a}_i=a_{i,0,k_0}^T,\ldots,a_{i,s-1,k_{s-1}}^T,a_{i,k}^H,a_{i,j}^Q$$ where $0\leq j\leq m$, $0\leq k\leq s-1$ and $k_0\ldots,k_{s-1}\in\{0,1,2\}$. Likewise, Bob’s moves for $a_i$ rounds is represented as $\bar{b}_i=b_{i,1},b_{i,2},\ldots,b_{i,s+2}$ for $0\leq i<s$ and $\bar{b}_s=b_{s,1},b_{s,2},\ldots,b_{s,s+1}$.
We denote the moves by Alice and Bob for $G(M,X)_P$ respectively as $$\bar{a}={\langle \bar{a}_0,\ldots, \bar{a}_s\rangle}\mbox{ and }
\bar{b}={\langle \bar{b}_0,\ldots, \bar{b}_s\rangle}$$
We sometimes ignore the type of the nodes of Alice’s move and denote by $a_{i,j}$ the $j$-th move of Alice in the $i$-round. Furthermore we define $$\bar{a}^{\leq i,j}=\bar{a}_1\ldots,\bar{a}_{i-1},a_{i,1},\ldots,a_{i,j}\mbox{ and }
\bar{a}^{<i,j}=\bar{a}_1\ldots,\bar{a}_{i-1},a_{i,1},\ldots,a_{i,j-1}.$$ $$\bar{a}^{\leq i}=\bar{a}_1\ldots,\bar{a}_{i}\mbox{ and }
\bar{a}^{<i}=\bar{a}_1\ldots,\bar{a}_{i-1}.$$ The sequences $\bar{b}^{\leq i,j}$, $\bar{b}^{<i,j}$, $\bar{b}^{\leq i}$ and $\bar{b}^{<i}$ are defined similarly.
For sequences $\bar{a}={\langle a_0,\ldots,a_k\rangle}$ and $\bar{b}={\langle b_0,\ldots,b_k\rangle}$ or ${\langle b_0,\ldots,b_{k-1}\rangle}$, we define the $V^0$-definable function $$merge(\bar{a},\bar{b})={\langle a_0,b_0,\ldots,a_k,b_k\rangle}
\mbox{ or }{\langle a_0,b_0,\ldots,a_k,b_k\rangle}.$$ respectively.
The proof of Lemma \[lemma:comp2\] is divided into a series of sublemmas. Define ${\Sigma^B_{0}}$ formulas $Init(r,z,M,X)$ and $Next(r,z,p,C,M)$ so that
$$\begin{array}{l}
Init(r,z,M,X)\Leftrightarrow z\mbox{ is the }r\mbox{th element of }C_{INIT}(M,X),\\
Next(r,z,p,C,M)\Leftrightarrow z\mbox{ is the }r\mbox{th element of }C'\mbox{ with }\delta_M(p,C,C').\\
\end{array}$$ A $A_{i,j}$-rule is a transition rule in $C_A$ whose succedent contains a node in $A_{i,j}$. We say that a legitimate subgraph $G'$ of $G(M,X)$ contains no $A_{i,j}$-rule if there is no node in $G'$ which belongs to $C_A$ and represents some $A_{i,j}$-rule. We also say that $G'$ contains no $A$-rules if for all $i\leq s$ and $j\leq s+1$, $G'$ contains no $A_{i,j}$-rules. Note that thess properties are formalized by a ${\Sigma^B_{0}}$ formula.
Let $\bar{z}={\langle z_0,\ldots,z_k\rangle}$ be a list of legitimate moves by Alice or Bob for $k\leq (s+1)(s+2)$. We define that $\bar{z}$ is a partial computation as $$\begin{array}{l}
PComp(\bar{a},P,M,X)\Leftrightarrow\\
A\mbox{-}Leg(\bar{a},G_P)\land
{\forall}k\leq Len(\bar{a})
\{(q_k=0\rightarrow Init(r_k,a_k,M,X))\land\\
(q_k>0\rightarrow Next(r_k,a_k,P(q_k-1),conf(\bar{a}_{q_k-1}),M)),
\end{array}$$ where $q_k$ and $r_k$ are such that $k=q_k(s+1)+r_k$ and $0\leq r_k\leq s+1$.
The next lemma states that moves by Alice or Bob must be consistent with the computation of $M$ in order to obtain legitimate options.
Let $G$ be either $G_A(M,X)$ or $G_R(M,X)$. Then ${V_{NK}}$ proves that $$\begin{array}{l}
{\forall}M,X{\forall}P{\forall}l\leq (s+1)l_0
{\forall}\bar{a}={\langle a_0,\ldots,a_l\rangle}{\forall}\bar{b}={\langle b_0,\ldots,b_l\rangle}\\
\biggl\{(|P|=s\land A\mbox{-}Leg(\bar{a})\land B\mbox{-}Leg(\bar{b}))\land Len(\bar{a})=Len(\bar{b})+1
\rightarrow\\
(PComp(\bar{a},P,M,X)\leftrightarrow
{\forall}k\leq l(G_{P*merge(\bar{a},\bar{b})}
\mbox{ contains no }A_{q_k,r_k}\mbox{-rule}))
\biggr\}.
\end{array}$$
(Proof). We prove the claim of the lemma for $A_{i,j}$-rules by induction on $l$. If $l=0$ then we have to do nothing. So suppose that $l\geq 0$ and by the inductive hypothesis assume that the claim holds for $l$. Let us denote the lefthand side of the subformula inside the brace $\{\cdots\}$ of the claim by $(*)_l$. Assume that $(*)_{l+1}$ holds, that is $${\forall}k\leq l+1((q_k=0\rightarrow Init(r_k,a_k,M,X))\land
(q_k>0\rightarrow\delta_M(p_{q_{k-1}},conf(\bar{a}_{q_{k-1}}),2r_k-1,a_k)).$$ By the inductive hypothesis we already have $${\forall}k\leq l((G_P)_{merge(\bar{a},\bar{b})}\mbox{ contains no }A_{q_k,r_k}\mbox{-rules}).$$ So it suffice to show that $(G_P)_{merge(\bar{a},\bar{b})}$ contains no $A_{q_{l+1},r_{l+1}}$-rules
If $q_{l+1}=0$ then we have $Init(r_{l+1},a_{l+1},M,X)$ and since $\rightarrow a_{l+1}$ is the only $L_{l+1}$-rule, we have the claim. Otherwise, we have $$Next(2r_{l+1}-1,a_{l+1},p_{q_{l+1}-1},conf(\bar{a}_{q_{l+1}-1}),M)$$ so there must be a rule in $C$ of the form $A\rightarrow a_{l+1}$ where $A$ represents a conjunction which is consistent with $conf(\bar{a}_{q_{l+1}-1})$. Furthermore, it is the only $A_{q_{l+1},r_{l+1}}$-rule which is in $(G_P)_{merge(\bar{a}_{\leq l},\bar{b}_{\leq l})}$. Thus again we have the claim.
Conversely, suppose that $(*)_{l+1}$ does not hold. If $(*)_l$ does not hold then we have the claim by the inductive hypothesis. So suppose that $$\begin{array}{l}
(q_{l+1}=0\land\neg Init(r_{l+1},a_{l+1},M,X))\\
\lor(q_{l+1}>0\land\neg Next(2r_{l+1}-1,a_{l+1},p_{q_{l+1}-1},conf(\bar{a}_{q_{l+1}-1}),M)).
\end{array}$$ If the first disjunct is true then there exists an initial rule $\rightarrow y_{l+1}$ where $y_{l+1}\in L_{l+1}$ and $y_{l+1}\neq a_{l+1}$ which is not eliminated by the move $a_{l+1}$ of Alice.
Otherwise if the second conjunct is true then we may assume that $(G_P)_{merge(\bar{a},\bar{b})}$ does not contain any $L_k$-rule for $k\leq l$. Since $$\neg Next(2r_{l+1},y_{l+1},p_{q_{l+1}-1},conf(\bar{a}_{q_{l+1}-1}),M)$$ there must be a rule of the form $A\rightarrow y_{l+1}$ such that $A$ is consistent with $conf(\bar{a}_{q_{l+1}-1})$ and so it remains in $(G_P)_{merge(\bar{a},\bar{b})}$. Since $A\rightarrow y_{l+1}$ is not eliminated by $a_{l+1}$ we have the claim. $\Box$
\[cor:accepting1\] Let $G$ be either $G_A(M,X)$ or $G_R(M,X)$. Then ${V_{NK}}$ proves that if Alice moves legitimately on $G_P$ then she removes all $A$-rules if and only if her moves are consistent with the computation of $M$ on $X$ along $P$: $$\begin{array}{l}
{\forall}M,X,P{\forall}\bar{a}
\biggl\{(|P|=s\land Leg(\bar{a})\land Len(\bar{a})=n_0)\rightarrow\\
(Comp(\bar{a},P,M,X)
\leftrightarrow(G_{P*{\langle e_0,e\rangle}*merge(\bar{a},\bar{b})}\mbox{ contains no $A$-rules of }M)
\biggr\}
\end{array}$$
(Proof). We argue inside ${V_{NK}}$. First we remark that
- the move $a_{i,j}$ by Alice removes all nodes in $C$ which contain $a_{i,j}$ in the succedent or $a'\in X_{i,j}$ with $a'\neq a_{i,j}$ in the antecedent and
- any move in $\bar{b}$ by Bob does not remove any node in $C$.
We say that a node in $C$ is a $i$-round rule if it is a $A_{i,j}$-rule for some $0\leq j\leq s+1$. We will prove that $$\begin{array}{l}
conf(\bar{a}_0)=C_{INIT}(M,X)\rightarrow
{\forall}k\leq N\{{\forall}i\leq k\delta_M(P(i),conf(\bar{a}_i),conf(\bar{a}_{i+1}))\\
\leftrightarrow
{\forall}i\leq k(G_P)_{merge(\bar{a}^{\leq i},\bar{b}^{\leq i})}\mbox{ contains no $i$-round rules}\}.
\end{array}$$ The proof is by induction on $k$. For $k=0$ we show that $$\begin{array}{l}
conf(\bar{a}_0)=C_{INIT}(M,X)\\
\leftrightarrow
(G_P)_{merge(\bar{a}_0,\bar{b}_0)}\mbox{ contains no initial rule of }M.
\end{array}$$ Suppose first that $conf(\bar{a}_0)=C_{INIT}(M,X)$. Then each move $a_{0,j}$ of Alice removes the initial rule $\rightarrow a_{0,j}$ in $L_{N_0}$. Such a rule exists since $conf(\bar{a}_0)=C_{INIT}(M,X)$.
Conversely, suppose that $conf(\bar{a}_0)\neq C_{INIT}(M,X)$. Then for some choice $a_{0,j}$ of Alice, $L_{N_0}$ contains the initial rule $\rightarrow z'_{0,j}$ with $a_{0,j}\neq z'_{0,j}$. Since $\rightarrow z'_{0,j}$ cannot be removed by any other moves in $a_0$-rounds, $(G_P)_{merge(\bar{a}_0,\bar{b}_0)}$ must contain it.
For induction step, suppose that for $k\leq s-1$ $$\begin{array}{l}
(conf(\bar{a}_0)=C_{INIT}(M,X)\land
{\forall}i<k\delta_M(P(i),conf(\bar{a}_i),conf(\bar{a}_{i+1}))\\
\leftrightarrow
{\forall}i<k(G_P)_{merge(\bar{a}^{\leq k},\bar{b}^{\leq k})}\mbox{ contains no $i$-round rules}.
\end{array}$$ and we show that $$\delta_M(P(i),conf(\bar{a}_i),conf(\bar{a}_{i+1}))\leftrightarrow
(G_P)_{merge(\bar{a}^{\leq k+1},\bar{b}^{\leq k+1})}\mbox{ contains no $k$-round rules}.$$
Suppose that $\delta_M(P(i),conf(\bar{a}_i),conf(\bar{a}_{i+1}))$ holds. By the construction of $G(M,X)$, antecedents of $k+1$ rules of $(G_P)_{merge(\bar{a}^{\leq k},\bar{b}^{\leq k})}$ form $conf(\bar{a}_k)$.
In $a_{k+1}$-rounds, Alice must choose nodes in order to remove all such nodes in $L_{N_0}$. Since each such node specifies a transition rule of $M$, we have the claim.
Also the induction step is easily seen by the above remarks. Since the claim is ${\Sigma^B_{0}}$, it is proved by ${\Sigma^B_{0}}$-IND in ${V_{NK}}$ and the claim of the lemma easily immediately follows. $\Box$
Let $G$ be a graph and $z_0\ldots,z_k\in V_G$. We say that ${\langle z_0,\ldots,z_k\rangle}$ is a winning sequent for $G$, denoted by $WSeq({\langle z_0,\ldots,z_k\rangle},G)$ if $$G_{{\langle z_0,\ldots,z_{k-1}\rangle}}\neq\emptyset\land G_{{\langle z_0,\ldots,z_k\rangle}}=\emptyset.$$
\[cor:accepting2-1\] ${V_{NK}}$ proves that Alice’s moves for $G_A(M,X)_P$ form an accepting computation if and only if Alice wins the game: $$\begin{array}{l}
{\forall}M,X,P{\forall}\bar{a}={\langle a_{0,0},\ldots a_{s,s+1}\rangle}
\biggl\{(|P|=s\land Leg(\bar{a})\land Len(\bar{a})=(s+1)(s+2))\rightarrow\\
(AComp(\bar{a},P,M,X)\leftrightarrow WSeq(merge(\bar{a},\bar{b}),G_A(M,X)_P)\biggr\}.
\end{array}$$
(Proof). First note that Bob cannot removes any nodes in $C_A$ unless he can move legitimately for a node in $C$. By Lemma \[cor:accepting1\], the only node in $C_A$ which may remain in $(G_P)_{merge(\bar{a},\bar{b})}$ is the acceptance node $Acc$. So we have $$conf(\bar{a}_s)=C_{ACCEPT}(M,X)
\leftrightarrow Acc\mbox{ is removed in $a_s$-rounds}".$$
\[cor:accepting2-2\] ${V_{NK}}$ proves that Alice’s moves for $G_R(M,X)_P$ form a rejecting computation if and only if Alice wins the game: $$\begin{array}{l}
{\forall}M,X,P{\forall}\bar{a}={\langle a_{0,0},\ldots a_{s,s+1}\rangle}
\biggl\{(|P|=s\land Leg(\bar{a})\land Len(\bar{a})=(s+1)(s+2))\rightarrow\\
(RComp(\bar{a},P,M,X)\leftrightarrow WSeq(merge(\bar{a},\bar{b}),G_R(M,X)_P)\biggr\}.
\end{array}$$
(Proof). The proof is almost identical to Corollary \[cor:accepting2-1\]. The only difference is if Alice moves in accordance with the computation of $M$ on $X$ along $P$ then she must remove the rejecting node $Rej$ by the last move. $\Box$
In order to show that the strategy function yields computations of $M$, we need to relate Sprague-Grundy number of $G=G_A(M,X)$ or $G_R(M,X)$ and the computation of $M$. The next lemma asserts that Alice can always chooses options $G'$ of $G_A(M,X)_P$ so that $sg(G')=0$ if and only if Alice’s moves form an accepting computation along $P$.
\[lemma:accepting\] ${V_{NK}}$ proves that $$\begin{array}{l}
{\forall}M,X,P{\forall}\bar{a}={\langle a_{0,0},\ldots a_{s,s+1}\rangle}
\biggl\{(|P|=s\land Leg(\bar{a}))\rightarrow\\
{\forall}k<Len(\bar{a})(sg(G_{P*merge(\bar{a}^{\leq k},\bar{b}^{< k})})=0)\leftrightarrow
AComp(\bar{a},P,M,X))\biggl\}\\
\end{array}$$
(Proof). Let $\bar{a}$ be as stated. Suppose that $$conf(\bar{a}_0)=C_{INIT}(M,X)\land
{\forall}i<s\delta_M(P(i),conf(\bar{a}_i),conf\bar{a}_{i+1}))\land
Accept(conf(\bar{a}_s),M,X)).$$ By induction on $k$ we show that ${\forall}k<l_0sg((G_P)_{merge(\bar{a}^{<l_0-k},\bar{b}^{<l_0-k})})\neq 0$. If $k=0$ then the claim follows from Corollary \[cor:accepting2-1\] since $$sg(((G_P)_{merge(\bar{a}^{<l_0},\bar{b}^{<l_0})})_{z^{l_0}})=sg((G_P)_{merge(\bar{a},\bar{b})})=0.$$
For $k<l_0-1$, suppose by the inductive hypothesis that $sg((G_P)_{merge(\bar{a}^{<l_0-k},\bar{b}^{<l_0-k})})\neq 0$. Then $$sg(((G_P)_{merge(\bar{a}^{<l_0-k},\bar{b}^{<l_0-k})})_{b_{l_0-k-1}})
=sg((G_P)_{merge(\bar{a}^{<l_0-k},\bar{b}^{<l_0-k})})\neq 0.$$ Thus by Corollary \[cor:legitimate2\], we have $sg((G_P)_{merge(\bar{a}^{<l_0-k},\bar{b}^{<l_0-k-1})})\neq 0$. Since $$sg(((G_P)_{merge(\bar{a}^{<l_0-k-1},\bar{b}^{<l_0-k-1})})_{a_{l_0-k-1}})
=sg((G_P)_{merge(\bar{a}^{<l_0-k},\bar{b}^{<l_0-k-1})}).$$ we have $sg((G_P)_{merge(\bar{a}^{<l_0-(k+1)},\bar{b}^{<l_0-(k+1)})})\neq 0$ as desired.
The converse direction is an immediate consequence of Corollary \[cor:accepting1\] and Corollary \[cor:accepting2-1\].$\Box$
Analogously, Alice always chooses options of $G_R(M,X)_P$ whose Sprague-Grundy number is equal to $0$ if and only if Bob’s moves form a rejecting computation along $P$.
\[lemma:rejecting\] ${V_{NK}}$ proves that $$\begin{array}{l}
{\forall}M,X,P{\forall}\bar{a}={\langle a_{0,0},\ldots a_{s,s+1}\rangle}
\biggl\{(|P|=s\land Leg(\bar{a}))\rightarrow\\
{\forall}k<Len(\bar{a})(sg(G_R(M,X)_{P*merge(\bar{a}^{\leq k},\bar{b}^{<k})})=0)\leftrightarrow
RComp(\bar{a},P,M,X))\biggl\}\\
\end{array}$$
(Proof). Suppose that $RComp(\bar{b},P,M,X)$ holds. By induction on $k$, we show that ${\forall}k<(s+1)(s+2)sg(G(M,X)_{P*merge(\bar{a}^{\leq (s+1)(s+2)-k},\bar{b}^{(s+1)(s+2)<k})})\neq 0$. If $k=0$ then the claim follows from Corollary \[cor:accepting2-2\] since $$sg(G(M,X)_{P*merge(\bar{a}^{\leq (s+1)(s+2)-k},\bar{b}^{(s+1)(s+2)<k})*b_{s,i}^Q})=0$$ for $i\neq 1$. The proof for $k>0$ is identical to the one for Lemma \[lemma:accepting\]. $\Box$
Finally we show that applying the strategy function $\tau_A$ to either $G_A(M,X)_P$ or $G_R(M,X)_P$ yields either accepting or rejecting computation respectively.
\[lemma:accepting2\] ${V_{NK}}$ proves that if $sg(G_A(M,X)_P)\neq 0$ then the application of $\tau_A$ to $G_A(M,X)_P$ yields an accepting computation along $P$: $$\begin{array}{l}
{\forall}M,X,P{\forall}\bar{a}={\langle a_{0,0},\ldots a_{s,s+1}\rangle}\\
\left\{(|P|=s\land sg(G_A(M,X)_P)\neq 0\land
\tau_A(\bar{b},G_A(M,X)_P)=merge(\bar{a},\bar{b}))
\rightarrow AComp(\bar{a},P,M,X)\right\}
\end{array}$$
(Proof). Suppose that $sg(G_A(M,X)_P)\neq 0$ and let $\tau_A(\bar{b},G_A(M,X)_P)=merge(\bar{a},\bar{b})$. By the definition of $\tau_A$, we have $${\forall}k\leq Len(\bar{a})(sg((G_P)_{merge(\bar{a}_{\leq k},\bar{b}_{<k})})=0).$$ So by Lemma \[lemma:accepting\], we have the claim. $\Box$
\[lemma:rejecting2\] ${V_{NK}}$ proves that if $sg(G_R(M,X)_P)\neq 0$ then the application of $\tau_A$ to $G_R(M,X)_P$ yields a rejecting computation: $$\begin{array}{l}
{\forall}M,X,P{\forall}\bar{a}={\langle a_{0,0},\ldots a_{s,s+1}\rangle}\\
\left\{(|P|=s\land sg(G_R(M,X)_P)\neq 0\land\tau_A(\bar{a},M,X)=merge(\bar{a},\bar{b}))
\rightarrow RComp(\bar{a},P,M,X)\right\}
\end{array}$$
(Proof). Suppose that $sg(G(M,X)_P\neq 0$. Then By Lemma \[lemma:rejecting\] we have the claim by a similar argument as for Lemma \[lemma:accepting2\]. $\Box$
Next lemma states that $G_A(M,X)_P$ and $G_R(M,X)_P$ play complementary roles to each other.
\[lemma:complement\] ${V_{NK}}$ proves that $${\forall}M,X,P(|P|=s\rightarrow
(sg(G_A(M,X)_P)\neq 0\leftrightarrow sg(G_A(M,X)_P)=0)).$$
(Proof). We argue in ${V_{NK}}$. Suppose that $sg(G_A(M,X)_P)\neq 0$. We show that $$\begin{array}{l}
{\forall}\bar{a}(Len(\bar{a})=(s+1)(s+2)\rightarrow\\
{\exists}\bar{b}(Len(\bar{b})\leq Len(\bar{a})\land
WSeq(merge(\bar{a}^{\leq Len(\bar{b})},\bar{b}),G_R(M,X)_P))).
\end{array}(*)$$ The proof is divided into cases. Let $\bar{a}$ be an arbitrary list of Alice’s moves with $Len(\bar{a})=(s+1)(s+2)$ and $\bar{b'}={\langle b_{0,0},\ldots,b_{s,s}\rangle}$.
If $A\mbox{-}Leg(\bar{a})\land Comp(\bar{a})$ then by Corollary \[cor:accepting2-1\], we have $$G_A(M,X)_{P*merge(\bar{a},\bar{b'})}=\emptyset\leftrightarrow AComp(\bar{a},M,X,P).$$ On the other hand, by Corollary \[cor:accepting2-2\], we have $$G_R(M,X)_{P*merge(\bar{a},\bar{b'})}=\emptyset\leftrightarrow RComp(\bar{a},M,X,P).$$ Thus we have $G_R(M,X)_{P*merge(\bar{a},\bar{b'})}\neq\emptyset$ and for any $c\in Node(G_R(M,X)_{P*merge(\bar{a},\bar{b'})}\subseteq C_R$, we have $G_R(M,X)_{P*merge(\bar{a},\bar{b'}*c)}=\emptyset$. Therefore we obtain $WSeq(merge(\bar{a},\bar{b}*c),G_R(M,X)_P)))$.
If $A\mbox{-}Leg(\bar{a})\land\neg Comp(\bar{a})$ then by Corollary \[cor:accepting1\] we have $$G_R(M,X)_{P*merge(\bar{a},\bar{b'})}\neq\emptyset\land
G_R(M,X)_{P*merge(\bar{a},\bar{b'})*c}=\emptyset.$$
Finally if $\neg A\mbox{-}Leg(a)$ then we can find the shortest initial part $\bar{a'}={\langle a_0,\ldots,a_k\rangle}$ of $\bar{a}$ such that $A\mbox{-}Leg(\bar{a'})\land a_{k+1}\not\in A_{q_{k+1},r_{k+1}}$. Then by Lemma \[lemma:legitimate1\], we have $x$ such that $$WSeq(merge(\bar{a'},\bar{b}^{\leq k})*a_{k+1}*x,G_R(M,X)_P).$$
Thus in any case we have $(*)$ and from this we readily have $sg(G_R(M,X))=0$. Conversely, if $sg(G_R(M,X))\neq 0$ the by a similar argument, we obtain $sg(G_A(M,X))=0$. $\Box$
(Proof of Lemma \[lemma:comp2\]). Suppose that $sg(G_A(M,X))\neq 0$. Then by Lemma \[lemma:accepting2\], we have the first part. If $sg(G_A(M,X))=0$ then by Lemma \[lemma:complement\], we have $sg(G_R(M,X))\neq 0$ and we can apply Lemma \[lemma:rejecting2\]
(Proof of Theorem \[theorem:main\]). We argue in ${V_{NK}}$. Let $M$ be an alternating Turing machine and $X$ be an input. We define $G(M,X)=G_A(M,X)$ For other functions, we set $$\begin{array}{l}
Path_A(P,M,X)=\tau_A(P,G_A(M,X))\\
Path_R(P,M,X)=\tau_B(P,G_A(M,X))\\
Comp_A(M,X,P)=ASeq(\tau_A(\bar{b'},G_A(M,X)_P))
Comp_R(M,X,P)=ASeq(\tau_A(\bar{b'},G_A(M,X)_P))
\end{array}$$ where the sequence $\bar{b}$ is defined by $\bar{b}={\langle b_{0,0},\ldots,b_{s,s}\rangle}$.
The condition (1) is trivial from the definition. Conditions (2) and (3) follows from the definition of the strategy functions $\tau_A$ and $\tau_B$.
Since we assume that $p(|X|)$ is even for all $X$, it follows that $${\forall}X,P(|P|=p(|X|)\rightarrow(sg(G(M,X)=0\leftrightarrow sg(G_A(M,X)_P)=0)).$$ Thus Lemma \[lemma:comp2\] implies 4. So the proof terminates.$\Box$
${V_{NK}}$ proves ${\Sigma^B_{\infty}}$-IND.
(Proof). For any $\varphi(X)\in{\Sigma^B_{\infty}}$ we can construct an alternating Turing machine which decides $\varphi$ in polynomial time. $\Box$
[99]{} C. Bouton, NIM, a game with a complete mathematical theory. Annals of Mathematics, 3, (1901), pp.35–39. S.R.Buss, Bounded Arithmetic. Ph.D. Dissertation, Princeton University (1985) S.A.Cook and P.Nguyen, Logical Foundations of Proof Complexity. ASL Perspectives in Logic Series. Cambridge University Press. (2010) N.Eguchi, Characterising Complexity Classes by Inductive Definitions in Bounded Arithmetic. arXiv:1306.5559 \[math.LO\] (2014) S.A.Fenner and J.Rogers, Combinatorial Game Complexity: An Introduction with Poset Games. Bulletin of the EATCS 116 (2015). P.M.Grundy, Mathematics and games. Eureka. 2 (1939) pp.6-–8 C.H.Papadimitriou, Computational Complexity. Addison-Wesley. (1993) T.J.Schaefer, On the complexity of some two-person perfect-information games. Journal of Computer and System Sciences, 16(2), (1978), pp.185–225. A.Skelley, Theories and Proof Systems for PSPACE and the EXP-Time Hierarchy. Ph.D. dissertation, University of Toronto, (2005) M.Soltys and C.Wilson, On the complexity of computing winning strategies for finite poset games, Theory of Computing Systems, 48(3), (2011), pp.680–692. R.P.Sprague, Über mathematische Kampfspiele. Tohoku Mathematical Journal. 41 (1935) pp.438-–444.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Call a permutation $k$-inflatable if it can be “blown up” into a convergent sequence of permutations by a uniform inflation construction, such that this sequence is symmetric with respect to densities of induced subpermutations of length $k$. We study properties of 3-inflatable permutations, finding a general formula for limit densities of pattern permutations in the uniform inflation of a given permutation. We also characterize and find examples of $3$-inflatable permutations of various lengths, including the shortest examples with length $17$.'
author:
- Tanya Khovanova
- Eric Zhang
bibliography:
- 'inflatable.bib'
title: 'On 3-Inflatable Permutations'
---
Introduction
============
In a broad sense, an object is called *quasirandom* if, asymptotically, it has similar properties to random objects of the class it belongs to. The notion of quasirandomness has been studied for a variety of objects, including groups [@gowers07], graphs [@chung89; @lovasz08], and $k$-uniform hypergraphs [@gowers06]. For permutations in particular [@cooper04; @cooper08], several different definitions of randomness are equivalent to a single concept of a *quasirandom* permutation sequence. One such definition states that in the limit, as the lengths of the permutations in a sequence grow toward infinity, the densities of all pattern permutations of length $k$ approach $1/k!$.
The study of regularity and quasirandomness is an active area of research, and there have been fruitful prior results [@sliacan08]. In 2012, Kral and Pikhurko [@kral12] proved that if a permutation has uniform densities for all permutations of length 4, then it is quasirandom. With this result in mind, it is natural to consider permutation sequences that have some set of uniform densities but are not quasirandom.
Inflatable permutations are one example of a general construction for permutation sequences. We define the inflation of two permutations $\tau$ and $\gamma$ with lengths $n$ and $m$ to be a new permutation with length equal to the product $nm$. For each index $i \in [n]$ (the range of integers from $1$ to $n$) in the first permutation $\tau$, we replace it with $m$ numbers $n(\tau_i - 1) + \gamma_j$ for each $j \in [m]$. In a sense, we are “inflating” each number in the first permutation by substituting for it $m$ numbers in the order of the second permutation.
We can use this definition to build convergent permutation sequences. The main idea is to start from a base permutation and take its inflation with a sequence of random permutations with lengths tending to infinity. The $n$-th term of this sequence is the inflation of the base permutation with a permutation chosen uniformly at random from all elements of $S_n$.
A permutation is called $k$-inflatable when this construction results in a permutation sequence that has uniform densities of all length-$k$ subpermutations. Cooper and Petrarca [@cooper08] previously studied this topic in 2008, finding examples of 3-inflatable permutations. The result from Kral and Pikhurko [@kral12] implies that there do not exist any nontrivial $4$-inflatable permutations. In general, nontrivial $k$-inflatable permutations do not exist for $k \ge 4$.
We decided to write this paper when we discovered via random sampling that the example of a 3-inflatable permutation of length 9 in [@cooper08] is wrong. We found an explicit formula that calculates densities in permutation inflations and used this to find constraints on admissible lengths of such permutations. This proved that the shortest length of a 3-inflatable permutation is 17, and we found many examples of this length with an optimized computer search. We also found a way to multiply 3-inflatable permutations that provides an infinite number of examples.
Code for calculating the limit densities of patterns within inflations, searching for 3-inflatable permutations, and running randomized tests is available at <https://github.com/ekzhang/inflatable>.
We start with basic definitions in Section \[sec:preliminaries\]. This includes formal definitions for permutation density and the convergence of permutation sequences, as well as a definition of quasirandomness for permutations.
In Section \[sec:inflation\], we prove a formula for the limit density of a permutation $\pi$ of length $k$ in the inflation of a permutation $\tau$ with permutations of a convergent sequence $\{ \gamma_j \}$, given the limit densities of patterns with lengths up to $k$ in that sequence. This allows the limit densities of induced patterns in inflations to be computed efficiently.
In Section \[sec:random-inflation\], we examine the asymptotic properties of fixed permutations when inflated by uniformly chosen random permutations of increasing length.
In Section \[sec:ksymmetry\] we apply the formula to densities of permutations of length 3. We calculate necessary and sufficient conditions for a permutation to be $3$-inflatable. From this, we deduce that the lengths of $3$-inflatable permutations must have remainders 0, 1, 17, 64, 80, or 81 modulo 144. By computational search, we find examples of $3$-inflatable permutations for each of these lengths, including the shortest possible examples, with length $17$.
In Section \[sec:structure3\] we examine the structure of $3$-inflatable permutations. We show that if two permutations are both $3$-inflatable, then the inflation of one with the other is also $3$-inflatable. We then discuss how a rotational symmetry helps us find examples of $3$-inflatable permutations by reducing the number of needed equivalences.
Preliminaries {#sec:preliminaries}
=============
A *permutation* is an ordering of the elements of a set. We represent a permutation of length $n$ as an $n$-tuple of distinct positive integers, representing the image of $[n]$ under application of the permutation. For example, $(1, 2, 3)$ is the identity permutation of length 3, while $\tau = (3, 2, 1)$ is the permutation that swaps 1 and 3. For convenience, we can leave out the parentheses and write $\tau = 321$. We denote the length of the permutation $\tau$ as $|\tau|$.
We are interested in patterns that are formed by subsets in a given permutation. Suppose we have an ordered $k$-tuple $(a_1, \ldots, a_k)$ of distinct positive integers. We say that this tuple is *order-isomorphic* to a permutation $\pi$ of size $k$, if and only if $a_i < a_j \iff \pi_{i} < \pi_{j}$ for all $i, j \in [k]$.
The *density* of a pattern permutation $\pi$ of length $k$ in a permutation $\tau$ is defined as the probability that a randomly-selected $k$-point subset of $\tau$ is order-isomorphic to $\pi$. We denote this density by $t(\pi, \tau)$. For example, $t(12, 132) = 2/3$.
A sequence of permutations $\{ \tau_j \}$ is *convergent* if and only if as $j$ increases, the length of the permutations $|\tau_j|$ approaches infinity, and for any permutation $\pi$, the sequence of densities $t(\pi, \tau_j)$ converges. The limit $\displaystyle \lim_{j \rightarrow\infty} t(\pi, \tau_j)$ is called the *limit density* of $\pi$ in the sequence $\{\tau_j\}$.
This notion of density can be used to describe the property of “randomness” in a permutation. Notice that given any pattern permutation $\pi$, if one selects a permutation $\lambda$ uniformly at random from $S_n$ for $n \ge |\pi|$, we have by symmetry that $\operatorname*{\mathbb{E}}t(\pi, \lambda) = \frac{1}{|\pi|!}$. This motivates the following definition of what it means for a convergent sequence of permutations to be *quasirandom*.
A convergent sequence of permutations $\{\tau_j\}$ is called *quasirandom* if for every permutation $\pi$, $$\lim_{j\rightarrow\infty} t(\pi, \tau_j) = \frac{1}{|\pi|!}.$$
Inflation {#sec:inflation}
=========
We define an inflation of one permutation with respect to another permutation. Cooper and Petrarca call this the *tensor product* of two permutations [@cooper08].
The *inflation* of $\tau \in S_n$ with respect to a permutation $\gamma \in S_m$ is defined as a permutation, denoted by $\operatorname*{inflate}(\tau, \gamma)$, of length $mn$ that consists of $n$ blocks of length $m$, which are each order-isomorphic to $\gamma$, and such that any restriction of the permutation to one element in each block is order-isomorphic to $\tau$.
For example, if $\tau = 12$ and $\gamma = 312$, then $\operatorname*{inflate}(\tau, \gamma) = 312645$. The resulting permutation consists of two blocks 312 and 645, each of which is order-isomorphic to $\gamma$. Also, each number in the second block is greater than every number in the first block, as dictated by permutation $\tau$. As another example, $\operatorname*{inflate}(\gamma, \tau) = 561234$.
Inflations of permutations are interesting objects of study, as it is possible to calculate densities in $\operatorname*{inflate}(\tau, \gamma)$ through densities in $\tau$ and $\gamma$. This later allows for the construction of interesting examples of permutation sequences with some set densities.
There is a more general definition of inflation, which turns out to be useful for computing densities. For some permutation $\tau \in S_n$, rather than inflating $\tau$ with respect to a single permutation $\pi$, we can define an operation on $\tau$ with respect to a sequence of $n$ permutations $\gamma_1, \ldots, \gamma_n$ as follows.
The *generalized inflation* of $\tau \in S_n$ with respect to a sequence of $n$ permutations $\gamma_1, \ldots, \gamma_n$ is defined as a permutation of length $|\gamma_1| + \cdots + |\gamma_n|$ that consists of $n$ blocks, such that the $i$-th block is order-isomorphic to $\gamma_i$, and any restriction of the permutation to one element in each block is order-isomorphic to $\tau$.
Note that inflation is a special case of generalized inflation, where all the $\gamma_i$ for $i \in [n]$ are equal to some single permutation $\gamma$.
For example, the generalized inflation of the permutation $\tau = 231$ with the sequence $\{\gamma_j\} = (12, 231, 1)$ is the length-6 permutation $23|546|1$. Note how each element in the original permutation $\tau$ corresponds to a “block” of elements in the new permutation.
Given a permutation $\pi$, it is useful to see how $\pi$ can be represented as a generalized inflation. To this end, consider pairs $(b, \sigma)$, where $\sigma$ is a permutation and $b$ is a sequence of permutations $(\sigma_1, \ldots, \sigma_k)$ so that $\pi$ is the generalized inflation of $\sigma$ with respect to $\sigma_1, \ldots, \sigma_k$. We call the pair $(b, \sigma)$ a *block-partition* of $\pi$, and define the function $B(\pi)$ to be the set of all *block-partitions* of $\pi$. Each block-partition of a permutation is a way of representing it as a generalized inflation.
For example, if $\pi = 132$, then we have three block-partitions: first, $1|3|2$ with $b=(1, 1, 1)$ and $\sigma = 132$; second, $1|32$ with $b=(1,21)$ and $\sigma = 12$; and third, $132$ with $b=(132)$ and $\sigma = 1$.
This definition allows us to derive an explicit formula for the limit densities of patterns in the inflation of a permutation with a convergent sequence $\{ \gamma_j \}$.
\[thm:inflate-density\] The limit density of $\pi$ in $\operatorname*{inflate}(\tau, \gamma_j)$ is $$\lim_{j\rightarrow\infty} t(\pi, \operatorname*{inflate}(\tau, \gamma_j)) = \frac{|\pi|!}{|\tau|^{|\pi|}} \sum_{(b, \sigma) \in B(\pi)} \left(\binom{|\tau|}{|\sigma|} t(\sigma, \tau) \cdot \prod_{\alpha \in b} \frac{\lim_{j\rightarrow\infty} t(\alpha, \gamma_j)}{|\alpha|!}\right).$$
There are $|\tau|^{|\pi|}$ ways of selecting (counting distinct orderings) $|\pi|$ blocks with replacement from the inflation of $\tau$.
For a given pair $(b, \sigma) \in B(\pi)$, let $k = |\sigma|$. There exist $\binom{|\tau|}{k} \cdot t(\sigma, \tau)$ order-respecting assignments of distinct blocks $a_1, a_2, \dots, a_{k}$ in the inflation of $\tau$ to the corresponding indices in $\sigma$. Furthermore, since we count distinct orderings, the number of ways to select these $k$ blocks with their respective multiplicities is given by the multinomial coefficient $$\binom{|\pi|}{|\sigma_1|; |\sigma_2|; \dots; |\sigma_k|} = |\pi|! \cdot \prod_{\alpha \in b} \frac{1}{|\alpha|!}.$$ The probability that a random $|\pi|$-point subset from the inflation of $\tau$ is order-isomorphic to $\pi$, and also blocked according to $a_1, a_2, \dots, a_k$, is thus equal to $$\frac 1{|\tau|^{|\pi|}} \left( |\pi|! \cdot \prod_{\alpha \in b} \frac{1}{|\alpha|!} \right) \left( \prod_{\alpha \in b} \lim_{j\rightarrow\infty} t(\alpha, \gamma_j) \right) = \frac{|\pi|!}{|\tau|^{|\pi|}} \prod_{\alpha \in b} \frac{\lim_{j\rightarrow\infty} t(\alpha, \gamma_j)}{|\alpha|!}.$$ Summing these probabilities for all assignments of $\{a_j\}$ mentioned above, as well as for each block-partition in $B(\pi)$, we arrive at the final formula.
For example, let $\pi = 12$. Then $B(\pi)$ consists of two elements $((1, 1), 12)$ and $((12), 1)$. Plugging this into the formula above, we get[^1] $$t(12, \operatorname*{inflate}(\tau, \sigma)) = \frac{2!}{|\tau|^2} \left( \binom{|\tau|}2 t(12, \tau)/2 + \frac{|\tau|}4 \cdot t(12, \sigma)\right).$$ In particular, if $t(12, \tau)= t(12, \sigma) = 1/2$, then $t(12, \operatorname*{inflate}(\tau, \sigma)) = 1/2$. However, it is possible that $t(12, \operatorname*{inflate}(\tau, \sigma)) = 1/2$, while $t(12, \tau) \neq t(12, \sigma)$.
Uniform inflation {#sec:random-inflation}
=================
A special case of inflation is of particular interest to us, when we inflate a fixed permutation with a random second permutation. In particular, given a permutation $\pi$ and a uniform random permutation $\gamma$ of length not less than $|\pi|$, the expected value of the density $\operatorname*{\mathbb{E}}t(\pi, \gamma) = \frac{1}{|\pi|!}$ for symmetry reasons.
Using this idea, we can define the *uniform inflation* of a single permutation as the convergent sequence of its inflations with random permutations of length tending to infinity.
The *uniform inflation* of $\tau$, denoted by $\operatorname*{inflate}(\tau)$, is a sequence of random permutations $\{ \operatorname*{inflate}(\tau, \lambda_j) \}$, where $\lambda_j$ is selected uniformly at random from $S_j$. In other words, the $j$-th term of the sequence is the inflation of $\tau$ with a uniform random permutation of length $j$.
It is not difficult to show that the densities of pattern permutations in the uniform inflation of $\tau$ converge for any permutation $\tau$, so the sequence is convergent. This convergence fact allows us to extend Theorem \[thm:inflate-density\] to the case of uniform inflation by plugging in $t(\alpha, \gamma) = \frac{1}{|\alpha|!}$, which yields the following result.
\[random-density\] If $\{ \lambda_j \}$ is a sequence of random permutations with lengths tending to infinity, then the limit density of $\pi$ in the sequence of permutations $\{\operatorname*{inflate}(\tau, \lambda_j)\}$ is $$t(\pi, \operatorname*{inflate}(\tau)) =
\frac{|\pi|!}{|\tau|^{|\pi|}} \sum_{(b, \sigma) \in B(\pi)} \left(\binom{|\tau|}{|\sigma|} t(\sigma, \tau) \cdot \prod_{\alpha \in b} \frac{1}{|\alpha|!^2}\right).$$
Example
-------
As we are primarily interested in the limit densities of length-3 pattern permutations, for our next example we choose $\pi = 132$. Consider some general permutation $\tau$ of length $n$.
The permutation 132 admits 3 block-partitions: $132$, $1|32$, and $1|3|2$. Our formula yields for the limit density of 132 in the inflation of $\tau$ with a random sequence:
$$\begin{aligned}
&\frac{3!}{n^3} \left(
\binom{n}{1}t(1, \tau) \cdot \frac{1}{3!^2} +
\binom{n}{2}t(12, \tau) \cdot \frac{1}{1!^2 2!^2} +
\binom{n}{3}t(132, \tau) \cdot \frac{1}{1!^2 1!^2 1!^2}
\right) \\
&\qquad= \frac{3!}{n^3}
\left(
\frac{1}{36}\binom{n}{1} +
\frac{1}{4}\binom{n}{2}t(12, \tau) +
\binom{n}{3}t(132, \tau)
\right).\end{aligned}$$
Notice from this expression that there exists additional structure if we focus our attention on length-3 pattern permutations. Let us denote first $\frac{3!}{n^3} \binom{n}{3}$ by $a(n)$, second $\frac{3!}{n^3}\frac{1}{4}\binom{n}{2}$ by $b(n)$, and third $\frac{3!}{n^3}\frac 1{36}\binom n1$ by $c(n)$. Substituting yields $$t(132,\operatorname*{inflate}(\tau))= a(n)t(132,\tau) + b(n) t(12,\tau)+ c(n).$$ Also, the limit densities of permutations 213, 312 and 231 in the inflation of $\tau$ are given by analogous formulas: $$t(213,\operatorname*{inflate}(\tau))= a(n)t(213,\tau) + b(n) t(12,\tau)+ c(n),$$ $$t(312,\operatorname*{inflate}(\tau))= a(n)t(312,\tau) + b(n) t(21,\tau)+ c(n),$$ $$t(231,\operatorname*{inflate}(\tau))= a(n)t(231,\tau) + b(n) t(21,\tau)+ c(n).$$ The reason for this similarity is that the block-partitions of $213$, $312$, and $132$ all follow the same pattern. Each has a partition into one block of size 3, two blocks of sizes 1 and 2, and three blocks of size 1.
Similarly, permutations 123 and 321 follow similar formulas, except the coefficient of the second term is doubled since there are two ways to partition each of $123$ and $321$ into two blocks. These limit densities are given by $$t(123,\operatorname*{inflate}(\tau))= a(n)t(123,\tau) + 2 b(n) t(12,\tau)+ c(n),$$ $$t(321,\operatorname*{inflate}(\tau))= a(n)t(321,\tau) + 2 b(n) t(12,\tau)+ c(n).$$ To see an example of this symmetry, consider the length-9 permutation $\tau = 472951836$. The density of $132$ in $\tau$ is $17/84$, and the density of $12$ in $\tau$ is $1/2$. Therefore, $$t(132, \operatorname*{inflate}(\tau)) = \frac{29}{162}.$$ Similarly, we then have $$t(213, \operatorname*{inflate}(\tau)) = t(231, \operatorname*{inflate}(\tau)) = t(312, \operatorname*{inflate}(\tau)) = \frac{29}{162}.$$ $$t(123, \operatorname*{inflate}(\tau)) = t(321, \operatorname*{inflate}(\tau)) = \frac{23}{162}.$$
Inflation and $k$-symmetry {#sec:ksymmetry}
==========================
We now turn our attention to symmetric permutation sequences.
A convergent permutation sequence $\{\tau_j\}$ is called *$k$-symmetric* (as defined in [@cooper08]) if for every permutation $\pi$ of length $k$, the limit density of $\pi$ in $\tau_j$ is $1/k!$.
A permutation $\tau$ is called *$k$-inflatable* if $\operatorname*{inflate}(\tau)$ is $k$-symmetric.
As a direct corollary of Theorem \[thm:inflate-density\] permutation $\tau$ is 2-inflatable if and only if $t(12, \tau) = 1/2$. We are interested in 3-inflatable permutations, that is that the $\operatorname*{inflate}(\tau)$ sequence has 3-point densities all equal to $1/3!$.
It was claimed in Cooper and Petrarca [@cooper08] that the smallest $3$-inflatable permutations are of length $9$. They provide as examples $472951836$ and its inverse. However, by our calculations above, this is not the case. We have also verified this empirically by estimating the limit densities in the inflation using a Monte Carlo method, and the results were consistent with our formula.
$k$-inflatable permutations
---------------------------
Theorem \[thm:inflate-density\] allows us to calculate the densities of the permutation we inflate, so that the result is 3-inflatable.
A permutation $\tau$ of length $n$ is $3$-inflatable if and only if $t(12, \tau) = 1/2,$ and $$t(123, \tau) = t(321, \tau) = (2 n - 7)/(12 (n - 2)),$$ $$t(132, \tau) = t(213, \tau) = t(231, \tau) = t(312, \tau) = (4 n - 5)/(24 (n - 2)).$$
First, a 3-inflatable permutation must also be 2-inflatable, so the density of 12 in $\tau$ must be 1/2.
A specific application of Corollary \[random-density\] for three-point densities yields $$t(123, \operatorname*{inflate}(\tau)) = \frac{9 \cdot 8 \cdot 7}{9^3} t(123, \tau) + \binom 92 \cdot \frac{3}{9^3} \cdot 2 \cdot t(12, \tau) \cdot \frac 12 + \frac{9}{9^3} \frac{1}{3!}.$$ Using this formula, we can now calculate the density of 123 in a 3-inflatable permutation. We have already showed that the density of inversions should be 1/2. Therefore, $$\frac{1}{6} = \frac{n(n-1)(n-2)}{n^3} t(123, \tau) + \binom n2 \cdot \frac{3}{n^3} \cdot 2 \cdot
\frac 12 \cdot \frac 12 + \frac{n}{n^3} \frac{1}{3!}.$$ Multiplying by $12n^2$ and rearranging yields $$2n^2 - 9n + 7 = 12(n-1)(n-2) t(123, \tau).$$ Then, we observe that the density is $$t(123, \tau) = \frac{2n^2 - 9n + 7}{12(n-1)(n-2)} = \frac{(2n-7)(n-1)}{12(n-1)(n-2)} = \frac{2n-7}{12(n-2)}.$$ Analogous applications of this formula for densities of the five other length-3 permutations yields the desired result.
For a given integer $n$, these densities might not be possible for any permutation $\tau$ due to divisibility. The conditions are described in the following corollary.
If a permutation of length $n$ is $3$-inflatable, then both:
1. $\binom n2$ is even.
2. $\binom n3$ is divisible by the reduced denominators of $\frac{2n-7}{12(n-2)}$ and $\frac{4 n - 5}{24 (n - 2)}$.
The above divisibility criteria can be more explicitly described by determining the complete set of admissible residue classes modulo 144.
For any 3-inflatable permutation $\tau$ with length $n$, $$n \equiv 0, 1, 17, 64, 80, 81 \pmod{144}.$$
Let $k$ be the number of occurrences of $132$ in the permutation $\tau$. By the above criteria, we have that $$\frac{k}{\binom{n}{3}} = \frac{4 n - 5}{24 (n - 2)},$$ $$144 k = n(n-1)(4 n - 5),$$ $$144 \mid n(n-1)(4n-5).$$ Also, $$\frac{k}{\binom{n}{3}} = \frac{2 n - 7}{12 (n - 2)},$$ $$72k = n(n - 1)(2 n - 7),$$ $$72 \mid n(n - 1)(2 n - 7).$$ Together, these two divisibility requirements are equivalent to $$n \equiv 0, 1, 17, 64, 80, 81 \pmod{144}.$$
The smallest possible nontrivial length for a 3-inflatable permutation is therefore 17. In this case, the binomials $\binom{17}{2} = 136$ and $\binom{17}{3} = 680$, while $(2n-7)/(12(n-2)) = 3/20$ and $(4 n - 5)/(24 (n - 2)) = 7/40$. Therefore, in a 3-inflatable permutation of length 17, the numbers of occurrences of the subpermutations 123 and 321 should both be 102, while the four other pattern permutations of length 3 should each occur 119 times.
unitsize(5cm); drawperm(“E534BGA9HC2D1687F”); label(“E534BGA9HC2D1687F”, (0.5, -0.03), S); drawperm(“G54ABC319HF678ED2”, 1.2); label(“G54ABC319HF678ED2”, (0.5 + 1.2, -0.03), S);
Examples of these minimum-length 3-inflatable permutations were found through an optimized computer search. For example, the permutations $$\text{E534BGA9HC2D1687F and G54ABC319HF678ED2}$$ are both 3-inflatable, where capital letters denote numbers greater than nine (A=10, B=11, C=12, etc.). Plots of these permutations are shown in Figure \[fig:plot-length17\].
Although it was intractable to check all $17!$ of such permutations for the inflation criterion through our available computational resources, we could reduce the search space by checking only those permutations which were *centrally symmetric* (further discussed in the next section). Of the $8 \cdot 2^8$ centrally symmetric permutations of length $17$, a computer search found that $750$ of them are $3$-inflatable.
Furthermore, additional computer searches found many more examples of centrally symmetric, 3-inflatable permutations of lengths $64$, $80$, $81$, $144$, and $145$. We have found 3-inflatable permutations of lengths belonging to each admissible residue class modulo 144. There also seems to be a large number of such permutations of each length. These empirical results suggest the following conjecture.
For any positive integer $x$ of length belonging to an admissible residue class modulo $144$, there exists a $3$-inflatable permutation of length $x$.
Structure of 3-inflatable permutations {#sec:structure3}
======================================
We start by proving that the inflation of two $k$-inflatable permutations is itself $k$-inflatable. This gives us an explicit construction that shows there exists an infinite number of $3$-inflatable permutations, and also allows us to construct examples of arbitrarily large length.
If $\tau_1$ and $\tau_2$ are two $k$-inflatable permutations, then $\operatorname*{inflate}(\tau_1, \tau_2)$ is also $k$-inflatable.
Note that inflation is associative, so for a random permutation $\gamma$, $$\operatorname*{inflate}(\operatorname*{inflate}(\tau_1, \tau_2), \gamma) = \operatorname*{inflate}(\tau_1, \operatorname*{inflate}(\tau_2, \gamma)).$$ Then, by the definition of a $k$-inflatable permutation, for any length-$k$ permutation $\pi$, $$\lim_{|\gamma| \rightarrow \infty} t(\pi, \operatorname*{inflate}(\tau_2, \gamma)) = \frac{1}{k!}.$$ Since our current discussion only considers densities of pattern permutations with length at most $k$, we can substitute $\operatorname*{inflate}(\tau_2, \gamma)$ for a random permutation $\gamma_2$ of the same length without changing the expected value of the expression. Thus, $$\lim_{|\gamma| \rightarrow \infty} t(\pi, \operatorname*{inflate}(\tau_1, \operatorname*{inflate}(\tau_2, \gamma))) = \lim_{|\gamma_2| \rightarrow \infty} t(\pi, \operatorname*{inflate}(\tau_1, \gamma_2)) = \frac{1}{k!},$$ So $\operatorname*{inflate}(\tau_1, \tau_2)$ is $k$-inflatable.
Given this result, it is natural to ask if the set of admissible lengths of 3-inflatable permutations modulo 144 are closed under multiplication — in other words, if the set of numbers with remainders 0, 1, 17, 64, 80, and 81 modulo 144 is closed under multiplication. This is true, as shown in Figure \[fig:mult\_table\].
$\times$ 0 1 17 64 80 81
---------- --- ---- ---- ---- ---- ----
0 0 0 0 0 0 0
1 0 1 17 64 80 81
17 0 17 1 80 64 81
64 0 64 80 64 80 0
80 0 80 64 80 64 0
81 0 81 81 0 0 81
For example, we can use the 3-inflatable permutations of size 17 that we found to build 3-inflatable permutations of sizes $17^n$.
Now we want to define a symmetry of permutations that is useful for increasing the likelihood of being symmetric. Let us define an operation $R$ on permutations which we call a rotation: $R(\pi)_i = n + 1 - \pi_{n + 1 - i}$. This is equivalent to drawing the permutation as a graph (on a square grid) then rotating it by 180 degrees about its center.
We call a permutation $\pi$ of length $n$ *centrally symmetric* if it is equal to its rotation (for example, see the right plot in Figure \[fig:plot-length17\]). In other words, $\pi_i + \pi_j = n+1$ whenever $i+j = n + 1$. The importance of centrally symmetric permutations is explained by the following lemma.
For permutations $\pi$ and $\gamma$, $$t(\pi,\gamma) = t(R(\pi),R(\gamma)).$$
Let $k = |\pi|$, and let $S_1$ be the set of $k$-point subsets of $\gamma$ that are order-isomorphic to $\pi$. Similarly, let $S_2$ be the set of $k$-point subsets of $R(\gamma)$ that are order-isomorphic to $R(\pi)$. There is a one-to-one correspondence between $S_1$ and $S_2$ given by the rotation operation.
More formally, for any set of indices $\{ i_1, i_2, \dots, i_k \}$ in $\gamma$, there is a corresponding set of indices, $\{ n + 1 - i_1, n + 1 - i_2, \dots, n + 1 - i_k \}$ in $R(\gamma)$. If the former indices induce some permutation $\pi$ in $\gamma$, then the latter set induces $R(\pi)$ in $R(\gamma)$ from the definition of rotation. Thus, we have $|S_1| = |S_2|$, so the result follows.
If $\pi$ is centrally symmetric, then $$t(\pi,\gamma) = t(\pi,R(\gamma)).$$
That means centrally symmetric permutations automatically give us some equalities among densities. For example, if $\pi$ is centrally symmetric, then $$t(132,\pi) = t(231,\pi) \quad \text{and} \quad t(312,\pi) = t(213,\pi).$$
Acknowledgements
================
We are grateful to Prof. Yufei Zhao for suggesting this project to us and for helpful discussions on the topic, as well as to the anonymous reviewer for their thorough review and suggestions. We would also like to thank the PRIMES program for giving us the opportunity to do this research.
[^1]: For clarity, here we abuse notation by taking $\operatorname*{inflate}(\tau, \sigma)$ to mean, for $\{ \sigma_j \}$ being a convergent permutation sequence, the sequence of inflations $\{ \operatorname*{inflate}(\tau, \sigma_j) \}$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We study the generation of a gravitational wave (GW) background produced from a population of core-collapse supernovae, which form black holes in scenarios of structure formation of the Universe. We obtain, for example, that a pre-galactic population of black holes, formed at redshifts $z\simeq 30-10$, could generate a stochastic GW background with a maximum amplitude of $h_{\rm
BG}\simeq 10^{-24}$ in the frequency band $\nu_{\rm obs}\simeq
30-470 {\rm Hz}$ (considering a maximum efficiency of generation of GWs, namely, $\varepsilon_{\rm GW}=7\times 10^{-4}$). In particular, we discuss what astrophysical information could be obtained from a positive, or even a negative, detection of such a GW background produced in scenarios such as those studied here. One of them is the possibility of obtaining the initial and final redshifts of the emission period from the observed spectrum of GWs.
address: |
Instituto Nacional de Pesquisas Espaciais - Divisão de Astrofísica\
Av. dos Astronautas 1758, São José dos Campos, 12227-010 SP, Brazil
author:
- 'J C N de Araujo, O D Miranda and O D Aguiar'
title: 'Background of gravitational waves from pre-galactic black hole formation'
---
Introduction
============
Because of the fact that gravitational waves (GWs) are produced by a large variety of astrophysical sources and cosmological phenomena, it is quite probable that the Universe is pervaded by a background of such waves. A variety of binary stars (ordinary, compact or combinations of them), Population III stars, phase transitions in the early Universe and cosmic strings are examples of sources that could produce such a putative background of GWs (Thorne 1987, Blair and Ju 1996, Owen 1998, Ferrari 1999a, 1999b, Schutz 1999, Giovannini 2000, Maggiore 2000, Schneider 2000, among others).
In the present study we have considered the background of GWs produced from a Population III black hole formation. The basic arguments in favour of the existence of these pre-galactic black holes are the following: (a) from the Gunn-Peterson effect (Gunn and Peterson 1965), it is widely accepted that the Universe underwent a reheating (or reionization) phase between the standard recombination epoch (at $z\sim 1000$) and $z
> 5$ as a result of the formation of the first structures of the Universe (see, Haiman and Loeb 1997, Loeb and Barkana 2001 for a review). However, at what redshift the reionization occured is still an open question (Loeb and Barkana 2001), although recent studies conclude that it occurred at redshifts in the range $6<z<30$ (Venkatesan 2000); (b) the metallicity of $\sim 10^{-2}
Z_{\odot}$ found in ${\rm high}-z$ Ly$\alpha$ forest clouds (Songaila and Cowie 1996, Ellison 2000) is consistent with a stellar population formed at $z > 5$ (Venkatesan 2000).
In the present paper we have adopted a stellar generation with a Salpeter initial mass function (IMF) as well as different stellar formation epochs. We then discuss what conclusions would be drawn from whether (or not) the stochastic background studied here is detected by forthcoming GW observatories such as LIGO and VIRGO.
The paper is organized as follows. In section 2 we describe how to calculate the background of GWs produced during the formation of the stellar black holes in this scenario (the reader finds a more detailed description in de Araujo 2002), in section 3 we present some numerical results and the discussions, in section 4 we consider the detectability of this putative GW background and finally in section 5 we present our conclusions.
The gravitational wave production from pre-galactic stars
=========================================================
Before going into detail of the calculation of the background of GWs produced by pre-galactic stars, it is import to consider in what ways the present study differs from previous ones.
Ferrari (1999a), for example, consider the background of GWs produced by the formation of black holes of galactic origin, i.e., those formed at redshifts $z<5$. Schneider (2000) study the collapse of Very Massive Objects (VMO), and the eventual formation of very massive black holes and the GWs they generate. Fryer (2002) study, among other issues, the collapse of very massive Population III stars ($>300{\rm M}_\odot$) and the GWs generated by them.
Here, we consider a different approach to the formation of pre-galactic objects. We take into account the formation of Population III stellar black holes, considering that the progenitor stars follow a Salpeter’s IMF. In this case, the progenitor stars have masses in the range $25-125 M_\odot$. We also consider that the formation of these black holes occurs at high redshifts, namely, $z>10$. It is worth noting that Sapeter’s Population III stars could account for the metallicity found in high-z Ly$\alpha$ forest clouds, and, at least in part, since the QSOs could also be important, for the reionization of the Universe.
Let us now focus on how to calculate the background of GWs produced by the Population III stellar black holes we propose exist.
The spectral energy density, the flux of GWs, received on Earth, $F_\nu$, in ${\rm erg}\, {\rm cm}^{-2}\,{\rm s}^{-1}\,{\rm
Hz}^{-1}$, reads (see, e.g. Douglass and Braginsky 1979, Hils 1990)
$$F_{\nu} = {c^{3}s_{\rm h}\omega_{\rm obs}^{2}\over 16{\rm \pi} G},
\label{fluxa}$$
where $\omega_{\rm obs}=2{\rm \pi} \nu_{\rm{obs}}$, with $\nu_{\rm{obs}}$ being the GW frequency (Hz) observed on Earth, $c$ is the velocity of light, $G$ is the gravitational constant and $\sqrt{s_{\rm h}}$ is the strain amplitude of the GW ($\rm
Hz^{-1/2}$).
The stochastic GW background produced by gravitational collapses that lead to black holes would have a spectral density of the flux of GWs and strain amplitude also related to equation (\[fluxa\]). The strain amplitude at a given frequency, at the present time, is the contribution of black holes with different masses at different redshifts. Thus, the ensemble of black holes formed produces a background whose characteristic strain amplitude at the present time is $\sqrt s_{\rm h}$.
On the other hand, the spectral density of the flux can be written as (Ferrari 1999a)
$$F_{\nu}=\int_{z_{\rm {cf}}}^{z_{\rm {ci}}} \int_{m_{\rm
{min}}}^{m_{\rm {u}}} f_{\nu}(\nu_{\rm{obs}}) dR_{\rm BH}(m,z)$$
where $f_{\nu}(\nu_{\rm{obs}})$ is the energy flux per unit of frequency (in ${\rm erg}\,{\rm cm}^{-2}\,{\rm Hz}^{-1}$) produced due to the formation of a unique black hole and $dR_{\rm
BH}$ is the differential rate of black hole formation.
The above equation takes into account the contribution of different masses that collapse to form black holes occurring between redshifts $z_{\rm ci}$ and $z_{\rm cf}$ (beginning and end of the star formation phase, respectively) that produce a signal at the same frequency $\nu_{\rm{obs}}$. On the other hand, we can write $f_{\nu}(\nu_{\rm{obs}})$ (Carr 1980) as
$$f_{\nu}(\nu_{\rm{obs}}) = {{\rm \pi} c^{3}\over 2G}h_{\rm BH}^{2}$$
where $h_{\rm BH}$ is the dimensionless amplitude produced by the collapse, to a black hole, of a given star with mass $m$ that generates at the present time a signal with frequency $\nu_{\rm{obs}}$. Then, the resulting equation for the spectral density of the flux is
$$F_{\nu} = {\pi c^{3}\over 2G} \int h_{\rm BH}^{2}dR_{\rm BH}.$$
From the above equations, we obtain for the strain amplitude
$$s_{\rm h} = {1 \over \nu_{\rm obs}^{2}}\int h_{\rm BH}^{2} dR_{\rm
BH}.$$
Then, the dimensionless amplitude of the background of GWs, $h_{\rm BG}$, reads
$$h_{\rm BG}^{2} = {1 \over \nu_{\rm obs}}\int h_{\rm BH}^{2}
dR_{\rm BH}$$
(see, de Araujo 2000 for details) where $\nu_{\rm obs}$, $h_{\rm BH}$ and $dR_{\rm BH}$ are defined below.
The dimensionless amplitude $h_{\rm BH}$ produced by the collapse of a star, or star cluster, to form a black hole is (Thorne 1987)
$$h_{\rm BH} \simeq 7.4\times 10^{-20}\, \varepsilon_{\rm
GW}^{1/2}\bigg({M_{\rm r}\over {\bf M}_{\odot}}\bigg)
\bigg({d_{\rm L}\over 1{\rm Mpc}}\bigg)^{-1} \label{hBH}$$
where $\varepsilon_{\rm GW}$ is the efficiency of generation of GWs, $M_{\rm r}$ is the remnant black hole mass and $d_{\rm L}$ is the luminosity distance to the source.
It is worth mentioning that equation (\[hBH\]) refers to the black hole ‘ringing’, which has to do with the de-excitation of the black hole quasi-normal modes. Note also that $\varepsilon_{\rm GW} \propto a^{4}$ (see, e.g. Stark and Piran 1986), where ‘$a$’ is the the dimensionless angular momentum. Thus, greater the GW efficiency, greater the dimensionless angular momentum.
We assume that the progenitor masses of the black holes range from $M=25-125{\rm M}_\odot$ (see Timmes 1995, Woosley and Timmes 1996). The remnant and the progenitor masses are related to $M_{\rm r}=\alpha M$ and, we assume $\alpha=0.1$ (see, e.g. Ferrari 1999a).
The collapse of a star to a black hole produces a signal with an observed frequency $\nu_{\rm{obs}}$ at the Earth (Thorne 1987)
$$\nu_{\rm{obs}} \simeq 1.3\times 10^{4}\,{\rm Hz}\bigg({{\rm
M}_{\odot}\over M_{\rm r}}\bigg)(1+z)^{-1}\label{freq}$$
where the factor $(1+z)^{-1}$ takes into account the redshift effect on the emission frequency.
The differential rate of black hole formation $dR_{\rm BH}$ reads
$$dR_{\rm BH} = \dot\rho_{\star}(z) {dV\over dz} \phi(m)dmdz$$
where $dV$ is the co-moving volume element, $\phi(m)$ is the stellar initial mass function (IMF) and $\dot\rho_{\star}(z)$ is the star formation rate (SFR) density.
The SFR density can be related to the reionization of the Universe. The amount of baryons necessary to participate in early star formation, to account for the reionization, would amount to a small fraction, $f_{\star}$, of all baryons of the Universe (see, e.g. Loeb and Barkana 2001). We then assume that
$$\dot\rho_\star \equiv {d\rho_{\star}\over dt}= {d\over dt}
[\Omega_{\star}\;\rho_{\rm c}\;(1+z)^{3}]$$
where the term in brackets represents the stellar mass density at redshift $z$, with $\rho_{\rm c}$ the present critical density, and $\Omega_{\star}$ the stellar density parameter. The latter can be written as a fraction of the baryonic density parameter, namely, $\Omega_{\star}=f_{\star}\Omega_{\rm B}$, which we assume to be independent of the redshift.
From the above equations, we obtain for the dimensionless amplitude
$$\begin{aligned}
h_{\rm BG}^{2} &=& {(7.4\times 10^{-20}\alpha)^{2}\varepsilon_{\rm
GW} \over \nu_{\rm{obs}}} \nonumber\\ && \times \bigg[\int_{z_{\rm
cf}}^{z_{\rm ci}} \int_{m_{\rm min}}^{m_{\rm u}}\bigg({m\over {\rm
M}_{\odot}}\bigg)^{2}\bigg({d_{\rm L}\over 1{\rm Mpc}}\bigg)^{-2}
\dot\rho_{\star}(z){dV\over
dz} \phi(m)dmdz\bigg]\label{hBG}\end{aligned}$$
where $m_{\rm min}=25{\rm M}_\odot$, $m_{\rm u}=125{\rm
M}_\odot$ and $z_{\rm ci}$ ($z_{\rm cf}$) is the beginning (end) of the black hole formation phase. Equation (\[hBG\]) is computed for each observed frequency. Also, looking at equation (\[hBG\]), one notes that to integrate it, one needs to choose the IMF, the cosmological parameters and set values for the following parameters: $z_{\rm ci}$, $z_{\rm cf}$, $\alpha$, $\varepsilon_{\rm GW}$, $f_{\star}$. In the next section we present the numerical results and discussions.
Numerical results and discussions
=================================
To evaluate the background of GWs produced by the formation of the Population III black holes, it is necessary to know in which redshifts they began and finished formation. This is a very difficult question to answer, since it involves knowledge of the role of the negative and positive feedbacks of star formation which are regulated by cooling and injection of energy processes.
Should the stochastic background of GWs studied here be significantly produced and detected at a reasonable confidence level, the present study can be used to obtain the redshift range where the Population III black holes were formed. We refer the reader to the paper by de Araujo (2002) for further discussions. In figure 1 an example is given of how one could get $z_{\rm ci}$ and $z_{\rm cf}$ from the curve $h_{\rm BG}$ versus $\nu_{\rm obs}$. Knowing the frequency band $\nu_{\rm
min}-\nu_{\rm max}$ detected from a cosmological source, and using equation (\[freq\]), one can obtain both $z_{\rm ci}$ and $z_{\rm cf}$ from figure 1. Thus, these redshifts are therefore observable. Note that we have assumed as did Ferrari (1999a) that $\alpha$ is a constant ($\alpha =0.1$).
Stars start forming at different redshifts, creating ionized bubbles (Strömgren spheres) around themselves, which expand into the intergalactic medium (IGM), at a rate dictated by the source luminosity and the background IGM density (Loeb and Barkana 2001). The reionization is complete when the bubbles overlap to fill the entire Universe. Thus the epoch of reionization is not the epoch of star formation. There is a non-negligible time span between them. Here, we have chosen different formation epochs to see their influence on the putative background of GWs and also to see if it could be detected by the forthcoming GW antennas.
To calculate $h_{\rm BG}$ we adopted the standard Salpeter IMF. For $\varepsilon_{\rm GW}$, the efficiency of production of GWs, whose distribution function is unknown, we have parametrized our results in terms of its maximum value, namely, $\varepsilon_{\rm
GW_{\rm max}}=7 \times 10^{-4}$. This figure is obtained from studies by Stark and Piran (1986) who simulated the axisymmetric collapse of a rotating star to a black hole.
To calculate $h_{\rm BG}$ we still need to know $\Omega_{\star}$, which has a key role in the definition of the SFR density. From different studies one can conclude that a few per cent, may be up to $\sim 10\%$, of the baryons must be condensed into stars in order for the reionization of the Universe to take place (see, e.g. Venkatesan 2000). Here we have set the value of $\Omega_{\star}$ in such a way that it amounts to $1\%$ of all baryons (our fiducial value).
Looking at equation (\[hBG\]), one could think it would depend critically on the cosmological parameters $H_0$, $\Omega_{\rm B}$, $\Omega_{\rm DM}$ (the density parameter for the dark matter) and $\Omega_\Lambda$ (the density parameter associated with the cosmological constant). But our results show that $h_{\rm BG}$ depends only on $H_{0}$ and $\Omega_{\rm B}$, the Hubble parameter and the baryonic density parameter, respectively.
The quantity $h_{100}^{2}\Omega_{\rm B}=0.019\pm 0.0024$ (where $h_{100}$ is the Hubble parameter given in terms of 100 ${\rm
km\;s^{-1}\;Mpc^{-1}}$) is adopted in the present model. This figure is obtained from Big Bang nucleosynthesis studies (see, e.g. Burles 1999).
In table 1 we present the redshift band, $z_{\rm ci}$ and $z_{\rm
cf}$, for the models studied and the corresponding GW frequency bands. For the cosmological parameters, we have adopted $h_{100}=0.65$, $\Omega_{\rm M}=0.3$, $\Omega_{\rm B}=0.045$ and $\Omega_{\Lambda}=0.7$. We have also adopted $\alpha=0.1$, $f_{\star}=0.01$ and the standard IMF.
It is worth noting that no structure formation model has been used to find the black holes formation epoch; instead, we have simply chosen the values of $z$ to see whether it is possible to obtain detectable GW signals. In the next section, it will be seen that unless $\varepsilon_{\rm GW}$ is negligible, the background of GWs we propose here can be detected. Our choices, however, can be understood as follows. The greater the redshift formation, the more power the masses related to the Population III objects have. Thus from our models A to D, our model D (A) has more (less) power when compared to the others. The models E, F and G would mean a more extended star formation epoch, which means that the feedback processes of star formation are such that they allow a more extended star formation epoch when compared to the models B, C and D, respectively.
Note that the reionization epoch occurred at lower redshifts as compared to the first stars formation redshifts. Loeb and Barkana (2001) found, for example, that if the stars were formed at $z
\simeq 10-30$, with standard IMF, they could have reionized the Universe at redshift $z \sim 6$. Our models A, B and E, for example, could account for such a reionization redshift.
If the process of structure formation of the Universe and the consequent star formation were well known, one could obtain the redshift formation epoch of the first stars. On the other hand, if the background of GWs really exists and is detected, one can obtain information about the formation epoch of the first stars.
[ccccccc]{} Model & $z_{\rm ci}$ & $z_{\rm cf}$ & $ \Delta\nu_{\rm obs} $ (Hz)\
A & 20 & 10 & 50-470\
B & 30 & 20 & 34-250\
C & 40 & 30 & 25-170\
D & 50 & 40 & 20-130\
E & 30 & 10 & 34-470\
F & 40 & 10 & 25-470\
G & 50 & 10 & 20-470\
A relevant question is whether the background we study here is continuous or not. The duty cycle indicates if the collective effect of the bursts of GWs generated during the collapse of a progenitor star generates a continuous background. For all the models studied here the duty cycle is $\gg 1$ (see de Araujo 2002 for details).
We find, for example, that the formation of a Population (III) of black holes, in the model D, could generate a stochastic background of GWs with amplitude $h_{\rm BG} \simeq (0.8-2)\times
10^{-24}$ and a corresponding closure density of $\Omega_{\rm{GW}}\simeq (0.7-1.4)\times 10^{-8}$, at the frequency band $\nu_{\rm{obs}} \simeq 20-130\, {\rm Hz}$ (assuming an efficiency of generation $\varepsilon_{\rm GW} \simeq 7\times
10^{-4}$, the maximum one).
In other paper to appear elsewhere (de Araujo 2002), we study in detail how the variations of the several parameters modify our results.
Detectability of the background of gravitational waves
======================================================
The background predicted in the present study cannot be detected by single forthcoming interferometric detectors, such as VIRGO and LIGO (even by advanced ones). However, it is possible to correlate the signal of two or more detectors to detect the background that we propose to exist.
To assess the detectability of a GW signal, one must evaluate the signal-to-noise ratio (SNR), which for a pair of interferometers is given by (see, e.g. Flanagan 1993)
$${\rm SNR}^2=\left[\left(\frac{9 H_0^4}{50\pi^4} \right) T
\int_0^\infty d\nu \frac{\gamma^2(\nu)\Omega^2_{GW}(\nu) } {\nu^6
S_{\rm h}^{(1)}(\nu) S_{\rm h}^{(2)}(\nu)} \right]$$
where $ S_{\rm h}^{(i)}$ is the spectral noise density, $T$ is the integration time and $\gamma(\nu)$ is the overlap reduction function, which depends on the relative positions and orientations of the two interferometers. The closure energy density is given by
$$\Omega_{\rm GW} = {1\over \rho_{\rm c}} {d\rho_{\rm GW}\over d\log
\nu_{\rm{obs}}}={4{\rm \pi}^{2}\over 3H^{2}_{0}}\nu_{\rm{obs}}^{2}
h_{\rm BG}^{2}.$$
Here we consider, in particular, the LIGO interferometers, and their spectral noise densities have been taken from a paper by Owen (1998).
In table 2 we present the SNR for 1 year of observation with $\alpha=0.1$, $\Omega_{\rm B}h^{2}_{100}=0.019$, $f_{\star}=0.01$ and $\varepsilon_{\rm GW_{\rm max}}=7\times 10^{-4}$ for the models in table 1, for the three LIGO interferometer configurations.
[cccc]{} & & SNR &\
&\
Model & LIGO I & LIGO II & LIGO III\
A & $8.3\times 10^{-3}$ & $1.6$ & $6.6$\
B & $8.5\times 10^{-3}$ & $2.3$ & $26 $\
C & $8.7\times 10^{-3}$ & $2.7$ & $47 $\
D & $8.1\times 10^{-3}$ & $2.5$ & $51 $\
E & $2.7\times 10^{-3}$ & $5.7$ & $37 $\
F & $5.0\times 10^{-3}$ & $12 $ & $120$\
G & $7.7\times 10^{-2}$ & $21 $ & $260$\
Note that for the ‘initial’ LIGO (LIGO I), there is no hope of detecting the background of GWs we propose here. For the ‘enhanced’ LIGO (LIGO II) there is some possibility of detecting the background, since ${\rm SNR}
> 1$, if $\varepsilon_{\rm GW}$ is around the maximum value. Even if the LIGO II interferometers cannot detect such a background, it will be possible to constrain the efficiency of GW production.
The prospect of detection with the ‘advanced’ LIGO (LIGO III) interferometers is much more optimistic, since the SNR for almost all models is significantly greater than unity. Only if the value of $\varepsilon_{\rm GW}$ were significantly lower than the maximum value would the detection not be possible. In fact, the signal-to-noise ratio is critically dependent on this parameter.
Note that the larger the star formation redshift band, the greater the SNR. Secondly, the earlier the star formation, the greater the SNR. It is worth recalling that, if one can obtain the curve ‘$h_{\rm BG}$ versus $\nu_{\rm obs}$’ and the value of $\alpha$ is known, one can find the redshift of star formation.
Conclusions
===========
We have shown that a background of GWs is produced from Population III black hole formation at high redshift. This background can in principle be detected by a pair of LIGO II (or more probably by a pair of LIGO III) interferometers. However, a relevant question should be considered: what astrophysical information can one obtain whether or not such a putative background is detected?
First, let us consider a non-detection of the GW background. The critical parameter to be constrained here is $\varepsilon_{\rm
GW}$. A non-detection would mean that the efficiency of GWs during the formation of black holes is not high enough. Another possibility is that the first generation of stars is such that the black holes formed had masses $> 100M_{\odot}$, and should they form at $z > 10$ the GW frequency band would be out of the LIGO frequency band.
Secondly, a detection of the background with a significant SNR would permit us to obtain the curve of $h_{\rm BG}$ versus $\nu_{\rm obs}$. From it, one can constrain $\alpha$ and the redshift formation epoch; and for a given IMF and $\Omega_{\rm
B}h^{2}$, one can also constrain the values of $f_{\star}$ and $\varepsilon_{\rm GW}$. On the other hand, using the curve of $h_{\rm BG}$ versus $\nu_{\rm obs}$ and in addition other astrophysical data, say CBR data, models of structure formation and reionization of the Universe, constraint on $\varepsilon_{\rm
GW}$ can also be imposed. We refer the reader to the paper by de Araujo (2002) for further discussions.
It is worth mentioning that a significant amount of GWs can also be produced during the formation of neutron stars and if such stars are r-mode unstable (Andersson 1998). We leave these issues for other studies to appear elsewhere.
The authors thank FAPESP (Brazil) for support (grant numbers 97/06024-4, 97/13720-7, 98/13468-9, 98/13735-7, 00/00116-9, 01/04086-0, and 01/04189-3). JCNA and ODA also thank CNPq (Brazil) for support (grant numbers 450994/01-5 and 300619/92-8, respectively). We also thank the referees for the suggestions and criticisms.
Andersson N 1998 [ *Astrophys. J.*]{} [**211**]{} 708
Blair D G and Ju L 1996 [*Mon. Not. R. Astron. Soc.*]{} [**283**]{} 618
Burles S, Nollett K M, Truran J N and Turner M S 1999 [*Phys.Rev.Lett.*]{} [**82**]{} 4176
Carr B J 1980 [*Astron. Astrophys.*]{} [**89**]{} 6
de Araujo J C N, Miranda O D and Aguiar O D 2000 [*Phys. Rev.*]{} D [**61**]{} 124015
de Araujo J C N, Miranda O D and Aguiar O D 2002 [*Mon. Not. R. Astron. Soc.*]{} at press
Douglass D H and Braginsky V G 1979 [*General Relativity: An Einstein Centenary Survey*]{} (Cambridge: Cambridge University Press) p 90
Ellison S, Songaila A, Schaye J and Petinni M 2000 [*Astron. J.*]{} [**120**]{} 1175
Ferrari V, Matarrese S and Schneider R 1999a [*Mon. Not. R. Astron. Soc.*]{} [**303**]{} 247
Ferrari V, Matarrese S and Schneider R 1999b [*Mon. Not. R. Astron. Soc.*]{} [**303**]{} 258
Flanagan E E 1993 [*Phys. Rev.*]{} D [**48**]{} 2389
Fryer C L, Holz D E and Hughes S A 2002 [ *Astrophys. J.*]{} [**565**]{} 430
Giovannini M 2000 [*Preprint*]{} gr-qc/0009101
Gunn J E and Peterson B A 1965 [ *Astrophys. J.*]{} [**142**]{} 1633
Haiman Z and Loeb A 1997 [*Astrophys. J.*]{} [**483**]{} 21
Hils D, Bender P L and Webbink R F 1990 [ *Astrophys. J.*]{} [**360**]{} 75
Loeb A and Barkana R 2001 [*Ann. Rev. Astron. Astrophys.*]{} [**39**]{} 19
Maggiore M 2000 [*Phys. Rep.*]{} [**331**]{} 283
Owen B J, Lindblom L, Cutler C, Schutz B F, Vecchio A and Andersson N 1998 [*Phys. Rev.*]{} D [**58**]{} 084020
Schneider R, Ferrara A, Ciardi B, Ferrari V and Matarrese S 2000 [*Mon. Not. R. Astron. Soc.*]{} [**317**]{} 385
Schutz B F 1999 [*Class. Quantum Grav.*]{} [**16**]{} A131
Songaila A and Cowie L L 1996 [*Astron. J.*]{} [**112**]{} 335
Stark R F and Piran T 1986 [*Proc. 4th Marcel Grossmann Meeting on General Relativity (Rome, Italy)*]{} (Amsterdam: Elsevier) p 327
Thorne K S 1987 [*300 years of Gravitation*]{} (Cambridge: Cambridge University Press) p 331
Timmes F X, Woosley S E and Weaver T A 1995 [*Astrophys. J. Suppl* ]{} [**98**]{} 617
Venkatesan A 2000 [*Astrophys. J.*]{} [**537**]{} 55
Woosley S E and Timmes F X 1996 [*Nucl. Phys.*]{} A [**606**]{} 137
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We construct an optimally local perfect lattice action for free scalars of arbitrary mass, and truncate its couplings to a unit hypercube. Spectral and thermodynamic properties of this “hypercube scalar” are drastically improved compared to the standard action. We also discuss new variants of perfect actions, using anisotropic or triangular lattices, or applying new types of RGTs. Finally we add a $\lambda \phi^{4}$ term and address perfect lattice perturbation theory. We report on a lattice action for the anharmonic oscillator, which is perfect to $O(\lambda )$.'
author:
- 'W. Bietenholz, HLRZ c/o Forschungszentrum Jülich, 52425 Jülich, Germany'
title: 'Perfect Actions for Scalar Theories [^1]'
---
Many examples have shown that the perfect action program to construct improved lattice actions works beautifully, if it can be properly implemented. However, a convincing application to QCD is still outstanding. Such attempts are a desperate struggle for locality and questions of parameterization and truncation are major issues. Hence it is motivated to study the properties of perfect action carefully in simple situations.
For free and perturbatively interacting fields, perfect action can be constructed conveniently by a technique that we call “blocking from the continuum” [@QuaGlu]. It corresponds to a blocking factor $n$ RGT in the limit $n\to \infty$, so that the RGT does not need to be iterated. For a scalar we can relate the continuum resp. lattice field $\varphi, \ \phi$ as $$\phi_{x} \sim \int \Pi (x-y) \varphi (y) dy \ ,$$ where $x \in {Z \!\!\! Z}^{d}$ and $\Pi (u) \doteq \prod_{\mu =1}^{d} \Theta (1/2- \vert u_{\mu}\vert )$. If we implement this relation by a Gaussian RGT term – with coefficient $1/\alpha$ – then we obtain in momentum space the perfect lattice action $$\begin{aligned}
S [ \phi ] &=& \frac{1}{(2\pi )^{d}} \int_{B} dk \frac{1}{2}
\phi (-k) G^{-1}(k) \phi (k) \ , \nonumber \\
G(k) &=& \sum_{l\in {Z \!\!\! Z}^{d}} \frac{\Pi^{2}(k+2\pi l)}{(k+2\pi l)^{2}
+m^{2}} + \alpha \label{perfact} \ ,
\vspace*{-4mm}\end{aligned}$$ where $\Pi (k) = \prod_{\mu} \hat k_{\mu}/k_{\mu}$, $\hat k_{\mu} = 2 \sin (k_{\mu}/2)$ and $B = ]-\pi ,\pi ]^{d}$. Remarkably, the spectrum of this action, $E^{2}(\vec k ) = (\vec k + 2\pi \vec l )^{2}+m^{2}$, is exactly the continuum spectrum, which shows that the perfect action can reproduce the full Poincaré symmetry in observables, even though this symmetry is not manifest in the action.
If we choose the RGT parameter $\alpha =
\bar \alpha (m) \doteq (\sinh (m) -m)/m^{3}$, then the couplings in $$S[\phi ] = \frac{1}{2} \sum_{x,y} \phi_{x} \rho (x-y) \phi_{y}$$ are restricted to nearest neighbors in $d=1$ [@AT], and they decay exponentially and very fast in $d>1$. [^2] The question arises, if this choice is really [*optimal*]{} for locality in $d=4$, as we postulated before for fermions [@QuaGlu]. We measure the decay by $\rho (i,0,0,0) \propto \exp \{
-c(\alpha )i \}$, and Fig. \[smdec\] shows $c(\alpha )$ for various masses. Indeed, the peaks are just at $\bar \alpha (m)$. [^3]
As in the case of fermions, locality becomes even better for increasing mass, if we use $\bar \alpha (m)$, see Fig. \[decm\].
To make this action applicable we need to truncate the couplings to a short range. We do so by imposing periodic boundary conditions over 3 lattice spacings, and use the resulting couplings then in any volume [@LAT96]. In contrast to a truncation in c-space, $\rho (r)$ vanishes smoothly at the edges of a unit hypercube, the normalization is automati-cally correct and the “decimation in p-space” is handy in perfect lattice perturbation theory.
The resulting couplings represent “smoothly smeared” lattice derivatives. Alternatively there are various ways to apply Symanzik’s program to cancel the standard $O(a^{2})$ artifacts. Symanzik himself suggested to use additional couplings at distance 2 on the axes. These couplings are completely different from the perfect truncated ones. If, however, one constructs a Symanzik improved hypercube scalar, then the couplings look very similar [@WB]. The same behavior has been observed for staggered fermions [@FK]. Other Symanzik improved fermions in the literature (D234, Naik) are constructed along the axes again. However, I would rather recommend to use couplings in the unit hypercube, which is e.g. more promising for the restoration of rotational invariance.
In Fig. \[spec0\] we compare the [*dispersion relations*]{} of our hypercube scalar, the standard formulation and the Symanzik scalar on the axes. As in the fermionic case [@LAT96], the latter is good at small momenta, until it is hit by a “ghost” and then the solutions become complex, i.e. useless. The hypercube scalar, on the other hand, behaves very well all the way up to the edge of the Brillouin zone. This behavior persists at $m>0$.
The hypercube scalar has also good [*thermodynamic*]{} properties. Fig. \[SB\] shows the ratio pressure/(temperature)$^{4}$, which is $\pi^{2}/90$ in the continuum. For a small number $N_{t}$ of discrete points in Euclidean time, this ratio is approximated well for the hypercube scalar, whereas the standard action requires a large $N_{t}$ to converge. Other quantities like the energy density look similar [@WB].
In thermodynamics it is fashionable to use [*anisotropic*]{} lattices. There is no additional problem to put a perfect action on an anisotropic lattice. In the most general case, lattice spacings $(a_{1},\dots ,a_{d})$, we just have to substitute in eq. (\[perfact\]) $l \to
(l_{1}/a_{1}, \dots , l_{d}/a_{d})$, $\hat k_{\mu} \to
(2/a_{\mu})\sin (k_{\mu}a_{\mu}/2)$, and with $\bar \alpha \to
\bar \alpha /a_{\nu}^{2}$ the mapping on the $\nu$ axis is ultralocal. In the typical case one has $a_{d} < a_{s} \equiv a_{spatial}$, and one might be afraid of time-like “ghosts”. One can avoid them by giving up perfection in the temporal direction, $$G_{ani}(k) \! = \!\!\!
\sum_{\vec l \in {Z \!\!\! Z}^{d-1}} \! \frac{\prod_{\nu =1}^{d-1}
\hat k_{\nu}^{2}/(k_{\nu}+2\pi l_{\nu} /a_{s})^{2}}
{(\vec k + 2\pi \vec l /a_{s})^{2}+ \hat k_{d}^{2}+m^{2}}
+ \frac{\alpha}{a_{s}^{2}} .$$
We can also put the perfect action on a [*triangular*]{} lattice. In $d=2$ we obtain a modified $\Pi$ function, $\Pi_{tria}(k) \ = $ $$\frac{8[
k_{1}\cos\frac{k_{1}}{2} + k_{2} \cos \frac{k_{2}}{2}
- (k_{1}+k_{2}) \cos \frac{k_{1}+k_{2}}{2}]}
{3k_{1}k_{2}(k_{1}+k_{2})} ,$$ where we refer to axes crossing under $\pi /3$. The blocking from the continuum then leads to $$\begin{aligned}
\phi (k) & \sim & \frac{3}{4} \sum_{l \in {Z \!\!\! Z}^{2}}
\varphi (k_{l}) \Pi_{tria}(k_{l}) \\
k_{l} &\doteq & k + \frac{4\pi}{3}
\left( \begin{array}{cc} 2 & -1 \\ -1 & 2 \end{array} \right) l \\
G_{tria}(k) &=& \!\!\!\! \sum_{l \in {Z \!\!\! Z}^{2}}
\frac{\Pi_{tria}^{2}(k_{l})}{k_{l,1}^{2}+k_{l,2}^{2}+
k_{l,1}k_{l,2}+m^{2}} + \alpha .\end{aligned}$$ Again the spectrum is perfect; we have e.g. full rotational invariance in the observables, although the lattice structure is visible in the action.
Back to the hypercubic lattice with spacing 1: we can also vary the convolution function in the blocking from the continuum, $$\phi_{x} = \int dy \Big[ \prod_{\mu} f_{n}(x_{\mu}-y_{\mu}) \Big]
\varphi (y) .$$ Sensible candidates are the B-spline functions $f_{0}(s)
= \delta (s)$, $f_{n+1}(s) = \int_{s-1/2}^{s+1/2}f_{n}(t)dt$. The appropriate convolution function for the gauge field $A_{\mu}$ then reads $f_{n+1}(u_{\mu}) \prod_{\nu \neq \mu} f_{n}(u_{\nu})$. This set of functions has the correct normalization and a “democracy of the continuum points”, which all contribute with the same weight to the lattice variables. In the propagator of eq. (\[perfact\]), the power of the $\Pi$ function generalizes to $2n$. Deci-mation ($n=0$) fails because the sum over $l$ diverges in $d>1$, and $n=1$ is what we had before. For higher $n$ we can still achieve ultralocality in $d=1$ by adding kinetic terms in $\alpha$, but the locality in $d>1$ is best for $n=1$. Generally, overlapping blocks seem to be unfavorable for locality. We can contract $f_{n}$ into \[-1/2,1/2\], and re-adjust the normalization (giving up “democracy”). Thus $f_{2}$ turns into an “Eiffel tower function” $\tilde f_{2}(s) = (2- 4 \vert s \vert) \Theta(1/2-\vert s \vert )$, which is in business in view of the induced locality [@WB]. For $n>2$ one obtains blocking schemes which are difficult to relate to finite blocking factors, hence they are not easily compatible with the multigrid improvement. So I recommend the “Eiffel tower function” $\tilde f_{2}$ as a promising alternative to $f_{1}$.\
Finally we proceed to the $\lambda \phi^{4}$ theory. If we block perturbatively from the continuum, we encounter divergent loop integrals, which can be regularized in the continuum. There is no divergence, however, for the [*anharmonic oscillator*]{} ($d=1$). There we constructed an $O(\lambda )$ perfect action. It involves 2- and 4-variable couplings $\propto \lambda$, which have an analytic form in momentum space and which we evaluated numerically in c-space. For $\bar \alpha (m)$ they do not couple any variables over distance $>2$ (the perfect action to $O(\lambda^{n})$ extends to maximal distances $2n$ ($n\geq 1$)).
The performance of this action is of interest in view of the direct application of the perfect quark-gluon vertex function in QCD [@KO]. We measured the first two energy gaps [@thor], $\Delta E_{1}$ and $\Delta E_{2}$, for our action and for the standard action, and observed that they are clearly closer to the continuum results for the new action up to $\tilde \lambda \doteq \lambda /m^{3}
\simeq 0.25$. This corresponds to an improved asymptotic scaling. However, for the scaling quantity $\Delta E_{2}/
\Delta E_{1}$ the improvement is difficult to demonstrate, because it is restricted to tiny $\tilde \lambda$. In both cases, the $O(\tilde \lambda )$ perfect action is successful up to the magnitude of $\tilde \lambda$, where also first order continuum perturbation theory – for the considered observable – collapses.
[9]{} W. Bietenholz and U.-J. Wiese, Nucl. Phys. B464 (1996) 319.
A. Tsapalis and U.-J. Wiese, Nucl. Phys. B (Proc. Suppl.) 53 (1997) 948.
T. Bell and K. Wilson, Phys. Rev. D11 (1975) 3431.
W. Bietenholz, R. Brower, S. Chandrasekharan and U.-J. Wiese, Nucl. Phys. B (Proc. Suppl.) 53 (1997) 921.
W. Bietenholz, in prep.
F. Karsch, hep-lat/9706006.
K. Orginos et al., hep-lat/9709100.
W. Bietenholz and T. Struckmann, in prep.
[^1]: Talk presented at LAT97.
[^2]: For a finite blocking factor $n$ and $m=0$, $\bar \alpha$ is replaced by $\bar \alpha_{n}
= (1-1/n^{2})/6 $, in agreement with [@BeWi].
[^3]: For heavy fermions, on the other hand, we noticed that locality can be improved slightly beyond the 1d formula.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The collective dynamics of a dipolar fermionic quantum gas confined in a one-dimensional double-well superlattice is explored. The fermionic gas resides in a paramagnetic-like ground state in the weak interaction regime, upon which a new type of collective dynamics is found when applying a local perturbation. This dynamics is composed of the local tunneling of fermions in separate supercells, and is a pure quantum effect, with no classical counterpart. Due to the presence of the dipolar interactions the local tunneling transports through the entire superlattice, giving rise to a collective dynamics. A well-defined momentum-energy dispersion relation is identified in the ab-initio simulations demonstrating the phonon-like behavior. The phonon-like characteristic is also confirmed by an analytical description of the dynamics within a semiclassical picture.'
author:
- Lushuai Cao
- 'Simeon I. Mistakidis'
- Xing Deng
- Peter Schmelcher
title: |
Collective excitations of dipolar gases based\
on local tunneling in superlattices
---
Introduction
============
Collective excitations constitute a fundamental concept in condensed matter physics which is at the origin of various phenomena in the field [@Anderson; @Simon]. Remarkable examples of collective excitations are phonons, magnons or plasmons. Among them, phonons describe the collective dynamics of the atomic vibrations in the crystal lattice, and play a key role for different fundamental effects in condensed matter physics, such as superconductivity [@Bardeen] or the thermal transport in solid matter [@Lepri; @Li].
Generalizations of the concept of a phonon can be found in ion traps [@Porras; @Bissbort] or ultracold dipolar quantum gases [@Pupillo; @Ortner]. Phonons in ion traps refer to the collective dynamics of ions’ motion around their equilibrium positions in e.g. Paul traps. In dipolar quantum gases, it describes the coupling between the local vibrations of dipolar atoms in a self-assembled chain or a lattice. Generally speaking, phonons in crystals, ion chains and dipolar lattices all refer to the collective dynamics of vibrations, which have a direct analogue to the motion of classical vibrators, and these phonons can be seen as a direct extension of the classical dynamics of a vibrating chain to the quantum regime. Here, we introduce a new type of collective dynamics in ultracold dipolar gases in a one-dimensional superlattice. The key ingredient for this collective dynamics is a local tunneling, which possesses no classical counterpart.
Our investigation is mainly based on ab-initio simulations, besides a semiclassical analytical treatment. The simulations are performed by employing the numerically exact Multi-Layer Multi-Configuration Time-Dependent Hartree method for identical particles and mixtures (ML-MCTDHX) [@MLX], which has been developed from MCTDH [@Meyer; @Beck], ML-MCTDH [@Wang; @Manthe] and ML-MCTDHB [@Kronke; @Cao], and has a close relation to MCTDHB(F) [@Alon; @Alon1; @Axel]. The ab-initio simulations of the corresponding ultracold quantum gases take into account all correlations, and can unravel new effects beyond the predictions of mean-field theory and for lattice systems beyond the single-band Bose-Hubbard model. Representative examples along this line are the loss of coherence and the decay of contrast of different types of solitons [@Streltsov; @Kronke1], higher band effects on the stationary [@Alon2] or dynamical properties [@Zollner; @Sakmann; @Cao1; @Mistakidis; @Mistakidis1; @Mistakidis2] in optical lattices. To investigate the collective dynamics in the double-well superlattice, we employ ML-MCTDHX which allows for a full description of the dynamics, and proves the robustness of the collective dynamics against higher order correlations and higher band effects.
This work is organized as follows: In section II, we present an introduction to the detailed setup (Sec. II.A), the initial state preparation (Sec. II.B), the local effect of the perturbation that drives the system out of equilibrium (Sec. II.C), and the global collective dynamics induced by the local perturbation (Sec. II.D). We also supply a semiclassical analytical description of the collective dynamics (Sec. II.E). The discussion of our results and the conclusions are provided in section III.
Collective excitations based on local correlation-induced tunneling
===================================================================
Setup
-----
We consider a dipolar superlattice quantum gas (DSG) composed of $N$ spin-polarized fermions confined in a one-dimensional double-well superlattice of $N$ supercells, $i.e.$ a unit filling per supercell. All the fermions interact with each other by dipolar interactions. The Hamiltonian read as follows $$\label{ham}
H=\sum_{i=1}^{N}(\frac{-\hbar^2}{2M}\partial^2_{x_i}+V_{sl}(x_i))+\sum_{i<j=1}^N\frac{D}{\mid x_i-x_j\mid^3+\delta}.
%+\sum_{i=1}^{N}V_{tr}(x_i,t).$$ The first term refers to the single-particle Hamiltonian, where $V_{sl}$ models the double-well superlattice with $V_{sl}(x)=V_0(sin^2(kx)+2cos^2(2kx))$. This superlattice can be formed by two pairs of counter-propagating laser beams of wave vectors $k$ and $2k$, and the strength of the lattice $V_0$ can be tuned by the amplitude of the laser beams. We consider a finite-length lattice of $N$ supercells, and hard-wall boundaries are applied at positions $x=\pm N\pi/(2k)$, to allow only $N$ supercells in our simulation. The second term in the Hamiltonian models the dipolar interaction between the fermions. In this work we consider the situation that all the dipoles are polarized along the same direction, perpendicular to the relative distance between the fermions, and $D$ denotes the strength of the interaction. To avoid in simulations the divergence of the interaction at $x_i=x_j$, an offset $\delta$ is added to the denominator. The offset $\delta$ takes a rather small value, which is about eight times smaller than the spacing of the discrete grid points chosen within our simulations. More specifically, in the present work we focus on the situation where all fermions reside in different wells for which case the distances where a significant overlap exists are much larger than $\delta$. Then, the offset is negligible.
In order to investigate the collective excitations, the DSG is firstly relaxed to the ground state of $H$, and at $t=0$ a local perturbation is applied to a single supercell of the lattice, $e.g.$ the outer most left cell is taken out of equilibrium. This perturbation is intended to induce a local dynamics in the left cell, and is applied only for a short time period, to avoid affecting the global dynamics on a long time scale. We model the local and temporal perturbation by $V_{pt}(x,t)=V_1~\theta(x+(N-1)\pi/(2k))~\theta(t-\tau)$, with the Heaviside step function $\theta(x)$. The perturbation is then modeled as a local step function applied to the outer most left supercell and lasts only for the temporal interval $[0,\tau]$. The double-well superlattice and the local perturbation are sketched in figure \[fig1\] for five fermions in a five-cell superlattice. In the simulation, we render the Hamiltonian dimensionless by setting $\hbar=M=k=1$, which is equivalent to rescaling the energy, space and time in the units of $E_R=\hbar^2k^2/2M$, $k^{-1}$ and $\hbar/E_R$, respectively.
The setup discussed above can be realized in ultracold atom experiments. Moreover, dipolar quantum gases have become a hot topic in the field of ultracold atoms and molecules [@Lahaye; @Baranov]. Their rich phase properties [@Yi; @Capogrosso; @Hauke; @Kadau; @Barbut] and perspectives in, for instance, quantum simulations [@Micheli; @Gorshkov; @Kaden] have inspired extensive studies on dipolar quantum gases. Experiments can nowadays prepare dipolarly interacting particles in lattices, due to the rapid development of cooling atoms [@Zhou; @Olmos; @Baier] with large magnetic dipole moments and polar molecules in optical lattices [@Ni; @Deiglmayr; @Yan; @Guo; @Frisch]. Specifically, the double-well superlattice has been realized in experiments, and has become a widely used testbed for various phenomena, such as correlated atomic tunneling [@Folling], generation of entanglement of ultracold atoms [@Dai] and the topological Thouless quantum pump [@Lohse]. The setup discussed and analyzed here is therefore well within experimental reach.
![Sketch of the dipolar fermionic gas in a double-well superlattice of five cells. The blue and black lines show the double-well superlattice and the local perturbation modeled as a step function applied only to the most left site of the lattice, respectively. Five fermions (red dashed gaussians) are loaded to the superlattice, each of which is localized in a different supercell and occupies the two sites in the cell for the initial state.[]{data-label="fig1"}](Fig1-eps-converted-to.pdf){width="8cm"}
Paramagnetic-like initial state {#initial}
-------------------------------
The dynamics investigated in the present work strongly depends on the choice of the initial state, which is prepared as the ground state of $H$ in a particular parameter regime. It has been shown [@Yin] that the DSG system can be mapped to an effective Ising spin chain model under the so-called pseudo-spin mapping. Then, within the Ising spin picture the ground state of the system undergoes a transition from a paramagnetic-like state to a single-kink state for increasing dipolar interaction. The paramagnetic-like state refers to the pseudo-spins polarized in the same direction due to an effective magnetic field, whereas the single-kink state is composed of two effective ferromagnetic domains aligning in opposite directions. In the present work, we focus on the dynamics in the weak interaction regime, $i.e.$ the DSG system initially resides in the paramagnetic-like state.
To comprehend and analyze qualitatively the initial particle configuration (being characterized by the many-body state $\ket{\Psi}$) we shall employ the notion of reduced densities. The one-body reduced density matrix $\rho_1(x,x')=\braket{x'|\hat\rho_1|x}$, is obtained by tracing out all fermions but one in the one-body density operator $\hat\rho_1\equiv tr_{2,...,N}\ket{\Psi} \bra{\Psi}$ of the $N$-body system, while the two body density $\rho_2(x_1,x_2)=\braket{x_1,x_2|\hat\rho_2|x_1,x_2}$ can be obtained by a partial trace over all but two fermions of the two-body density operator $\hat\rho_2\equiv tr_{3,...,N}\ket{\Psi} \bra{\Psi}$. Subsequently, the initial state can be characterized by the two-body and one-body correlations in the superlattice, as shown in figure 2. Figure 2($a$) presents the two-body correlation of five fermions in a five-cell superlattice. The vanishing occupation along the diagonal direction in the two-body correlation illustrates that no two fermions (or more) occupy the same supercell, and each supercell hosts only one fermion, which is a Mott-like configuration. Being localized in a separate supercell, the fermions can occupy the left and right sites of the cell simultaneously, giving rise to particle number fluctuations in these sites and in particular to a non-vanishing off-diagonal one-body correlation, as shown in figure 2($b$). This non-vanishing one-body correlation plays a key role in the collective dynamics investigated in the present work.
To a good approximation, the paramagnetic-like ground state (see also Appendix B) can be expressed as $$\label{ground}
|G\rangle=\sqrt{2^{-N}}\prod_{i=1}^N(|L\rangle_i+|R\rangle_i),$$ where $|L\rangle_i$ and $|R\rangle_i$ denote the lowest-band Wannier states in the left and right site of the $i$-th supercell, respectively.
![($a$) The two-body correlations and ($b$) one-body correlations of the initial state of five fermions in a five-cell superlattice. The two-body correlation illustrates that any two fermions cannot occupy the same supercell, $i.e.$ each supercell hosts a single fermion. The off-diagonal of the one-body correlations indicates the delocalization of the fermion between the left and right sites of the supercell. The parameters used here are $(V_0,D)=(10,0.3)$, which correspond to a weakly interacting dipolar gas in a deep superlattice. []{data-label="fig2"}](Fig2-eps-converted-to.pdf){width="7cm"}
Correlation-induced tunneling in a single supercell
---------------------------------------------------
To drive the system out of equilibrium from the initial state $|G\rangle$, we apply a local perturbation $V_{pt}$ to the most left supercell. The perturbation is intended to induce a local tunneling of the fermion in this supercell and is modeled by a step function applied to the left site of the supercell. In this subsection we describe the local dynamics in this supercell under the perturbation $V_{pt}$. The step function introduces an energy offset between the left and right sites of the cell. Normally, the energy offset inhibits the tunneling between the two sites, of which the amplitude is reduced, by increasing the amplitude of the offset. When a particle is initially prepared in a superposition state involving the two sites equally, however, the offset can enhance the tunneling of the particle in a narrow parameter window of the offset strength. The tunneling amplitude becomes maximal when the strength of the perturbation matches that of the hopping between the two sites.
The explanation of such an unusual tunneling is as follows: In the normal case, the energy offset breaks the resonance between the two sites in terms of the potential energy, and thus it inhibits the tunneling between the two sites. When the initial state is chosen as a superposition state of the particle occupying the two sites equally, a finite kinetic energy is stored in the system. The finite kinetic energy can then compensate the resonance breaking of the potential energy and promote the tunneling. The maximum compensation is reached when the energy offset matches the initial kinetic energy, which can be realized when the strength of the offset equals the hopping strength. In the double well system, the kinetic energy coincides with the one-body correlation between the two sites, up to a factor determined by the hopping strength, and we term this unusual tunneling as correlation induced tunneling (CIT) [@Cao2], to indicate the connection between the kinetic energy and the one-body spatial correlation. Moreover, the CIT can also be viewed as a Rabi oscillation between the two states $(|L\rangle\pm|R\rangle)/\sqrt{2}$, where the tilt couples these two states and determines the corresponding Rabi frequency. In figure \[fig3\] we illustrate the CIT of a single particle confined in a double well potential with a temporal energy offset. To proceed we calculate the population of each well, e.g. for the right well $\rho_R(t)=\int_{0}^{\pi}dx \rho_1(x,t)$ (with $\rho_1$ being the one-body density). As shown in the figure, initially the particle is occupying both sites with equal probability, and after the perturbation (applied at $t=0$), the probability oscillates from the right to the left well, indicating a tunneling between the two wells. When the perturbation is turned off (the turn-off time is marked by the dashed red line in figure 3), we observe that the tunneling persists. Turning to the whole superlattice, it can be expected that the CIT also takes place in the left supercell when the same perturbation is applied to a double-well supercell.
![Density oscillation of a single particle in a double well with a local perturbation applied to the left well. The local perturbation is applied for a short time period, and the red dashed line marks the time when it is turned off. The double well is taken from a single unit cell of the superlattice with $V_0=10$, and the height of the perturbation potential is $h_1=0.1$.[]{data-label="fig3"}](Fig3-eps-converted-to.pdf){width="8cm"}
Collective dynamics of local CIT
--------------------------------
Having introduced the initial state and the local dynamics of the CIT, let us proceed to the global dynamics of the entire DSG system being subjected to a local perturbation. Our main finding can be summarized as follows: Once the local perturbation induces the CIT in a single supercell, $e.g.$ the most left one as considered here, the dipolar interaction can transport the local CIT to other cells. In this manner, all the fermions, while remaining well localized in their separate supercells, perform local CIT between the two sites of their supercells, giving rise to a collective dynamics of local CIT in the DSG system. Moreover, the collective dynamics resembles phonon-like excitations, with a well defined momentum-energy dispersion relation. In following, we shall demonstrate the collective dynamics of local CIT employing ab-initio simulations.
![($a$) The density evolution $\rho_1(x,t)$ of a three-fermion system in a three-cell superlattice, and ($b$) in a plain triple well, under a corresponding local perturbation. The dashed lines attached to the right of the figures illustrate the corresponding trapping potentials. The arrows mark the position where the local perturbation is applied for both cases. The parameters used here and in figure 5 are $(V_0,D,V_1)=(10,0.3,0.1)$.[]{data-label="fig4"}](Fig4-eps-converted-to.pdf){width="8cm"}
Firstly, we simulate the collective dynamics of $N=3$ fermions confined in a three-cell superlattice, $i.e.$ a 3F3C (3 fermions in 3 cells) system, and compare it with the phonon of three dipolar-interacting fermions in a plain triple well. In figure 4($a$) we show the one-body density oscillation $\rho_1(x,t)$ of the 3F3C system under the perturbation $V_{pt}$. We observe that CIT takes place in all the three supercells, with no inter-cell tunneling between neighboring supercells. This collective dynamics of CITs is different from the dipolar phonon as well as the ion phonon, which refer to the collective dynamics of local classical vibrations of dipolar atoms or ions confined in a lattice respectively. In figure 4($b$) we also present the one-body density of the dipolar phonon of three fermions in a plain triple well. In the dipolar phonon case, a local tilt induces a dipole oscillation of the fermion in the left well, and the dipolar interaction transports the local density oscillation to fermions in remote wells, giving rise to the collective phonon dynamics. Firstly, a similarity can be drawn between the collective CIT and the dipolar phonon, where both cases are composed of local dynamics coupled by the dipolar interaction. On the other hand, the distinction of the two collective dynamics is also obvious: The dipolar phonon (as well as the ion phonon) is composed of local oscillations of particles and can be seen as a direct extension of the classical phonons to the quantum regime. Meanwhile, the collective dynamics of CIT has no counterpart in the classical world and is a pure quantum effect.
![The density evolution $\rho_1(x,t)$ of ($a$) a five-fermion and ($b$) eleven-fermion system of unit-filling, under the local perturbation. The dashed yellow lines in both figures illustrate the finite transport velocity of the local CIT through the superlattice. The dashed lines (see the corresponding slopes) also suggest an equal transport velocity of the CIT in both systems, implying that the transport velocity is independent of the system size. []{data-label="fig5"}](Fig5-eps-converted-to.pdf){width="8cm"}
To demonstrate the generality of such collective dynamics with respect to the size of the superlattice we show that the same behavior is evident in 5F5C (5 fermions in 5 cells) and 11F11C (11 fermions in 11 cells) systems, as shown in figures 5($a$) and 5($b$), respectively. In both figures we observe that the collective dynamics of local CIT indeed takes place in bigger systems, indicating that it is not restricted to a particular size. Moreover, in the longer lattices, we observe more clearly how the local CIT transport through the whole system: They are not simultaneously excited along the lattice once the perturbation is applied, but the CIT are transported with a finite velocity from the left supercell to remote ones. The transport of local CIT with a finite velocity is illustrated in both figures with the yellow dashed lines, where one can even observe the reflection at the edges of the lattice. In this way, the collective dynamics of local CIT in the DSG systems also serve as a test bed for the light-cone like behavior of two-body correlations.
It is known that all phonon-like collective excitations share a common property of well defined momentum-energy dispersion relation, where the collective dynamics can be decomposed into a set of momentum modes and each mode has a well defined energy, $i.e.$ characteristic frequency. It is interesting to investigate whether the collective dynamics of DSG systems is also associated with a dispersion relation. For this purpose, we calculate the density difference between the left and right sites of each supercell $\delta\rho(i,t)$ ($i \in [1,N]$), and further define a set of $k$-modes as $$\delta\tilde\rho(k,t)=\sum_{n=1}^N \sin(\frac{nk\pi}{N+1})\delta\rho(n,t), k\in[1,N]$$ To verify the corresponding dispersion relation we then calculate the spectra of $\delta\rho(i,t)$ and $\delta\tilde\rho(k,t)$. We show the spectra of $\delta\rho(1,t)$ for 5F5C and 11F11C in figures 6($a_1$) and 6($b_1$), respectively, and the corresponding spectra of $\delta\tilde\rho(k,t)$ in figures 6($a_2$) and 6($b_2$). These figures demonstrate that, firstly the spectra of $\delta\rho(i,t)$ show $N$ main peaks for the $N$-fermion system, each of which corresponds to one $k$-mode, indicating that the collective dynamics can be indeed decomposed into $N$ $k$-modes. More importantly, each $k$-mode is associated with a dominant frequency peak, as shown in the spectra of $\delta\tilde\rho(k,t)$, and this directly verifies a well-defined momentum-energy dispersion relation in the collective dynamics of the CIT. Further, we also observe some weakly pronounced peaks lying near zero in the spectra, which are close to the values of the frequency difference between the corresponding peaks. These peaks are attributed to a weak nonlinear effect similar to phonon-phonon interactions.
{width="14cm"}
Semiclassical description of the collective CIT dynamics
--------------------------------------------------------
In this section, we supply a semiclassical description of the collective CIT excitation, in terms of $\langle\delta\rho(i,t)\rangle$. The starting point is the second-order time derivative equation $$\label{eqrho}
-\partial^2_t\langle\delta\hat\rho(i,t)\rangle=\langle[[\delta\hat\rho(i),\hat H],\hat H]\rangle,$$ which is derived simply by applying $i\partial_t\langle\delta\hat\rho(i,t)\rangle=\langle[\delta\hat\rho(i),\hat H]\rangle$ twice, while the notation $\langle...\rangle$ denotes the expectation value $\langle\Psi(t)|...|\Psi(t)\rangle$. Then the major task of solving equation (4) is to find proper expressions of the Hamiltonian $\hat H$ (for more details see Appendix B and in particular equation (B1)) and to solve for $|\Psi(t)\rangle$.
We adopt the lowest-band Hubbard model for $\hat H$ and apply degenerate perturbation theory to solve for $|\Psi(t)\rangle$ and derive a set of closed equations for $\langle\delta\hat\rho(i,t)\rangle$: a detailed derivation is given in Appendix B. The final form of the equations that $\langle\delta\hat\rho(i,t)\rangle$ obeys, reads $$\label{rho}
\begin{split}
-\partial^2_t\langle\delta\hat\rho(i,t)\rangle=4J^2\langle\delta\hat\rho(i,t)\rangle+4JV\big(\langle\delta\hat\rho(i-1,t)\rangle\\
+\langle\delta\hat\rho(i+1,t)\rangle\big),
\end{split}$$ where $J$ and $V$ refer to the intra-cell hopping and the dipolar interaction strength, respectively. Equation (5) is the semiclassical version of equation (4), and it is clear that equation (5) resembles that of classical vibrating chains, where $\langle\delta\rho(i,t)\rangle$ plays the role of the local displacement of the $i$-th vibrator. The general solutions of equation (5) correspond to a set of eigenmodes, and a particular solution is given by the superposition of these eigenmodes (equation corrected) $$\label{kmode}
\langle\delta\rho(i,t)\rangle=\sum_{k=1}^N \big(C_ksin(\omega_k t)+D_kcos(\omega_k t)\big)sin(\frac{ki\pi}{N+1}),$$ where $\omega_k=\sqrt{4J[J+2Vcos(k\pi/(N+1))]}$, and $C_k$, $D_k$ are determined by the initial state. The semiclassical equations (5) and their eigenmode solutions (see equation (6)) directly illustrate the phonon-like behavior of the collective dynamics of the local CIT in the DSG system.
Discussion and conclusions
==========================
In this work we demonstrate a new type of collective excitations in dipolar quantum gases confined in the double-well superlattice with a unit filling factor. The collective excitations manifest themselves as the coupling and transport of local CIT within each supercell. The local CIT are a pure quantum effect and have no classical counterpart, which endows the dynamics composed of these collective excitations with a pure quantum nature, instead of being a quantum correction to any classical dynamics.
These collective excitations can also be generalized from the double-well superlattice to more complicate superlattices, where new properties of the collective dynamics can be engineered. For instance, the CIT in a double well possesses a single characteristic frequency, and in the spectrum of the collective dynamics a single band arises from this characteristic frequency. When the supercell is expanded to multiple wells, the corresponding characteristic frequencies of the local CIT will also increase, and each of these frequencies seeds a band in the spectrum of the collective dynamics of the local CIT, resulting in a multi-band spectrum. The tunability of the band structure by the supercell properties indicate a high flexibility in designing and engineering new properties of such collective excitations. Meanwhile, in the relatively strong interaction regime, one can expect more pronounced nonlinear effects, such as the scattering of the collective excitations, which, however, is beyond the scope of the current work.
It is in place to discuss the realizability and robustness of the collective excitations under realistic conditions. Firstly, these excitations are not restricted to fermionic dipolar gases, but can also be realized with bosonic dipolar gases, as the particles are localized in separate cells and the particle statistics plays almost no role here. For realistic implementations, the collective excitations may be blurred by effects due to finite temperature, an additional external potential and the imperfectness of the filling factor. To observe collective excitations, it is required to cool the particles to the lowest band of the lattice. In previous experiments on double well superlattices, this condition has been fulfilled for contact interacting atoms, and with the fast progress in cooling dipolar lattice gases we expect this condition will become also feasible for our setup. In experiments, the confinement of lattice gases to a finite spatial domain is realized by an external harmonic trap, which will also introduce some constraints on the realization. In the bottom of the harmonic trap, it is possible to prepare a paramagnetic-like state, while at the edge deviations from the perfect paramagnetic-like configuration can arise. It has been shown that this edge effect will not change the global paramagnetic-like configuration [@Yin], and we also note that it is now possible to compensate the extra harmonic trap with a dipole trap in experiments [@Will], which can further release the constraints. Finally if the filling deviates from unit filling per supercell, holes or doublons can arise in the superlattice, which can scatter and couple to the collective excitations. New phenomena can be generated due to such scattering and coupling, and we refer the reader for possible new phenomena to future investigations.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is dedicated to Prof. Lorenz Cederbaum on the occasion of his 70th birthday. The authors acknowledge the efforts of Sven Schmidt and Xiangguo Yin in the initial stage of the work. L. Cao is also grateful to Antonio Negretti for inspiring discussions on ion phonons and the conditions of realistic implementations. S.M and P.S gratefully acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG) in the framework of the SFB 925 ”Light induced dynamics and control of correlated quantum systems”.
ML-MCTDHX
=========
The Multi-Layer Multi-Configuration Time-Dependent Hartree method for multicomponent quantum gases (ML-MCTDHX) [@MLX] constitutes a variational numerical ab-initio method for investigating both the stationary properties and in particular the non-equilibrium quantum dynamics of mixture ensembles covering the weak and strong correlation regimes. Its multi-layer feature enables us to deal with multispecies systems (e.g. Bose-Bose, Fermi-Fermi or Bose-Fermi mixtures), multidimensional or mixed dimensional systems in an efficient manner. The multiconfigurational expansion of the wavefunction in the ML-MCTDHX method takes into account higher band effects which renders this approach suitable for the investigation of systems governed by temporally varying Hamiltonians, where the system can be excited to higher bands especially during the dynamics. Finally within the ML-MCTDHX approach the representation of the wavefunction is performed by variationally optimal (time-dependent) single particle functions (SPFs) and expansion coefficients ${A_{{i_1}...{i_S}}}(t)$ which makes the truncation of the Hilbert space optimal when employing the optimal time-dependent moving basis. The requirement for convergence demands a sufficient number of SPFs such that the numerical exactness of the method is guaranteed. Therefore, the number of SPFs has to be increased until the quantities of interest acquire the corresponding numerical accuracy.
In a generic mixture system consisting of ${N_\sigma }$ atoms (bosons or fermions) of species $\sigma = 1,2,...,S$ the main concept of the ML-MCTDHX method is to solve the time-dependent Schrödinger equation $i {|\dot{\Psi}}\rangle = \widehat {\rm H}\left| \Psi \right\rangle$ as an initial value problem, $|{\Psi (0)}\rangle = \left| {{\Psi _0}} \right\rangle$, by expanding the total wave-function in terms of Hartree products $$\begin{split}
\label{eq:3}\left| {\Psi (t)} \right\rangle = \sum\limits_{{i_1} =
1}^{{M_1}} {\sum\limits_{{i_2} = 1}^{{M_2}} {...\sum\limits_{{i_S} =
1}^{{M_S}} {{A_{{i_1}...i{}_S}}(t)} } } \\\times\ket{{\psi
_{{i_1}}^{(1)}(t)}} ... \ket{{\psi _{{i_S}}^{(S)}(t)}}.
\end{split}$$ Here each species state $\ket{\psi _i^{(\sigma )}}$ ($i =
1,2,...,{M_\sigma }$) corresponds to a system of ${N_\sigma }$ indistinguishable atoms (bosons or fermions) and describes a many-body state of a subsystem composed of $\sigma$-species particles. The expansion of each species state in terms of bosonic or fermionic number states ${\ket{{\vec n (t)}}^\sigma }$ reads $$\label{eq:10}\ket{\psi _i^{(\sigma )}} = \sum\limits_{\vec n
\parallel \sigma } {C_{i;\vec n }^\sigma (t)} {\left| {\vec n (t)}
\right\rangle ^\sigma },$$ where each $\sigma $ atom can occupy ${m_\sigma }$ time-dependent SPFs $\ket{\varphi _j^{(\sigma )}}$. The vector $\left| {\vec n } \right\rangle = \left|
{{n_1},{n_2},...,{n_{{m_\sigma }}}} \right\rangle$ contains the occupation number ${n_j}$ of the $j - th$ SPF that obeys the constraint ${n_1} + {n_2} + ... + {n_{{m_\sigma
}}} = {N_\sigma }$. Note that for the bosonic case $n_j=0,1,2,...,N$ while for the fermionic case only $n_j=0,1$ are permitted due to the Pauli exclusion principle.
In the present work, we focus on the case of a single fermionic species in one spatial dimension where the ML-MCTDHX is equivalent to MCTDHF. To be self-contained, let us briefly discuss the ansatz for the many-body wavefunction and the procedure for the derivation of the equations of motion. The many-body wavefunction which is a linear combination of time-dependent Slater determinants reads $$\label{eq:10}\left| {\Psi (t)} \right\rangle = \sum\limits_{\vec n
} {{C_{\vec n }}(t)\left| {{n_1},{n_2},...,{n_M};t} \right\rangle }.$$ Here $M$ denotes the total number of SPFs and the summation is performed over all possible combinations which retain the total number of fermions. In the limit in which $M$ approaches the number of grid points the above expansion becomes numerically exact in the sense of a full configuration interaction approach. Another limiting case of the used expansion refers to the case that $M$ equals the number of particles, being referred to in the literature as Time-Dependent Hartree Fock (TDHF). The Slater determinants in (A3) can be expanded in terms of the creation operators $a_j^\dag (t)$ for the $j - th$ orbital ${\varphi _j}(t)$ as follows $$\begin{split}
\label{eq:4}\left| {{n_1},{n_2},...,{n_M};t} \right\rangle =
\frac{1}{{\sqrt {{n_1}!{n_2}!...{n_M}!} }}{\left( {a_1^\dag }
\right)^{{n_1}}}{\left( {a_2^\dag } \right)^{{n_2}}}\\\times...{\left(
{a_M^\dag } \right)^{{n_M}}}\left| {vac} \right\rangle,
\end{split}$$ satisfying the standard fermionic anticommutation relations $\left[
{{a_i}(t),{a_j}^{\dag}(t)} \right]_- = {\delta _{ij}}$, etc. To determine the time-dependent wave function $\left|
\Psi \right\rangle$, we have to find the equations of motion for the coefficients ${{C_{\vec n }}(t)}$ and the orbitals (which are both time-dependent). To derive the equations of motion for the mixture system one can employ various approaches such as the Lagrangian, McLachlan or the Dirac-Frenkel variational principle, each of them leading to the same result. Following the Dirac-Frenkel variational principle $$\label{eq:5}{\bra{\delta \Psi}}{i{\partial _t} - \hat{ H}\ket{\Psi
}}=0,$$ we can determine the time evolution of all the coefficients ${{C_{\vec n }}(t)}$ in the ansatz (A3) and the time dependence of the orbitals $\left| {{\varphi _j}} \right\rangle $. In this manner, we end up with a set of $M$ non-linear integro-differential equations of motion for the orbitals $\varphi_{j}(t)$, which are coupled to the $\frac{M!}{N!(M-N)!}$ linear equations of motion for the coefficients $C_{\vec{n}}(t)$ . These equations are the well-known MCTDHF equations of motion [@Alon; @Alon1].
Within our implementation, a discrete variable representation (DVR) scheme is applied, and in particular we adopt the sin-DVR, which intrinsically implements hard-wall boundaries conditions. Furthermore, for the cases of three and five fermions, six and ten SPFs have been used, respectively, i.e. the number of SPFs being twice the number of the particles. As it turned out, the number of major occupied natural orbitals for both cases, reflecting the convergence of the simulation with respect to the number of SPFs, is equal to the number of particles. This indicates that one just needs to use as many SPFs as there are fermions in order to reach a converged simulation. Finally, in the simulation of the eleven-fermion case, only eleven SPFs have been used.
Semiclassical equations
=======================
We firstly re write the Hamiltonian of equation (1) in the Hubbard form. Upon the lowest-band Wannier states $\{|L\rangle_i,|R\rangle_i\}_{i=1}^N$, we define a set of basis vectors of $\{|S\rangle_i,|A\rangle_i\}_{i=1}^N$, where $|S(A)\rangle_i\equiv(|L\rangle_i+(-)|R\rangle_i)/\sqrt{2}$ denote the corresponding symmetric (anti-symmetric) superposition within the $i$-th supercell. By regarding $|S\rangle_i$/$|A\rangle_i$ as a pseudo-spin state of $|\downarrow\rangle$/$|\uparrow\rangle$, we introduce the Pauli matrices $\sigma_\alpha$, with $\alpha=x,y,z$ for these basis vectors. We focus on the evolution of the system after the tilt is removed, and the corresponding Hamiltonian can then be expressed in this basis as $$\label{hamhub}
\begin{split}
\hat H = J\sum_{i=1}^N\sigma_z(i)+\sum_{i=1}^{N-1}V(S^{+}_iS^{-}_{i-1}+h.c.)\\+\sum_{i=1}^{N-1}V(S^{+}_iS^{+}_{i-1}+H.c.)
+U(\sigma_x(1)-\sigma_x(N)),
\end{split}$$ where J refers to the intra-cell hopping strength, and V, U are determined by the interaction strength. In this reduced Hamiltonian, we approximate the dipolar interaction by a nearest-neighbor interaction and neglect the inter-cell hopping, which is valid within the weak interaction regime considered in this work [@Yin].
Based on equation (B1), $|\Psi(t)\rangle$ can be solved analytically by perturbation theory. It turns out to be enough to use the first order perturbation. In the perturbation treatment, we take the last two terms in equation (B.1) as a perturbation. To zero-th order, the ground state is given by equation (2). The first-order correction of degenerate perturbation theory gives that a set of low-lying excited states bunch into a band on top of the ground state, and the eigenstates in this first excited band can be expressed as $$\label{k-state}
|k\rangle = \sum_{i=1}^{N}\sin(\frac{ki\pi}{N+1})|i\rangle_{-},$$ where $|i\rangle_{-}=\big(\prod_{n\in [1,i-1]\bigcup[i+1,N]}|S\rangle_n\big)\times|A\rangle_i$. It can be shown that for the collective dynamics considered here, it is enough to focus on the ground state and first excited band. Without loss of generality, we can assume the wave function at time $\tau$, $i.e.$ when $V_{pt}$ vanishes, as $|\Psi(\tau)\rangle=(\alpha|S\rangle_1+\beta|A\rangle_1)\times\prod_{i=2}^N|S\rangle_i$ with $|\alpha|^2+|\beta|^2=1$. Then $|\Psi(t-\tau)\rangle$ becomes $$\label{wfunc}
\begin{split}
|\Psi(t-\tau)\rangle=&e^{-i\epsilon_0 (t-\tau)}|G\rangle\langle G|\Psi(\tau)\rangle
\\&+\sum_{k=1}^N e^{-i\epsilon_k (t-\tau)}|k\rangle\langle k|\Psi(\tau)\rangle
\\=&\alpha e^{-i\epsilon_0 (t-\tau)}|G\rangle\\
&+\sum_{k=1}^N\beta sin(\frac{k\pi}{N+1})e^{-i\epsilon_k (t-\tau)}|k\rangle.
\end{split}$$
In the basis of $\{|S\rangle_i,|A\rangle_i\}_{i=1}^N$, $\delta\hat\rho_i$ becomes $\sigma_x(i)$. Substituting this expression and equation (B1) into equation (4), we obtain $$\label{commut}
\begin{split}
[[\delta\hat\rho_i,\hat H],\hat H]=4J^2\sigma_x(i)-4JV\sigma_z(i)\\\times(\sigma_x(i+1)+\sigma_x(i-1)).
\end{split}$$ Further using equation (B3) for the average $<...>$, it can be proven that $\langle\sigma_z(i)\sigma_x(i\pm 1)\rangle=\langle\sigma_x(i\pm 1)\rangle$. It is then straightforward to obtain the time derivative equation (5).
[60]{}
P. W. Anderson, Concepts in Solids: Lectures on the Theory of Solids. World Scientific Lecture Notes in Physics. World Scientific Pub Co Inc, (1998).
S.H. Simon, The Oxford solid state basics. Oxford: Oxford University Press., 1st ed. edition, (2013).
J. Bardeen, Cooperative Phenomena. Springer-Verlag, Berlin, Heidelberg, New York, (1973).
S. Lepri, R. Livi, and A. Politi, Thermal conduction in classical low-dimensional lattices, Phys. Rep., **377**, 1 (2003).
N. Li, J. Ren, L. Wang, G. Zhang, P. H[ä]{}nggi, and B. Li, Colloquium : Phononics: Manipulating heat flow with electronic analogs and beyond, Rev. Mod. Phys., **84**, 1045 (2012).
D. Porras, and J. I. Cirac, Bose-Einstein condensation and strong-correlation behavior of phonons in ion traps., Phys. Rev. Lett. **93**, 263602 (2004).
U. Bissbort, D. Cocks, A. Negretti, Z. Idziaszek, T. Calarco, F. Schmidt-Kaler, W. Hofstetter, and R. Gerritsma, Emulating solid-state physics with a hybrid system of ultracold ions and atoms, Phys. Rev. Lett., **111**, 080501 (2013).
G. Pupillo, A. Griessner, A. Micheli, M. Ortner, D.-W. Wang, and P. Zoller, Cold atoms and molecules in self-assembled dipolar lattices, Phys. Rev. Lett., **100**, 050402 (2008).
M. Ortner, A. Micheli, G. Pupillo, and P. Zoller, Quantum simulations of extended hubbard models with dipolar crystals, New J. Phys., **11**, 055045 (2009).
L. Cao, V. Bolsinger, S.I. Mistakidis, G. Koutentakis, S. Kr[ö]{}nke, J.M. Schurer and P. Schmelcher, An all-in-all ab-initio approach to multi-component quantum gases of fermions and bosons, **in preparation**.
H.-D. Meyer, U. Manthe, and L.S. Cederbaum, The multi-configurational time-dependent hartree approach, Chem. Phys. Lett., **165**, 73 (1990).
M. H. Beck, A. J[ä]{}ckle, G. A. Worth, and H. D. Meyer, The multiconfiguration time-dependent hartree (mctdh) method: a highly efficient algorithm for propagating wavepackets, Phys. Rep., **324**, 1 (2000).
H. Wang and M. Thoss, Multilayer formulation of the multiconfiguration time-dependent hartree theory, J. Chem. Phys., **119**, 1289 (2003).
U. Manthe, A multilayer multiconfigurational time-dependent hartree approach for quantum dynamics on general potential energy surfaces, J. Chem. Phys., **128**, 164116 (2008).
S. Kr[ö]{}nke, L. Cao, O. Vendrell, and P. Schmelcher, Non-equilibrium quantum dynamics of ultra-cold atomic mixtures: the multi-layer multi-configuration time-dependent hartree method for bosons, New J. Phys., **15**, 063018 (2013).
L. Cao, S. Kr[ö]{}nke, O. Vendrell, and P. Schmelcher, The multi-layer multi-configuration time-dependent hartree method for bosons: Theory, implementation, and applications, J. Chem. Phys., **139**, 134103 (2013).
O. E. Alon, A. I. Streltsov, and L. S. Cederbaum, Multiconfigurational time-dependent hartree method for bosons: Many-body dynamics of bosonic systems, Phys. Rev. A, **77**, 033613 (2008).
O. E. Alon, A. I. Streltsov, K. Sakmann, A. U. J. Lode, J. Grond, and L. S. Cederbaum, Recursive formulation of the multiconfigurational time-dependent hartree method for fermions, bosons and mixtures thereof in terms of one-body density operators, Chem. Phys., **401**, 2 (2012).
E. Fasshauer and A. U. J. Axel, Multiconfigurational time-dependent Hartree method for fermions: Implementation, exactness, and few-fermion tunneling to open space, Phys. Rev. A, **93**, 033635 (2015).
A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Swift loss of coherence of soliton trains in attractive bose-einstein condensates, Phys. Rev. Lett., **106**, 240401 (2011).
S. Kr[ö]{}nke and P. Schmelcher, Many-body processes in black and gray matter-wave solitons, Phys. Rev. A, **91**, 053614 (2015).
O. E. Alon, A. I. Streltsov, and L. S. Cederbaum, Zoo of quantum phases and excitations of cold bosonic atoms in optical lattices, Phys. Rev. Lett., **95**, 030405 (2005).
S. Z[ö]{}llner, H.-D. Meyer, and P. Schmelcher, Few-boson dynamics in double wells: From single-atom to correlated pair tunneling, Phys. Rev. Lett., **100**, 040401 (2008).
K. Sakmann, A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Exact quantum dynamics of a bosonic josephson junction, Phys. Rev. Lett., **103**, 220601 (2009).
L. Cao, I. Brouzos, S. Z[ö]{}llner, and P. Schmelcher, Interaction-driven interband tunneling of bosons in the triple well, New J. Phys., **13**, 033032 (2011).
S. I. Mistakidis, L. Cao, and P. Schmelcher, Interaction quench induced multimode dynamics of finite atomic ensembles, J. Phys. B: At. Mol. Opt. Phys. **47**, 225303 (2014).
S. I. Mistakidis, L. Cao, and P. Schmelcher, Negative-quench-induced excitation dynamics for ultracold bosons in one-dimensional lattices, Phys. Rev. A **91**, 033611 (2014).
S. I. Mistakidis, T. Wulf, A. Negretti, and P. Schmelcher, Resonant quantum dynamics of few ultracold bosons in periodically driven finite lattices, J. Phys. B: At. Mol. Opt. Phys. **48**, 244004 (2015).
T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, The physics of dipolar bosonic quantum gases. Rep. Prog. Phys., **72** 126401, (2009).
M. A. Baranov, M. Dalmonte, G. Pupillo, and P. Zoller, Condensed matter theory of dipolar quantum gases, Chem. Rev., **112** 5012, 2012.
S. Yi, T. Li, and C. P. Sun, Novel quantum phases of dipolar bose gases in optical lattices, Phys. Rev. Lett., **98** 260405, (2007).
B. Capogrosso-Sansone, C. Trefzger, M. Lewenstein, P. Zoller, and G. Pupillo, Quantum phases of cold polar molecules in 2d optical lattices, Phys. Rev. Lett., **104** 125301, (2010).
P. Hauke, F. M. Cucchietti, A. M[ü]{}ller-Hermes, M.-C. Banuls, J. I. Cirac, and M. Lewenstein, Complete devil’s staircase and crystalsuperfluid transitions in a dipolar xxz spin chain: a trapped ion quantum simulation, New J. Phys., **12** 113037, (2010).
H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, I. Ferrier-Barbut, and T. Pfau, Observing the rosensweig instability of a quantum ferrofluid. Nature, **530**, 7589 (2016).
I. Ferrier-Barbut, H. Kadau, M. Schmitt, M. Wenzel, and T. Pfau, Observation of quantum droplets in a strongly dipolar bose gas, Phys. Rev. Lett., **116**, 215301 (2016).
A. Micheli, G. K. Brennen, and P. Zoller, A toolbox for lattice-spin models with polar molecules, Nat. Phys., **2**, 341 (2006).
A. V. Gorshkov, S. R. Manmana, G. Chen, J. Ye, E. Demler, M. D. Lukin, and A. M. Rey, Tunable superfluidity and quantum magnetism with ultracold polar molecules, Phys. Rev. Lett., **107**, 115301 (2011).
R. A. H. Kaden, S. R. Manmana, M. Foss-Feig, and A. M. Rey, Far-from-equilibrium quantum magnetism with ultracold polar molecules, Phys. Rev. Lett., **110**, 075301 (2013).
X. Zhou, X. Xu, X. Chen, and J. Chen, Magic wavelengths for terahertz clock transitions, Phys. Rev. A, **81**, 012115 (2010).
B. Olmos, D. Yu, Y. Singh, F. Schreck, K. Bongs, and I. Lesanovsky, Long-range interacting many-body systems with alkaline-earth-metal atoms, Phys. Rev. Lett., **110**, 143602 (2013).
S. Baier, M. J. Mark, D. Petter, K. Aikawa, L. Chomaz, Z. Cai, M. Baranov, P. Zoller, and F. Ferlaino, Extended bose-hubbard models with ultracold magnetic atoms. Science, **352**, 6282 (2016).
K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye, A high phase-space-density gas of polar molecules, Science, **322**, 5899 (2008).
J. Deiglmayr, A. Grochola, M. Repp, K. M[ö]{}rtlbauer, C. Gl[ü]{}ck, J. Lange, O. Dulieu, R. Wester, and M. Weidem[ü]{}ller, Formation of ultracold polar molecules in the rovibrational ground state, Phys. Rev. Lett., **101**, 133004 (2008).
B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. A. Hazzard, A. M. Rey, D. S. Jin, and J. Ye, Observation of dipolar spin-exchange interactions with lattice-confined polar molecules, Nature, **501**, 7468 (2013).
M. Guo, B. Zhu, B. Lu, X. Ye, F. Wang, R. Vexiau, N. Bouloufa-Maafa, G. Quemener, O. Dulieu, and D. Wang, Creation of an ultracold gas of ground-state dipolar$\;^{23}$Na$\;^{87}$Rb molecules, Phys. Rev. Lett., **116**, 205303 (2016).
A. Frisch, M. Mark, K. Aikawa, S. Baier, R. Grimm, A. Petrov, S. Kotochigova, G. Quemener, M. Lepers, O. Dulieu, and F. Ferlaino, Ultracold dipolar molecules composed of strongly magnetic atoms, Phys. Rev. Lett., **115**, 203201 (2015).
S. F[ö]{}lling, S. Trotzky, P. Cheinet, M. Feld, R. Saers, A. Widera, T. Muller, and I. Bloch, Direct observation of second-order atom tunnelling, Nature, **448**, 1029 (2007).
H.-N. Dai, B. Yang, A. Reingruber, X.-F. Xu, X. Jiang, Yu-Ao Chen, Z.-S. Yuan, and J.-W. Pan, Generation and detection of atomic spin entanglement in optical lattices, Nature Phys., **3705** (2016).
M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, A thouless quantum pump with ultracold bosonic atoms in an optical superlattice, Nature Phys., **12**, 350 (2016).
X. Yin, L. Cao, and P. Schmelcher, Magnetic kink states emulated with dipolar superlattice gases, EPL, **110**, 26004 (2015).
L. Cao, I. Brouzos, B. Chatterjee, and P. Schmelcher, The impact of spatial correlation on the tunneling dynamics of few-boson mixtures in a combined triple well and harmonic trap, New J. Phys., **14**, 093011 (2012).
S. Will, T. Best, U. Schneider, L. Hackermuller, D.-S. Luhmann, and I. Bloch, Time-resolved observation of coherent multi-body interactions in quantum phase revivals, Nature, **465** 7295 (2010).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We discuss the two point functions for the real and imaginary parts of the Polyakov loop in a pure $SU(3)$ gauge theory. The behavior of these correlation functions in the Polyakov Loop Model is markedly different from that in perturbation theory.'
address: |
Department of Physics\
Brookhaven National Laboratory\
Upton, NY 11973 USA\
author:
- 'Adrian Dumitru and Robert D. Pisarski [^1]'
title: Test of the Polyakov Loop Model
---
Consider the behavior of an $SU(3)$ gauge theory without dynamical quarks. The usual quantity measured is the pressure, as a function of the temperature. While important, and indeed the only thing which one needs for thermodynamics, there are many other things to measure, such as the correlation functions of gauge-invariant operators.
In this note we discuss what certain correlation functions may tell us about the behavior of the deconfined phase. We work within the context of the Polyakov Loop Model [@rp1; @wirstam; @ogilvie]. For reasons which will become clear later, however, it is probably imperative to think of how to parametrize these correlation functions in a manner [*in*]{}dependent of any theoretical prejudice.
At a nonzero temperature $T$, a fundamental quantity is the thermal Wilson Line, $${\bf L}(\vec{x}) \; = \;
{\cal P} \exp \left( i g \int^{1/T}_0 A_0(\vec{x},\tau) \,
d\tau \right) \; .$$ This transforms as an adjoint field under the local $SU(3)/Z(3)$ gauge symmetry, and as a field with charge one under the global $Z(3)$ symmetry. To obtain a gauge invariant operator, the simplest thing to do is to take the trace, forming the Polyakov loop, $$\ell_1 = \frac{1}{3} \; {\rm tr}\left( {\bf L} \right) \; .$$ This transforms under the global $Z(3)$ as a field with charge one. The expectation value of $\ell_1$ is only nonzero above $T_c$, which is the temperature for the deconfining phase transition.
In fact $\ell_1$ is only the first in an infinite series of gauge-invariant operators. For example, consider $$\ell_2 = \frac{1}{3} {\rm tr}\left( {\bf L}^2 \right) - \ell_1^2 \; .$$ Under the global $Z(3)$ symmetry, this Polyakov loop has charge two. The charge one part of ${\rm tr}( {\bf L}^2)$, $\ell_1^2$, is subtracted off to obtain an independent field. In this note we concentrate only upon the charge one Polyakov loop, $\ell \equiv \ell_1$, and drop all Polyakov loops with other charges, such as $\ell_2$, the singlet field $\ell_3 \sim {\rm tr}({\bf L}^3) + \ldots$, [*etc.*]{}
We begin with a potential for the Polyakov loop, taking the simplest form consistent with the global $Z(3)$ symmetry: $${\cal V}(\ell) = - \frac{b_2}{2} |\ell|^2 - \frac{b_3}{3}
\left( \ell^3 + (\ell^*)^3 \right) + \frac{1}{4}\left( |\ell|\right)^2 \;$$ The coefficient of the quartic term is chosen to simplify further results. At the minimum of the potential, which we assume occurs for real $\ell$, $\partial{\cal V}/\partial \ell = 0$, $$\ell_0 \equiv \langle \ell \rangle = b_3 + \sqrt{b_2 + b_3^2} \; .$$ In the Polyakov Loop Model, the pressure is related to the potential as: $$p(T) = - {\cal V}(\ell_0) b_4 T^4 \; .$$ At high temperatures, $b_2$ is adjusted so that $\ell_0 \approx 1$; then $b_4$ is adjusted to give the proper value of the ideal gas term. Away from infinite temperature, in the spirit of mean field we take the quantities $b_3$ and $b_4$ to be approximately constant with temperature. Given the pressure, the dependence of $b_2$ upon the temperature is then fixed.
While $\ell_0$ is the standard variable measured on the lattice, this is the bare value. Single insertions of the Polyakov loop are regularized by introducing a renormalization constant; the natural condition to fix the value of that constant is to require that the renormalized Polyakov loop is unity at infinite temperature [@yaffe]. If a lattice regulator is used instead of dimensional regularization, though, one has to deal with divergences $\sim g^2/(a T)$, [*etc.*]{}, which are most singular as the lattice spacing $a \rightarrow 0$.
Thus at present, we cannot easily relate the one point function of the Polyakov loop, as measured on the lattice, to the pressure. However, we now show that for three colors, one can relate certain two point functions of the Polyakov loop, to the pressure, in an unambiguous fashion.
For $SU(3)$, the Polyakov loop is a complex number, with real, $\ell_r = {\rm Re} \ell$, and imaginary, $\ell_i = {\rm Im} \ell$, parts. By a global $Z(3)$ rotation, we can assume that the vacuum expectation value of $\ell$, $\ell_0$, is real. Computing second derivatives, the mass squared for the real part is: $$m^2_r = \frac{\partial^2 {\cal V}}{\partial \ell_r^2}
= - b_2 - 4 b_3 \ell_0 + 3 \ell_0^2 \; ,$$ while that for the imaginary part is: $$m^2_i = \frac{\partial^2 {\cal V} }{\partial \ell_i^2}
= - b_2 + 4 b_3 \ell_0 + \ell_0^2 \; .$$
When $b_3 \neq 0$, the transition is necessarily of first order. The transition occurs when the nontrivial minimum is degenerate with the trivial minimum; [*i.e.*]{}, when ${\cal V}(\ell_0) = 0$. Putting in the expression for $\ell_0$, we find $$b_2(T_c^+) = - \frac{8}{9} b_3^2 \;\;\; , \;\;\;
\ell_0(T_c^+) = \frac{4}{3} b_3 \; .$$
This is all trivial algebra, done in detail to avoid any possible confusion. The full effective lagrangian can then be computed. Besides the potential term, given above, there is also the kinetic term, with a nonstandard normalization: $${\cal Z}_W T^2 |\vec{\partial} \ell|^2 \;\;\;, \;\;\;
{\cal Z}_W = \frac{3}{g^2} \left( 1 - .08 \frac{g^2}{4 \pi}
+\ldots \right) \; .$$ The first term in ${\cal Z}_W$ appears at the classical level, while the second arises from one loop corrections, as computed by Wirstam [@wirstam].
Over large distances, $x\rightarrow \infty$, the two point functions of the Polyakov loop are $$\langle \ell_{r}(x) \ell_{r}(0) \rangle - \langle \ell \rangle^2
\sim \frac{\exp(- \widetilde{m}_{r} x)}{x} \;\;\;
\; ,$$ $$\langle \ell_{i}(x) \ell_{i}(0) \rangle \sim
\frac{\exp(- \widetilde{m}_{i} x)}{x} \;.$$ The two fields, $\ell_r$ and $\ell_i$, don’t mix to the order at which we work. The masses which enter into the correlation functions are $$\widetilde{m}_{r,i}^2 = \frac{b_4}{{\cal Z}_W} m_{r,i}^2 T^2 \; .$$
For two colors, the Polyakov loop is real, and one can only measure one mass. Then, without knowing both the coupling constant and the wave function renormalization constant ${\cal Z}_W$, there is no firm prediction.
This is not true for three colors. Then one can form the [*ratio*]{} of the masses for the real and imaginary parts of the Polyakov loop. The constants $b_4$ and ${\cal Z}_W$ drop out, and one has a unique relation between this ratio of masses and the pressure. In particular, at the point of transition, using the previous results we find that $$\frac{\widetilde{m}_i}{\widetilde{m}_r} = 3 \;\;\; , \;\;\;
T=T_c^+ \; .
\label{e1}$$ This is our principal result. It is dependent upon the assumed form of the potential for ${\cal V}(\ell)$, and would change if terms such as $\sim (|\ell|^2)^3$ were included.
These two point functions in the Polyakov Loop Model are very different from those of ordinary perturbation theory. In perturbation theory, $\ell_0$ is near unity, and correlations are determined by multiple exchanges of $A_0$ fields. The mass of the $A_0$ field is the Debye mass, $m^2_D \sim g^2 T^2$. Expanding the exponentials, the real part of the Polyakov loop couples to $\sim {\rm tr}A_0^2$, while that for the imaginary part couples to $\sim {\rm tr}A_0^3$. Thus over large distances, $x \rightarrow \infty$, $$\langle \ell_{r}(x) \ell_{r}(0) \rangle - \langle \ell \rangle^2 \sim
\frac{\exp(- 2 m_D x)}{x^2} \; ,$$ $$\langle \ell_{i}(x) \ell_{i}(0) \rangle \sim
\frac{\exp(- 3 m_D x)}{x^3} \; .$$ Notice that the prefactors in front differ markedly from those of the Polyakov Loop Model; instead of $1/x$, they are $1/x^2$ and $1/x^3$, respectively, with the power of $1/x$ measuring the number of quanta exchanged.
If we ignore the difference in prefactors, even so the perturbative result for the mass ratio of (\[e1\]) is not $3$, but $3/2$.
Measurements of the two point function of the real part of the Polyakov Loop have been carried out by Kaczmarek [*et al.*]{} [@potential]. From the two point function of Polyakov loops, which is presumably dominated by that for the real part, the mass drops by about a factor of ten, from $m/T \sim 2.5$ at $T=2 T_c$, to perhaps $m/T \sim .25$ at $T_c^+$. We are not aware of any measurements of the imaginary part close to $T_c$.
There are also measurements by Bialas [*et al.*]{} for a $SU(3)$ gauge theory in $2+1$ dimensions [@pi]. While the critical behavior in this model is that of a two dimensional system, and so can have characteristics special to a low dimension, for the Polyakov Loop Model in mean field theory, our predictions remain the same. These authors find that the ratio $\widetilde{m}_i/\widetilde{m}_r$ does increase from $3/2$ as the temperature approaches $T_c$.
In fact, the Polyakov Loop Model must be inapplicable at some temperature not too far above $T_c$. At high temperature, where $\ell_0 \approx 1$, the above formula give $b_2 = 1 - 2 b_3$, and $$\frac{\widetilde{m}_i}{\widetilde{m}_r} =
\sqrt{ \frac{3 b_3}{1 - b_3} } \;\;\; , \;\;\;
T \rightarrow \infty \; .$$ The constant $b_3$ is not well determined, but for $b_3 < 3/7$, the above ratio is less than the perturbative value of $3/2$.
Thus we propose that the two point function of Polyakov loops can be used as a measure of the regime in which the Polyakov Loop Model applies, and the regime where perturbation theory applies.
What if the ratio of masses, (\[e1\]), is wrong even at $T_c$? Besides including terms of higher order in the potential, it may also be necessary to include the charge two Polyakov loop, $\ell_2$. Since $Z(3)$ is a cyclic symmetry, a field with charge two is the same as one with charge minus one. The couplings of this loop with itself are identical to the couplings of the usual Polyakov loop, since the sign of the charge doesn’t matter. Unlike the charge one Polyakov loop, however, the charge two loop should always have a positive mass squared, in order to avoid condensation which breaks $SU(3) \rightarrow SU(2)$ [@rp1]. Thus one would assume that the charge two field, as a massive field, can be ignored. Nevertheless, the following coupling is $Z(3)$ symmetric: $$\ell_1 \ell_2 + \ell_1^* \ell_2^*$$ This term mixes the charge one and charge two Polyakov loops, $\sim {\rm tr}{\bf L} {\rm tr}{\bf L}^2 + \ldots$. Its coupling constant is directly measurable; if small, the charge two Polyakov loop can be ignored, and our prediction should hold.
Lastly, we note that the Polyakov loop may well couple weakly to other operators. Thus while in principle it should dominate all correlation functions at large distances (if it is the lightest state), this may be very difficult to see unless the operator couples strongly.
[9]{} R. D. Pisarski, Phys. Rev. D62 (2000) 111501; A. Dumitru and R. D. Pisarski, Phys. Lett. B504 (2001) 282; hep-ph/0106176. J. Wirstam, hep-ph/0106141. P. N. Meisinger, T. R. Miller, and M. C. Ogilvie, hep-ph/0108009; P. N. Meisinger and M. C. Ogilvie, hep-ph/0108026. L. G. Yaffe, private communication. O. Kaczmarek, F. Karsch, E. Laermann, and M. Lutgemeier, Phys.Rev. D62 (2000) 034021. P. Bialas, A. Morel, B. Petersson, K. Petrov, and T. Reisz, these proceedings.
[^1]: This research was supported by DOE grant DE-AC-02-98CH-10886.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The first author constructed a $q$-parameterized spherical category ${\mathscr{C}}$ over $\mathbb{C}(q)$ in [@LiuYB], whose simple objects are labelled by all Young diagrams. In this paper, we compute closed-form expressions for the fusion rule of ${\mathscr{C}}$, using Littlewood-Richardson coefficients, as well as the characters (including a generating function), using symmetric functions with infinite variables.'
address:
- 'Yau Mathematical Science Center and Department of Mathematics, Tsinghua University, Beijing, 100084, China'
- 'Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA'
author:
- Zhengwei Liu
- Christopher Ryba
title: The Grothendieck Ring of a Family of Spherical Categories
---
\[section\] \[theorem\][Lemma ]{} \[theorem\][Proposition ]{} \[theorem\][Corollary ]{} \[theorem\][Definition ]{} \[theorem\][Notation ]{} \[theorem\][Example ]{} \[theorem\][Remark ]{}
Introduction
============
Jones introduced planar algebras in [@Jon99] inspired by subfactor theory and knot theory. The topological notion of (spherical) planar algebra is parallel to the algebraic notion of (spherical) pivotal, monoidal category. The first author investigated the skein-theoretical classification of planar algebras and discovered a continuous family of unshaded planar algebras over $\mathbb{C} $ from the classification of Yang-Baxter relation planar algebras in [@LiuYB]. This family was constructed in terms of $q$-parameterized generators and relations in linear skein theory. This family could be regarded as a planar algebra ${\mathscr{C}}$ over $\mathbb{C}(q)$ with a generic parameter $q$. The canonical idempotent category of this unshaded planar algebra ${\mathscr{C}}$ is a $\mathbb{Z}_2$-graded spherical monoidal category. It was shown in [@LiuYB] that the Grothendick ring $\mathcal{G}$ of ${\mathscr{C}}$ has simple objects $X_{\lambda}$ labelled by all Young diagrams $\lambda$ with an explicit construction of a minimal idempotent ${\tilde{y}}_{\lambda}$ in ${\mathscr{C}}$ which represents $X_{\lambda}$.
The main purpose of this paper is computing the fusion rule of $\mathcal{G}$, $$\begin{aligned}
\label{Equ: fusion rule}
X_{\mu}X_{\nu}=\sum_{\lambda} R_{\mu, \nu}^ \lambda X_{\lambda}, \end{aligned}$$ where $R_{\mu, \nu}^ \lambda \in \mathbb{N}$ is called the fusion coefficient for Young diagrams $\mu$, $\nu$, $\lambda$. We compute the fusion coefficient $R_{\mu, \nu}^ \lambda$ in a closed-form expression in Theorem \[Thm: Main3\], $$\begin{aligned}
R_{\mu, \nu}^ \lambda &= \sum_{\alpha, \beta, \gamma} c_{\alpha, \beta}^\mu c_{\beta', \gamma}^\nu c_{\alpha, \gamma}^\lambda,\end{aligned}$$ where $\gamma^\prime$ is the Young diagram dual to $\gamma$ and $c_{\cdot,\cdot}^{\cdot}$ is the Littlewood-Richardson coefficient.
The paper is organized as follows. In §\[Sec: Yang-Baxter relation planar algebras and spherical categories\], we recall some basic properties of ${\mathscr{C}}$ and its type $A$ Hecke subalgebra $H$. We prove in Theorem \[Thm: ring iso X\] that the Grothendieck ring $\mathcal{G}$ of ${\mathscr{C}}$ is the polynomial ring freely generated by the fundamental representations $\{X_{1^n} : n\in\mathbb{N} \}$, where $1^n$ is the Young diagram with one column and $n$ cells. In particular, $\mathcal{G}$ is commutative.
In §\[Sec: Fusion Rules of Fundamental Representations for the Generic Case\], we compute the fusion rule of $\mathcal{G}$ with respect to the fundamental representations in Theorem \[Thm: fusion with column\]: $$\begin{aligned}
X_{(1^r)} X_{\mu}&=\sum_{i=0}^{r} \sum_{\nu \in \mu- i} \sum_{\lambda \in \nu+1^{r-i}}X_{\lambda}.\end{aligned}$$ The multiplicity of $X_{\lambda}$ is the number of ways of constructing $\lambda$ from $\mu$ by removing $i$ cells, no two in the same column, and then adding $r-i$ cells, no two in the same row. The proof follows from an explicit construction of the basis of $\hom(X_{(1^r)} X_{\mu}, X_{\lambda})$ in ${\mathscr{C}}$ through the linear skein theory of the Yang-Baxter relation planar algebra.
In §\[Sec: Characters, Generating Functions and Fusion Rules for the Generic Case\], we compute fusion rules of $\mathcal{G}$. In principle, one can compute the fusion rule of $\mathcal{G}$ recursively using the fusion rule of fundamental representations. However, the complexity grows exponentially w.r.t. the size of the Young diagrams. We observe that $\mathcal{G}$ is isomorphic to the ring $\Lambda$ of symmetric polynomial with infinite variables. We establish a ring isomorphism $\Phi: \mathcal{G} \to \Lambda$ in Definition \[Def: Iso\] and consider $\Phi(X_\lambda)$ as the character of the simple object $X_{\lambda}$ of $\mathcal{G}$. We prove in Theorem \[Thm: Main1\] that $$\begin{aligned}
\label{Equ: Weyl Character}
\Phi(X_\lambda) & = \sum_{\mu} (-1)^{|\mu|} s_{\lambda/2\mu} \;. \end{aligned}$$ where $s_{\lambda/2\mu}$ is a skew-Schur polynomial. Moreover, we compute the generating function of the characters in closed form in Theorem \[Thm: Main2\], $$\begin{aligned}
\sum_{\lambda} s_\lambda(x) \Phi(X_\lambda)(y) &= \prod_{i_1 \leq i_2}\frac{1}{1 + x_ix_j} \prod_{i, j}\frac{1}{1-x_i y_j}.\end{aligned}$$ Using the generating function, we compute the fusion coefficient in a closed form, namely Equation , in Theorem \[Thm: Main3\]. Our computational tools on the characters and the generating function come from the theory of symmetric functions [@Macdonald], which we recall in §\[Sec: Characters, Generating Functions and Fusion Rules for the Generic Case\].
In this paper, we compute the fusion rule of $\mathscr{C}$ over $\mathbb{C}(q)$. Unitary fusion categories $\mathscr{C}^{N,k,\ell}$, $N,k,\ell \in \mathbb{N}$, were constructed in [@LiuYB] as quotients of $\mathscr{C}$ over $\mathbb{C}$ at $q=e^{\frac{\pi i}{2N+2}}$. In particular, $\mathscr{C}^{N,0,1}$ is an exceptional quantum subgroup of $SU(N)_{N+2}$, conjectured to be isomorphic to the exceptional quantum subgroup constructed by Xu in [@Xu98] in 1998 through the $\alpha$-induction of the conformal inclusion $SU(N)_{N+2} \subseteq SU(N(N+1)/2)_1$. Xu asked the question to compute the fusion rules of these exceptional quantum subgroups [@Xu], which we will compute in the future. From this point of view, $\mathscr{C}$ can be regarded as the parameterization of a family of exceptional quantum subgroups. It was conjecture in [@LiuYB] that there is a continuous family of monoidal categories parameterizing the exceptional quantum subgroups from the $\alpha$-induction of any family of conformal inclusions of quantum groups. We believe that our methods in this paper also apply to the other continuous families of monoidal categories, if they exist.
Acknowledgements {#acknowledgements .unnumbered}
----------------
Zhengwei Liu was partially supported by Grant 100301004 from Tsinghua University and by Templeton Religion Trust under the grant TRT 159. Zhengwei Liu would like to thank Pavel Etingof and Feng Xu for helpful discussions and to thank Arthur Jaffe for the hospitality at Harvard University. Christopher Ryba would like to thank Pavel Etingof for useful conversations.
Yang-Baxter relation planar algebras and spherical categories {#Sec: Yang-Baxter relation planar algebras and spherical categories}
=============================================================
The first author constructed the following continuous family of Yang-Baxter relation planar algebras ${\mathscr{C}}$ in terms of generators and relations in linear skein theory in [@LiuYB].
\[Def:Centralizer algebra\] Let $\mathscr{C}_{\bullet}$ be the unshaded planar algebra over $\mathbb{C}(q)$ with circle parameter $$\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.6,yscale=.6]
\draw[thick] (0,0) arc (0:360:.5);
\end{scope}
\end{tikzpicture}
}=\delta =q+q^{-1} \;,$$ which is generated by $R=\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.6,yscale=.6]
\draw[thick] (0,0)--(1,1);
\draw[thick] (1,0)--(0,1);
\node at (0,1/2) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}$ with the following relations:
$$\begin{aligned}
\label{YBrelation1}
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.6,yscale=.6]
\draw[thick] (0,0)--(1,1);
\draw[thick] (1,0)--(0,1);
\node at (0,1/2) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}&=i
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.6,yscale=.6]
\draw[thick] (0,0)--(1,1);
\draw[thick] (1,0)--(0,1);
\node at (1/2,1) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}, \\\label{YBrelation2}
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.6,yscale=.6]
\draw[thick] (0,0)--(2/3,2/3);
\draw[thick] (2/3,2/3) arc (135:-135: 1.414/6);
\draw[thick] (0,1)--(2/3,1/3);
\node at (0,1/2) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}&=0, \\\label{YBrelation3}
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.6,yscale=.4]
\draw[thick] (0,0)--(1,1)--(0,2);
\draw[thick] (1,0)--(0,1)--(1,2);
\node at (0,1/2) {\tiny{$R$}};
\node at (0,1+1/2) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
&= {
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (0,0)--(0,1);
\draw[thick] (2/3,0)--(2/3,1);
\end{scope}
\end{tikzpicture}
}
}-\frac{1}{\delta} {
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (0,0) arc (180:0:1/3);
\draw[thick] (0,1) arc (180:360:1/3);
\end{scope}
\end{tikzpicture}
}
}, \\
\label{YBrelation4}
{
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (0,0)--(1,1);
\draw[thick] (1,0)--(0,1);
\draw[thick] (.3,0)--(.3,1);
\node at (.1,.65) {\tiny{$R$}};
\node at (.45,.15) {\tiny{$R$}};
\node at (.55,.75) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
}&=\frac{i}{\delta^2}({
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (1,0)--(1,1);
\draw[thick] (0,0)--(.5,1);
\draw[thick] (.5,0)--(0,1);
\node at (0,.5) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
}+{
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (.5,0)--(1,1);
\draw[thick] (0,0) arc (180:0:.5 and .4);
\draw[thick] (0,1) arc (180:360:.25 and .4);
\node at (.75,.15) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
}+{
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (1,0)--(.5,1);
\draw[thick] (0,0) arc (180:0:.25 and .4);
\draw[thick] (0,1) arc (180:360:.5 and .4);
\node at (.75,.85) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
})-\frac{1}{\delta^2}({
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (0,0)--(0,1);
\draw[thick] (1,0)--(.5,1);
\draw[thick] (.5,0)--(1,1);
\node at (.75,.15) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
}+{
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (0,0)--(.5,1);
\draw[thick] (.5,0) arc (180:0:.25 and .4);
\draw[thick] (0,1) arc (180:360:.5 and .4);
\node at (.65,.85) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
}+{
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (.5,0)--(0,1);
\draw[thick] (0,0) arc (180:0:.5 and .4);
\draw[thick] (.5,1) arc (180:360:.25 and .4);
\node at (0,.5) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
})+i{
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (0,0)--(1,1);
\draw[thick] (1,0)--(0,1);
\draw[thick] (.6,0)--(.6,1);
\node at (.1,.5) {\tiny{$R$}};
\node at (.75,.05) {\tiny{$R$}};
\node at (.75,.95) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
}.\end{aligned}$$
The vector space ${\mathscr{C}}_{n}$ consists of linear sums of $R$-labelled planar diagrams with $2n$ boundary points modulo the above relations. Consider a disc with $2n$ boundary points numbered by $\{1,2,\ldots, 2n\}$ clockwise. A pairing $p$ of $\{1,2,\ldots, 2n\}$ is a bijection on $\{1,2,\ldots, 2n\}$, such that $p^2$ is the identity and $p(i) \neq i$, $\forall 1\leq i \leq 2n$. We call $\{i,p(i)\}$ a pair of the paring $p$. Let $P_{n}$ be the set of pairings of $2n$ boundary points.
We can construct a diagram in the disc which connects the $n$ pairs of boundary points by $n$ strings with a minimal number of crossings. (The minimal condition is equivalent to that any two strings either intersect at one point transversally or do not intersect.) Such diagrams have been used by Brauer to construct the Brauer algebras. We label each crossing of the diagram by the generator $R$, then we obtain an element in ${\mathscr{C}}_{n}$, denoted by $\hat{p}$. Note that there are four choices to label $R$ at each crossing, and the corresponding elements in ${\mathscr{C}}_{n}$ differ by a phase due to Relation . We fix a choice at the beginning to define $\hat{p}$.
\[Prop: Bp Basis\] The set $\mathcal{B}_n=\{\hat{p} : p\in P_{n}\}$ is a basis of the vector space ${\mathscr{C}}_{n}$ over $\mathbb{C}(q)$.
Applying the Yang-Baxter relation, any element in ${\mathscr{C}}_{n}$ is a linear sum of such $\hat{p}$’s. On the other hand, $\dim{\mathscr{C}}_n=(2n-1)!!$ by Corollary 6.6 in [@LiuYB], and $\#\{\hat{p} : p\in P_{n}\}=(2n-1)!!$. Therefore, $\{\hat{p} : p\in P_{n}\}$ is a basis of ${\mathscr{C}}_{n}$.
Note that ${
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (0,0)--(1,1);
\draw[thick] (1,0)--(0,1);
\draw[thick] (.3,0)--(.3,1);
\node at (.1,.65) {\tiny{$R$}};
\node at (.45,.15) {\tiny{$R$}};
\node at (.55,.75) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
}$ and ${
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (0,0)--(1,1);
\draw[thick] (1,0)--(0,1);
\draw[thick] (.6,0)--(.6,1);
\node at (.1,.5) {\tiny{$R$}};
\node at (.75,.05) {\tiny{$R$}};
\node at (.75,.95) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
}$ correspond to the same pairing. When we define the element $\hat{p}$, we fix a choice. Either ${
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (0,0)--(1,1);
\draw[thick] (1,0)--(0,1);
\draw[thick] (.3,0)--(.3,1);
\node at (.1,.65) {\tiny{$R$}};
\node at (.45,.15) {\tiny{$R$}};
\node at (.55,.75) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
}$ or ${
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (0,0)--(1,1);
\draw[thick] (1,0)--(0,1);
\draw[thick] (.6,0)--(.6,1);
\node at (.1,.5) {\tiny{$R$}};
\node at (.75,.05) {\tiny{$R$}};
\node at (.75,.95) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
}$ with the 14 lower terms $$\begin{aligned}
& {
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (0,0)--(0,1);
\draw[thick] (.5,0) arc (180:0:.25 and .4);
\draw[thick] (.5,1) arc (180:360:.25 and .4);
\end{scope}
\end{tikzpicture}
}
},{
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (1,0)--(1,1);
\draw[thick] (0,0) arc (180:0:.25 and .4);
\draw[thick] (0,1) arc (180:360:.25 and .4);
\end{scope}
\end{tikzpicture}
}
},{
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (0,0)--(0,1);
\draw[thick] (.5,0)--(.5,1);
\draw[thick] (1,0)--(1,1);
\end{scope}
\end{tikzpicture}
}
},{
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (0,0)--(1,1);
\draw[thick] (.5,0) arc (180:0:.25 and .4);
\draw[thick] (0,1) arc (180:360:.25 and .4);
\end{scope}
\end{tikzpicture}
}
},{
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (1,0)--(0,1);
\draw[thick] (0,0) arc (180:0:.25 and .4);
\draw[thick] (.5,1) arc (180:360:.25 and .4);
\end{scope}
\end{tikzpicture}
}
}, \\
& {
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (1,0)--(1,1);
\draw[thick] (0,0)--(.5,1);
\draw[thick] (.5,0)--(0,1);
\node at (0,.5) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
},{
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (.5,0)--(1,1);
\draw[thick] (0,0) arc (180:0:.5 and .4);
\draw[thick] (0,1) arc (180:360:.25 and .4);
\node at (.75,.15) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
},{
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (1,0)--(.5,1);
\draw[thick] (0,0) arc (180:0:.25 and .4);
\draw[thick] (0,1) arc (180:360:.5 and .4);
\node at (.75,.85) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
},{
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (0,0)--(0,1);
\draw[thick] (1,0)--(.5,1);
\draw[thick] (.5,0)--(1,1);
\node at (.75,.15) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
},{
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (0,0)--(.5,1);
\draw[thick] (.5,0) arc (180:0:.25 and .4);
\draw[thick] (0,1) arc (180:360:.5 and .4);
\node at (.65,.85) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
},{
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (.5,0)--(0,1);
\draw[thick] (0,0) arc (180:0:.5 and .4);
\draw[thick] (.5,1) arc (180:360:.25 and .4);
\node at (0,.5) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
}, \\
& {
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (.5,0)--(.5,1);
\draw[thick] (0,0) arc (180:0:.5 and .4);
\draw[thick] (0,1) arc (180:360:.5 and .4);
\node at (.7,.85) {\tiny{$R$}};
\node at (.7,.15) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
},{
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (0,0)--(1,1);
\draw[thick] (.5,0)--(0,1);
\draw[thick] (1,0)--(.5,1);
\node at (.1,.35) {\tiny{$R$}};
\node at (.75,.95) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
},{
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (1,0)--(0,1);
\draw[thick] (0,0)--(.5,1);
\draw[thick] (.5,0)--(1,1);
\node at (.1,.65) {\tiny{$R$}};
\node at (.75,.05) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
},\end{aligned}$$ form a basis of ${\mathscr{C}}_{3}$.
A planar algebra canonically has a vertical multiplication, a horizontal tensor product and a Markov trace, see [@Jon99]. For a diagram in ${\mathscr{C}}_{n}$, we draw the first $n$ boundary points on the top and the last $n$ boundary points at the bottom. Then ${\mathscr{C}}_{n}$ forms an algebra whose multiplication is gluing the diagrams vertically. The tensor product $\otimes: {\mathscr{C}}_n \otimes {\mathscr{C}}_m \to {\mathscr{C}}_{n+m}$ is a horizontal union of two diagrams. In particular, the algebra ${\mathscr{C}}_{n}$ can be embedded in ${\mathscr{C}}_{n+1}$ by adding a through string on the right. So ${\mathscr{C}}_{\bullet}$ is a filtered algebra.
The planar algebra ${\mathscr{C}}_{\bullet}$ has a has a type $A$ Hecke subalgebra $H_{\bullet}$ generated by $$\alpha=
\frac{q-q^{-1}}{2}{
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (0,0)--(0,1);
\draw[thick] (2/3,0)--(2/3,1);
\end{scope}
\end{tikzpicture}
}
}+\frac{q-q^{-1}}{2i}{
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (0,0) arc (180:0:1/3);
\draw[thick] (0,1) arc (180:360:1/3);
\end{scope}
\end{tikzpicture}
}
}+ \frac{q+q^{-1}}{2} {
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (0,0)--(.8,1);
\draw[thick] (.8,0)--(0,1);
\node at (.1,1/2) {\tiny{$R$}};
\end{scope}
\end{tikzpicture}
}
}.$$ The generic type $A$ Hecke algebra has two parameters $q$ and $r$. Here $qr=\sqrt{-1}$. The planar algebra has a Markov trace by gluing the top and the bottom from the right. The Markov trace of ${\mathscr{C}}_{\bullet}$ extends the Markov trace of the Hecke algebra in [@Jon87].
The idempotent $e=\frac{1}{\delta}{
\raisebox{-.3cm}{
\begin{tikzpicture}
\begin{scope}[xscale=.7,yscale=.7]
\draw[thick] (0,0) arc (180:0:1/3);
\draw[thick] (0,1) arc (180:360:1/3);
\end{scope}
\end{tikzpicture}
}
}$ in ${\mathscr{C}}_2$ is called the Jones idempotent [^1]. The two-sided ideal of ${\mathscr{C}}_n$ generated by $e$ is denoted by ${\mathscr{I}}_n$, called the basic construction ideal. The complement of the maximal idempotent of ${\mathscr{I}}_n$ is denoted by $s_n$. Then, $s_n$ is central in ${\mathscr{C}}_n$ and $$\begin{aligned}
\label{Equ: sn} x s_n&=0, ~\forall~ x \in {\mathscr{I}}_n. \\
\label{Equ: sn0} s_n(s_m \otimes s_{n-m}) &=s_n, ~\forall~ m \leq n.\end{aligned}$$
The following proposition is a consequence of Theorem 6.5 in [@LiuYB]:
\[Prop: sn\] For any $n \geq 0$, $$\begin{aligned}
\label{Equ: sn1} H_n &\cong s_nH_n =s_n {\mathscr{C}}_n \;, \\
\label{Equ: sn2} {\mathscr{C}}_n &={\mathscr{I}}_n \oplus H_n = {\mathscr{I}}_n \oplus s_n H_n\;.\end{aligned}$$
For each Young diagram $\lambda$, let $|\lambda|=n$ be the number of cells of $\lambda$. A minimal idempotent $y_{\lambda}$ in $\hom_{{\mathscr{C}}}(X^{n}, X^{n})$ was constructed in Section 2.5 in [@LiuYB]. These $y_{\lambda}$’s are representatives of the equivalent minimal idempotents of $H_{n}$. Furthermore, ${\tilde{y}}_{\lambda}=s_n y_{\lambda}$ are representatives of the equivalent minimal idempotents of $s_nH_{n}$. We refer the readers to [@LiuYB] for the explicit construction of $s_n$, $y_{\lambda}$ and ${\tilde{y}}_{\lambda}$. Following the construction in Theorem 6.5 in [@LiuYB], we have that $$\label{Equ: y=ty}
y_{(1^n)}={\tilde{y}}_{(1^n)},$$ where $(1^n)$ is the Young diagram with one column and $n$ cells. The Young diagram with one row and $n$ cells is denoted by $(n)$. We write them as $1^n$ and $n$ for short, if there is no risk of confusion.
We recall the above properties of ${\mathscr{C}}$, which we will apply in this paper. We refer the readers to [@LiuYB] for the construction of $s_n$, $y_{\lambda}$ and ${\tilde{y}}_{\lambda}$, which we do not repeat here.
\[Prop: iso\] Note that $\hom_H (y_{\mu} \otimes y_{\nu}, y_{\lambda} ) \subseteq H_n \subseteq {\mathscr{C}}_n$, when $|\mu|+|\nu|=|\lambda|=n$. We have that $$s_n \hom_{H} (y_{\mu} \otimes y_{\nu}, y_{\lambda} ) =\hom_{{\mathscr{C}}} ({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{\nu}, {\tilde{y}}_{\lambda} ) .$$
Note that $s_n y_{\mu} \otimes y_{\nu} ={\tilde{y}}_{\mu} \otimes {\tilde{y}}_{\nu}$ and $s_n y_{\lambda}={\tilde{y}}_{\lambda}$, so $$s_n \hom_{H} (y_{\mu} \otimes y_{\nu}, y_{\lambda} ) \subseteq \hom_{{\mathscr{C}}} ({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{\nu}, {\tilde{y}}_{\lambda} ) .$$ On the other hand, by Equation \[Equ: sn2\], for any element $x$ in $\hom_{{\mathscr{C}}} ({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{\nu}, {\tilde{y}}_{\lambda} ) \subseteq {\mathscr{C}}_n$, we have a unique decomposition $x=y+z$, such that $y\in {\mathscr{I}}_n$ and $z \in H_n$. Note that $$\begin{aligned}
y_{\lambda} x (y_{\mu} \otimes y_{\nu}) &=x, \\
y_{\lambda} y (y_{\mu} \otimes y_{\nu})& \in {\mathscr{I}}_n ,\\
y_{\lambda} z (y_{\mu} \otimes y_{\nu}) &\in H_n .\\\end{aligned}$$ So $y_{\lambda} z (y_{\mu} \otimes y_{\nu})=z$, and $z \in \hom_{H} (y_{\mu} \otimes y_{\nu}, y_{\lambda} )$. Moreover, $s_n z = s_n x =x$. So $$s_n \hom_{H} (y_{\mu} \otimes y_{\nu}, y_{\lambda} ) = \hom_{{\mathscr{C}}} ({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{\nu}, {\tilde{y}}_{\lambda} ) .$$
From the semi-simple, spherical, unshaded planar algebra ${\mathscr{C}}$, we obtain an $\mathbb{N}$-graded monoidal category ${\mathscr{C}}^{q,0}$ whose degree $n$ objects are idempotents in ${\mathscr{C}}_n$. In subfactor theory, we usually consider the Jones idempotent to be equivalent to the unit in ${\mathscr{C}}_0$, namely the empty diagram $\emptyset$. The isometries between the two idempotents are given by the the diagrams $\cap$ and $\cup$. Modulo this relation, we obtain a $\mathbb{Z}_2$-graded spherical category ${\mathscr{C}}^{q,1}$, the canonical one associated with the spherical planar algebra ${\mathscr{C}}$.
\[Not: Y\] Let $\mathcal{G}$ be the Grothendieck ring of ${\mathscr{C}}^{q,1}$. It has a basis $X_{\lambda}$ corresponding to the minimal idempotents ${\tilde{y}}_{\lambda}$ of $s_n{\mathscr{C}}_n$, $n\in \mathbb{N}$, for all Young diagrams $\lambda$.
Let $1^r$ be the Young diagram with one column and $r$ cells. In particular, $X=X_1$, corresponds to the Young diagram with one cell. Then the identity map $1_X$ is a through string, and $$\begin{aligned}
(\cup \otimes 1_X)(1_X \otimes \cap) &=1_X \; , \\
(1_X \otimes \cup)(\cap \otimes 1_X) &=1_X \; .\end{aligned}$$ The morphism space $\hom_{{\mathscr{C}}}(X^{n}, X^{m})$ consists of linear combinations of $R$-labelled planar diagrams in ${\mathscr{C}}$ with $n$ boundary points on the top and $m$ boundary points at the bottom. For Young diagrams $\mu, \nu, \lambda$, the morphisms of ${\mathscr{C}}^{q,1}$ are given by $$\hom_{{\mathscr{C}}}(X_{\mu} \otimes X_{\nu}, X_{\lambda})= {\tilde{y}}_{\lambda} (\hom_{{\mathscr{C}}}(X^{|\mu|+|\nu|}, X^{|\lambda|} ) ({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{\nu}) \;.$$
\[Not: Y\] Let $Y_{\lambda}$ be the element of $\mathcal{G}$ corresponding to the idempotent $y_{\lambda}$. Note that ${\tilde{y}}_{\lambda}=s_{|\lambda|} y_{\lambda}$, so $y_{\lambda}-{\tilde{y}}_{\lambda}$ is an idempotent in ${\mathscr{I}}_{|\lambda|}$. Therefore, $$\begin{aligned}
\label{Equ: Y=X+}
Y_{\lambda}=X_{\lambda}+\sum_{|\mu| < |\lambda|} n_{\lambda, \mu} X_{\mu}, \text{ for some } n_{\lambda, \mu} \in \mathbb{N}.\end{aligned}$$ We call $n_{\lambda, \mu}$ the [**extended constants**]{}. Then we can solve for the $ X_{\lambda}$ recursively in terms of the $Y_{\lambda}$, and $$\begin{aligned}
\label{Equ: X=Y+}
X_{\lambda}=Y_{\lambda}+\sum_{|\mu| < |\lambda|} z_{\lambda, \mu} Y_{\mu}, \text{ for some } z_{\lambda, \mu} \in \mathbb{Z}.\end{aligned}$$ We call $z_{\lambda, \mu}$ the [**inverse extended constants**]{}. By Equation , for any $n\geq 0$, $$\begin{aligned}
\label{Equ: Yn=Xn}
X_{1^n}=Y_{1^n}.\end{aligned}$$
\[Thm: ring iso X\] The Grothendieck ring $\mathcal{G}$ of ${\mathscr{C}}$ is the polynomial ring in the generators $\{ X_{1^n}: n > 0 \}$. In particular, $\mathcal{G}$ is commutative.
Note that $\{X_\lambda\}$ forms a basis of the Grothendieck ring $\mathcal{G}$. By Equations and , $\{Y_\lambda\}$ also forms a basis of $\mathcal{G}$. It is known that the set $\{Y_{\lambda} \}$ is a basis of the polynomial ring in the generators $\{Y_{1^n}: n > 0 \}$. By Equation , $\mathcal{G}$ is the polynomial ring in the generators $\{ X_{1^n}: n > 0 \}$.
Based on the algebraic structure on ${\mathscr{C}}_n \cong \hom_{{\mathscr{C}}}(X^n,X^n)$, We decompose the partitions $P_n$ into two subset sets $I_n$ and $T_n$. A pairing is in $I_n$ if there is a pair among the first $n$ points. On the other hand, a pairing $p$ in $T_n$ pairs the first $n$ points with the last $n$ points. For any pairing $p \in T_n$, we can identify $p$ with an element $p'$ in the permutation group $S_n$, such that $p'(i)=2n+1-p(i)$, for $1\leq i \leq n$.
\[Prop: IS\] For a pairing $p \in P_n$, we have that $p\in I_n$ iff $\hat{p} \in {\mathscr{I}}_n$. Moreover, $\{s_n \hat{p} : p \in T_n\}$ is a basis of $s_nH_n$.
Obviously if $p \in P_n$, then $\hat{p} \in {\mathscr{I}}_n$ and $s_n \hat{p}=0$. By Equation , $\{s_n \hat{p} : p \in T_n\}$ is a spanning set of $s_n H_n$. By Equation , $\dim s_n H_n= \dim H_n = \#T_n$, so $\{s_n \hat{p} : p \in T_n\}$ is a basis of $s_nH_n$, and for any $p \in T_n$, $s_n \hat{p} \neq 0$, namely $\hat{p} \notin {\mathscr{I}}_n$.
We define $T_{i,n-i}$ to be a subset of $P_n$ as follows: $T_{0,n}=T_{n,0}=T_n$, and for any $1\leq i \leq n-1$, $$\begin{aligned}
T_{i,n-i}&=\{p \in P_n: 2n+1-j \leq p(j) \leq 2n, ~\forall~ 1\leq j \leq i; n+1 \leq p(j) \leq 2n-i, ~\forall~ i+1 \leq j \leq n\} . \\\end{aligned}$$ We can consider $T_{i,n-i}$ as $T_i \times T_{n-i}$. Note that $T_{i,n-i} \subseteq T_n$.
For any Young diagram $\lambda$, $|\lambda|=n$, we express ${\tilde{y}}_{\lambda}$ in terms of the basis $\mathcal{B}_n$, $${\tilde{y}}_{\lambda}=\sum_{p\in P_n} c_p \hat{p} \;, ~c_p\in \mathbb{C}(q).$$ We define $$\begin{aligned}
\label{Equ: yi}
{\tilde{y}}_{\lambda,i}=\sum_{p\in T_{i,n-i}} c_p \hat{p}.\end{aligned}$$
For any Young diagram $\lambda$, we have that ${\tilde{y}}_{\lambda,0} {\tilde{y}}_{\lambda}={\tilde{y}}_{\lambda} {\tilde{y}}_{\lambda,0}={\tilde{y}}_{\lambda}$.
Note that $s_n \hat{p}=0$, $\forall p \in I_n$, and ${\tilde{y}}_{\lambda}=s_n{\tilde{y}}_{\lambda}$. So $$\begin{aligned}
{\tilde{y}}_{\lambda} {\tilde{y}}_{\lambda,0} &={\tilde{y}}_{\lambda} s_n \sum_{p\in T_{n}} c_p \hat{p}
={\tilde{y}}_{\lambda} s_n \sum_{p\in P_{n}} c_p \hat{p}
={\tilde{y}}_{\lambda} s_n {\tilde{y}}_{\lambda}
={\tilde{y}}_{\lambda} .\end{aligned}$$ Similarly, ${\tilde{y}}_{\lambda,0} {\tilde{y}}_{\lambda}={\tilde{y}}_{\lambda}$.
\[Lem: non-zero\] For any $0\leq i \leq n$, we have that $${\tilde{y}}_{1^n,i} ({\tilde{y}}_{1^i}\otimes {\tilde{y}}_{1^{n-i}})= c {\tilde{y}}_{1^i}\otimes {\tilde{y}}_{1^{n-i}},$$ for some $c\neq 0$ in $\mathbb{C}(q)$, and $\displaystyle \lim_{q \to 1} c= {n\choose i}^{-1}$.
By Proposition \[Prop: sn\], $ {\tilde{y}}_{1^k}$ is a minimal central projection in ${\mathscr{C}}_k$, so for any $p \in T_{i,n-i}$, $\hat{p} ({\tilde{y}}_{1^i} \otimes {\tilde{y}}_{1^{n-i}})$ is a multiple of ${\tilde{y}}_{1^i}\otimes {\tilde{y}}_{1^{n-i}}$. Therefore, $${\tilde{y}}_{1^n,i} ({\tilde{y}}_{1^i} \otimes {\tilde{y}}_{1^{n-i}})= c {\tilde{y}}_{1^i}\otimes {\tilde{y}}_{1^{n-i}},$$ for some $c \in \mathbb{C}(q)$. Moreover, ${\tilde{y}}_{1^n,i} {\tilde{y}}_{1^n}= c {\tilde{y}}_{1^n}$. We need to show that $c\neq 0$. For any $p \in T_n$, we consider $p$ as a permutation. Without loss of generality, we assume that the strings of $\hat{p}$ move vertically and the generator $R$’s of $\hat{p}$ are all labelled on the left side of the crossings. Note that $R {\tilde{y}}_{1^2}=-{\tilde{y}}_{1^2}$, so $$\hat{p} {\tilde{y}}_{1^n} = (-1)^{|p|} {\tilde{y}}_{1^n},$$ where $|p|$ is the number of the crossing $R$’s in $\hat{p}$. We express ${\tilde{y}}_{1^n}$, $y_{1^n}$ and $y_{1^i} \otimes y_{1^{n-i}}$ in terms of the basis $\mathcal{B}_n$ as $$\begin{aligned}
{\tilde{y}}_{1^n}&=\sum_{p\in P_n} c_p \hat{p} \;, \\
y_{1^n}&=\sum_{p\in P_n} c'_p \hat{p} \;.\end{aligned}$$ Then $$c =\sum_{p\in T_{i,n-i}} \frac{(-1)^{|p|}}{n!} c_p \;.$$ Recall that ${\tilde{y}}_{\lambda}=s_n y_{\lambda}$, so by Proposition \[Prop: IS\], $$c_p=c'_p, ~\forall~ p\in T_n.$$
When $q \to 1$, the Hecke algebra $H$ specializes to the symmetric group algebra; the generator $\alpha$ becomes the symmetric braiding; $\alpha-R \to 0$; and $n! y_{1^n} $ becomes the alternating sum of permutations of the symmetric groups $S_n$. So for any $p\in T_n$, $$\begin{aligned}
\lim_{q \to 1} c'_p = \frac{(-1)^{|p|}}{n!} .\end{aligned}$$ Then $$\begin{aligned}
\lim_{q \to 1} c =\lim_{q \to 1} \sum_{p\in T_{i,n-1}} \frac{(-1)^{|p|}}{n!} c_p = \sum_{p\in T_{i,n-i}} \frac{1}{n!}={n\choose i}^{-1}.\end{aligned}$$ Therefore, $C \neq 0$ in $\mathbb{C}(q)$.
Fusion Rules of Fundamental Representations for the Generic Case {#Sec: Fusion Rules of Fundamental Representations for the Generic Case}
================================================================
In this section, we compute the fusion rule for $X_{1^n} \otimes $ in the Grothendieck ring $\mathcal{G}$, and construct a basis for the hom space. We apply this to study the characters of the simple objects using symmetric functions. Recall that $\{X_\lambda\}$ indexed by Young diagrams are a basis of $\mathcal{G}$. The [**structure constants**]{} of the multiplication $R_{\mu, \nu}^{\lambda}$ are defined by: $$X_\mu X_\nu = \sum_\lambda R_{\mu, \nu}^\lambda X_\lambda$$
We define the morphism $\cup_n$ in $\hom_{{\mathscr{C}}}(X^{2n},\emptyset)$ as $$\cup_n=
\raisebox{-.5cm}{
\begin{tikzpicture}
\draw[thick] (4,2) arc (-180:0:1);
\node at (3.8,1.8) {n};
\end{tikzpicture}}
=
\raisebox{-.5cm}{
\begin{tikzpicture}
\draw (4,2) arc (-180:0:.5 and .25);
\draw (3,2) arc (-180:0:1.5 and .75);
\node at (3.5,1.8) {$\cdots$};
\node at (3,2.2) {1};
\node at (4,2.2) {n};
\node at (5,2.2) {n+1};
\node at (6,2.2) {2n};
\end{tikzpicture}}$$ where the label $n$ in the first picture indicates the number of parallel strings.
By Proposition 9.2 in [@LiuYB], the dual object (or the $180^{\circ}$ rotation) of ${\tilde{y}}_{1^n}$ is ${\tilde{y}}_{n}$. In particular, $\cup_n({\tilde{y}}_{1^n} \otimes {\tilde{y}}_{n} )$ is a non-zero morphism in $\hom_{{\mathscr{C}}}({\tilde{y}}_{n} \otimes {\tilde{y}}_{1^n},\emptyset)$.
\[Prop: dual object\] The dual object of ${\tilde{y}}_{\lambda}$ is ${\tilde{y}}_{\lambda'}$, where $\lambda'$ is the reflection of $\lambda$ in the diagonal, called the Young diagram dual to $\lambda$.
This is a consequence of Proposition 9.6 in [@LiuYB]. We give a quick proof here. The duality map $\lambda \to \lambda'$ is a $\mathbb{Z}_2$ automorphism of the principal graph of the planar algebra ${\mathscr{C}}$, which is Young’s lattice. This $\mathbb{Z}_2$ fixes the the Young diagrams $\emptyset$ and $1$, and switches $1^2$ and $2$. Therefore it has to be the reflection in the diagonal.
For any Young diagram $\lambda$, we define the following sets of Young diagrams:
1. $\lambda-1^n$ are Young diagrams that removes $n$ cells from $\lambda$, and no two cells in the same row;
2. $\lambda+1^n$ are Young diagrams that adds $n$ cells to $\lambda$, and no two cells in the same row;
3. $\lambda-n$ are Young diagrams that removes $n$ cells from $\lambda$, and no two cells in the same column;
4. $\lambda+n$ are Young diagrams that adds $n$ cells to $\lambda$, and no two cells in the same column.
The following result is well-known for the type $A$ Hecke algebra. It can be derived from the fusion rule of fundamental representations of (quantum) $SU(N)$, as $N \to \infty$. The fusion rule can be characterized by Schur polynomials.
\[Lem: fusion 1\] Suppose $\lambda$ and $\mu$ are Young diagrams. If $n=|\mu|-|\lambda|\geq 0$, then $$\begin{aligned}
\dim \hom_{H}(y_{\lambda} \otimes y_{1^{n}}, y_{\mu})&=\left\{
\begin{aligned}
1, & ~\forall~ \mu \in \lambda+1^n; \\
0, & ~\forall~ \mu \notin \lambda+1^n.
\end{aligned}
\right.\end{aligned}$$
We give an explicit construction of a non-zero morphism $\rho$ in $\hom_{H}(y_{\lambda} \otimes y_{1^{n}}, y_{\mu})$.
\[Lem: dim 1\] Suppose $\lambda$ and $\mu$ are Young diagrams. If $n=|\mu|-|\lambda|\geq 0$, then $$\begin{aligned}
\dim \hom_{{\mathscr{C}}}({\tilde{y}}_{\lambda} \otimes {\tilde{y}}_{1^{n}}, {\tilde{y}}_{\mu})&=
\left\{
\begin{aligned}
1, & ~\forall~ \mu \in \lambda+1^n; \\
0, & ~\forall~ \mu \notin \lambda+1^n.
\end{aligned}
\right. \\
\dim \hom_{{\mathscr{C}}}({\tilde{y}}_{\lambda} \otimes {\tilde{y}}_{n} , {\tilde{y}}_{\mu})&=
\left\{
\begin{aligned}
1, & ~\forall~ \mu \in \lambda+n; \\
0, & ~\forall~ \mu \notin \lambda+n;
\end{aligned}
\right.\end{aligned}$$ If $n=|\lambda|-|\mu|\geq 0$, then $$\begin{aligned}
\dim \hom_{{\mathscr{C}}}({\tilde{y}}_{\lambda} \otimes {\tilde{y}}_{1^n} , {\tilde{y}}_{\mu})&=
\left\{
\begin{aligned}
1, & ~\forall~ \mu \in \lambda-n; \\
0, & ~\forall~ \mu \notin \lambda-n;
\end{aligned}
\right. \\
\dim \hom_{{\mathscr{C}}}({\tilde{y}}_{\lambda} \otimes {\tilde{y}}_{n} , {\tilde{y}}_{\mu})&=
\left\{
\begin{aligned}
1, & ~\forall~ \mu \in \lambda-1^n; \\
0, & ~\forall~ \mu \notin \lambda-1^n.
\end{aligned}
\right.\end{aligned}$$
If $n=|\mu|-|\lambda|\geq 0$, then by Equation , Proposition \[Prop: iso\] and Lemma , we have $$\begin{aligned}
\dim \hom_{{\mathscr{C}}}({\tilde{y}}_{\lambda} \otimes {\tilde{y}}_{1^{n}}, {\tilde{y}}_{\mu})&= \dim \hom_{H}(y_{\lambda} \otimes y_{1^{n}}, y_{\mu}) =
\left\{
\begin{aligned}
1, & ~\forall~ \mu \in \lambda+1^n; \\
0, & ~\forall~ \mu \notin \lambda+1^n.
\end{aligned}
\right.\end{aligned}$$
The planar algebra ${\mathscr{C}}$ has a $\mathbb{Z}_2$ automorphism $\Omega$ mapping the generator $R$ to $-R$. By Proposition 9.5 in [@LiuYB], the idempotent $\Omega({\tilde{y}}_{\lambda})$ is equivalent to ${\tilde{y}}_{\lambda^\prime}$. (The dual Young diagram $\lambda'$ is denoted by $\Omega(\lambda)$ in [@LiuYB].)
In particular, $n^\prime=1^n$. Note that $\mu \in \lambda+n$ iff $\mu^\prime \in \lambda^\prime+1^n$. So $$\begin{aligned}
\dim \hom_{{\mathscr{C}}}({\tilde{y}}_{\lambda} \otimes {\tilde{y}}_{n} , {\tilde{y}}_{\mu})&= \dim \hom_{{\mathscr{C}}}({\tilde{y}}_{\lambda^\prime} \otimes {\tilde{y}}_{1^n} , {\tilde{y}}_{\mu^\prime}) =
\left\{
\begin{aligned}
1, & ~\forall~ \mu \in \lambda+n; \\
0, & ~\forall~ \mu \notin \lambda+n.
\end{aligned}
\right.\end{aligned}$$
By Proposition 9.2 in [@LiuYB], the dual object (or $180^{\circ}$ rotation) of ${\tilde{y}}_{1^n}$ is ${\tilde{y}}_{n}$. If $n=|\lambda|-|\mu|\geq 0$, then by Frobenius reciprocity, $$\begin{aligned}
\dim \hom_{{\mathscr{C}}}({\tilde{y}}_{\lambda} \otimes {\tilde{y}}_{1^n} , {\tilde{y}}_{\mu})&= \dim \hom_{{\mathscr{C}}}({\tilde{y}}_{\lambda}, {\tilde{y}}_{\mu} \otimes {\tilde{y}}_{n} )=
\left\{
\begin{aligned}
1, & ~\forall~ \mu \in \lambda-n; \\
0, & ~\forall~ \mu \notin \lambda-n;
\end{aligned}
\right.\end{aligned}$$ $$\begin{aligned}
\dim \hom_{{\mathscr{C}}}({\tilde{y}}_{\lambda} \otimes {\tilde{y}}_{n} , {\tilde{y}}_{\mu})= \dim \hom_{{\mathscr{C}}}({\tilde{y}}_{\lambda}, {\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^n} )=
\left\{
\begin{aligned}
1, & ~\forall~ \mu \in \lambda-1^n; \\
0, & ~\forall~ \mu \notin \lambda-1^n.
\end{aligned}
\right.\end{aligned}$$
Suppose $a,b,c \in \mathbb{N}$, and $n=a+b+c$. Let $p_{a,b,c}\in P_n$ be the pairing $$\begin{aligned}
\label{Equ: p_{a,b,c}}
p_{a,b,c}(k)=\left\{
\begin{aligned}
&(2n+1-k), &&\forall~ 1\leq k \leq a \text{ or } 2n-a< k \leq 2n ; \\
&(2a+2b+1-k), &&\forall~ a < k \leq a+2b; \\
&(2n+2b+1-k), &&\forall~ n+b < k \leq 2n-a; \\
\end{aligned}
\right.\end{aligned}$$ We can identify $\hat{p}_{a,b,c} \in {\mathscr{C}}_{n}$ as a morphism in $\hom_{{\mathscr{C}}}(X^{a+b} \otimes X^{b+c}, X^{a+c})$, illustrated as $$\hat{p}_{a,b,c}=\raisebox{-1cm}{
\begin{tikzpicture}
\draw[thick] (0,0) --++(0,2);
\draw[thick] (1,2) arc (-180:0:.5);
\draw[thick] (1,0) --++(2,2);
\node at (-.2,1) {a};
\node at (1-.2,1.8) {b};
\node at (1.7,1) {c};
\end{tikzpicture}}
=
\raisebox{-1.5cm}{
\begin{tikzpicture}
\draw (1,0) --++(0,2);
\draw (2,0) --++(0,2);
\draw (3.25,0) --++(4,2);
\draw (4.5,0) --++(4,2);
\draw (4,2) arc (-180:0:.5 and .25);
\draw (3,2) arc (-180:0:1.5 and .75);
\node at (1.5,1) {$\cdots$};
\node at (6,1) {$\cdots$};
\node at (3.5,1.8) {$\cdots$};
\node at (1,2.2) {1};
\node at (2,2.2) {a};
\node at (3,2.2) {a+1};
\node at (4,2.2) {a+b};
\node at (5,2.2) {a+b+1};
\node at (6,2.2) {a+2b};
\node at (7.25,2.2) {a+2b+1};
\node at (8.5,2.2) {n+b};
\node at (1,-.2) {2n};
\node at (2,-.2) {2n+1-a};
\node at (3.25,-.2) {2n-a};
\node at (4.5,-.2) {n+b+1};
\end{tikzpicture}}$$ where $a,b,c$ in the first picture indicate the number of parallel strings.
Suppose $\mu$ is a Young diagram, $|\mu|=a+b$. Take Young diagrams $\nu \in \mu-b$ and $\lambda \in \nu +1^{c}$. By Lemma \[Lem: dim 1\], there are non-zero morphisms $\rho_{1,\nu} \in \hom_{{\mathscr{C}}}({\tilde{y}}_{\mu}, {\tilde{y}}_{\nu} \otimes {\tilde{y}}_{b} )$ and $\rho_{2,\nu} \in \hom_{{\mathscr{C}}}({\tilde{y}}_{\nu} \otimes {\tilde{y}}_{1^{c}}, {\tilde{y}}_{\lambda})$. We construct a morphism $\rho'_{\mu,\nu,\lambda} \in \hom_{{\mathscr{C}}}({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^b} \otimes {\tilde{y}}_{1^{c}}, {\tilde{y}}_{\lambda})$ as $$\begin{aligned}
\label{Equ: rho}
\rho'_{\mu,\nu,\lambda}&:=\rho_{2,\nu} \hat{p}_{a,b,c} (\rho_{1,\nu} \otimes {\tilde{y}}_{1^b} \otimes {\tilde{y}}_{1^{c}}) \;.\end{aligned}$$ We identify ${\tilde{y}}_{b+c}$ with a morphism in $\hom_{{\mathscr{C}}}({\tilde{y}}_{1^{b+c}}, {\tilde{y}}_{1^{b}} \otimes {\tilde{y}}_{1^{c}} )$ and construct a morphism $\rho'_{\mu,\nu,\lambda} \in \hom_{{\mathscr{C}}}({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^{b+c}}, {\tilde{y}}_{\lambda})$: $$\begin{aligned}
\label{Equ: rho'}
\rho_{\mu,\nu,\lambda}&:=\rho'_{\mu,\nu,\lambda}({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{b+c}) =\rho_{2,\nu} \hat{p}_{a,b,c} (\rho_{1,\nu} \otimes {\tilde{y}}_{b+c}) \;.\end{aligned}$$
Their pictorial representations are $$\begin{aligned}
\rho'_{\mu,\nu,\lambda}=
\raisebox{-2.5cm}{
\begin{tikzpicture}
\draw[thick] (0,0) --++(0,2);
\draw[thick] (1,2) arc (-180:0:.5);
\draw[thick] (1,0) --++(2,2);
\node at (-.2,1) {${\tilde{y}}_{\nu}$};
\node at (1-.2,1.8) {${\tilde{y}}_{b}$};
\node at (1.7,3.2) {${\tilde{y}}_{1^{b}}$};
\node at (2.7,3.2) {${\tilde{y}}_{1^{c}}$};
\node at (0,3.2) {${\tilde{y}}_{\mu}$};
\node at (0,2.5) {$\rho_{1,\nu}$};
\node at (0,-1.2) {${\tilde{y}}_{\lambda}$};
\node at (0,-.5) {$\rho_{2,\nu}$};
\draw[thick] (2,2) --++(0,1.5);
\draw[thick] (3,2) --++(0,1.5);
\draw[thick] (0,2) --(.5,2.5)--++(0,1);
\draw[thick] (1,2) --(.5,2.5);
\draw[thick] (0,0) --(.5,-.5)--++(0,-1);
\draw[thick] (1,0) --(.5,-.5);
\end{tikzpicture}
}
\quad \text{and} \quad
\rho_{\mu,\nu,\lambda}=
\raisebox{-2.5cm}{
\begin{tikzpicture}
\draw[thick] (0,0) --++(0,2);
\draw[thick] (1,2) arc (-180:0:.5);
\draw[thick] (1,0) --++(2,2);
\node at (-.2,1) {${\tilde{y}}_{\nu}$};
\node at (1-.2,1.8) {${\tilde{y}}_{b}$};
\node at (1.7,1.8) {${\tilde{y}}_{1^b}$};
\node at (1.7,1) {${\tilde{y}}_{1^c}$};
\node at (2,2.5) {${\tilde{y}}_{1^{b+c}}$};
\node at (0,3.2) {${\tilde{y}}_{\mu}$};
\node at (0,2.5) {$\rho_{1,\nu}$};
\node at (0,-1.2) {${\tilde{y}}_{\lambda}$};
\node at (0,-.5) {$\rho_{2,\nu}$};
\draw[thick] (2,2) --(2.5,2.5)--++(0,1);
\draw[thick] (3,2) --(2.5,2.5);
\draw[thick] (0,2) --(.5,2.5)--++(0,1);
\draw[thick] (1,2) --(.5,2.5);
\draw[thick] (0,0) --(.5,-.5)--++(0,-1);
\draw[thick] (1,0) --(.5,-.5);
\end{tikzpicture}
}\end{aligned}$$
\[Lem: Basis 0\] Suppose $a,b,c \in \mathbb{N}$ and $r=b+c$. For any Young diagrams $\mu$ and $\lambda$, $|\mu|=a+b$, $|\lambda|=a+c$, the elements $\{\rho'_{\mu,\nu, \lambda} : \nu \in \mu-b, \nu\in \lambda -1^{c} \}$ are linearly independent in $\hom_{{\mathscr{C}}}({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^b} \otimes {\tilde{y}}_{1^c}, {\tilde{y}}_{\lambda})$.
By Frobenius reciprocity, for any $\nu$, $({\tilde{y}}_{\nu} \otimes \cup_{b} ) (\rho_{\mu,\nu} \otimes {\tilde{y}}_{1^b})\neq 0$ in $\hom_{{\mathscr{C}}}({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^b}, {\tilde{y}}_{\nu} )$. As ${\mathscr{C}}$ is semi-simple, there is a morphism $\rho_{3,\nu} \in \hom_{{\mathscr{C}}}({\tilde{y}}_{\nu}, {\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^b})$, such that $$({\tilde{y}}_{\nu} \otimes \cup_{b} ) (\rho_{\mu,\nu} \otimes {\tilde{y}}_{1^b}) \rho_{3,\nu}={\tilde{y}}_{\nu}.$$ If $$\sum_{\nu \in \mu-b, \nu\in \lambda -1^{c}} c_{\nu} \rho'_{\mu,\nu, \lambda}=0, ~c_{\nu} \in \mathbb{C}(q),$$ then for any $\nu' \in \mu-b, \nu'\in \lambda -1^{c}$, $$\rho_{3,\nu'} \sum_{\nu \in \mu-b, \nu\in \lambda -1^{c}} \rho'_{\mu,\nu, \lambda}=c_{\nu'} \rho_{2,\nu'}=0.$$ So $c_{\nu'}=0$. Therefore, $\{\rho'_{\mu,\nu, \lambda} : \nu \in \mu-b, \nu\in \lambda -1^{c} \}$ are linearly independent.
\[Lem: Basis 1\] Suppose $a,b,c \in \mathbb{N}$ and $r=b+c$. For any Young diagrams $\mu$ and $\lambda$ with $|\mu|=a+b$, $|\lambda|=a+c$, the morphisms $\{\rho_{\mu,\nu, \lambda} : \nu \in \mu-b, \nu\in \lambda -1^{c} \}$ form a spanning set of $\hom_{{\mathscr{C}}}({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^r}, {\tilde{y}}_{\lambda})$.
For any $p_1 \in P_{a+b}$, $p_2 \in P_{b+c}$ and $p_3 \in P_{a+c}$. we define $$\label{Equ: xp}
x_{p_1,p_2,p_3}=\hat{p}_3 \hat{p}_{a,b,c} (\hat{p}_1 \otimes \hat{p}_2) .$$ By Proposition \[Prop: Bp Basis\], $\{x_{p_1,p_2,p_3} : p_1 \in P_{a+b}, p_2 \in P_{b+c}, p_3 \in P_{a+c}.\}$ is a spanning set of ${\mathscr{C}}_{n}$, because any pairing in $P_n$ can be implemented by some diagram $x_{p_1,p_2,p_3}$ with a minimal number of crossings. Note that $y_{1^{b+c}}$ is a central minimal idempotent in ${\mathscr{C}}_{b+c}$. By Equation , ${\tilde{y}}_{1^{b+c}}$ is a central minimal idempotent in $H_{b+c}$.
By Lemma \[Lem: dim 1\], $\dim \hom_{{\mathscr{C}}}({\tilde{y}}_{1^{b+c}}, {\tilde{y}}_{1^b}\otimes {\tilde{y}}_{1^c})=1$, so ${\tilde{y}}_{1^{b+c}} \in \hom_{{\mathscr{C}}}({\tilde{y}}_{1^{b+c}}, {\tilde{y}}_{1^b}\otimes {\tilde{y}}_{1^c})$ and $({\tilde{y}}_{1^b}\otimes {\tilde{y}}_{1^c}) {\tilde{y}}_{1^{b+c}}={\tilde{y}}_{1^{b+c}}$. We define $$\tilde{x}_{p_1,p_2,p_3}={\tilde{y}}_{\lambda} x_{p_1,p_2,p_3} ({\tilde{y}}_{\mu}\otimes {\tilde{y}}_{1^{b+c}}).$$ Then $\{\tilde{x}_{p_1,p_2,p_3} : p_1 \in P_{a+b}, p_2 \in P_{b+c}, p_3 \in P_{a+c}.\}$ is a spanning set of $\hom_{{\mathscr{C}}}({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^r}, {\tilde{y}}_{\lambda})$. Recall that the $180^{\circ}$ rotation of ${\tilde{y}}_{1^b}$ is ${\tilde{y}}_{b}$. So $$\tilde{x}_{p_1,p_2,p_3}=
\raisebox{-2.5cm}{
\begin{tikzpicture}
\draw[thick] (0,0) --++(0,2);
\draw[thick] (1,2) arc (-180:0:.5);
\draw[thick] (1,0) --++(2,2);
\node at (-.2,1) {$s_a$};
\node at (1-.2,1.8) {${\tilde{y}}_{b}$};
\node at (1.7,1.8) {${\tilde{y}}_{1^b}$};
\node at (1.7,1) {${\tilde{y}}_{1^c}$};
\node at (2,3.2) {${\tilde{y}}_{1^{b+c}}$};
\node at (2,2.5) {$\hat{p}_2$};
\node at (0,3.2) {${\tilde{y}}_{\mu}$};
\node at (0,2.5) {$\hat{p}_1$};
\node at (0,-1.2) {${\tilde{y}}_{\lambda}$};
\node at (0,-.5) {$\hat{p}_3$};
\draw[thick] (2,2) --(2.5,2.5)--++(0,1);
\draw[thick] (3,2) --(2.5,2.5);
\draw[thick] (0,2) --(.5,2.5)--++(0,1);
\draw[thick] (1,2) --(.5,2.5);
\draw[thick] (0,0) --(.5,-.5)--++(0,-1);
\draw[thick] (1,0) --(.5,-.5);
\end{tikzpicture}
}
=\sum_{|\mu|=a} \sum_{j}
\raisebox{-2.5cm}{
\begin{tikzpicture}
\draw[thick] (0,0) --++(0,2);
\draw[thick] (1,2) arc (-180:0:.5);
\draw[thick] (1,0) --++(2,2);
\node at (-.2,1) {${\tilde{y}}_{\nu}$};
\node at (1-.2,1.8) {${\tilde{y}}_{b}$};
\node at (1.7,1.8) {${\tilde{y}}_{1^b}$};
\node at (1.7,1) {${\tilde{y}}_{1^c}$};
\node at (2,3.2) {${\tilde{y}}_{1^{b+c}}$};
\node at (2,2.5) {${\tilde{y}}_{\hat{p}_2}$};
\node at (0,3.2) {${\tilde{y}}_{\mu}$};
\node at (0,2.5) {$\rho_{1,j}$};
\node at (0,-1.2) {${\tilde{y}}_{\lambda}$};
\node at (0,-.5) {$\rho_{2,j}$};
\draw[thick] (2,2) --(2.5,2.5)--++(0,1);
\draw[thick] (3,2) --(2.5,2.5);
\draw[thick] (0,2) --(.5,2.5)--++(0,1);
\draw[thick] (1,2) --(.5,2.5);
\draw[thick] (0,0) --(.5,-.5)--++(0,-1);
\draw[thick] (1,0) --(.5,-.5);
\end{tikzpicture}
}
=\sum_{\nu \in \mu-b, \nu\in \lambda -1^{c} } \sum_{j} c_j \rho_{\mu,\nu,\lambda}
,$$ for some $\rho_{1,j} \in \hom_{{\tilde{y}}_{\mu}, {\tilde{y}}_{\nu}\otimes {\tilde{y}}_{b}}$, $\rho_{2,j} \in \hom_{{\tilde{y}}_{\nu} \otimes {\tilde{y}}_{1^c}, {\tilde{y}}_{\lambda} }$, and $c_j \in \mathbb{C}(q)$. Precisely, the label $a$ is replaced by $s_a$ in the first equality by Equation . Then $s_a$ is replaced by ${\tilde{y}}_{\nu}$ in the second equality by Equation . Then we obtain the third equality by Lemma \[Lem: dim 1\]. Therefore, $\{\rho_{\mu,\nu, \lambda} : \nu \in \mu-b, \nu\in \lambda -1^{c} \}$ is a spanning set of $\hom_{{\mathscr{C}}}({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^r}, {\tilde{y}}_{\lambda})$.
\[Lem: Basis 2\] Suppose $a,b,c \in \mathbb{N}$ and $r=b+c$. For any Young diagrams $\mu$ and $\lambda$ with $|\mu|=a+b$, $|\lambda|=a+c$, the morphisms $\{\rho_{\mu,\nu, \lambda} : \nu \in \mu-b, \nu\in \lambda -1^{c} \}$ are linearly independent over $\mathbb{C}(q)$.
Take $n=a+b+c$, and define
- $S_1=\{k \in \mathbb{N} : 1\leq k \leq a+b \}$,
- $S_2=\{k \in \mathbb{N} : a+b< k \leq a+2b+c \}$,
- $S_3=\{k \in \mathbb{N} : a+2b+c < k \leq 2n \} $,
- $S=\{p \in \ P_n: p \text{ has no pair in } S_i, ~i=1,2,3 \}$.
Note that for any pairing $p \in S$, $p$ has $a$ pairs between $S_1$ and $S_3$; $b$ pairs between $S_1$ and $S_2$; and $c$ pairs between $S_2$ and $S_3$. So we obtain a bijection $\iota: T_{a+b} \times T_{b+c} \times T_{a+c} \to S$ via $$\begin{aligned}
\iota(p_1,p_2,p_3)=p_{a,b,c} \circ (p_1\otimes p_2 \otimes p_3),\end{aligned}$$ where $p_{a,b,c}$ is defined in Equation , and $p_1\otimes p_2 \otimes p_3$ is a permutation on $2n$ points, $$\begin{aligned}
(p_1\otimes p_2 \otimes p_3)(k)
=\left\{
\begin{aligned}
&p_1(k), &&\forall~ 1\leq k \leq a+b, \\
&p_2(k-a-b)+a+b, &&\forall~ a+b < k \leq n+b, \\
&p_3(k-n-b)+n+b, &&\forall~ n+b < k \leq 2n. \\
\end{aligned}
\right.\end{aligned}$$ For any $p \in S$, we can choose $\hat{p} \in \mathcal{B}_n$ as $$\hat{p}=x_{\iota^{-1}(p)},$$ where $x_{p_1,p_2,p_3}$ is defined in Equation .
Assume that $$\sum_{\nu} c_{\mu,\nu,\lambda} \rho_{\mu,\nu, \lambda} =0$$ for some $c_{\mu,\nu,\lambda} \in \mathbb{C}(q)$.
Recall that $\rho' \in \hom_{{\mathscr{C}}}({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^{b}} \otimes {\tilde{y}}_{1^c} , {\tilde{y}}_{\lambda})$ and $\rho \in \hom_{{\mathscr{C}}}({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^{b+c}}, {\tilde{y}}_{\lambda})$ are defined in Equations and . We identify the two hom spaces with subspaces of ${\mathscr{C}}_{n}$. By Proposition \[Prop: Bp Basis\], $$\begin{aligned}
\rho_{\mu,\nu, \lambda} &=\sum_{p \in P_n} b_{\mu,\nu,\lambda}(p) \hat{p} \;; \\
\rho'_{\mu,\nu, \lambda} ({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^{b+c},b}) &=\sum_{p \in P_n} b'_{\mu,\nu,\lambda}(p) \hat{p} \;,\end{aligned}$$ for some $b_{\mu,\nu,\lambda}(p), b'_{\mu,\nu,\lambda}(p) \in \mathbb{C}(q)$, and $$\sum_{\nu} c_{\mu,\nu,\lambda} b_{\mu,\nu,\lambda}(p) =0, ~\forall p\in P_n \;.$$
Take $S_0=\iota(T_{a+b} \times T_{b,c} \times T_{a+c})$. Note that $$\begin{aligned}
\rho_{\mu,\nu,\lambda}=
\raisebox{-2.5cm}{
\begin{tikzpicture}
\draw[thick] (0,0) --++(0,2);
\draw[thick] (1,2) arc (-180:0:.5);
\draw[thick] (1,0) --++(2,2);
\node at (-.2,1) {${\tilde{y}}_{\nu}$};
\node at (1-.2,1.8) {${\tilde{y}}_{b}$};
\node at (1.7,1.8) {${\tilde{y}}_{1^b}$};
\node at (1.7,1) {${\tilde{y}}_{1^c}$};
\node at (2,2.5) {${\tilde{y}}_{1^{b+c}}$};
\node at (0,3.2) {${\tilde{y}}_{\mu}$};
\node at (0,2.5) {$\rho_{1,\nu}$};
\node at (0,-1.2) {${\tilde{y}}_{\lambda}$};
\node at (0,-.5) {$\rho_{2,\nu}$};
\draw[thick] (2,2) --(2.5,2.5)--++(0,1);
\draw[thick] (3,2) --(2.5,2.5);
\draw[thick] (0,2) --(.5,2.5)--++(0,1);
\draw[thick] (1,2) --(.5,2.5);
\draw[thick] (0,0) --(.5,-.5)--++(0,-1);
\draw[thick] (1,0) --(.5,-.5);
\end{tikzpicture}
}
=
\raisebox{-2.5cm}{
\begin{tikzpicture}
\draw[thick] (0,0) --++(0,2);
\draw[thick] (1,2) arc (-180:0:.5);
\draw[thick] (1,0) --++(2,2);
\node at (-.2,1) {$a$};
\node at (1-.2,1.8) {$b$};
\node at (1.7,1) {$c$};
\node at (2,2.5) {${\tilde{y}}_{1^{b+c}}$};
\node at (2,3.2) {$b+c$};
\node at (0,3.2) {$a+b$};
\node at (0,2.5) {$\rho_{1,\nu}$};
\node at (0,-1.2) {$a+c$};
\node at (0,-.5) {$\rho_{2,\nu}$};
\draw[thick] (2,2) --(2.5,2.5)--++(0,1);
\draw[thick] (3,2) --(2.5,2.5);
\draw[thick] (0,2) --(.5,2.5)--++(0,1);
\draw[thick] (1,2) --(.5,2.5);
\draw[thick] (0,0) --(.5,-.5)--++(0,-1);
\draw[thick] (1,0) --(.5,-.5);
\end{tikzpicture}
}\end{aligned}$$ On the other hand, ${\tilde{y}}_{1^{b+c},b}=\sum_{k}c_j \hat{p}_{1,j} \otimes \hat{p}_{2,j}$ for some $c_j \in \mathbb{C}(q)$, $p_{1,j} \in T_{b}$ and $p_{2,j} \in T_{c}$ as defined in Equation . So $$\begin{aligned}
\rho'_{\mu,\nu, \lambda} ({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^{b+c},b})
=\sum_{j}
\raisebox{-2.5cm}{
\begin{tikzpicture}
\draw[thick] (0,0) --++(0,2);
\draw[thick] (1,2) arc (-180:0:.5);
\draw[thick] (1,0) --++(2,2);
\node at (-.2,1) {$a$};
\node at (1-.2,1.8) {$b$};
\node at (1.7,1) {$c$};
\node at (2-.3,2.5) {$\hat{p}_{1,j}$};
\node at (3-.3,2.5) {$\hat{p}_{2,j}$};
\node at (1.8,3.2) {$b$};
\node at (2.8,3.2) {$c$};
\node at (0,3.2) {$a+b$};
\node at (0,2.5) {$\rho_{1,\nu}$};
\node at (0,-1.2) {$a+c$};
\node at (0,-.5) {$\rho_{2,\nu}$};
\draw[thick] (2,2) --++(0,1.5);
\draw[thick] (3,2) --++(0,1.5);
\draw[thick] (0,2) --(.5,2.5)--++(0,1);
\draw[thick] (1,2) --(.5,2.5);
\draw[thick] (0,0) --(.5,-.5)--++(0,-1);
\draw[thick] (1,0) --(.5,-.5);
\end{tikzpicture}
}\end{aligned}$$ Note that for any $p\in S_0$, if we express ${\tilde{y}}_{1^{b+c}}$ in terms of the basis $\mathcal{B}_{b+c}$, then only the components in $T_{b}\times T_{c}$ contribute non-zero coefficients of $p$. Recall that ${\tilde{y}}_{1^{b+c},b}$ is the sum of such components of ${\tilde{y}}_{1^{b+c}}$ in $T_{b}\times T_{c}$, so $$\begin{aligned}
b'_{\mu,\nu,\lambda}(p)&= b_{\mu,\nu,\lambda}(p) \;, &&~\forall~ p\in S_0 \;, \\
b'_{\mu,\nu,\lambda}(p)&=0 \;, &&~\forall~ p \in S\setminus S_0 \;.\end{aligned}$$ Then $$\sum_{p\in S_0} \sum_{\nu} c_{\mu,\nu,\lambda} b'_{\mu,\nu,\lambda}(p) \hat{p} =\sum_{p\in S_0} \sum_{\nu} c_{\mu,\nu,\lambda} b_{\mu,\nu,\lambda}(p) \hat{p} =0 .$$ Note that if $b'_{\mu,\nu,\lambda}(p) \neq 0$, $p$ has no pair in $S_1$ or $S_2$, then $p$ has no pair between the first $b$ points and the last $c$ points in $S_3$. So $$b'_{\mu,\nu,\lambda}(p) {\tilde{y}}_{\lambda} \hat{p} ({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^b} \otimes {\tilde{y}}_{1^c}) \neq 0, \text{ only when } p \in S_0.$$ By Lemma \[Lem: non-zero\], ${\tilde{y}}_{1^{b+c},b}({\tilde{y}}_{1^b} \otimes {\tilde{y}}_{1^c})= c_0 {\tilde{y}}_{1^b} \otimes {\tilde{y}}_{1^c}$ for some $c_0 \neq 0$ in $\mathbb{C}(q)$. So $$\begin{aligned}
&c_0 \sum_{\nu} c_{\mu,\nu,\lambda} \rho'(\tilde{\mu,\nu,\lambda}) \\
=&c_0 \sum_{\nu} c_{\mu,\nu,\lambda} {\tilde{y}}_{\lambda} \rho'(\tilde{\mu,\nu,\lambda}) ({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^b} \otimes {\tilde{y}}_{1^c}) \\
=&\sum_{\nu} c_{\mu,\nu,\lambda} {\tilde{y}}_{\lambda} \rho'(\tilde{\mu,\nu,\lambda}) ({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^{b+c},b}) ({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^b} \otimes {\tilde{y}}_{1^c}) \\
=&\sum_{\nu} c_{\mu,\nu,\lambda} {\tilde{y}}_{\lambda} \left( \sum_{p\in P_n} b'_{\mu,\nu,\lambda}(p) \hat{p} \right) ({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^b} \otimes {\tilde{y}}_{1^c}) \\
=& \sum_{\nu} c_{\mu,\nu,\lambda} \sum_{p\in S_0} b'_{\mu,\nu,\lambda}(p) {\tilde{y}}_{\lambda} \hat{p} ({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^b} \otimes {\tilde{y}}_{1^c}) \\
=& {\tilde{y}}_{\lambda} \left( \sum_{p\in S_0} \sum_{\nu} c_{\mu,\nu,\lambda} b'_{\mu,\nu,\lambda}(p) \hat{p} \right) ({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^b} \otimes {\tilde{y}}_{1^c}) \\
=&0 \;.\end{aligned}$$ By Lemma \[Lem: Basis 0\], we have that $c_{\mu,\nu,\lambda} =0$ for all $\nu$, Therefore, the elements $\{\rho'_{\mu,\nu, \lambda} : \nu \in \mu-b, \nu\in \lambda -1^{c} \}$ are linear independent in ${\mathscr{C}}_n$. In particular, $\rho'_{\mu,\nu, \lambda} \neq 0$, whenever $\nu \in \mu-b, \nu\in \lambda -1^{c}$.
We consider $\hat{p}$ as a morphism in $\hom_{{\mathscr{C}}}(X^{n+b}, X^{a+c})$. Then $${\tilde{y}}_{\lambda} \left( \sum_{p\in S} \sum_{\nu} c_{\mu,\nu,\lambda} b_{\mu,\nu,\lambda}(p) \hat{p} \right) ({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^b} \otimes {\tilde{y}}_{1^c})=0.$$ Note that the set $S$ of pairings are the same as the pairings implemented by
$$=\sum_{\nu} c_{\mu,\nu,\lambda} c_{r,i} \rho_{\nu,\lambda} ({\tilde{y}}_{\nu} \otimes m_{k} \otimes {\tilde{y}}_{1^{n-k}}) (\rho_{\mu,\nu} \otimes {\tilde{y}}_{1^i} \otimes {\tilde{y}}_{1^{r-i}}) \; .$$ Note that $\rho_{\nu,\lambda} ({\tilde{y}}_{\nu} \otimes m_{k} \otimes {\tilde{y}}_{1^{n-k}}) (\rho_{\mu,\nu} \otimes {\tilde{y}}_{1^i} \otimes {\tilde{y}}_{1^{r-i}})\neq 0$ and they are linearly independent for different $\tau$. Therefore, $c_{\mu,\nu,\lambda} c_{r,i}=0$. Recall that $c_{r,i} \neq 0$, so $c_{\mu,\nu,\lambda} =0$. Therefore, the morphisms $\{\rho_{\mu,\nu, \lambda} : \nu \in \mu-b, \nu\in \lambda -1^{c} \}$ are linearly independent.
\[Thm: fusion with column\] Suppose $a,b,c \in \mathbb{N}$ and $r=b+c$. For any Young diagrams $\mu$ and $\lambda$, $|\mu|=a+b$, $|\lambda|=a+c$, the elements $\{\rho_{\mu,\nu, \lambda} : \nu \in \mu-b, \nu\in \lambda -1^{c} \}$ form a basis of $\hom_{{\mathscr{C}}}({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^n}, {\tilde{y}}_{\lambda})$. In particular, we obtain the fusion for $X_{1^r}$ in a closed form: $$X_{(1^r)} X_{\mu}=\sum_{i=0}^{r} \sum_{\nu \in \mu- i} \sum_{\lambda \in \nu+1^{r-i}}X_{\lambda}.$$
By Lemmas \[Lem: Basis 1\], \[Lem: Basis 2\], $\{\rho_{\mu,\nu, \lambda} : \nu \in \mu-b, \nu\in \lambda -1^{c} \}$ form a basis of $\hom_{{\mathscr{C}}}({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{1^n}, {\tilde{y}}_{\lambda})$.
We remove $i$ cells from $\mu$ (no two in the same column), and then we add $r-i$ cells (no two in the same row).
\[Cor: fusion with row\] Applying the automorphism $\Omega$, we obtain the fusion with $X_{r}$ in a closed form: $$X_{(r)} X_{\mu}=\sum_{i=0}^{r} \sum_{\nu \in \mu- 1^i} \sum_{\lambda \in \nu+(r-i)}X_{\lambda}.$$
We remove $i$ cells from $\lambda$ (no two in the same row), and then we add $n-i$ cells (no two in the same column).
The morphisms can be constructed explicitly following the construction in [@LiuYB]. They are essentially used in the proof of Theorem \[Thm: fusion with column\]. We are going to compute the characters and the generating functions in the next section using Theorem \[Thm: fusion with column\].
Characters, Generating Functions and Fusion Rules for the Generic Case {#Sec: Characters, Generating Functions and Fusion Rules for the Generic Case}
======================================================================
We begin by introducing the tools we will need from the theory of symmetric functions. All the material we use can be found in the first chapter of [@Macdonald].
Symmetric Functions
-------------------
Recall that the ring of symmetric functions, $\Lambda$, is defined in the following way.
Let $n$ be a natural number, and $R_n = \mathbb{Z}[x_1, x_2, \ldots, x_n]^{S_n}$ be the ring of symmetric polynomial in $n$ variables. We write $R_n^k$ for the degree $k$ component of $R_n$. For each $k$, we have maps $\rho_n: R_n^k \to R_{n-1}^k$ defined by setting $x_n = 0$; these form an inverse system, so we may take the inverse limit $\displaystyle \lim_\leftarrow R_n^k$. Then, as an abelian group, we define: $$\Lambda = \bigoplus_{k \geq 0} \lim_\leftarrow R_n^k.$$ The multiplication on $\Lambda$ is inherited from the multiplication $R_n^{k_1} \otimes R_n^{k_2} \to R_n^{k_1+k_2}$. We may complete $\Lambda$ with respect to the grading. In this case we obtain $$\hat{\Lambda} = \prod_{k \geq 0} \lim_\leftarrow R_n^k.$$
We introduce some important elements of the ring of symmetric functions.
We have the following facts about $\Lambda$:
1. The polynomials $\displaystyle \sum_{i_1 < i_2 < \cdots < i_r} x_{i_1}x_{i_2} \cdots x_{i_r} \in R_n^{r}$ define an element $e_r \in \displaystyle \lim_\leftarrow R_n^k$ called the $r$-th elementary symmetric function. These $e_r$ freely generate $\Lambda$ as a polynomial ring: $\Lambda = \mathbb{Z}[e_1, e_2, \ldots]$. We have the generating function $\displaystyle E(t) = \sum_r e_r t^r = \prod_i (1 + x_it)$.
2. Similarly, the polynomials $\displaystyle \sum_{i_1 \leq i_2 \leq \cdots \leq i_r} x_{i_1}x_{i_2} \cdots x_{i_r} \in R_n^{r}$ define an element $h_r \in \displaystyle \lim_\leftarrow R_n^k$ called the $r$-th complete symmetric function. These $h_r$ also freely generate $\Lambda$ as a polynomial ring: $\Lambda = \mathbb{Z}[h_1, h_2, \ldots]$. We have the generating function $\displaystyle H(t) = \sum_r h_r t^r = \prod_i (1 - x_it)^{-1}$.
3. The polynomials $\displaystyle \sum_i x_i^r \in R_n^{r}$ define an element $p_r \in \displaystyle \lim_\leftarrow R_n^k$ called the $r$-th power-sum symmetric function. They freely generate $\mathbb{Q} \otimes \Lambda$ as a polynomial ring over $\mathbb{Q}$ (but they do not generate $\Lambda$ over $\mathbb{Z}$). We have the generating function $\displaystyle P(t) = \sum_r p_{r+1} t^r = \sum_i \frac{x_i}{1 - x_it}$.
4. The generating functions $E(t)$ and $H(t)$ satisfy the relation $H(t)E(-t) = 1$, and this equation encodes how to express the elementary symmetric functions in terms of the complete symmetric functions and vice versa. Similarly, we have $H^\prime(t)/H(t) = P(t)$, and $E^\prime(t)/E(t) = P(-t)$. In particular, we have the equations $$\begin{aligned}
\sum_{r \geq 0} h_r t^r &=& \exp \left( \sum_{i \geq 1} \frac{p_i}{i}t^i \right),\\
\sum_{r \geq 0} e_r t^r &=& \exp \left( \sum_{i \geq 1} \frac{(-1)^{i-1}p_i}{i}t^i \right).\end{aligned}$$
5. Elements of $\Lambda \otimes \Lambda$ may be viewed as polynomials in two sets of variables, say $x_i$ and $y_j$, symmetric in each separately. To indicate which variable set is being considered, we write $f(x)$ or $f(y)$. Given $f \in \Lambda$, we write $f(x, y)$ for the element of $\Lambda \otimes \Lambda$ defined by the symmetric function $f$ in the variable set $\{x_i\} \cup \{y_j\}$. (This operation defines a comultiplication $\Lambda \to \Lambda \otimes \Lambda$.)
6. Fix a Young diagram $\lambda = (\lambda_1, \lambda_2, \ldots)$, adding trailing zeros if needed, so that $\lambda$ has $n$ parts (usually we do not distinguish between Young diagrams that differ by trailing zeros). The polynomials $\det(x_i^{\lambda_j+n-j}) / \det(x_i^{n-j}) \in R_{n}^{|\lambda|}$ define an element $s_\lambda \in \displaystyle \lim_\leftarrow R_n^{|\lambda|}$, called the Schur function associated to $\lambda$.
7. We have that $s_{1^r} = e_r$ and $s_{r} = h_r$.
We have: $$\frac{p_1^2 + p_2}{2} = \frac{1}{2}\left(\sum_{i \neq j} x_i x_j + 2 \sum_i x_i^2\right) = \sum_{i \leq j}x_ix_j = h_2.$$
Schur functions may be viewed as the characters of irreducible representations of $GL_n(\mathbb{C})$ in the following sense. If $M \in GL_n(\mathbb{C})$ has eigenvalues $x_i$, then the trace of the action of $M$ on the irreducible representation of $GL_n(\mathbb{C})$ corresponding to the Young diagram $\lambda$ is $s_\lambda(x_i)$: the Schur function corresponding to $\lambda$ evaluated at the eigenvalues $x_i$. Note that this quantity is zero unless the Young diagram $\lambda$ has at most $n$ nonzero parts. This means we have a homomorphism $\Lambda \to R_n$ whose kernel has basis $s_\mu$ for Young diagrams $\mu$ with more than $n$ nonzero parts.
We now discuss a bilinear form on $\Lambda$.
The ring $\Lambda$ satisfies the following properties:
1. The Schur functions form a $\mathbb{Z}$-basis of $\Lambda$: $\Lambda = \mathbb{Z}\{s_\lambda \mid \mbox{$\lambda$ a Young diagram}\}$. In particular, there is a bilinear form $\langle -, - \rangle : \Lambda \otimes \Lambda \to \mathbb{Z}$ for which the Schur functions are orthonormal.
2. The adjoint to multiplication by $p_i$ is $\displaystyle i \frac{\partial}{\partial p_i}$ (where elements of $\Lambda$ are viewed as polynomials in the $p_i$ with possibly rational coefficients).
3. The adjoint to multiplication by $s_\mu$ (with respect to $\langle -,- \rangle$) is denoted $s_\mu^\perp$. The symmetric function $s_\mu^\perp(s_\lambda)$ is called a skew-Schur function, and denoted $s_{\lambda / \mu}$. It is nonzero if and only if $\mu_i \leq \lambda_i$ for all $i$.
4. Schur functions satisfy the following multiplication rule $\displaystyle s_\mu s_\nu = \sum_{\mu, \nu} c_{\mu, \nu}^\lambda s_\lambda$, where $c_{\mu, \nu}^\lambda$ are the Littlewood-Richardson coefficients (which are zero unless $|\mu| + |\nu| = |\lambda|$). They also satisfy $\displaystyle s_\lambda(x, y) = \sum_{\mu, \nu} c_{\mu, \nu}^\lambda s_\mu(x)s_\nu(y)$.
5. The identity $e_r(x, y) = \sum_{i=0}^r e_i(x)e_{r-i}(y)$ shows that $c_{\mu, \nu}^{1^r}$ is zero unless $\mu = 1^i$ and $\nu = 1^{r-i}$ for some $0 \leq i \leq r$, in which case it is equal to 1.
6. The Littlewood-Richardson coefficient $c_{\mu, r}^\lambda$ is zero unless the diagram of $\lambda$ can be obtained by adding $r$ cells to the diagram of $\mu$, with no two cells in the same column; this is the Pieri rule. Similarly, $c_{\mu, 1^r}^\lambda$ is zero unless the diagram of $\lambda$ can be obtained by adding $r$ cells to the diagram of $\mu$, with no two cells in the same row; this is the dual Pieri rule.
There are two identities that will be important to us, which we now state.
We have the following equations:
1. The following equality of series holds in a completion of $R_n \otimes R_n$ for each $n$, and therefore in a completion of $\Lambda$ (note that the homogeneous components of the right-hand side define elements of the inverse limits used to define the ring of symmetric functions): $$\sum_{\lambda} s_\lambda(x) s_\lambda(y) = \prod_{i,j} \frac{1}{1-x_iy_j}.$$ This is called the Cauchy Identity.
2. Similarly, we have the Dual Cauchy Identity: $$\sum_{\lambda} s_\lambda(x) s_{\lambda^\prime}(y) = \prod_{i,j} (1+x_iy_j).$$ Here, $\lambda^\prime$ is the Young diagram dual to $\lambda$.
There is another operation on symmetric functions called plethysm.
Given symmetric functions $f$ and $g$ which are sums of monomials in the variables $x_i$ with coefficients in $\mathbb{Z}_{\geq 0}$, the plethysm of $g$ with $f$ is a symmetric function denoted $g[f]$. It may be calculated in the following way. Express $f(x_1, x_2, \ldots )$ as a sum of monomials (repeated according to their multiplicity) $f = \sum_i x_1^{\alpha_1^{(i)}}x_2^{\alpha_2^{(i)}} \cdots $. Then $g[f]$ is the symmetric function obtained by evaluating $g$ on the variable set given by the monomials $x_1^{\alpha_1^{(i)}}x_2^{\alpha_2^{(i)}} \cdots $. It immediately follows that the map $\Lambda \to \Lambda$ defined by $g \mapsto g[f]$ is an algebra homomorphism (but this is not true for $f \mapsto g[f]$).
There is a way of generalising the above definition to $f$ and $g$ for which are not necessarily a positive (or even integral, if one is prepared to base change $\Lambda$) sum of monomials. The most general definition is the one given in Chapter 1, Section 8 of [@Macdonald].
Let $f = \sum_{\mu} m_\mu s_{\mu}$ be the character of a representation $V$ of $GL_n(\mathbb{C})$ (where $n$ is taken to be sufficiently large), so $m_\mu \in \mathbb{Z}_{\geq 0}$, and all but finitely many $m_\mu$ are zero. Thus, $f$ encodes a homomorphism $\varphi_f: GL_n(\mathbb{C}) \to GL(V)$. Similarly, fix $\displaystyle g = \sum_{\nu} n_\nu s_\nu$ (with the same conditions on $n_\nu$ as on $m_\mu$), which uniquely defines a representation $W$ of $GL(V) = GL_{\dim(V)}(\mathbb{C})$, encoding a homomorphism $\varphi_g : GL(V) \to GL(W)$. Then, $W$ is a representation of $GL_n(\mathbb{C})$ via the composition $\varphi_g \circ \varphi_f$: $$GL_n(\mathbb{C}) \xrightarrow{\varphi_f} GL(V) \xrightarrow{\varphi_g} GL(W).$$ The character of this representation is the plethysm $g[f]$. The value of $n$ used in this construction does not affect $g[f]$, provided it is large enough (e.g. $n=\deg(f) \deg(g)$ will suffice).
We show that $\displaystyle e_1 = \sum_{i} x_i$ is a two-sided identity for plethysm. Note that by definition, $e_1[f]$ recovers the sum of the monomials of $f$, namely $f$ itself. On the other hand, $f[e_1]$ is the evaluation of $f$ on the variable set $\{x_i\}$ (the monomials of $e_1$), which again is $f$ itself. This is consistent with the formulation in terms of representations of general linear groups, where $\varphi_{e_1}$ represents the identity map $GL_n(\mathbb{C}) \to GL_n(\mathbb{C})$ (for any $n$).
\[power\_sum\_plethysm\_remark\] For power-sum symmetric functions $p_r$, plethysm has some useful properties. In particular, $p_r[f] = f[p_r]$ for arbitrary $f$, because both sides are equal to the symmetric function obtained by multiplying the exponents of all monomials of $f$ by $r$. As a special case, we obtain $p_{r_1}[p_{r_2}] = p_{r_2}[p_{r_1}] = p_{r_1r_2}$.
Ultimately, the result we need about plethysm is the following.
\[plethysm\_theorem\] We have the following equation: $$h_r[h_2] = \sum_{|\lambda| = r} s_{2\lambda},$$ where, if $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_k)$, then $2\lambda = (2\lambda_1, 2\lambda_2, \ldots, 2\lambda_k)$.
This can be found in Chapter 1, Section 8, Example 6 of [@Macdonald]. Alternatively, see Example $A2.9$ of [@Stanley].
We now introduce a linear operator which will play an important role in what follows, and prove some properties that it satisfies.
Let $L$ be the linear operator on the completion of $\Lambda$ defined by multiplication by $\displaystyle \prod_{i \leq j}\frac{1}{1 + x_ix_j}$.
\[adjoint\_formula\_prop\] The adjoint of $L$ with respect to $\langle -, - \rangle$ is: $$L^\dagger = \sum_{\mu}(-1)^{|\mu|} s_{2\mu}^\perp.$$
We recognise the product defining $L$ as the generating function of complete symmetric functions evaluated at $-1$, with variable set $\{x_{i}x_{j} \}_{i \leq j}$ (these are the monomials in $h_2$); the degree $2r$ component of this sum is precisely what is computed in Theorem \[plethysm\_theorem\]. Thus, $$\prod_{i \leq j}\frac{1}{1 + x_ix_j} = H(-1)[h_2] = \sum_{r \geq 0}(-1)^r h_r[h_2] = \sum_{r \geq 0} (-1)^r \sum_{|\mu| = r} s_{2 \mu} = \sum_{\mu}(-1)^{|\mu|} s_{2\mu}.$$ Noting that the adjoint of multiplication by $s_{2\mu}$ is $s_{2\mu}^\perp$, the proposition follows.
\[Not: Branching Coefficient\] Let $\phi_{2}: GL_{n} \to GL_{n(n+1)/2}$ be the symmetric square representation of $GL_n$ and $\phi_{1^r}$ be the $r$-th antisymmetric power representation of $GL_{n(n+1)/2}$. Then $\phi_{1^r}\phi_{2}$ is a representation of $GL_n$. The multiplicity of the irreducible representation of $GL_n$ with highest weight $\lambda$ in $\phi_{1^r}\phi_{2}$ is denoted by $b_{n,r,\lambda}$. We define $\displaystyle b_{r,\lambda}= \lim_{n \to \infty} b_{n,r,\lambda}$. Then $$\begin{aligned}
e_r[h_2]&=\sum_{\lambda} b_{r,\lambda} s_\lambda ;\\
\label{Equ: L-1}
L^{-1}&=\prod_{i \leq j} (1 + x_ix_j)=\sum_{r\geq 0} e_r[h_2] =\sum_{r\geq 0,\lambda} b_{r,\lambda} s_\lambda.\end{aligned}$$
\[power\_sum\_L\_lemma\] Let $\theta_i = \frac{1 + (-1)^i}{2}$, so that $\theta_i$ is equal to $0$ when $i$ is odd, and equal to $1$ when $i$ is even. When expressed in terms of power-sum symmetric functions, $L$ has the following form: $$L = \exp \left( \sum_i \frac{(-1)^i p_i^2 + 2(-1)^{i/2} \theta_i p_{i}}{2i} \right).$$
We write $L = H(-1)[h_2]$ (as in the proof of Proposition \[adjoint\_formula\_prop\]), where we express $H(-1)$ and $h_2$ in terms of power-sum symmetric functions. We use Remark \[power\_sum\_plethysm\_remark\] to manipulate the plethysm: $$\begin{aligned}
L &=& \exp \left( \sum_i \frac{(-1)^ip_i}{i} \right)[\frac{p_1^2 + p_2}{2}] \\
&=& \exp \left( \sum_i \frac{(-1)^ip_i[\frac{p_1^2 + p_2}{2}] }{i} \right) \\
&=& \exp \left( \sum_i \frac{(-1)^i\frac{p_1^2 + p_2}{2}[p_i] }{i} \right) \\
&=& \exp \left( \sum_i \frac{(-1)^i\frac{p_i^2 + p_{2i}}{2}}{i} \right) \\
&=& \exp \left( \sum_i \frac{(-1)^i(p_i^2 + p_{2i})}{2i} \right).\end{aligned}$$ We rearrange the sum so that all instances of $p_i$ occur in the $i$-th summand. This means moving the term $(-1)^i p_{2i}/2i$ from the $i$-th summand to the $2i$-th summand. Upon noting that only even index summands obtain a contribution in this way, we obtain the stated formula.
\[comultiplication\_L\_prop\] Consider $\Lambda \otimes \Lambda$ as the set of symmetric functions in two sets of variables $\{x_i^{(1)}\}$ and $\{x_i^{(2)}\}$. Suppose that the symmetric function $f$ satisfies $f(x^{(1)}, x^{(2)}) = \sum_i g_i(x^{(1)})h_i(x^{(2)})$. Then, we have the following equation: $$\left( \sum_\lambda s_\lambda(x^{(1)})s_{\lambda^\prime}(x^{(2)})\right) L(f)(x^{(1)}, x^{(2)}) = L(g_i)(x^{(1)}) L(h_i)(x^{(2)}).$$
In the definition of $L$ (considered to have variable set $\{x_i^{(1)}\} \cup \{ x_i^{(2)}\}$), products of pairs of variables take one of three forms: either both variables come from $\{x_i^{(1)}\}$, or both variables come from $\{x_i^{(2)}\}$, or one variable comes from each. Giving $L$ a subscript to show its variable set, we obtain: $$L_{\{x_i^{(1)}\} \cup \{ x_i^{(2)}\}} = \prod_{i_1 \leq i_2} \frac{1}{1 + x_{i_1}^{(1)}x_{i_2}^{(1)}} \prod_{i_1 \leq i_2} \frac{1}{1 + x_{i_1}^{(2)}x_{i_2}^{(2)}} \prod_{i_1, i_2} \frac{1}{1 + x_{i_1}^{(1)}x_{i_2}^{(2)}}.$$ Moving the last factor to the left-hand side, and using the Dual Cauchy Identity, $$\left( \sum_\lambda s_\lambda(x^{(1)})s_{\lambda^\prime}(x^{(2)})\right) L_{\{x_i^{(1)}\} \cup \{ x_i^{(2)}\}} = L_{\{x_i^{(1)}\}}L_{\{x_i^{(2)}\}}.$$ This is equivalent to the statement of the proposition.
Characters, Generating Functions and Fusion Rules
-------------------------------------------------
In this section, we recall some properties of the Grothendieck ring $\mathcal{G}$, and then study its structure using symmetric functions. Recall that $\mathcal{G}$ has basis $\{Y_\lambda\}$ indexed by Young diagrams.
By Schur-Weyl duality, we obtain a ring isomorphism $\Phi: \mathcal{G} \to \Lambda$, such that $\Phi(Y_{\lambda})=s_{\lambda}$. Moreover, $$Y_{\mu}Y_{\nu}=\sum_{\lambda} c_{\mu, \nu}^{\lambda}Y_{\lambda},$$ where $c_{\mu, \nu}^{\lambda}$ are the Littlewood-Richardson coefficients.
Recall that $s_{\lambda/2\mu}$ is a skew-Schur function, and $2\mu$ is the Young diagram obtained by doubling each part of $\mu$.
\[Def: Iso\] By Theorem \[Thm: ring iso X\], we define a ring isomorphism $\Phi: \mathcal{G} \to \Lambda$, such that $$\Phi(Y_{\lambda})=s_{\lambda}, ~\forall~ \lambda.$$ In particular, $$\Phi(X_{1^r})=\Phi(Y_{1^r})=s_{1^r}, ~\forall~ r\geq 0.$$
\[Thm: Main1\] For any Young diagram $\lambda$, we call $\Phi(X_\lambda)$ the character of $X_{\lambda}$ in $\mathcal{G}$. Then $$\begin{aligned}
\Phi(X_\lambda) &= L^\dagger s_\lambda = \sum_{\mu} (-1)^{|\mu|} s_{\lambda/2\mu} \;; \\
\label{Equ: X=Y+2}
X_{\lambda}&=\sum_{\substack{\mu,\nu \\ 2|\mu| +|\nu|= |\lambda|}} (-1)^{|\mu|} c_{2\mu,\nu}^{\lambda} Y_{\nu} \;.\end{aligned}$$
To prove the first statement, it suffices (by induction on $\lambda$) to show that the claimed expressions for the $\Phi(X_\lambda)$ multiply according to the rule defined by Theorem \[Thm: fusion with column\]. When we encode the operations of removing and adding cells via the Pieri rule and dual Pieri rule, what we must prove becomes $$e_r L^\dagger (s_\lambda) = \sum_{i=0}^r L^\dagger (e_{r-i} h_i^\perp s_\lambda).$$ This is precisely the assertion of the following equality of operators: $e_r L^\dagger = \sum_{i=0}^r L^\dagger e_{r-i} h_i^\perp$. We prove the adjoint of this equality, namely $L e_r^\perp = \sum_{i=0}^r h_i e_{r-i}^\perp L$. To prove this statement for all $r$ simultaneously, we multiply by $t^r$ and sum over $r \geq 0$; it is equivalent to prove the following identity of (operator-valued) generating functions: $$L E(t)^\perp = H(t)E(t)^\perp L.$$ We rewrite all quantities in terms of power-sum symmetric functions. We have: $$\begin{aligned}
E(t)^\perp &=& \exp \left( \sum_i \frac{(-1)^{i-1}p_i^\perp}{i} t^i \right) = \exp \left( \sum_i (-1)^{i-1}\frac{\partial}{\partial p_i} t^i \right), \\
H(t) &=& \exp \left( \sum_i \frac{p_i}{i} t^i \right), \\
L &=& \exp \left( \sum_i \frac{(-1)^i p_i^2 + 2(-1)^{i/2} \theta_i p_{i}}{2i} \right).\end{aligned}$$ (Recall from Lemma \[power\_sum\_L\_lemma\] that $\theta_i$ is equal to $0$ if $i$ is odd, and equal to $1$ if $i$ is even.) We use an operator-theoretic version of Taylor’s theorem, namely $$\exp \left(a \frac{\partial}{\partial x} \right) f(x) = f(x+a).$$ Applying this termwise to the composition of operators $E(t)^\perp L$, we obtain: $$\begin{aligned}
E(t)^\perp L &=& \exp \left( \sum_i (-1)^{i-1}\frac{\partial}{\partial p_i} t^i \right)\exp \left( \sum_i \frac{(-1)^i p_i^2 + 2(-1)^{i/2} \theta_i p_{i}}{2i} \right) \\
&=& \exp \left( \sum_i \frac{(-1)^i (p_i + (-1)^{i-1}t^i)^2 + 2(-1)^{i/2} \theta_i (p_{i}+ (-1)^{i-1}t^i)}{2i} \right)\exp \left( \sum_i (-1)^{i-1}\frac{\partial}{\partial p_i} t^i \right) \\
&=& \exp \left( \sum_i \frac{(-1)^i p_i^2 -2t^ip_i + (-1)^i t^{2i} + 2(-1)^{i/2} \theta_i p_{i}+ 2(-1)^{i/2} \theta_i (-1)^{i-1}t^i}{2i} \right)E(t)^\perp \\
&=& \exp\left(-\sum_i \frac{p_i t^i}{i}\right) \exp \left( \sum_i \frac{(-1)^i p_i^2 + 2(-1)^{i/2} \theta_i p_{i}}{2i} \right) \\
& & \times \exp \left( \sum_i \frac{(-1)^i t^{2i}+ 2(-1)^{i/2} \theta_i (-1)^{i-1}t^i}{2i} \right) E(t)^\perp .\end{aligned}$$ We recognise the first term as $H(t)^{-1}$, the second term as $L$, and the third term as $1$ (noting that all powers of $t$ cancel out). Thus we have: $$H(t)^{-1}LE(t)^\perp = E(t)^\perp L,$$ which is equivalent to the statement $$\Phi(X_\lambda) = L^\dagger s_\lambda = \sum_{\mu} (-1)^{|\mu|} s_{\lambda/2\mu}.$$ Furthermore, $$\Phi(X_{\lambda})=\sum_{\mu} (-1)^{|\mu|} s_{\lambda/2\mu}=\sum_{\substack{\mu,\nu \\ 2|\mu| +|\nu|= |\lambda|}} (-1)^{|\mu|} c_{2\mu,\nu}^{\lambda} s_{\nu}=\sum_{\substack{\mu,\nu \\ 2|\mu| +|\nu|= |\lambda|}} (-1)^{|\mu|} c_{2\mu,\nu}^{\lambda} \Phi(Y_{\nu}).$$ Recall that $\Phi$ is an isomorphism, so $$X_{\lambda}=\sum_{\substack{\mu,\nu \\ 2|\mu| +|\nu|= |\lambda|}} (-1)^{|\mu|} c_{2\mu,\nu}^{\lambda} Y_{\nu},$$
\[Thm: Y=X+2\] For a Young diagram $\lambda$, let us define $\lambda_{<}$ to be set of proper sub Young diagrams $\mu$, such that $|\lambda|-|\mu| \in 2\mathbb{N}^+$. Then $$\begin{aligned}
\label{Equ: Y=X+2}
Y_{\lambda}&=X_{\lambda}+\sum_{\mu \in \lambda_{<}} n_{\lambda, \mu} X_{\mu}, \\
\sum_{\lambda} n_{\lambda, \mu} s_{\lambda}&=L^{-1}s_{\mu}= s_\mu \prod_{i \leq j} (1 + x_ix_j), \\
n_{\lambda, \mu}&=\sum_{r\geq0,\nu} b_{r,\nu} c_{\mu,\nu}^{\lambda}.\end{aligned}$$
We assume that $$Y_{\lambda}=\sum_{\mu} n_{\lambda, \mu} X_{\mu}, \text{ for some } n_{\lambda, \mu} \in \mathbb{N}.$$ By Theorem \[Thm: Main1\], $$s_{\lambda}=\sum_{\mu} n_{\lambda, \mu} L^{\dagger}s_{\mu}.$$ Then $$\langle L^{-1} s_\nu, s_{\lambda} \rangle =\sum_{\mu} n_{\lambda, \mu} \langle L^{-1}s_\nu, L^{\dagger}s_{\mu} \rangle=\sum_{\mu} n_{\lambda, \mu} \langle s_\nu, s_{\mu} \rangle=n_{\lambda, \nu}.$$ By Equation , $$\sum_{\lambda} n_{\lambda, \mu} s_{\lambda}= \sum_{\lambda} \langle L^{-1} s_\mu, s_{\lambda} \rangle s_{\lambda}= L^{-1}s_{\mu}= s_\mu \prod_{i \leq j} (1 + x_ix_j).$$ Moreover, $$n_{\lambda, \mu}=\langle L^{-1} s_\mu, s_{\lambda} \rangle =\langle \sum_{r\geq0, \nu} b_{r,\nu} s_\nu s_\mu, s_{\lambda} \rangle =\sum_{r\geq0,\nu} b_{r,\nu} c_{\mu,\nu}^{\lambda}.$$
\[Thm: Main2\] We have the following generating function for $\Phi(X_{\lambda})$, $$\sum_{\lambda} s_\lambda(x) \Phi(X_\lambda)(y) = \prod_{i_1 \leq i_2}\frac{1}{1 + x_ix_j} \prod_{i, j}\frac{1}{1-x_i y_j}.$$ (Here $\Phi(X_\lambda)(y)$ means that the symmetric function $\Phi(X_\lambda)$ has variable set $\{y_j \}$.)
Now we apply Theorem \[Thm: Main1\] to prove this theorem. We consider the first equation of Theorem \[Thm: Main1\] as a having symmetric function variables $\{y_j\}$, and multiply by $s_\lambda(x)$. Summing over $\lambda$, we are required to show: $$\sum_{\lambda} s_\lambda(x) \sum_{\mu} (-1)^{|\mu|} s_{\lambda / 2\mu}(y) = \prod_{i_1 \leq i_2}\frac{1}{1 + x_ix_j} \prod_{i, j}\frac{1}{1-x_i y_j}$$ We now calculate: $$\begin{aligned}
\sum_\lambda s_\lambda(x) \sum_\mu (-1)^{|\mu|} s_{\lambda / 2\mu}(y)
&=& \sum_\lambda s_\lambda(x) L^\dagger(s_\lambda)(y) \\
&=& \sum_\lambda \sum_\rho \langle s_\rho, L^\dagger(s_\lambda) \rangle s_\lambda(x) s_\rho(y) \\
&=& \sum_\rho \sum_\lambda \langle L (s_\rho), s_\lambda \rangle s_\lambda(x) s_\rho(y) \\
&=& \sum_\rho L (s_\rho)(x) s_\rho(y) \\
&=& \prod_{i_1 \leq i_2}\frac{1}{1 + x_ix_j} \sum_\rho s_\rho(x) s_\rho(y) \\
&=& \prod_{i_1 \leq i_2}\frac{1}{1 + x_ix_j} \prod_{i, j}\frac{1}{1-x_i y_j}.\end{aligned}$$
\[Thm: Main3\] We have the following fusion rules: $$R_{\mu, \nu}^ \lambda = \sum_{\alpha, \beta, \gamma} c_{\alpha, \beta}^\mu c_{\beta', \gamma}^\nu c_{\alpha, \gamma}^\lambda.$$ (Here $\beta^\prime$ is the Young diagram dual to $\beta$.)
Now we apply Theorem \[Thm: Main2\] to prove this theorem. To do this, we consider a suitable generating function for the $R_{\mu, \nu}^\lambda$, and express it in terms of two instances of the generating function in the second part of the theorem. We work with three variable sets: $\{x_i^{(1)}\}$, $\{x_i^{(2)}\}$, and $\{y_j\}$, and use Proposition \[comultiplication\_L\_prop\]. $$\begin{aligned}
& &\sum_{\mu, \nu, \lambda} R_{\mu, \nu}^{\lambda} s_\mu(x^{(1)}) s_\nu(x^{(2)}) \Phi(X_\lambda)(y) \\
&=& \sum_{\mu, \nu} s_\mu(x^{(1)}) s_\nu(x^{(2)}) \Phi(X_\mu)(y) \Phi(X_\nu)(y) \\
&=& \sum_{\mu} s_\mu(x^{(1)})\Phi(X_\mu)(y) \sum_{\nu} s_\nu(x^{(2)}) \Phi(X_\nu)(y) \\
&=& \sum_{\mu} L(s_\mu)(x^{(1)}) s_\mu(y) \sum_{\nu} L(s_\nu)(x^{(2)}) s_\nu(y) \\
&=& \sum_{\mu, \nu} L(s_\mu)(x^{(1)}) L(s_\nu)(x^{(2)}) \sum_{\lambda}c_{\mu,\nu}^\lambda s_\lambda(y) \\
&=& \sum_{\mu, \nu} L_{\{x^{(1)}\}}L_{\{x^{(2)}\}} s_\mu(x^{(1)}) s_\nu(x^{(2)}) \sum_{\lambda}c_{\mu,\nu}^\lambda s_\lambda(y) \\
&=& \sum_{\mu, \nu} \left( \sum_\beta s_\beta(x^{(1)})s_{\beta^\prime}(x^{(2)})\right) L_{\{x^{(1)}\}\cup \{x^{(2)}\}} s_\mu(x^{(1)}) s_\nu(x^{(2)}) \sum_{\lambda}c_{\mu,\nu}^\lambda s_\lambda(y) \\
&=& \left( \sum_\beta s_\beta(x^{(1)})s_{\beta^\prime}(x^{(2)})\right) L_{\{x^{(1)}\}\cup \{x^{(2)}\}} \sum_{\lambda}s_\lambda(y) \sum_{\mu, \nu}c_{\mu,\nu}^\lambda s_\mu(x^{(1)}) s_\nu(x^{(2)}) \\
&=& \left( \sum_\beta s_\beta(x^{(1)})s_{\beta^\prime}(x^{(2)})\right) L_{\{x^{(1)}\}\cup \{x^{(2)}\}} \sum_{\lambda}s_\lambda(y) s_\lambda(x^{(1)}, x^{(2)}) \\
&=& \left( \sum_\beta s_\beta(x^{(1)})s_{\beta^\prime}(x^{(2)})\right) \sum_\lambda s_\lambda(x^{(1)}, x^{(2)}) \Phi(X_\lambda)(y). \\\end{aligned}$$ At this point, we may take the coefficient of $\Phi(X_\lambda)(y)$ (these form a basis of $\Lambda$) to deduce $$\begin{aligned}
\sum_{\mu, \nu} R_{\mu, \nu}^{\lambda} s_\mu(x^{(1)}) s_\nu(x^{(2)})
&=& \left( \sum_\beta s_{\beta}(x^{(1)})s_{\beta^\prime}(x^{(2)})\right) s_\lambda(x^{(1)}, x^{(2)}) \\
&=& \left( \sum_\beta s_{\beta}(x^{(1)})s_{\beta^\prime}(x^{(2)})\right) \sum_{\alpha, \gamma} c_{\alpha, \gamma}^\lambda s_\alpha(x^{(1)}) s_\gamma(x^{(2)}) \\
&=& \sum_{\mu, \nu} \sum_{\alpha, \beta, \gamma} c_{\alpha, \beta}^\mu c_{\beta^\prime, \gamma}^\nu c_{\alpha, \gamma}^\lambda s_\mu(x^{(1)}) s_\nu(x^{(2)}).\end{aligned}$$ Taking coefficient of $s_\mu(x^{(1)}) s_\nu(x^{(2)})$, we recover the formula for $R_{\mu, \nu}^\lambda$.
Note that if $|\alpha|=a$, $|\beta|=b$, $|\gamma|=c$, and $$c_{\alpha, \beta}^\mu c_{\beta', \gamma}^\nu c_{\alpha, \gamma}^\lambda \neq 0,$$ then $|\mu|=a+b$, $|\nu|=b+c$, $|\lambda|=a+c$. Conversely, $a$, $b$, $c$ are determined by $|\mu|$, $|\nu|$, $|\lambda|$. Thus the equation in Theorem \[Thm: Main3\] is a finite sum.
Recall that the simple objects ${\tilde{y}}_{\gamma}$ and ${\tilde{y}}_{\gamma'}$ are dual to each other in ${\mathscr{C}}$. We denote $\cup_{\beta}$ to be the evaluation map in the hom space $\hom_{{\mathscr{C}}}({\tilde{y}}_{\beta} \otimes {\tilde{y}}_{\beta^\prime}, \emptyset)$. For Young diagrams $\mu$, $\nu$, $\lambda$, $\alpha$, $\beta$, $\gamma$, with $|\mu|=a+b$, $|\nu|=b+c$, $|\lambda|=a+c$, $|\alpha|=a$, $|\beta|=b$, $|\gamma|=c$, we define the triangle map $\bigtriangledown: \hom_{{\mathscr{C}}}({\tilde{y}}_{\mu}, {\tilde{y}}_{\alpha} \otimes {\tilde{y}}_{\beta'}) \otimes \hom_{{\mathscr{C}}}({\tilde{y}}_{\mu}, {\tilde{y}}_{\beta} \otimes {\tilde{y}}_{\gamma}) \otimes \hom_{{\mathscr{C}}}({\tilde{y}}_{\alpha} \otimes {\tilde{y}}_{\gamma}, {\tilde{y}}_{\lambda}) \to \hom_{{\mathscr{C}}}({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{\nu}, {\tilde{y}}_{\lambda})$ as $$\begin{aligned}
\bigtriangledown(\rho_1 \otimes \rho_2 \otimes \rho_3)&= \rho_3( {\tilde{y}}_{\mu} \otimes \cup_{\beta} \otimes {\tilde{y}}_{\lambda}) (\rho_1\otimes \rho_2) \\
&=
\raisebox{-2.5cm}{
\begin{tikzpicture}
\draw[thick] (0,0) --++(0,2);
\draw[thick] (1,2) arc (-180:0:.5);
\draw[thick] (1,0) --++(2,2);
\node at (-.2,1) {${\tilde{y}}_{\alpha}$};
\node at (1-.2,1.8) {${\tilde{y}}_{\beta}$};
\node at (1.7,1.8) {${\tilde{y}}_{\beta^\prime}$};
\node at (1.7,1) {${\tilde{y}}_{\gamma}$};
\node at (2,3.2) {${\tilde{y}}_{\nu}$};
\node at (2,2.5) {$\rho_2$};
\node at (0,3.2) {${\tilde{y}}_{\mu}$};
\node at (0,2.5) {$\rho_{1}$};
\node at (0,-1.2) {${\tilde{y}}_{\lambda}$};
\node at (0,-.5) {$\rho_{3}$};
\draw[thick] (2,2) --(2.5,2.5)--++(0,1);
\draw[thick] (3,2) --(2.5,2.5);
\draw[thick] (0,2) --(.5,2.5)--++(0,1);
\draw[thick] (1,2) --(.5,2.5);
\draw[thick] (0,0) --(.5,-.5)--++(0,-1);
\draw[thick] (1,0) --(.5,-.5);
\end{tikzpicture}
}.\end{aligned}$$
\[Thm: Basis generic\] For any Young diagrams $\mu$, $\nu$, $\lambda$, the triangle map $\bigtriangledown$ is an embedding map and $$\begin{aligned}
\hom_{{\mathscr{C}}}({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{\nu}, {\tilde{y}}_{\lambda})&=\bigoplus_{\alpha,\beta,\gamma}
\bigtriangledown\left(\hom_{{\mathscr{C}}}({\tilde{y}}_{\mu}, {\tilde{y}}_{\alpha} \otimes {\tilde{y}}_{\beta'}) \otimes \hom_{{\mathscr{C}}}({\tilde{y}}_{\mu}, {\tilde{y}}_{\beta} \otimes {\tilde{y}}_{\gamma}) \otimes \hom_{{\mathscr{C}}}({\tilde{y}}_{\alpha} \otimes {\tilde{y}}_{\gamma}, {\tilde{y}}_{\lambda}) \right).\end{aligned}$$
Similarly to the proof of Lemma \[Lem: Basis 1\], we take $$\tilde{x}_{p_1,p_2,p_3}={\tilde{y}}_{\lambda} x_{p_1,p_2,p_3} ({\tilde{y}}_{\mu}\otimes {\tilde{y}}_{\nu}).$$ Then $\{\tilde{x}_{p_1,p_2,p_3} : p_1 \in P_{a+b}, p_2 \in P_{b+c}, p_3 \in P_{a+c}.\}$ is a spanning set of $\hom_{{\mathscr{C}}}({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{\nu}, {\tilde{y}}_{\lambda})$. Note that the $180^{\circ}$ rotation of $s_{b}$ is $s_{b}$. So $$\begin{aligned}
\tilde{x}_{p_1,p_2,p_3}&=
\raisebox{-2.5cm}{
\begin{tikzpicture}
\draw[thick] (0,0) --++(0,2);
\draw[thick] (1,2) arc (-180:0:.5);
\draw[thick] (1,0) --++(2,2);
\node at (-.2,1) {$s_a$};
\node at (1-.2,1.8) {$s_b$};
\node at (1.7,1.8) {$s_b$};
\node at (1.7,1) {$s_c$};
\node at (2,3.2) {${\tilde{y}}_{\nu}$};
\node at (2,2.5) {$\hat{p}_2$};
\node at (0,3.2) {${\tilde{y}}_{\mu}$};
\node at (0,2.5) {$\hat{p}_1$};
\node at (0,-1.2) {${\tilde{y}}_{\lambda}$};
\node at (0,-.5) {$\hat{p}_3$};
\draw[thick] (2,2) --(2.5,2.5)--++(0,1);
\draw[thick] (3,2) --(2.5,2.5);
\draw[thick] (0,2) --(.5,2.5)--++(0,1);
\draw[thick] (1,2) --(.5,2.5);
\draw[thick] (0,0) --(.5,-.5)--++(0,-1);
\draw[thick] (1,0) --(.5,-.5);
\end{tikzpicture}
}
=\sum_{\alpha,\beta,\gamma} \sum_{\rho_1,\rho_2,\rho_3}
\raisebox{-2.5cm}{
\begin{tikzpicture}
\draw[thick] (0,0) --++(0,2);
\draw[thick] (1,2) arc (-180:0:.5);
\draw[thick] (1,0) --++(2,2);
\node at (-.2,1) {${\tilde{y}}_{\alpha}$};
\node at (1-.2,1.8) {${\tilde{y}}_{\beta}$};
\node at (1.7,1.8) {${\tilde{y}}_{\beta^\prime}$};
\node at (1.7,1) {${\tilde{y}}_{\gamma}$};
\node at (2,3.2) {${\tilde{y}}_{\nu}$};
\node at (2,2.5) {$\rho_2$};
\node at (0,3.2) {${\tilde{y}}_{\mu}$};
\node at (0,2.5) {$\rho_{1}$};
\node at (0,-1.2) {${\tilde{y}}_{\lambda}$};
\node at (0,-.5) {$\rho_{3}$};
\draw[thick] (2,2) --(2.5,2.5)--++(0,1);
\draw[thick] (3,2) --(2.5,2.5);
\draw[thick] (0,2) --(.5,2.5)--++(0,1);
\draw[thick] (1,2) --(.5,2.5);
\draw[thick] (0,0) --(.5,-.5)--++(0,-1);
\draw[thick] (1,0) --(.5,-.5);
\end{tikzpicture}
} \\
&=\sum_{\alpha,\beta,\gamma} \sum_{\rho_1,\rho_2,\rho_3}\bigtriangledown(\rho_1,\rho_2,\rho_3).\end{aligned}$$ for some Young diagrams $\alpha$, $\beta$, $\gamma$, with $|\alpha|=a$, $|\beta|=b$, $|\gamma|=c$, and some morphisms $\rho_{1}\otimes \rho_2 \otimes \rho_3 \in \hom_{{\mathscr{C}}}({\tilde{y}}_{\mu}, {\tilde{y}}_{\alpha} \otimes {\tilde{y}}_{\beta'}) \otimes \hom_{{\mathscr{C}}}({\tilde{y}}_{\mu}, {\tilde{y}}_{\beta} \otimes {\tilde{y}}_{\gamma}) \otimes \hom_{{\mathscr{C}}}({\tilde{y}}_{\alpha} \otimes {\tilde{y}}_{\gamma}, {\tilde{y}}_{\lambda})$. Therefore $$\bigcup_{\alpha,\beta,\gamma}
\bigtriangledown\left(\hom_{{\mathscr{C}}}({\tilde{y}}_{\mu}, {\tilde{y}}_{\alpha} \otimes {\tilde{y}}_{\beta'}) \otimes \hom_{{\mathscr{C}}}({\tilde{y}}_{\mu}, {\tilde{y}}_{\beta} \otimes {\tilde{y}}_{\gamma}) \otimes \hom_{{\mathscr{C}}}({\tilde{y}}_{\alpha} \otimes {\tilde{y}}_{\gamma}, {\tilde{y}}_{\lambda}) \right)$$ is a spanning set of $\hom_{{\mathscr{C}}}({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{\nu}, {\tilde{y}}_{\lambda})$. By Proposition \[Prop: iso\], $$\begin{aligned}
R_{\mu, \nu}^ \lambda &= \dim \hom_{{\mathscr{C}}}({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{\nu}, {\tilde{y}}_{\lambda}) \\
&\leq \sum_{\alpha,\beta,\gamma} \dim \hom_{{\mathscr{C}}}({\tilde{y}}_{\mu}, {\tilde{y}}_{\alpha} \otimes {\tilde{y}}_{\beta'}) \times \dim \hom_{{\mathscr{C}}}({\tilde{y}}_{\mu}, {\tilde{y}}_{\beta} \otimes {\tilde{y}}_{\gamma}) \times \dim \hom_{{\mathscr{C}}}({\tilde{y}}_{\alpha} \otimes {\tilde{y}}_{\gamma}, {\tilde{y}}_{\lambda}) \\
&= \sum_{\alpha,\beta,\gamma} \dim \hom_{H}(y_{\mu}, y_{\alpha} \otimes y_{\beta'}) \times \dim \hom_{H}(y_{\mu}, y_{\beta} \otimes y_{\gamma}) \times \dim \hom_{H}(y_{\alpha} \otimes y_{\gamma}, y_{\lambda}) \\
&= \sum_{\alpha, \beta, \gamma} c_{\alpha, \beta}^\mu c_{\beta', \gamma}^\nu c_{\alpha, \gamma}^\lambda.\end{aligned}$$ By Theorem \[Thm: Main3\], the equality holds. So the triangle map $\bigtriangledown$ is an embedding map and $$\begin{aligned}
\hom_{{\mathscr{C}}}({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{\nu}, {\tilde{y}}_{\lambda})&=\bigoplus_{\alpha,\beta,\gamma}
\bigtriangledown\left(\hom_{{\mathscr{C}}}({\tilde{y}}_{\mu}, {\tilde{y}}_{\alpha} \otimes {\tilde{y}}_{\beta'}) \otimes \hom_{{\mathscr{C}}}({\tilde{y}}_{\mu}, {\tilde{y}}_{\beta} \otimes {\tilde{y}}_{\gamma}) \otimes \hom_{{\mathscr{C}}}({\tilde{y}}_{\alpha} \otimes {\tilde{y}}_{\gamma}, {\tilde{y}}_{\lambda}) \right).\end{aligned}$$
Combining Theorem \[Thm: Basis generic\] and Proposition \[Prop: iso\], we can construct an explicit basis of $\hom_{{\mathscr{C}}}({\tilde{y}}_{\mu} \otimes {\tilde{y}}_{\nu}, {\tilde{y}}_{\lambda})$ using $s_n$ and the basis of the hom spaces $\hom_{H}({\tilde{y}}_{\mu}, y_{\alpha} \otimes y_{\beta'})$, $\hom_{H}(y_{\mu}, y_{\beta}$, $ y_{\gamma})$, $\hom_{H}(y_{\alpha} \otimes y_{\gamma}, y_{\lambda})$ in the Hecke algebra $H$. Applying the evaluation algorithm of the Yang-Baxter relation, we obtain the 6j-symbols of ${\mathscr{C}}$.
When the Young diagrams are small, the 6j-symbols can be computed by hand or by computer. We do not expect to compute 6j-symbols for large Young diagrams in this way, the complexity of this algorithm grows exponentially w. r. t. the size of the Young diagrams. Even computing the 6j-symbols for $Rep(H(q))$ remains challenging.
[1]{}
V. F. R. Jones, [*Hecke algebra representations of braid groups and link polynomials,*]{} Annals of Math. 126 (1987) 335-388.
V. F. R. Jones, [*Planar algebras, I*]{} [https://arxiv.org/abs/math/9909027]{}[https://arxiv.org/abs/math/9909027]{}.
Z. Liu. . <https://arxiv.org/abs/1507.06030>, 2015.
I.G. Macdonald. . Oxford classic texts in the physical sciences. Clarendon Press, 1998.
R.P. Stanley. . Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2001.
F. Xu, private discussions
F. Xu *New Braided Endomorphisms from Conformal Inclusions*, Communications in Mathematical Physics **192**.2 (1998), 249–403.
[^1]: It was called the Jones projection is the operator algebraic setting.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'This letter proposes two novel proactive cooperative caching approaches using deep learning (DL) to predict users’ content demand in a mobile edge caching network. In the first approach, a (central) content server takes responsibilities to collect information from all mobile edge nodes (MENs) in the network and then performs our proposed deep learning (DL) algorithm to predict the content demand for the whole network. However, such a centralized approach may disclose the private information because MENs have to share their local users’ data with the content server. Thus, in the second approach, we propose a novel distributed deep learning (DDL) based framework. The DDL allows MENs in the network to collaborate and exchange information to reduce the error of content demand prediction without revealing the private information of mobile users. Through simulation results, we show that our proposed approaches can enhance the accuracy by reducing the root mean squared error (RMSE) up to $33.7$% and reduce the service delay by $36.1$% compared with other machine learning algorithms.'
author:
- '[^1] [^2] [^3]'
title: 'Distributed Deep Learning at the Edge: A Novel Proactive and Cooperative Caching Framework for Mobile Edge Networks'
---
Mobile edge caching, deep learning, distributed deep learning, proactive and cooperative caching.
Introduction
============
edge caching (MEC) has been emerging as one of the most effective solutions to deal with the ever-increasing traffic demand for new services (e.g., video streaming, IoT, and virtual reality applications) in mobile networks. The key idea of an MEC network is to distribute popular contents closer to the mobile users via mobile edge nodes (MENs) [@Mao:2017] to reduce the service delay for the mobile users. As a result, the deployment of MEC network helps to improve the users’ experiences (e.g., trustworthy wireless connections, fast data transfer, and low energy consumption) and thus maximize the revenues for the MEC service providers [@Hoang:2018].
To efficiently cache the popular contents in the MEC network, proactive caching is one of the most effective methods to predict the mobile users’ demands (i.e., content requests). In particular, the proactive caching can provide optimal caching decisions to increase the cache hit rate and reduce the operational as well as service costs on the backhaul link for the MEC service providers [@Hoang:2018]. In [@Zeydan:2016], a learning based proactive caching using singular value decomposition (SVD) to cache data at the base stations was investigated. In this work, the data is first collected from the base stations and then trained in a big data platform. Nevertheless, the SVD technique sets all empty entries to be zero, leading to a poor prediction accuracy, especially when a dataset is extremely sparse. Furthermore, the SVD observes approximated ranks of elements and thus may produce negative numbers which provide no information about real users’ demands. To address this problem, the authors in [@Ahn:2018] adopted the non-negative matrix factorization (NMF) to predict the demand probability along with an implicit feedback of the users’ social context. As such, the NMF technique applies the additive parts-based representation with non-negative elements to enhance the interpretability of the elements when the dataset is sparse. However, the NMF is a linear model which considers only two-factor correlation (i.e., the user-content relationship) without learning multi-level correlation. Given that, deep learning seems a suitable solution that relies on deep neural networks (DNN) to learn multiple levels of processing layers. Each layer of the DNN provides nonlinear transformations of the complex hidden features to obtain correlations between the mobile users and the content demands hierarchically (i.e., a layer learns and aggregates a set of features according to the previous layer’s results) [@Zhang4:2018].
In this letter, we introduce two novel proactive cooperative caching approaches using DL algorithms to improve the accuracy of content demand prediction for the MEC network. In the first approach, we develop a model utilizing the content server (CS) as a centralized node to collect information from all the MENs. We then use the DL to predict the demands for the whole network. However, such an approach may raise the concerns on information privacy and communication overhead. To address these problems, we propose the novel approach using DDL-based framework. In this framework, the CS only needs to collect the trained models from MENs and update the global model accordingly [@Dean:2012]. After that, the global model will be sent back to the MENs for further updates. Through simulation results, we demonstrate that both proposed approaches can improve the accuracy of prediction up to $33.7$% and reduce the service delay by $36.1$% compared with other well-known proactive caching algorithms at MENs, i.e., SVD and NMF.
System Model {#Sec.System}
============
![Network architecture.[]{data-label="fig:Edge_Architecture"}](Figures/SystemModel2)
Network Architecture
--------------------
The proposed network architecture is illustrated in Fig. \[fig:Edge\_Architecture\]. Mobile users are connected to MENs within their service area. All MENs are also connected to the CS through the backhaul links by using either wireless (i.e., cellular networks) or wired connections. Each MEN is equipped with a finite storage capacity to cache popular contents locally according to the decision of proactive cooperative caching framework. When a user sends a content request to an MEN, the content will be sent to the user instantly if the content is stored locally at the MEN. Otherwise, the MEN downloads the content from the CS or from one of its directly connected MENs, and sends it to the requesting user. We denote $\mathcal{N} = \{1,\ldots,n,\ldots,N\}$ as the set of MENs, $\mathcal{U} = \{\mathcal{U}_1,\ldots,\mathcal{U}_n,\ldots,\mathcal{U}_N\}$ as the set of mobile users in the MEC network, and $U$ as the total number of mobile users in the network. In this way, $\mathcal{U}_n = \{1,\ldots,u_n,\ldots,U_n\}$ represents the set of mobile users at MEN-$n$’s coverage area. Each MEN-$n$ has the storage capacity denoted by $S_n$. Note that mobile users can move and download their requested contents from any MEN in the network. Thus, the set of users $\mathcal{U}_n$ captures all users who visit and download contents at MEN-$n$. In addition, the set $\mathcal{U}_n$ also captures the case when a user, e.g., user $u_n$, downloads a content at MEN-$n$ via another MEN. Then, we denote $\mathcal{I} = \{1,\ldots,i,\ldots,I\}$ as the set of contents.
Proactive Cooperative Caching Mechanism
---------------------------------------
To cache popular contents, each MEN, e.g., MEN-$n$, collects from mobile users in its serving area and sets up a dataset, i.e., $\mathbf{X}_n$, containing popularity factors $f^i_{u_n}$ of user $u_n$ over the content $i$. For the first approach (i.e., using DL in the CS), the CS collects $\mathbf{X}_n, \forall n \in \mathcal{N}$ from the MENs cooperatively and then concatenates them into dataset $\mathbf{X}_{cs}$ vertically with popularity factor $f^i_u$ of the user $u$ (where $u \in \mathcal{U}$) over the content $i$. In this way, MENs can share the model information to improve the prediction accuracy for the whole network. We use $\mathbf{X}_{\emph{\mbox{cs}}}$ to predict the content demands and then generate dataset $\mathbf{\hat Y}_{\emph{\mbox{cs}}}$ containing predicted popularity factors ${\hat f}^i_u$ at the CS. This $\mathbf{\hat Y}_{\emph{\mbox{cs}}}$ is then sent back to the MENs for content placement decision. Specifically, each MEN-$n$ obtains ${\hat f}^i_n = \underset{u \in \mathcal{U}}{\sum} {\hat f}^i_{u}$ as the predicted popularity factor aggregation of content $i$.
In the second approach with the DDL, each MEN-$n$ can predict the demands locally using $\mathbf{X}_n$. Then, the CS only needs to collect the trained models from MENs and update the global model cooperatively (explained in Section \[Sec.PF2\]) and create $\mathbf{\hat Y}_{\emph{\mbox{n}}}$ which contains predicted popularity factors ${\hat f}^i_{u_n}$. To perform the content placement decision, each MEN-$n$ aggregates the predicted popularity factors of content $i$ as ${\hat f}^i_n = \underset{u_n \in \mathcal{U}_n}{\sum} {\hat f}^i_{u_n}$. Based on ${\hat f}^i_n$ of the first and second approaches, we can obtain specific largest numbers of ${\hat f}^i_n$ at MEN-$n$ in descending order. In particular, we select the contents with top-$R$ of ${\hat f}^i_n$ which are likely to be cached at MEN-$n$.
DL-Based Proactive Cooperative Caching {#Sec.PF}
======================================
In this approach, the CS needs to learn $\mathbf{X}_{\emph{\mbox{cs}}}$ through the DNN by partitioning $\mathbf{X}_{\emph{\mbox{cs}}}$ into smaller subsets (referred to as mini-batch size $\beta$). For DNN, each layer $\ell$ produces an output matrix containing global weight matrix $\mathbf{W}_\ell$ to control how strong the influence of a layer’s each neuron to the other, and global bias vector $\mathbf{v}_\ell$ to fit the dataset as follows: $$\label{eqn1}
\begin{aligned}
\mathbf{Y}_{\emph{\mbox{cs}}}^\ell = \alpha_{\emph{\mbox{cs}}} \big(\mathbf{W}_\ell\mathbf{X}_{\emph{\mbox{cs}}}^\ell + \mathbf{v}_\ell\big),
\end{aligned}$$ where $\mathbf{X}_{\emph{\mbox{cs}}}^\ell$ is the input matrix (i.e., training dataset) of layer $\ell$ in the CS (with $\mathbf{X}_{\emph{\mbox{cs}}}^1 = \mathbf{X}_{\emph{\mbox{cs}}}$) and $\alpha_{\emph{\mbox{cs}}}$ is the *rectified linear unit (ReLU)* activation function to transform the input of the layer into a nonlinear form for learning more complex feature interaction patterns. In this case, $\alpha_{\emph{\mbox{cs}}}$ returns $\mathbf{X}_{\emph{\mbox{cs}}}^\ell$ if it receives any positive input, and zero otherwise. As the DNN contains several layers including the hidden layers, we can express $\mathbf{X}_{\emph{\mbox{cs}}}^{\ell+1} = \mathbf{Y}_{\emph{\mbox{cs}}}^\ell$. To prevent the overfitting problem and the generalization error [@Srivastava:2014], we augment a dropout layer $\ell_{\emph{\mbox{drop}}}$ just after the last hidden layer. This additional layer randomly drops the input $\mathbf{X}_{\emph{\mbox{cs}}}^{\ell_{\emph{\mbox{drop}}}}$ by a fraction rate $r$, and thus the rest of the input elements are scaled by $\frac{1}{1-r}$. Then, the output layer $L$ will generate $\mathbf{Y}_{\emph{\mbox{cs}}}^L$ which is used to find the prediction loss for each mini-batch iteration $\tau$. In particular, if we consider $\mathbf{\omega} = (\mathbf{W}, \mathbf{v})$, where $\mathbf{W} = [\mathbf{W}_1,\ldots,\mathbf{W}\ell,\ldots,\mathbf{W}_L]$ and $\mathbf{v} = [\mathbf{v}_1,\ldots,\mathbf{v}_\ell,\ldots,\mathbf{v}_L]$, as the global model for all DNN layers, the prediction loss $p(\mathbf{\omega_\tau})$ for one $\tau$ in the CS is expressed by the mean-squared error (MSE) $p(\mathbf{\omega_\tau}) = \frac{1}{\beta}{\underset{u=1}{\overset{\beta}{\sum}}} p_u(\mathbf{\omega_\tau})$, where $p_u(\mathbf{\omega_\tau}) = (y_{\emph{\mbox{cs}}}^u - x_{\emph{\mbox{cs}}}^u)^2$. Here, $x_{\emph{\mbox{cs}}}^u$ and $y_{\emph{\mbox{cs}}}^u$ are the elements of matrices $\mathbf{X}_{\emph{\mbox{cs}}}^1$ and $\mathbf{Y}_{\emph{\mbox{cs}}}^L$, respectively. Then, we can compute the global gradient of using DL by $G_{\tau} = \nabla \mathbf{\omega}_\tau = \frac{\partial p(\mathbf{\omega_\tau})}{\partial \mathbf{\omega}_\tau}$.
After $G_\tau$ is obtained, the CS updates the global model $\mathbf{\omega}_{\tau}$ with the aim to minimize the prediction loss function, i.e., $\underset{\mathbf{\omega}}{\text{\bf min }}p(\mathbf{\omega})$. As such, we adopt the adaptive learning rate optimizer *Adam* to provide fast convergence and profound robustness to the model [@Kingma:2015]. Consider $\eta_\tau$ and $\delta_\tau$ to be the exponential moving average (to estimate the mean) of the $G_\tau$ and the squared $G_\tau$ to predict the variance at $\tau$, respectively. Then, the update rules of $\eta_{\tau+1}$ and $\delta_{\tau+1}$ can be expressed by: $$\label{eqn6}
\eta_{\tau+1} = \gamma_\eta^\tau \eta_{\tau} + (1 - \gamma_\eta^\tau)G_\tau, \mbox{and } \delta_{\tau+1} = \gamma_\delta^\tau \delta_{\tau} + (1 - \gamma_\delta^\tau)G_\tau^2,$$ where $\gamma_\eta^\tau$ and $\gamma_\delta^\tau \in [0,1)$ represent the exponential decay steps of $\eta_\tau$ and $\delta_\tau$ at $\tau$, respectively. To update the global model, we also consider the learning step $\lambda$ to decide how fast the global model will be updated at each $\tau$. In particular, the update rule for $\lambda$ follows this expression: $$\label{eqn7}
\begin{aligned}
\lambda_{\tau+1} = \lambda\frac{\sqrt{1 - \gamma_\delta^{\tau+1}}}{1 - \gamma_\eta^{\tau+1}}.
\end{aligned}$$ Then, the global model $\mathbf{\omega}_{\tau+1}$ for the next $\tau+1$ is updated by: $$\label{eqn8}
\begin{aligned}
\mathbf{\omega}_{\tau+1} = \mathbf{\omega}_{\tau} - \lambda_{\tau+1}\frac{\eta_{\tau+1}}{\sqrt{\delta_{\tau+1}} + \epsilon},
\end{aligned}$$ where $\epsilon$ indicates a constant to avoid zero division when the $\sqrt{\delta_{\tau+1}}$ is almost zero. For this approach, $\mathbf{\omega}_{\tau+1}$ is used to learn the dataset for the next $\tau+1$ in the CS. The same process is repeated until each sample $u$ of $\mathbf{X}_{\emph{\mbox{cs}}}$ has been observed referred to as epoch time $t$. Then, the process terminates when the prediction loss converges or the certain number of epoch time $T$ is reached. In this case, we can obtain the final global model $\mathbf{\omega}^*$ to predict $\mathbf{\hat Y}_{\emph{\mbox{cs}}}$ of training dataset $\mathbf{X}_{\emph{\mbox{cs}}}$ and new dataset $\mathbf{\hat X}_{\emph{\mbox{cs}}}$ using Eq. (\[eqn1\]). The algorithm for proactive cooperative caching using DL is shown in Fig. \[DDL-PC\] in which the process inside the dotted block (A) is executed at the CS.
DDL-Based Proactive Cooperative Caching {#Sec.PF2}
=======================================
In this approach, each MEN distributedly implements the DL technique to learn from its dataset $\mathbf{X}_{n}$ locally. The $\mathbf{X}_{n}$ is then divided into smaller subsets with mini-batch size $\frac{\beta}{N}$. For DNN, each MEN-$n$ generates the output matrix $$\label{eqn1b}
\begin{aligned}
\mathbf{Y}_n^\ell = \alpha_n \big(\mathbf{W}_\ell\mathbf{X}_n^\ell + \mathbf{v}_\ell\big),
\end{aligned}$$ where $\mathbf{X}_n^\ell$ is the input matrix of layer $\ell$ at MEN-$n$ (with $\mathbf{X}_{\emph{\mbox{n}}}^1 = \mathbf{X}_{\emph{\mbox{n}}}$) and $\alpha_n$ is the *ReLU* activation function at MEN-$n$. We also drop the input $\mathbf{X}_n^{\ell_{\emph{\mbox{drop}}}}$ in the dropout layer by a fraction rate $r$. In the output layer, we can generate $\mathbf{Y}_n^L$ and find the prediction loss for each $\tau$ by $p_n(\mathbf{\omega_\tau}) = \frac{N}{\beta}{\underset{u=1}{\overset{\frac{N}{\beta}}{\sum}}} p_n^u(\mathbf{\omega_\tau})$, where $p_n^u(\mathbf{\omega_\tau}) = (y_n^u - x_n^u)^2$. Here, $x_n^u$ and $y_n^u$ are the element of matrices $\mathbf{X}_n^1$ and $\mathbf{Y}_n^L$ at MEN-$n$, respectively. Next, we can compute the local gradient by $g_n^\tau = \nabla \mathbf{\omega}_\tau = \frac{\partial p_n(\mathbf{\omega_\tau})}{\partial \mathbf{\omega}_\tau}$. When $g_n^\tau$ computation is completed for each $\tau$, each MEN will send this local gradient to the CS for global gradient aggregation $G_\tau$. Specifically, the CS acts as a parameter server to aggregate the gradients of the models from all connected MENs and then update the global model $\mathbf{\omega}_\tau$ by using Eq. (\[eqn8\]) before sending back to the MENs. Doing so allows all MENs to collaborate by sharing local model information to each other to further improve the prediction accuracy through the CS. To guarantee that the gradient staleness is 0, the gradient averaging process is enabled right after $N$ local gradients, i.e., $g_n^\tau$ are received by the CS synchronously. Here, the gradient staleness happens when local gradients are computed using an obsolete/non-latest global model. Then, the global gradient $G_\tau$ of the DDL is $G_\tau = \frac{1}{N}{\underset{n=1}{\overset{N}{\sum}}} g_n^\tau$.
To minimize the prediction loss function, i.e., $\underset{\mathbf{\omega}}{\text{\bf min }}p_n(\mathbf{\omega})$, at each MEN-$n$, we also adopt the Adam optimizer and update the global model $\mathbf{\omega}_{\tau+1}$ as expressed in Eqs. (\[eqn6\])-(\[eqn8\]). This $\mathbf{\omega}_{\tau+1}$ is then sent back to the MENs for the next local learning process. The aforementioned process continues until the prediction loss converges or $T$ is reached. We then can predict $\mathbf{\hat Y}_{\emph{\mbox{n}}}$ of training dataset $\mathbf{X}_n$ and new dataset $\mathbf{\hat X}_n$ at each MEN using $\mathbf{\omega}^*$ through Eq. (\[eqn1b\]). The algorithm for proactive cooperative caching using DDL is summarized in Fig. \[DDL-PC\]. The process inside the dotted block (B) is implemented at the CS.
![Flowchart of DL and DDL approaches.[]{data-label="DDL-PC"}](Figures/Flowchart_DDL)
Performance Evaluation
======================
[0.33]{} 
[0.33]{} 
[0.33]{} 
![Learning 2000 epoch time for various methods.[]{data-label="fig:Learning_time"}](Figures/learning_time_result2)
Experimental Setup
------------------
We evaluate the performance of the content prediction on the MEC network with one CS and six MENs using *TensorFlow CPU* library in Linux platform of an Intel Xeon Gold 6150 2.7GHz 18 cores with 180GB RAM. We compare our proposed frameworks with two well-known machine learning methods including SVD [@Zeydan:2016] and NMF [@Mao:2006]. We use Movielens 1M-dataset with more than 1M ratings from 6040 users with 3952 movies. Then, we split the dataset into 80% training dataset and 20% testing dataset. From the training dataset, we divide the number of samples equally with respect to the number of MENs when DDL is implemented. Each MEN runs the testing dataset for the popularity factor prediction to compute the performance metrics. For DNN, we use two hidden layers with $64$ neurons per layer and one dropout layer with a fraction rate $0.8$. We also apply the adaptive learning rate optimizer Adam with initial step size $0.001$ and $2000$ epoch time during the learning process. Furthermore, for the content placement algorithm, we consider the same size for each content at 200MB. The bandwidth between an MEN and the CS is set at 60Mbps.
Simulation Results
------------------
Fig. \[fig:performance\] shows the comparison between conventional baseline and proposed methods. We first evaluate the prediction accuracy, i.e., RMSE, as the learning epoch increases in Fig. \[fig:performance\](a). In particular, the RMSE obtained by the DDL is 33.7% lower than those of SVD and NMF. The reason is that DDL can deeply learn the meaningful features from the subset of the whole dataset independently at different MENs, and thus the sensitivity to learn new testing dataset becomes better when the local models obtained by the MENs are aggregated together. In other words, the average prediction of all MENs will produce less variance and lower error regarding the number of MENs [@Guo:1999]. In contrast, SVD and NMF only generate linear assumptions of two factors based on the low-rank approximation [@Mao:2006] without deeply learning the representations, and thus the RMSE cannot be minimized properly. For the DL, although the RMSE is higher than that of the DDL, the DL can improve the RMSE by as much as 25.1% compared with those of the SVD and NMF.
We then observe the average delay to download contents from the CS and cache hit rate when the storage capacity increases in Figs. \[fig:performance\](b) and \[fig:performance\](c), respectively. Align with the trend of the RMSE, the DL and DDL approaches can reduce average delay up to 21.5% and 36.1% and increase the cache hit rate by 10% and 14.4%, respectively, compared with those of SVD and NMF. The reason is that the proposed approaches can optimize the use of hyperparameter settings to improve the accuracy of content demand prediction. Examples of the hyperparameters settings include the number of hidden layers and neurons, the regularization methods, the activation functions, and the size of mini-batch. We also observe in Fig. \[fig:Learning\_time\] that the DDL can learn the dataset faster than the DL as the number of the MENs increases. This interesting trend can provide useful information for MEC service providers to tradeoff between the learning time of the users’ demands and the implementation costs in the MEC network.
Summary
=======
In this letter, we have presented two novel proactive cooperative caching approaches leveraging deep learning (DL) and distributed deep learning (DDL) algorithms for the MEC network. In the first approach, the CS collects the information from all MENs and uses the DL technique to predict the users’ demands for the network. Then, to further minimize the communication overhead and address the privacy concern, we proposed the DDL-based scheme in which the DL can be executed at the edge. This scheme allows MENs to only exchange the gradient information, not the complete information of the users, and perform the DL to predict users demand without revealing the private information of the mobile users.
[100]{}
Y. Mao, C. You, J. Zhang, K. Huang, and K. B. Letaief, “A survey on mobile edge computing: the communication perspective", *IEEE Communications Surveys & Tutorials*, vol. 19, no. 4, Fourth Quarter 2017, pp. 2322-2358.
D. T. Hoang, D. Niyato, D. N. Nguyen, E. Dutkiewicz, P. Wang, and Z. Han, “A dynamic edge caching framework for mobile 5G networks", *IEEE Wireless Communications*, 2018.
E. Zeydan, E. Bastug, M. Bennis, M. A. Kader, I. A. Karatepe, A. S. Er, and M. Debbah, “Big data caching for networking: moving from cloud to edge", *IEEE Communications Magazine*, vol. 54, no. 9, Sept. 2016, pp. 36-42.
J. Ahn, S. H. Jeon, and H. Park, “A novel proactive caching strategy with community-aware learning in [CoMP]{}-enabled small-cell networks", *IEEE Communications Letters*, vol. 22, no. 9, Sept. 2018, pp. 1918-1921.
S. Zhang, L. Yao, A. Sun, and Y. Tay, “Deep learning based recommender system: a survey and new perspectives", *ACM Computing Surveys*, vol. 1, no. 1, Jul. 2018, pp. 1-35.
J. Dean et al., “Large scale distributed deep networks", *ACM NIPS 2012*, Dec. 2012, pp. 1223-1231.
N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: A simple way to prevent neural networks from overfitting", *The Journal of Machine Learning Research*, vol. 15, no. 1, Jan. 2014, pp. 1532-4435.
D. Kingma and J. Ba, “Adam: a method for stochastic optimization", *ICLR 2015*, May 2015.
Y. Mao, L. K. Saul, and J. M. Smith,“IDES: an internet distance estimation service for large networks", *IEEE Journal on Selected Areas in Communications*, vol. 24, no. 12, Dec. 2006, pp. 2273-2284.
Y. Guo and J. Sutiwaraphun, “Probing knowledge in distributed data mining", *PAKDD 1999: Methodologies for Knowledge Discovery and Data Mining*, vol 1574, pp. 443-452, Springer, Berlin, 1999.
[^1]: Y. M. Saputra and D. T. Hoang, D. N. Nguyen, and E. Dutkiewicz are with University of Technology Sydney, Australia (email: yurismulya.saputra@student.uts.edu.au, Hoang.Dinh, Diep.Nguyen, and Eryk.Dutkiewicz@uts.edu.au).
[^2]: D. Niyato is with Nanyang Technological University, Singapore (email: dniyato@ntu.edu.sg).
[^3]: D. I. Kim is with Sungkyunkwan University, South Korea (e-mail: dikim@skku.ac.kr).
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'V. P. Maslov[^1]'
title: Nonlinear Averaging in Economics
---
Kolmogorov nonlinear averaging
==============================
A sequence of functions $M_n$ defines a [*regular*]{} type of average if the following conditions are satisfied (Kolmogorov):
[I.]{} $M(x_1,x_2,\dots,x_n)$ is continuous and monotone in each variable; to be definite, we assume that $M$ is increasing in each variable;
[II.]{} $M(x_1,x_2,\dots,x_n)$ is a symmetric function[^2];
[III.]{} the average of identical numbers is equal to their common value: $M(x,x,\dots,x)= x$;
[IV.]{} a group of values can be replaced by their own average, without changing the overall average: $$M(x_1,\dots,x_m,y_1,\dots,y_n)
=M_{n+m}(x,\dots,x,y_1,\dots,y_n),$$ where $x=M(x_1,\dots,x_m)$.
If conditions [I–IV]{} are satisfied, the average $M(x_1,x_2,\dots, x_n)$ is of the form $$M(x_1,x_2,\dots,x_n)
=\psi\biggl(\frac{\varphi(x_1)+\varphi(x_2)+\dots+
\varphi(x_n)}n\biggr),
\tag{1}$$ where $\varphi$ is a continuous, strictly monotone function and $\psi$ is its inverse.
For the proof of Theorem 1, see [@1].
Main axiom of averaging
=======================
For a stable system, it is fairly obvious that the following axiom must be satisfied:
[V.]{} if the same quantity $\omega$ is added to each $x_k$, then the average will increase by the same amount $\omega$.
I have had a detailed discussion of this axiom with the practicing economist V. N. Baturin.
Obviously, in normal conditions, the nonlinear average of $x_i$ must also increase by this amount. We take this fact as [*Axiom*]{} V.
This axiom leads to a unique solution in the nonlinear case, i.e., it is naturally satisfied in the linear case (the arithmetic mean) and by a unique (up to an identical constant by which we can multiply all the incomes $x_i$) nonlinear function.
In fact, the incomes $x_i$ are calculated in some currency and, in general, must be multiplied by some constant $\beta$ corresponding to the purchasing power of this currency, so that this constant (the parameter $\beta$) must be incorporated into the definition of an income. Therefore, we assert that there exists a unique nonlinear function satisfying Axiom V.
The function $\varphi(x)$ is of the form $$\varphi(x)=C\exp(Dx)+B,
\tag{2}$$ where $C,D\ne 0$, and $B$ are numbers independent of $x$.
Semiring: an example of a linear self-adjoint operator
======================================================
Consider the semiring generated by the nonlinear average and the space $L_2$ with values in this semiring.
First, consider the heat equation $$\frac{\partial u}{\partial t}
=\frac h2\,\frac{\partial^2u}{\partial x^2}.
\tag{3}$$ Here $h$ is a small parameter, whose smallness is not needed at this point.
Equation is linear. As is well known, this means that if $u_1$ and $u_2$ are its solutions, then their linear combination $$u=\lambda_1u_1+\lambda_2u_2
\tag{4}$$ is also its solution. Here $\lambda_1$ and $\lambda_2$ are constants.
Next, we make the following substitution. Let $$u=e^{-M/h}.
\tag{5}$$ Then the new unknown function $M(x,t)$ satisfies the nonlinear equation $$\frac{\partial M}{\partial t}
+\frac12\biggl(\frac{\partial M}{\partial x}\biggr)^2
-\frac h2\,\frac{\partial^2M}{\partial x^2}=0.
\tag{6}$$ This well known equation is sometimes called the [*Bürgers equation*]{}[^3].
To the solution $u_1$ of Eq. there corresponds the solution $M_1=-h\ln u_1$ of Eq. and to the solution $u_2$ of Eq. there corresponds the solution $M_2=-h\ln u_2$ of Eq. . To the solution of Eq. there corresponds the solution $$M=-h\ln\bigl(e^{-(M_1+\mu_1)/h}+ e^{-(M_2+\mu_2)/h}\bigr),$$ where $\mu_i=-h\ln\lambda_i$, $i=1,2$.
Hence Eq. is also linear, but it is linear in the function space with the following operations:
[$\bullet$]{} the operation of addition $a\oplus b=-h\ln(e^{-a/h}+ e^{-b/h})$;
[$\bullet$]{} the operation of multiplication $a\odot\lambda=a+ \lambda$.
Now, the substitution $M=-h\ln u$ takes zero to infinity and one to zero. Thus, in this new space, the generalized zero is $\infty$: $\bold0=\infty$, and the generalized unit is the ordinary zero: $\bold1= 0$. The function space in which the operations $\oplus$ and $\odot$ have been introduced together with the zero $\bold0$ and the unit $\bold1$ is isomorphic to the ordinary function space with ordinary multiplication and addition.
We can thus imagine that, somewhere on another planet, people have grown accustomed to the newly introduced operations $\oplus$ and $\odot$ and, from their point of view, Eq. is linear.
All this, of course, is trivial and there is no need for people living on our planet to learn new arithmetical operations, because we can, by a change of function, pass from Eq. to Eq. , which is linear in the accepted meaning. However, it turns out that the “kingdom of crooked mirrors,” which this semiring yields, is related to “capitalist” economics [@2].
In the function space with values in the ring $a\oplus b=-h\ln(e^{-a/h}+ e^{-b/h})$, $\lambda\odot b=\lambda+ b$, we introduce the inner product $$(M_1,M_2)=-h\ln\int e^{(M_1+W_2)/h}\,dx.$$ Let us show that it possesses bilinear properties in this space; namely, $$(a\oplus b,c)=(a,c)\oplus(b,c),
\qquad
(\lambda\odot a,c)=\lambda\odot(a,c).$$ Indeed, $$\begin{aligned}
(a\oplus b,c)
&=-h\ln\biggl(\int\exp\biggl(\frac{-(-h\ln(e^{-a/h}+e^{-b/h})+
c)}h\biggr)
dx\biggr)\tag{7}
\\ &
=-h\ln\biggl(\int(e^{-a/h}+e^{-b/h})e^{-c/h}\,dx\biggr)\notag
\\ &
=-h\ln\biggl(\int e^{-(a+c)/h}\,dx
+\int e^{-(b+c)/h}\,dx\biggr)
=(a,c)\oplus(b,c),\notag
\\
(\lambda\odot a,c)
&=-h\ln\int e^{-(a+\lambda)/h}e^{-c/h}\,dx\notag
\\ &
=-h\ln\biggl(e^{-\lambda/h}\int e^{-(a+c)/h}\,dx\biggr)
=\lambda+\ln\int e^{-(a+c)/h}\,dx
=\lambda\odot(a,c).\notag\end{aligned}$$ Let us give an example of a self-adjoint operator in this space. Consider the operator $$L\:W\to W\odot\biggl(-h\ln\biggl(\frac{(W')^2}{h^2}
-\frac{W''}h\biggr)\biggr).$$ And now let us verify its self-adjointness: $$\begin{aligned}
&
(W_1,LW_2)
=-h\ln\int e^{-(W_1+LW_2)/h}\,dx
\tag{8}
\\ &\qquad
=-h\ln\int\exp\biggl[-\biggl(W_1+W_2-h\ln\biggl(\frac{W'_2)^2}{h^2}
-\frac{W''_2}h\biggr)\biggr)\bigg/h\biggr]dx\notag
\\ &\qquad
=-h\ln\int e^{-W_1/h}e^{-W_2/h}\biggl(\frac{(W_2')^2}{h^2}
-\frac{W''_2}h\biggr)dx
=-h\ln\int e^{-W_1/h}\frac{d^2}{dx^2}e^{-W_2/h}\,dx\notag
\\ &\qquad
=-h\ln\int\frac{d^2}{dx^2}
e^{-W_1/h}e^{-W_2/h}\,dx
=-h\ln\int e^{-W_1/h}\biggl(\frac{(W'_1)^2}{h^2}
-\frac{W''_1}h\biggr)e^{-W_2/h}\,dx\notag
\\ &\qquad
=-h\ln\int\exp\biggl[-\biggl(W_1-h\ln\biggl(\frac{(W'_1)^2}{h^2}
-\frac{W'_2}h\biggr)\biggr)\bigg/h\biggr]dx\notag
\\ &\qquad
=-h\ln\int e^{-(LW_1+W_2)/h}\,dx
=(LW_1,W_2).\notag\end{aligned}$$ It is also easy to verify linearity.
We construct the resolving operator of the Bürgers equation: $L\:W_0\to W$, where $W$ is the solution of Eq. satisfying the initial condition $W|_{t=0}= 0$.
The solution of Eq. satisfying the condition $u|_{t=0}= u_0$ is of the form $$u=\frac1{\sqrt{2\pi h}}
\int e^{-(x-\xi)^2/2th}u_0(\xi)\,d\xi.$$ Taking into account the fact that $u= e^{-W/h}$, $W=-h\ln u$, we obtain the resolving operator $L_t$ of the Bürgers equation: $$L_tW_0=-\frac h{\sqrt{2\pi h}}
\ln\int e^{-((x-\xi)^2/2th+M(\xi)/h)}\,d\xi.
\tag{9}$$ The operator $L_t$ is self-adjoint in terms of the new inner product.
Theorem on the nonlinear average
================================
Consider a collection of prices $\lambda_i$, where $i=1,\dots,n$, and a collection of numbers $g_i$ equal to the number of financial instruments (FI), which are shares, bonds, etc., of [*different types*]{} having the price $\lambda_i$. The prices $\lambda_i$ are, by definition, positive numbers; without loss of generality, we can further assume that the prices are numbered so as to satisfy the inequalities $$0<\lambda_1<\lambda_i<\lambda_n
\qquad \text{for
all}\quad i=2,\dots,n-1.
\tag{10}$$ The total number of FIs of different type is denoted by $G$. This number is $$G=\sum_{i=1}^ng_i.
\tag{11}$$ Let $k_i$ denote the number of FIs purchased at the price $\lambda_i$. Since $g_i$ different FIs are sold at the price $\lambda_i$, it follows that the number of different methods of buying $k_i$ FI’s at the price $\lambda_i$ can be expressed by $$\gamma_i(k_i)=\frac{(k_i+g_i-1)!}{k_i!\,(g_i-1)!}.
\tag{12}$$ The number of different methods of buying the collection $\{k\}$ of FIs consisting of $k_1,\dots,k_n$ FIs bought at corresponding prices $\lambda_1,\dots,\lambda_n$, is expressed by and is equal to $$\gamma(\{k\})=\prod_{i=1}^n\gamma_i(k_i)
=\prod_{i=1}^n\frac{(k_i+g_i-1)!}{k_i!\,(g_i-1)!}.
\tag{13}$$ The expenditure of the buyer in the purchase of the collection FI $\{k\}$ is equal to $${\mathcal{B}}(\{k\})=\sum_{i=1}^n\lambda_ik_i.
\tag{14}$$
Let us carry out nonlinear averaging of the expenditure over budget restraints: $$M(\beta,N)
=-\frac1{\beta N}\ln\biggl(\frac{N!\,(G-1)!}{(N+G-1)!}
\sum_{\{k_i=N\}}\gamma(\{k\})\exp\bigl(-\beta{\mathcal{B}}(\{k\})\bigr)\biggr),
\tag{15}$$ where $\beta$ is a positive parameter.
We split the FIs arbitrarily into $m$ nonintersecting groups, where $m\le n$. This implies that, by employing some method, we choose two sequences $i_\alpha$ and $j_\alpha$, where $\alpha=1,\dots,m$, satisfying the conditions $$i_\alpha\le j_\alpha,\; \;
i_{\alpha+1}=j_\alpha+1, \quad \alpha=1,\dots,m,
\qquad
i_1=1,\; \;j_m=n,
\tag{16}$$ and assume that the FIs are contained in the group indexed by $\alpha$ if the number $i$ of its price $\lambda_i$ satisfies the condition $i_\alpha\le i\le j_\alpha$. Note that there are a number of ways by which we can choose the sequences $i_\alpha$ and $j_\alpha$ satisfying conditions .
The number of FIs contained in the group indexed by $\alpha$ is expressed by the formula $$G_\alpha=\sum_{i=i_\alpha}^{j_\alpha}g_i,
\tag{17}$$ while the number of FIs bought from the group with number $\alpha$ in the purchase of the collection $\{k\}=k_1,\dots,k_n$, is expressed by the formula $$N_\alpha=\sum_{i=i_\alpha}^{j_\alpha}k_i.
\tag{18}$$ The collection $\{k\}$ satisfies condition ; therefore, the collection of numbers $N_\alpha$ satisfies the equality $$\sum_{\alpha=1}^mN_\alpha=N;
\tag{19}$$ also, it follows from that the number $G_\alpha$ satisfies the relation $$\sum_{\alpha=1}^mG_\alpha=G.
\tag{20}$$ Here $G$ depends on $N$ so that the following relation holds: $$\lim_{N\to\infty}\frac GN={\widetilde}g>0.
\tag{21}$$ In addition, we assume that the partition of the FIs into the groups satisfies the following condition: $m$ is independent of $N$, and the $G_\alpha$ depend on $N$ so that the following relations hold: $$\lim_{N\to\infty}\frac{G_\alpha}N
={\widetilde}g_\alpha>0
\qquad \text{for
all}\quad
\alpha=1,\dots,m.
\tag{22}$$ Hence $$\lim_{N\to\infty}\frac{N_\alpha}N\approx{\overline}n_\alpha>0,
\qquad \sum{\overline}n_\alpha=1.$$ By , the quantities ${\widetilde}g_\alpha$ and ${\widetilde}g$ are related by $$\sum_{\alpha=1}^m{\widetilde}g_\alpha={\widetilde}g.
\tag{23}$$ We denote $${\mathcal{N}}_\alpha(\beta,N)
=\sum_{i=i_\alpha}^{j_\alpha}
\frac{g_i}{\exp(\beta(\lambda_i+\nu))-1},
\tag{24}$$ where $\delta> 0$ is an arbitrary parameter and $\nu$ is specified by the equation $$N=\sum_{i=1}^n\frac{g_i}{\exp(\beta(\lambda_i+\nu))-1}.
\tag{25}$$ Further, we use the notation $$\Gamma(\beta,N)
=\frac{(N+G-1)!}{N!\,(G-1)!}
\exp\bigl(-\beta NM(\beta,N)\bigr).
\tag{26}$$
We can also consider the case $\beta< 0$; moreover, for $\beta< 0$, we choose the solution of Eq. satisfying the condition $\nu< -\lambda_n$ and, for $\beta> 0$, we choose the solution satisfying the condition $\nu> -\lambda_1$.
Let condition be satisfied, and let $\Delta=aN^{3/4+\delta}$, where $a$ and $\delta<1/3$ are positive parameters independent of $N$. Then, for any $\varepsilon> 0$, the following relation holds as $N\to \infty$, regardless of the value of $N$[:]{} $$\begin{aligned}
&
\frac1{\Gamma(\beta,N)}
\sum_{\sum\{k_i\}=N,\,\sum^m_{\alpha=1}
(N_\alpha(\{k\})-{\mathcal{N}}_\alpha(\beta,N))^2\ge\Delta}
\gamma(\{k\})\exp\bigl(-\beta{\mathcal{B}}(\{k\})\bigr)
\tag{27}
\\ &\qquad
=O\biggl(\exp\biggl(-\frac{(1-\varepsilon)a^2N^{1/2+2\delta}}
{2{\widetilde}gd}\biggr)\biggr),
\notag\end{aligned}$$ where $d$ is defined by $$d=\begin{cases}
\dfrac{\exp(-\beta(\lambda_1+\nu))}{(\exp(-\beta(\lambda_1+\nu))-1)^2}
& \text{for}\ \beta<0, \\
\dfrac{\exp(-\beta(\lambda_n+\nu))}{(\exp(-\beta(\lambda_n+\nu))-1)^2}
& \text{for}\ \beta>0.
\end{cases}
\tag{28}$$ In other words, the contribution to the average expenditure of the buyer from the number $N_\alpha$ of the purchased FI’s [(]{}of the order of $N$[)]{}, which differs from ${\mathcal{N}}_\alpha(\beta,N)$ by a value $O(N^{3/4+\delta})$, is of exponentially small value.
Thus, for the case in which the FIs can be divided into sufficiently large groups (condition ), given a fixed nonlinear average expenditure of the buyer, the number of purchased FIs belonging to the group $\alpha$ is, in most cases, approximately equal to ${\mathcal{N}}_\alpha(\beta,N)$, which is an analog of the law of large numbers.
To find $\beta$, we propose the following method. Suppose we are given a collection $N_\alpha^a$ of actual purchases from the group $\alpha$, $\alpha=1,2,\dotsb,m$, of financial instruments. Consider the following function of $\beta$: $$DN(\beta)=\sum_{\alpha=1}^m
\bigl(N_\alpha^a-{\mathcal{N}}_\alpha(\beta,\nu_\alpha)\bigr)^2,
\tag{29}$$ where ${\mathcal{N}}_\alpha(\beta,\nu_\alpha)$ is given by . For $\beta$ we take the value for which the function attains its minimum.
The minimum of this value, which can be specified, for example, at the stock exchange, shows the degree of randomness of the distribution of FIs over the portfolios of the traders. If the minimum is zero, then the distribution over the portfolios is random (see [@3]).
Before proving the theorem, we shall prove several assertions and lemmas.
Suppose that the buyer possesses a sum of money $B$ (budget restraint), but not all of this sum is necessarily spent on purchases; suppose that the buyer purchases $N$ FIs. The total number of different types of purchase corresponding to the $N$ purchased FIs when the sum of money spent does not exceed $B$ can be expressed by the formula [@4] $$\sigma(B,N)
=\sum_{\{k=N\}}\gamma(\{k\})\Theta\bigl(B-{\mathcal{B}}(\{k\})\bigr).
\tag{30}$$ Here ${\mathcal{B}}(\{k\})\le B$, ${\mathcal{B}}(\{k\})$ is defined by , $\Theta(x)$ is the Heaviside function: $$\Theta(x)=\begin{cases}
0& \text{for}\ x<0,\\
1& \text{for}\ x\ge1,
\end{cases}$$ and $\sum_{\{k\}}^N$ denotes the sum over all collections $\{k\}$ of nonnegative integers $k_1,\dots,k_n$ satisfying the condition $$\sum_{i=1}^nk_i=N.
\tag{31}$$ Note that, by , the expenditure for any collection $\{k\}$ fulfilling condition satisfies the inequalities $$0<N\lambda_1\le {\mathcal{B}}(\{k\})\le N\lambda_n.
\tag{32}$$
The number $\sigma(B,N)$ of different types of purchase is a piecewise constant function of the variable $B$ and possesses the following properties: $$\alignedat2
\sigma(B,N)
&\le\sigma(B',N) &\qquad \text{for}\quad B&\le B',
\\
\sigma(B,N)
&=0 &\qquad \text{for}\quad B&<N\lambda_1,
\\
\sigma(B,N)
&=\sigma(N\lambda_n,N) &\qquad \text{for}\quad B&\ge N\lambda_n.
\endalignedat$$ These properties of $\sigma(B,N)$ follow immediately from the definition and inequality .
For any positive $\beta$ and any $N$, the following identity holds[:]{} $$\int_0^\infty e^{-\beta B}\,d\sigma(B,N)
=\Gamma(\beta,N).
\tag{33}$$
[**Proof.**]{} Substituting into the left-hand side of relation and changing the order of summation and integration, we obtain $$\int_0^\infty e^{-\beta B}\,d\sigma(B,N)
=\sum_{\{k\}}^N\gamma(\{k\})\int_0^\infty
e^{-\beta B}\,d\Theta\bigl(B-{\mathcal{B}}(\{k\})\bigr).
\tag{34}$$ Next, we use the equality $$\int_0^\infty e^{-\beta B}\,d\Theta(B-B_0)
=e^{-\beta B_0},
\tag{35}$$ which is valid for all positive $\beta$ and $B_0$. Taking into account, from we obtain the following equality: $$\int_0^\infty e^{-\beta B}\,d\sigma(B,N)
=\sum_{\{k\}}^N\gamma(\{k\})
\exp\bigl(-\beta{\mathcal{B}}(\{k\})\bigr).
\tag{36}$$ It follows from and that the expression on the right-hand side of relation is equal to $\Gamma(\beta,N)$. Thus, the proposition is proved.
We introduce the following function $\zeta(\beta,\nu)$ defined for all $\nu> -\lambda_1$ and $\beta> 0$: $$\zeta(\beta,\nu)
=\sum_{N=0}^\infty\Gamma(\beta,N)e^{-\beta N\nu}.
\tag{37}$$
For all $\nu> -\lambda_1$ and $\beta> 0$, the following relation holds[:]{} $$\zeta(\beta,\nu)
=\prod_{i=1}^n\frac1{(1-\exp(-\beta(\lambda_i+\nu)))^{g_i}}.
\tag{38}$$
[**Proof.**]{} By formulas , , we have $$\sum_{N=0}^\infty\Gamma(\beta,N)e^{-\beta N\nu}
=\sum_{N=0}^\infty e^{-\beta N\nu}
\sum_{\{k\}}^N\gamma(\{k\})\exp\bigl(-\beta{\mathcal{B}}(\{k\})\bigr).
\tag{39}$$ Taking into account the explicit form of $\gamma(\{k\})$ and ${\mathcal{B}}(\{k\})$ (see ), we find that the sum in relation splits into the following product of sums: $$\sum_{N=0}^\infty e^{-\beta N\nu}
\sum_{\{k\}}^N\gamma(\{k\})\exp\bigl(-\beta{\mathcal{B}}(\{k\})\bigr)
=\prod_{i=1}\biggl(\sum_{k_i=1}^\infty
\frac{(k_i+g_i-1)!}{k_i!\,(g_i-1)!}
e^{-\beta(\lambda_i+\nu)k_i}\biggr).
\tag{40}$$ We now use the equality $$\sum_{k=0}^\infty\frac{(k+g-1)!}{k!\,(g-1)!}x^k
=\frac1{(1-x)^g},
\tag{41}$$ valid for any natural number $g$ and any $x$ such that $|x|< 1$. Since $\nu> -\lambda_1$ and $\beta> 0$, by from , , we obtain . The assertion is proved.
In order to prove certain properties of the function $\Gamma(\beta,N)$ (see ), we shall use the following lemma.
The functions $z_n(x_1,\dots,x_n;N)$ given by $$z_l(x_1,\dots,x_l;N)
=\sum_{\{k\}}^N\exp\biggl(\sum_{i=1}^lx_ik_i\biggr),
\tag{42}$$ where $l$ and $N$ are arbitrary integers satisfying the conditions $l\ge 2$ and $N\ge 1$, satisfy, for all real numbers $x_1,\dots,x_n$, the inequality $$z_l(x_1,\dots,x_l,N)^2
>z_l(x_1,\dots,x_l,N-1)z_l(x_1,\dots,x_l,N+1).
\tag{43}$$
[**Proof.**]{} Let us prove the lemma by induction. First, we prove that is valid for $l= 2$. If $l= 2$ and $x_2= x_1$, then, for all nonnegative integers $N$, the function is equal to $$z_2(x_1,x_1;N)
=\sum_{k=0}^Ne^{x_1N}=(N+1)e^{x_1N},$$ and it can be verified by elementary means that, in that case, inequality holds. If $l= 2$ and $x_2\ne x_1$, then, for all nonnegative integers $N$, the function is of the form $$z_2(x_1,x_2;N)
=\sum_{k=0}^Ne^{x_1k+x_2(N-k)}
=e^{x_2N}\frac{1-e^{(x_1-x_2)(N+1)}}{1-e^{x_1-x_2}}.
\tag{44}$$ Substituting into , after a few manipulations we find that, in the case under consideration, inequality is equivalent to the following inequality: $$\frac12\bigl(e^{x_1-x_2}+e^{x_2-x_1}\bigr)>1.
\tag{45}$$ However, since the function $y(x)=\exp(x)$ is convex, inequality holds for all numbers $x_1$ and $x_2$ not equal to one another. Thus, inequality is proved for $l= 2$. Next, suppose that inequality is proved for $l= k$, where $k\ge 2$, and let us prove that this inequality holds for $n=l+ 1$. Note that the functions possess the following property, which is a trivial consequence of their explicit form: $$z_{k+1}(x_1,\dots,x_{k+1};N)
=e^{x_{k+1}N}\sum_{m=0}^Nz_k({\widetilde}x_1,\dots,{\widetilde}x_k;m),
\tag{46}$$ where ${\widetilde}x_i=x_i- x_{k+1}$, $i=1,\dots,n$. Substituting into inequality and carrying out a few manipulations, we see that inequality for $n=k+ 1$ is equivalent to the inequality $$z_k({\widetilde}x;0)z_k({\widetilde}x;N)
+\sum_{m=0}^{N-1}\bigl(z_k({\widetilde}x;k+1)z_k({\widetilde}x;N)
-z_k({\widetilde}x;k)z({\widetilde}x;N+1)\bigr)
>0,
\tag{47}$$ where $z_k({\widetilde}x;m)$ denotes $z_k({\widetilde}x_1,\dots,{\widetilde}x_k;m)$. By assumption, inequality holds for $n= k$; this implies that, for all $k=0,\dots,N- 1$, we have $$z_k({\widetilde}x;k+1)z_k({\widetilde}x;N)-z_k({\widetilde}x;k)z({\widetilde}x;N+1)>0,
\tag{48}$$ and hence all summands on the left-hand side of inequality are positive, i.e., inequality holds. Thus, the validity of for $l=k+ 1$ is proved and, therefore, the assertion of the lemma is also proved by induction.
Lemma 1 implies the convexity of the logarithm of the function $\Gamma(\beta,N)$ (given by ) with respect to the discrete variable $N$.
For all natural numbers $N$ and all positive numbers $\beta$, the following inequalities hold[:]{} $$\begin{gathered}
\Gamma(\beta,N)^2
>\Gamma(\beta,N-1)\Gamma(\beta,N+1),
\tag{49}
\\
\Gamma(\beta,N-1)
<e^{\beta\lambda_1}\Gamma(\beta,N).
\tag{50}\end{gathered}$$
[**Proof.**]{} To prove these inequalities, it suffices to note that, in view of formula , $\Gamma(\beta,N)$ can be written in the form $$\Gamma(\beta,N)
=\sum_{\{k\}}^N\gamma(\{k\})\exp\bigl(-\beta{\mathcal{B}}(\{k\})\bigr).
\tag{51}$$ The right-hand side of relation is equal to the function for $$\begin{gathered}
l=G,
\qquad
x_1=\dots=x_{g_1}=-\beta\lambda_1,
\quad
x_{g_1+1}=\dots=x_{g_1+g_2}=-\beta\lambda_2,
\quad \dots\,,\tag{52}
\\
x_{G-g_n+1}=\dots=x_G=-\beta\lambda_G;\notag\end{gathered}$$ this equality follows from the formula $$\sum_{0\le m_1,\dots,m_g}^{m_1+\dots+m_g=k}1
=\frac{(k+g-1)!}{k!\,(g-1)!}.$$ Thus, inequality is a consequence of and, to prove , it suffices to use relation .
Property implies another property of $\Gamma(\beta,N)$.
For any given positive $\beta$ and for any nonnegative integer $N$, there exists a number $\nu> -\lambda_1$ such that, for all $N'\ne N$, $$\Gamma(\beta,N)e^{-\beta N\nu}
>\Gamma(\beta,N')e^{-\beta N'\nu}.
\tag{53}$$
[**Proof.**]{} Using , we choose any $\nu$ satisfying the inequality $$\frac1\beta\ln\biggl(\frac{\Gamma(\beta,N+1)}{\Gamma(\beta,N)}\biggr)
<\nu
<\frac1\beta\ln\biggl(\frac{\Gamma(\beta,N)}{\Gamma(\beta,N-1)}\biggr).
\tag{54}$$ It follows from that such a $\nu$ also satisfies the inequality $\nu> -\lambda_1$. Moreover, inequalities and imply the inequalities $$\begin{aligned}
e^{-\beta(N'+1)\nu}\Gamma(\beta,N'+1)
&>e^{-\beta N'\nu}\Gamma(\beta,N')
&\qquad \text{for
all}\quad N', \quad &0\le N'\le N-1,
\tag{55}
\\
e^{-\beta(N'+1)\nu}\Gamma(\beta,N'+1)
&<e^{-\beta N'\nu}\Gamma(\beta,N')
&\qquad \text{for
all}\quad N', \quad &N\le N'.\notag\end{aligned}$$ However, the validity of inequalities implies that of inequality for all $N'\ne N$. The proposition is proved.
Next, let us prove another lemma, which gives an estimate of the nonlinear average via relation . Suppose that $p_l$, $l=0,1,\dots$, is a sequence of numbers satisfying the conditions $$p_l\ge0 \quad \text{for
all}\ l=0,1,\dots,
\qquad
\sum_{l=0}^\infty p_l=1,
\tag{56}$$ and also $$\begin{aligned}
{\overline}l
&\equiv\sum_{l=0}^\infty lp_l<\infty,
\tag{57}
\\
Dl
&\equiv\sum_{l=0}^\infty(l-{\overline}l)^2p_l<\infty.
\tag{58}\end{aligned}$$ It follows from conditions that the numbers $p_l$ are bounded above and we can choose one of the maximal numbers among them. In other words, there exists an index, which will be denoted by $L$, such that the following condition is satisfied: $$p_L\ge p_l \qquad \text{for all}\quad l=0,1,\dotsc.
\tag{59}$$
For an arbitrary sequence of numbers $p_l$, $l=0,1,\dots$, satisfying conditions – the following inequalities hold[:]{} $$\begin{gathered}
|L-{\overline}l|\le(3Dl)^{3/4},
\tag{60}
\\
p_L\ge\frac1{\sqrt{27Dl}}.
\tag{61}\end{gathered}$$
\*\*\* [**Proof.**]{} For any $\Delta> 0$, the following Chebyshev inequality holds: $$\sum_{l=0}^\infty p_l\Theta(|l-{\overline}l|-\Delta)
\le\sum_{l=0}^\infty p_l\frac{(l-{\overline}l)^2}{\Delta^2}
=\frac{Dl}{\Delta^2}.
\tag{62}$$ It follows from , , and that, for any $\Delta> 0$, we have the chain of inequalities $$2\Delta p_L
\ge\sum_{|l-{\overline}l|<\Delta}p_l
\ge1-\frac{Dl}{\Delta^2}.
\tag{63}$$ From we obtain $$p_L\ge\frac1{2\Delta}\biggl(1-\frac{Dl}{\Delta^2}\biggr)
\qquad \text{for any}\quad \Delta>0.
\tag{64}$$ It can be easily shown that the function on the right-hand side of inequality attains its maximum value (with respect to the variable $\Delta$) equal to $1/\sqrt{27Dl}$ at $\Delta=\sqrt{3Dn}$. Inequality is thus proved. Now, take $\Delta=|L-{\overline}l|$; then it follows from and that $$p_L
\le\sum_{l=0}^\infty p_l\Theta(|l-{\overline}l|-|L-{\overline}l|)
\le\frac{Dl}{(L-{\overline}l)^2}.
\tag{65}$$ Taking into account, from we obtain inequality .
To each group there corresponds a particular nonlinear averaging of the expenditure; as in , the averaging of the expenditure for FIs from the group with index $\alpha$ can be expressed by the formula $$M_\alpha(\beta,N_\alpha)
=-\frac1{\beta N_\alpha}
\ln\biggl(\frac{N_\alpha!\,(G_\alpha-1)!}{(N_\alpha+G_\alpha-1)!}
\sum_{\{k\}_\alpha}^{N_\alpha}
\prod_{i=i_\alpha}^{j_\alpha}
\bigl(\gamma_i(k_i)e^{-\beta\lambda_ik_i}\bigr)\biggr),
\tag{66}$$ where $\gamma_i(k_i)$ is given by formula and $\sum_{\{k\}_\alpha}^{N_\alpha}$ denotes the sum over all collections $\{k\}_\alpha$ of nonnegative integers $k_{i_\alpha},\dots,k_{j_\alpha}$ satisfying condition . As in and , for all nonnegative integers $N_\alpha$, for all $\beta> 0$, and all $\nu> -\lambda_1$, we define the functions $$\begin{aligned}
\Gamma_\alpha(\beta,N_\alpha)
&=\frac{(N_\alpha+G_\alpha-1)!}{N_\alpha!\,(G_\alpha-1)!}
\exp\bigl(-\beta N_\alpha M_\alpha(\beta,N_\alpha)\bigr),
\tag{67}
\\
\zeta_\alpha(\beta,\nu)
&=\sum_{N_\alpha=0}^\infty
\Gamma_\alpha(\beta,N_\alpha)e^{-\beta N_\alpha\nu}.
\tag{68}\end{aligned}$$
The following relations hold[:]{} $$\begin{aligned}
\Gamma(\beta,N)
&=\sum_{\{N_\alpha\}}^N\prod_{\alpha=1}^m
\Gamma_\alpha(\beta,N_\alpha),
\tag{69}
\\
\zeta(\beta,\nu)
&=\prod_{\alpha=1}^m\zeta_\alpha(\beta,\nu),
\tag{70}
\\
\zeta_\alpha(\beta,\nu)
&=\prod_{i=i_\alpha}^{j_\alpha}
\frac1{(1-\exp(-\beta(\lambda_i+\nu)))^{g_i}},
\tag{71}\end{aligned}$$ where $\sum_{\{N_\alpha\}}^N$ denotes the sum over all collections $\{N_\alpha\}$ of nonnegative integers $N_1,\dots,N_m$ satisfying condition .
[**Proof.**]{} Relations and follow from . To prove , it suffices to use formula and take into account the fact that, for all $\Gamma_\alpha(\beta,N_\alpha)$, $\alpha=1,\dots,m$, similar formulas hold, and also use the fact that, in formula , the summands of the sum split into products, which follows from the explicit forms of $\gamma(\{k\})$ and ${\mathcal{B}}(\{k\})$ .
The average number of FIs belonging to the group with index $\alpha$ in the overall purchase of $N$ FIs is determined as a function of $\beta$ as follows: $${\overline}N_\alpha(\beta,N)
=\frac1{\Gamma(\beta,N)}\sum_{\{k\}}^N
N_\alpha(\{k\})\gamma(\{k\})
\exp\bigl(-\beta{\mathcal{B}}(\{k\})\bigr),
\tag{72}$$ where $N_\alpha(\{k\})$ depends on the collection $\{k\}$ by formula . In studying the averages , we also need the following functions: $$\begin{aligned}
{\widetilde}N(\beta,\nu)
&=\frac1{\zeta(\beta,\nu)}
\sum_{N=0}^\infty N\Gamma(\beta,N)
e^{-\beta N\nu},
\tag{73}
\\
D{\widetilde}N(\beta,\nu)
&=\frac1{\zeta(\beta,\nu)}\sum_{N=0}^\infty
\bigl(N-{\overline}N(\beta,\nu)\bigr)^2
\Gamma(\beta,N)e^{-\beta N\nu},
\tag{74}
\\
{\widetilde}N_\alpha(\beta,\nu)
&=\frac1{\zeta(\beta,\nu)}\sum_{\{k\}}N_\alpha(\{k\})
\gamma(\{k\})\exp\bigl(-\beta({\mathcal{B}}(\{k\})+\nu N(\{k\}))\bigr),
\tag{75}
\\
D{\widetilde}N_\alpha(\beta,\nu)
&=\frac1{\zeta(\beta,\nu)}\sum_{\{k\}}
\bigl(N_\alpha(\{k\})-{\widetilde}N_\alpha(\beta,\nu)\bigr)^2
\gamma(\{k\})
\exp\bigl(-\beta({\mathcal{B}}(\{k\})+\nu N(\{k\}))\bigr),
\tag{76}\end{aligned}$$ where $N(\{k\})$ depends on the collection $\{k\}$ by formula and $\sum_{\{k\}}$ denotes the sum over all collections $\{k\}$ of nonnegative integers $k_1,\dots,k_n$.
The functions – satisfy the identities $$\begin{aligned}
{\widetilde}N(\beta,\nu)
&=\sum_{i=1}^n\frac{g_i}{\exp(\beta(\lambda_i+\nu))-1},
\tag{77}
\\
D{\widetilde}N(\beta,\nu)
&=\sum_{i=1}^n\frac{g_i\exp(\beta(\lambda_i+\nu))}
{(\exp(\beta(\lambda_i+\nu))-1)^2},
\tag{78}
\\
{\widetilde}N_\alpha(\beta,\nu)
&=\sum_{i=i_\alpha}^{j_\alpha}
\frac{g_i}{\exp(\beta(\lambda_i+\nu))-1},
\tag{79}
\\
D{\widetilde}N_\alpha(\beta,\nu)
&=\sum_{i=i_\alpha}^{j_\alpha}
\frac{g_i\exp(\beta(\lambda_i+\nu))}
{(\exp(\beta(\lambda_i+\nu))-1)^2}.
\tag{80}\end{aligned}$$
[**Proof.**]{} First, note that the functions and can be written as $$\begin{aligned}
{\widetilde}N(\beta,\nu)
&=-\frac1\beta\frac\partial{\partial\nu}
\ln\bigl(\zeta(\beta,\nu)\bigr),
\tag{81}
\\
D{\widetilde}N(\beta,\nu)
&=-\frac1\beta\frac\partial{\partial\nu}{\widetilde}N(\beta,\nu);
\tag{82}\end{aligned}$$ this can be verified by differentiating $\zeta(\beta,\nu)$ . Next, substituting into and , we obtain and . Now, taking into account the explicit form of $\gamma(\{k\})$ and ${\mathcal{B}}(\{k\})$ as well as formulas , , and , we can write expressions , in a form similar to , : $$\begin{aligned}
{\widetilde}N_\alpha(\beta,\nu)
&=\frac1{\zeta_\alpha(\beta,\nu)}
\sum_{N_\alpha=0}^\infty N_\alpha\Gamma_\alpha(\beta,N_\alpha)
e^{-\beta N_\alpha\nu},
\tag{83}
\\
D{\widetilde}N(\beta,\nu)
&=\frac1{\zeta_\alpha(\beta,\nu)}\sum_{N_\alpha=0}^\infty
\bigl(N_\alpha-{\overline}N_\alpha(\beta,\nu)\bigr)^2
\Gamma_\alpha(\beta,N_\alpha)e^{-\beta N_\alpha\nu}.
\tag{84}\end{aligned}$$ Therefore, the functions and satisfy formulas similar to and , only $\zeta(\beta,\nu)$ must be replaced by $\zeta_\alpha(\beta,\nu)$; substituting into these formulas, we obtain and . The proposition is proved.
In order to prove the exponential estimate in the assertion of Theorem 2, we shall use the following assertion.
For all positive $\Delta$, the following inequality holds[:]{} $$\begin{aligned}
&
\frac1{\zeta(\beta,\nu)}\sum_{\{k\}}
\Theta\biggl(\sum_{\alpha=1}^m
|N_\alpha-{\widetilde}N_\alpha(\beta,\nu)|-\Delta\biggr)\gamma(\{k\})
\exp\bigl(\beta({\mathcal{B}}(\{k\})+\nu N(\{k\}))\bigr)
\tag{85}
\\ &\qquad
\le2^m\exp\biggl(-\frac{\Delta^2}{2Gd}\biggr),
\notag\end{aligned}$$ where $d$ is defined by formula .
[**Proof.**]{} We use the following properties of the hyperbolic cosine $\cosh(x)=(e^x+e^{-x})/2$: $$\prod_{\alpha=1}^m\cosh(x_\alpha)
\ge\biggl(\cosh\biggl(\frac\Delta m\biggr)\biggr)^m
\quad \forall x_\alpha:
\sum_{\alpha=1}^m|x_\alpha|\ge\Delta,
\qquad
\frac1{\cosh(x)}\le2e^{-x}.
\tag{86}$$ Inequalities , together with formula , lead to the following (exponential) Chebyshev inequality: $$\begin{aligned}
&
\sum_{\{k\}}\Theta\biggl(\sum_{\alpha=1}^m
|N_\alpha-{\widetilde}{N}_\alpha(\beta,\nu)|-\Delta\biggr)\gamma(\{k\})
\exp\bigl(\beta({\mathcal{B}}(\{k\})+\nu N(\{k\}))\bigr)
\tag{87}
\\ &\qquad
\le2^me^{-c\Delta}\prod_{\alpha=1}^m\biggl(\sum_{N_\alpha=0}^\infty
\Gamma_\alpha(\beta,N_\alpha)e^{\beta\nu N_\alpha}
\cosh\bigl(c(N_\alpha-{\widetilde}{N}_\alpha(\beta,\nu))\bigr)\biggr),
\notag\end{aligned}$$ where $c$ is an arbitrary positive number. Taking and into account, we rewrite in the form $$\begin{aligned}
&
\frac1{\zeta(\beta,\nu)}\sum_{\{k\}}\Theta\biggl(\sum_{\alpha=1}^m
|N_\alpha-{\widetilde}{N}_\alpha(\beta,\nu)|-\Delta\biggr)\gamma(\{k\})
\exp\bigl(\beta({\mathcal{B}}(\{k\})+\nu N(\{k\}))\bigr)
\\ &\qquad
\le2^me^{-c\Delta}\prod_{\alpha=1}^m\frac1{2\zeta_\alpha(\beta,\nu)}
\biggl(\exp(-c{\widetilde}{N}_\alpha(\beta,\nu))
\zeta_\alpha\biggl(\beta,\nu+\frac c\beta\biggr)
\tag{88}
\\ &\qquad \kern30mm
+\exp(c{\widetilde}{N}_\alpha(\beta,\nu))
\zeta_\alpha\biggl(\beta,\nu-\frac c\beta\biggr)\biggr).
\notag\end{aligned}$$
Further, we use the identity $$\begin{aligned}
&
\ln\biggl(\zeta_\alpha\biggl(\beta,\nu+\frac c\beta\biggr)\biggr)
=\ln(\zeta_\alpha(\beta,\nu))
+\frac c\beta\frac\partial{\partial\nu}
\ln(\zeta_\alpha(\beta,\nu))
\tag{89}
\\ &\qquad
+\int_\nu^{\nu+c/\beta}d\nu'\,\biggl(\nu+\frac c\beta-\nu'\biggr)
\frac{\partial^2}{{\partial\nu'}^2}\ln(\zeta_\alpha(\beta,\nu')),
\notag\end{aligned}$$ which holds because the function can be differentiated twice with respect to the variable $\nu$. From the explicit form of the function $\zeta_\alpha(\beta,\nu)$ , we obtain $$\frac{\partial^2}{\partial\nu^2}\ln(\zeta_\alpha(\beta,\nu))
=\beta^2\sum_{i=i_\alpha}^{j_\alpha}
\frac{g_i\exp(-\beta(\lambda_i+\nu))}
{(\exp(-\beta(\lambda_i+\nu))-1)^2}
\le\beta^2G_\alpha d,
\tag{90}$$ where $d$ is expressed by . Using inequality and formulas and , from we obtain the inequality $$-c{\widetilde}N_\alpha(\beta,\nu)
+\ln\biggl(\zeta_\alpha\biggl(\beta,\nu+\frac c\beta\biggr)\biggr)
-\ln(\zeta_\alpha(\beta,\nu))
\le\frac{G_\alpha}2dc^2.
\tag{91}$$ By and , from we obtain the inequality $$\begin{aligned}
&
\frac1{\zeta(\beta,\nu)}\sum_{\{k\}}\Theta\biggl(\sum_{\alpha=1}^m
|N_\alpha-{\widetilde}N_\alpha(\beta,\nu)|-\Delta\biggr)\gamma(\{k\})
\exp\bigl(\beta({\mathcal{B}}(\{k\})+\nu N(\{k\}))\bigr)
\tag{92}
\\ &\qquad
\le2^m\exp\biggl(-c\Delta+\frac{Gdc^2}2\biggr).
\notag\end{aligned}$$ The expression on the right-hand side of inequality contains an arbitrary positive parameter $c$; it attains its minimum equal to $2^m\exp(-\Delta^2/2d)$ at $c=\Delta/d$, and hence inequality is a consequence of . The proposition is proved.
Let us suppose that $\lambda_1$ and $\lambda_n$ are independent of $N$; by , this implies that the prices $\lambda_i$, $i=1,\dots,n$, lie in an interval independent of $N$. The form of the dependence of $n$ on $N$ is not specified, since it does not affect the results obtained. It should only be noted that, by , we have $n\le G$. The conditions formulated above essentially restrict neither the choice of $\lambda_i$ and $g_i$ nor that of the partition of FIs into groups. Because of this, we cannot write the asymptotics of the nonlinear averages and and of the averages in the limit as $N\to \infty$ in the general case; however, it turns out that we can prove some properties of these averages in such a limiting case.
[**Proof of Theorem 2.**]{} By Proposition 4, for given $\beta$ and $N$, we choose a $\nu'$ such that inequality holds. By formulas , and , , , , the sequence $$p_l=\frac{\Gamma(\beta,l)e^{\beta l\nu'}}{\zeta(\beta,\nu')}$$ is a sequence for which Lemma 2 is applicable. This means that for given $\beta$, $N$, and the corresponding $\nu'$, the following inequalities hold: $$\begin{gathered}
|{\widetilde}N(\beta,\nu')-N|
\le\bigl(3D{\widetilde}N(\beta,\nu')\bigr)^{3/4},
\tag{93}
\\
1>\frac{\Gamma(\beta,N)e^{\beta N\nu'}}{\zeta(\beta,\nu')}
\ge\frac1{\sqrt{27D{\widetilde}N(\beta,\nu')}}.
\tag{94}\end{gathered}$$
Now, note that the function ${\widetilde}N(\beta,\nu)$ possesses the following properties for all $\nu> -\lambda_1$ if $\beta< 0$ and all $\nu< -\lambda_n$ if $\beta> 0$: $$\begin{gathered}
{\widetilde}N(\beta,\nu)\ge0,
\qquad
\lim_{\nu\to\infty}{\widetilde}N(\beta,\nu)=0,
\qquad
\lim_{\nu\to-\lambda_1+0}{\widetilde}N(\beta,\nu)
=\lim_{\nu\to-\lambda_n-0}{\widetilde}N(\beta,\nu)
=+\infty,
\tag{95}
\\
\frac1\beta\frac{\partial{\widetilde}N}{\partial\nu}(\beta,\nu)
=\sum_{i=1}^n\frac{g_i\exp(-\beta(\lambda_i+\nu))}
{(\exp(-\beta(\lambda_i+\nu))-1)^2}>0,
\notag
\\
\biggl|\frac{\partial{\widetilde}N}{\partial\nu}(\beta,\nu)\biggr|
\le\begin{cases}
-\dfrac{\beta G\exp(-\beta(\lambda_1+\nu))}
{(\exp(-\beta(\lambda_1+\nu))-1)^2}
& \text{for}\ \beta<0, \\
\dfrac{\beta G\exp(-\beta(\lambda_n+\nu))}
{(\exp(-\beta(\lambda_n+\nu))-1)^2}
& \text{for}\ \beta>0.
\end{cases}
\notag\end{gathered}$$ By , first, equation has a solution $\nu> -\lambda_1$ for all $\beta< 0$ and $\nu< -\lambda_n$ for all $\beta> 0$; we denote this solution by $\nu(\beta,N)$. Second, it follows from and that $$\nu'=\nu(\beta,N)+O\biggl(\frac1{N^{1/4}}\biggr),
\tag{96}$$ because, by , $D{\widetilde}N(\beta,\nu)$ satisfies the estimate $$\begin{aligned}
\frac{-\beta G\exp(-\beta(\lambda_n+\nu))}
{(\exp(-\beta(\lambda_n+\nu))-1)^2}
&\le D{\widetilde}N(\beta,\nu)
\le\frac{-\beta G\exp(-\beta(\lambda_1+\nu))}
{(\exp(-\beta(\lambda_1+\nu))-1)^2}
\qquad \text{for}\quad \beta<0,
\tag{97}
\\
\frac{\beta G\exp(-\beta(\lambda_1+\nu))}
{(\exp(-\beta(\lambda_1+\nu))-1)^2}
&\le D{\widetilde}N(\beta,\nu)
\le\frac{\beta G\exp(-\beta(\lambda_n+\nu))}
{(\exp(-\beta(\lambda_n+\nu))-1)^2}
\qquad \text{for}\quad \beta>0
\notag\end{aligned}$$ and $G$ increases in the same way as $N$ by condition . Let us now take into account the fact that the functions ${\widetilde}N_\alpha(\beta,\nu)$ satisfy identity , and, therefore, they possess properties similar to . Then it follows from , , and the definitions of ${\mathcal{N}}_\alpha(\beta,N)$ that $$|{\mathcal{N}}_\alpha(\beta,N)-{\widetilde}N_\alpha(\beta,\nu')|
=O(N^{3/4}).
\tag{98}$$
Next, suppose that $\Delta=aN^{3/4+\delta}$, where $a$ and $\delta$ are arbitrary positive parameters independent of $N$. If the condition $$|{\widetilde}N_\alpha(\beta,\nu')-N_\alpha|\ge\Delta,$$ is satisfied, then the following condition is satisfied for sufficiently large $N$: $$\sum_{\alpha=1}^m|{\mathcal{N}}_\alpha(\beta,N)-N_\alpha|
\ge\Delta'
=\Delta-\sum_{\alpha=1}^m
|{\mathcal{N}}_\alpha(\beta,N)-{\widetilde}N_\alpha(\beta,\nu')|.$$ This yields the inequality $$\begin{aligned}
&
\sum_{\{k\}}\Theta\biggl(\sum_{\alpha=1}^m
|N_\alpha-{\mathcal{N}}_\alpha(\beta,\nu)|-\Delta\biggr)\gamma(\{k\})
\exp\bigl(\beta({\mathcal{B}}(\{k\})+\nu N(\{k\}))\bigr)
\tag{99}
\\ &\qquad
\le\sum_{\{k\}}\Theta\biggl(\sum_{\alpha=1}^m
|N_\alpha-{\widetilde}N_\alpha(\beta,\nu')|-\Delta'\biggr)\gamma(\{k\})
\exp\bigl(\beta({\mathcal{B}}(\{k\})+\nu N(\{k\}))\bigr).
\notag\end{aligned}$$ In view of and , the following estimate holds for any parameter $\varepsilon> 0$ independent of $N$ and for all positive $a$ and $\delta$ as $N\to \infty$: $$\sqrt N\exp\biggl(-\frac{{\Delta'}^2}{2Gd}\biggr)
=O\biggl(\exp\biggl(-\frac{(1-\varepsilon)a^2N^{1/2+2\delta}}
{2{\widetilde}gd}\biggr)\biggr).
\tag{100}$$ Relations , , and yield . The theorem is proved.
Here we have studied the case $\beta> 0$. In exactly the same way, we may consider the case $\beta< 0$. If we take $\nu< -\alpha_n$ everywhere, then we shall obtain the same results.
The tunnel canonical operator in economics
==========================================
Equilibrium prices are determined from the condition of the equality between supply and demand in each commodity and resource. Similarly, the following pairs are determined: “flows of commodities and services–prices”; “flows of labor resources–level of wages”; “flows of natural resources–rents”; and “loan interest rate–loan volume.”
The asymptotics of $M$ and ${\widetilde}M$ is given by the tunnel canonical operator in the phase space of pairs [@5].
We consider the phase space $\mathbb R^{2n}$, where the intensive quantities play the role of coordinates and the extensive ones, of momenta. In economics, the role of the values of the random variable $\lambda_i$ can be played by the prices of the corresponding commodities and $N_i$ is, for example, the number of commodities sold, i.e., the number of people who have bought of this commodity or the bank rate of the $i$th bank, etc. Obviously, the price depends on the demand, i.e., $\lambda_i(N_i)$ is a curve in two-dimensional phase space. In two-dimensional phase space, to each point (vector) $\lambda_i$, $i=1,\dots,n$, there corresponds a vector $N_i(\lambda_1,\dots,\lambda_n)$, $i=1,\dots,n$. In a more general case, we deal with an $N$@-dimensional manifold (surface), where the “coordinates” and the “momenta” locally depend on $n$ parameters and, moreover, a certain condition holds: the Lagrange brackets of the “coordinates” and of “the momenta” with respect to these parameters are zero. Therefore, the author has called such a manifold [*Lagrangian*]{}. In other words, we can say that the form $\sum N_i\,d\lambda_i$ is closed (see the concluding remarks in [@6] and [@7]). Hence $\int N_id\lambda_i$ is independent of the path and, just as in mechanics, $\int p\,dq$ ($p$ is the momentum, $q$ is the coordinate), can be called an [*action*]{}.
The producer buys resources and transforms a resource expenditure vector into a commodity production vector. The consumer buys these commodities. Accordingly, the equilibrium prices for resources and for consumer goods are determined by the equalities given above (see also [@8]–[@17]).
In addition to such equilibrium prices, vertical pairs of isolated consumer goods and pairs “seller–buyer,” i.e., “permanent seller–permanent buyer,” can also be formed and the corresponding prices related to this pairs are generated. The analog in quantum statistics are Cooper pairs.
This construction requires the use of the ultrasecond quantization method in an abstract algebraic form which could be applicable in economics. Such a theory leads to the formation of “vertical” clusters. The method in question is developed in another paper.
[99]{}
A. N. Kolmogorov, in: [*Selected Works in Mathematics and Mechanics*]{} \[in Russian\], 1985, pp. 136–137.
V. P. Maslov, “Axioms of nonlinear averaging in financial mathematics and stock-price dynamics,” [*Teor. Veroyatnost. i Primenen.*]{} \[[*Theory Probab. Appl.*]{}\], [**48**]{} (2003), no. 4, 800–810.
“’Zero Intelligence’ Trading Closely Mimics Stock Market,” in: [http://www.newscientist.com/ article.ns?id=dn6948]{}, Katharine Davis, 05/02/01; see also other sites with reference to “Zero Intelligence.”
V. V. V’yugin and V. P. Maslov, “A concentration theorem for entropy and free energy,” [*Problemy Peredachi Informatsii*]{} \[[*Problems Inform. Transmission*]{}\], [**41**]{} (2005), no. 2, 72–88.
V. P. Maslov, “Geometric “quantization” of thermodynamics and statistical corrections at critical points,” [*Teoret. Mat. Fiz.*]{} \[[*Theoret. and Math. Phys.*]{}\], [**101**]{} (1994), no. 3, 433–441.
J. Heading, “Introduction to the phase-interval method,” in: [*Afterword to the book: V. P. Maslov, The WKB method in the multidimensional case*]{}, Mir, Moscow, 1965.
V. P. Maslov, [*Perturbation Theory and Asymptotic Methods*]{} \[in Russian\], Izd. Moskov. Univ., Moscow, 1965.
V. P. Maslov, “Approximative probabilities, the law of a quasistable market, and the phase transition from “condensate” state,” [*Dokl. Ross. Akad. Nauk*]{} \[[*Russian Acad. Sci. Dokl. Math.*]{}\], [**392**]{} (2003), no. 6, 727–732.
V. P. Maslov, “Integral equations and phase transitions in probabilistic games: Analogy with statistical physics,” [*Teor. Veroyatnost. i Primenen.*]{} \[[*Theory Probab. Appl.*]{}\], [**48**]{} (2003), no. 2, 482–502.
V. P. Maslov, “Econophysics and quantum statistics,” [*Mat. Zametki*]{} \[[*Math. Notes*]{}\], [**72**]{} (2002), no. 6, 883–891.
V. P. Maslov, “Dependence of the purchasing power and the average income of the population on the number of buyers on a specialized market and in a region: The laws of econophysics,” [*Dokl. Ross. Akad. Nauk*]{} \[[*Russian Acad. Sci. Dokl. Math.*]{}\], [**395**]{} (2004), no. 2, 164–168.
V. P. Maslov, “Consumer expenditure and turnover rate under nonlinear financial averaging: The laws of econophysics,” [*Dokl. Ross. Akad. Nauk*]{} \[[*Russian Acad. Sci. Dokl. Math.*]{}\], [**396**]{} (2004), no. 2.
V. P. Maslov, “Nonlinear financial averaging, the evolution process, and the laws of econophysics,” [*Teor. Veroyatnost. i Primenen.*]{} \[[*Theory Probab. Appl.*]{}\], [**49**]{} (2004), no. 2, 34.
V. P. Maslov, “A quasistable economy and its connection with the thermodynamics of a superfluid: A default as a phase transition of zeroth kind. 1,” [*Survey of Applied and Industrial Mathematics*]{}, [**11**]{} (2004), no. 4, 690-732; [**12**]{} (2005), no. 1, 3–40.
V. N. Baturin, S. G. Lebedev, V. P. Maslov, B. I. Sadovnikov, and A. Chebotarev, “Reconstruction of the Pareto distribution in the domain of high incomes,” [*Russian Economic Science of Current Interest*]{} (2005), no. 3.
V. P. Maslov, “Quantum economics,” [*Russian J. Math. Phys.*]{}, [**12**]{} (2005), no. 2, 219–231.
V. P. Maslov, “Capitalistic mathematics,” in: [*Manuscript at [www.viktor-maslov.narod.ru]{}*]{}, 2005.
[^1]: Moscow Institute of Electornics and Mathematics, pm@miem.edu.ru
[^2]: In our case, symmetry follows from the Bose statistics of banknotes.
[^3]: The ordinary Bürgers equation is obtained from (6) by differentiating with respect to $x$ and substituting $v=\partial M/\partial x$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We compute the higher $\Sigma$-invariants $\Sigma^m(F_{n,\infty})$ of the generalized Thompson groups $F_{n,\infty}$, for all $m,n\ge 2$. This extends the $n=2$ case done by Bieri, Geoghegan and Kochloukova, and the $m=2$ case done by Kochloukova. Our approach differs from those used in the $n=2$ and $m=2$ cases; we look at the action of $F_{n,\infty}$ on a ${\operatorname{CAT}}(0)$ cube complex, and use Morse theory to compute all the $\Sigma^m(F_{n,\infty})$.
We also obtain lower bounds on $\Sigma^m(H_n)$, for the Houghton groups $H_n$, again using actions on ${\operatorname{CAT}}(0)$ cube complexes, and discuss evidence that these bounds are sharp.
address: 'Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902'
author:
- 'Matthew C. B. Zaremsky'
title: 'On the $\Sigma$-invariants of generalized Thompson groups and Houghton groups'
---
Introduction {#introduction .unnumbered}
============
A group is of *type ${\operatorname{F}}_m$* if it has a classifying space with compact $m$-skeleton. These *finiteness properties* of groups are natural generalizations of finite generation (${\operatorname{F}}_1$) and finite presentability (${\operatorname{F}}_2$). In 1987 and 1988, Bieri, Neumann, Strebel and Renz introduced a family of geometric invariants $\Sigma^m(G)$ ($m\in{\mathbb{N}}$), defined whenever $G$ is of type ${\operatorname{F}}_m$, which reveal a wealth of information about $G$ and ${\operatorname{Hom}}(G,{\mathbb{R}})$. However, since the $\Sigma^m(G)$ contain so much information, e.g., they serve as a complete catalog of precisely which subgroups of $G$ containing $[G,G]$ have which finiteness properties, they are in general quite difficult to compute.
Thanks to this difficulty, there are very few groups whose higher $\Sigma$-invariants are completely known. If ${\operatorname{Hom}}(G,{\mathbb{R}})$ is trivial then all $\Sigma^m(G)$ are empty, so in that case the question is uninteresting, e.g., for groups with finite abelianization. Focusing on groups for which ${\operatorname{Hom}}(G,{\mathbb{R}})$ is sufficiently large, the only really robust family of groups for which the question of all the higher $\Sigma$-invariants is 100% solved is the family of right-angled Artin groups, done independently by Bux–Gonzalez [@bux99] and Meier–Meinert–VanWyk [@meier98]. Other interesting families of groups for which there are substantial partial results about the higher $\Sigma$-invariants include Artin groups [@meier01], solvable $S$-arithmetic groups [@bux04], and metabelian groups [@meinert96; @meinert97; @kochloukova99]. The question of the higher $\Sigma$-invariants of a direct product, in terms of the invariants of the factors, is also solved [@bieri10a].
The generalized Thompson groups $F_{n,\infty}$ ($n\ge2$), **which we will just denote by $F_n$ from now on**, can be quickly defined by their standard presentations $$F_n\cong{\langle x_i~(i\in{\mathbb{N}}_0)\mid x_j x_i = x_i x_{j+(n-1)} \text{ for all }i<j \rangle}\text{.}$$ These groups were first introduced by Brown in [@brown87] as an “$F$-like” version of the Higman–Thompson groups $V_{n,r}$. They generalize Thompson’s group $F$, namely $F=F_2$. The $F_n$ are all of type ${\operatorname{F}}_\infty$ [@brown87]. The group $F_n$ can also be described as the group of orientation preserving piecewise linear self homeomorphisms of $[0,1]$ with slopes powers of $n$ and breakpoints in ${\mathbb{Z}}[1/n]$. These groups are interesting for many reasons; from the perspective of $\Sigma$-invariants they are interesting for instance since every proper quotient of $F_n$ is abelian [@brown87; @brin98], and so the $\Sigma$-invariants reveal the finiteness properties of *every* normal subgroup of $F_n$. Also, $F_n$ abelianizes to ${\mathbb{Z}}^n$, and so homomorphisms to ${\mathbb{R}}$ become more and more prevalent as $n$ goes up. In contrast, the “type $V$” Higman–Thompson groups $V_{n,r}$ are virtually simple [@higman74], so have no non-trivial maps to ${\mathbb{R}}$ (and their $\Sigma$-invariants are empty).
The main result of the present work is a complete computation of $\Sigma^m(F_n)$ for all relevant $m$ and $n$. The previously known results are as follows. First, $\Sigma^1(F_2)$ was computed in the original Bieri–Neumann–Strebel paper [@bieri87]. In [@bieri10], Bieri, Geoghegan and Kochloukova computed $\Sigma^m(F_2)$ for all $m$. In the other “variable”, $n$, Kochloukova computed $\Sigma^2(F_n)$ for all $n$ in [@kochloukova12]. The techniques used there however proved difficult to extend to the cases when $n$ and $m$ are both greater than $2$. Our approach differs from those in [@bieri10] and [@kochloukova12]. We look at the action of $F_n$ on a proper ${\operatorname{CAT}}(0)$ cube complex $X_n$, and use topological and combinatorial tools to compute all the $\Sigma^m(F_n)$. This builds off work of the author and Witzel, in [@witzel15], where the $\Sigma^m(F_2)$ computations from [@bieri10] were redone using such an action of $F=F_2$.
Taking Kochloukova’s computation of $\Sigma^2(F_n)$ for granted, our main result can be phrased succinctly as:
For any $n,m \ge 2$, we have $\Sigma^m(F_n)=\Sigma^2(F_n)$.
Note that for any group $G$ of type ${\operatorname{F}}_\infty$ one always has $$\Sigma^1(G)\supseteq \Sigma^2(G) \supseteq \cdots \supseteq \Sigma^\infty(G) \text{.}$$
A more detailed description of $\Sigma^m(F_n)$ requires a lot of terminology and notation: we show that for $2\le n,m$, if $\chi=a\chi_0+c_0\psi_0+\cdots+c_{n-3}\psi_{n-3}+b\chi_1$ is a character of $F_n$ then $[\chi]$ fails to lie in $\Sigma^m(F_n)$ if and only if all $c_i$ are zero, and both $a$ and $b$ are non-negative. The reader will have to consult Section \[sec:groups\_and\_chars\] to see what all this means.
Computing $\Sigma$-invariants has historically proved difficult, and here one difficulty is in finding a way to realize an arbitrary character of $F_n$ as a height function on $X_n$. We do this by first introducing some measurements (“proto-characters”) on $n$-ary trees and forests, and extrapolating these to characters on $F_n$ and height functions on $X_n$. Once all the characters are cataloged, we use Morse theory and combinatorial arguments to compute all the $\Sigma^m(F_n)$. One key tool, Lemma \[lem:popular\_simplex\], is a new technique for proving higher connectivity properties of a simplicial complex, building off of recent work of Belk and Forrest.
A pleasant consequence of Theorem A is the following, which is immediate from Citation \[cit:bnsr\_fin\_props\] below, plus the aforementioned fact that every proper quotient of $F_n$ is abelian.
Let $N$ be any normal subgroup of $F_n$. Then as soon as $N$ is finitely presented, it is already of type $F_\infty$.
It should be noted that it is possible to find subgroups of $F_n$ that are finitely presented but not of type ${\operatorname{FP}}_3$, and hence not of type ${\operatorname{F}}_\infty$ [@bieri10 Theorem B]. However, the corollary says that for normal subgroups this cannot happen.
Another immediate application of Theorem A comes from [@kochloukova14], namely Kochloukova’s Theorem C in that paper holds for all $F_n$. In words, not only is the deficiency gradient of $F_n$ zero with respect to any chain of finite index subgroups with index going to infinity, but so too are all the higher dimensional analogs. This can be viewed as a strong finiteness property. For more details and background, see [@kochloukova14].
At the end of the present work, we discuss the problem of computing the higher $\Sigma$-invariants of the *Houghton groups* $H_n$. The group $H_n$ is of type ${\operatorname{F}}_{n-1}$ but not ${\operatorname{F}}_n$ [@brown87 Theorem 5.1], so one can ask what $\Sigma^m(H_n)$ is for $1\le m\le n-1$. We compute large parts of each $\Sigma^m(H_n)$ (Theorem \[thrm:houghton\_pos\]), using the action of $H_n$ on a ${\operatorname{CAT}}(0)$ cube complex, and conjecture that anything not accounted for by the theorem must lie outside $\Sigma^m(H_n)$ (Conjecture \[conj:houghton\_neg\]). The conjecture holds for $m=1,2$, but it seems that proving it for higher $m$ will require new ideas.
The paper is organized as follows. After some topological setup in Section \[sec:prelims\], we define the groups $F_n$ and their characters in Section \[sec:groups\_and\_chars\]. In Section \[sec:stein\_farley\] we discuss a ${\operatorname{CAT}}(0)$ cube complex $X_n$ on which $F_n$ acts, and in Section \[sec:links\_matchings\] we provide a combinatorial model for links in $X_n$. In Section \[sec:computations\] we prove Theorem A. Section \[sec:houghton\] is devoted to the Houghton groups $H_n$; we compute lower bounds on $\Sigma^m(H_n)$, and discuss the problem of trying to make this bound sharp.
Acknowledgments {#acknowledgments .unnumbered}
---------------
I would first like to acknowledge Stefan Witzel, my coauthor on [@witzel15]; some of the tools used here (e.g., Lemma \[lem:morse\]) were developed there, and working on that paper spurred me to attempt this problem. I am grateful to Robert Bieri and Desi Kochloukova for first suggesting I try this problem and for helpful conversations along the way, and to Matt Brin for many fruitful discussions as well.
Topological setup {#sec:prelims}
=================
Let $G$ be a finitely generated group. A *character* of $G$ is a homomorphism $\chi\colon G\to {\mathbb{R}}$. If $\chi(G)\cong{\mathbb{Z}}$, then $\chi$ is *discrete*. The *character sphere* of $G$, denoted $S(G)$, is ${\operatorname{Hom}}(G,{\mathbb{R}}) \cong {\mathbb{R}}^d$ with $0$ removed and modulo positive scaling, so $S(G)\cong S^{d-1}$, where $d$ is the rank of $G/[G,G]$. The *Bieri–Neumann–Strebel (BNS) invariant* $\Sigma^1(G)$ of $G$ is the subset of $S(G)$ defined by: $$\Sigma^1(G) {\mathrel{\mathop{:}}=}\{[\chi]\in S(G)\mid \Gamma_{0\le\chi} \text{ is connected}\} \text{.}$$ Here $\Gamma$ is the Cayley graph of $G$ with respect to some finite generating set, and $\Gamma_{0\le\chi}$ is the full subgraph spanned by those vertices $g$ with $0\le\chi(g)$. We write $[\chi]$ for the equivalence class of $\chi$ in $S(G)$.
The *Bieri–Neumann–Strebel–Renz (BNSR) invariants*, also called *$\Sigma$-invariants* $\Sigma^m(G)$ ($m\in{\mathbb{N}}\cup\{\infty\}$), introduced in [@bieri88], are defined for groups $G$ of type ${\operatorname{F}}_m$. Our working definition for $\Sigma^m(G)$ is almost identical to Definition 8.1 in [@bux04]:
\[def:bnsr\] Let $G$ be of type ${\operatorname{F}}_m$, and let $Y$ be an $(m-1)$-connected $G$-CW complex. Suppose $Y^{(m)}$ is $G$-cocompact and the stabilizer of any $k$-cell is of type ${\operatorname{F}}_{m-k}$. For $0\ne\chi\in{\operatorname{Hom}}(G,{\mathbb{R}})$, there is a *character height function*, denoted $h_\chi$, i.e., a continuous map $h_\chi\colon Y\to{\mathbb{R}}$, such that $h_\chi(gy)=\chi(g)+h_\chi(y)$ for all $y\in Y$ and $g\in G$. Then $[\chi]\in\Sigma^m(G)$ if and only if the filtration $(Y^{t\le h_\chi})_{t\in{\mathbb{R}}}$ is essentially $(m-1)$-connected[^1].
Here $Y^{t\le h_\chi}$ is defined to be the full[^2] subcomplex of $Y$ supported on those vertices $y$ with $t\le h_\chi(y)$. The only difference between our definition and [@bux04 Definition 8.1] is that we use $Y^{t\le h_\chi}$ instead of $h_\chi^{-1}([t,\infty))$. However, the first filtration is essentially $(m-1)$-connected if and only if the second is, so our definition is equivalent.
As mentioned in [@bux04], this definition of $\Sigma^m(G)$ is independent of the choices of $Y$ and $h_\chi$. We will sometimes abuse notation and write $\chi$ instead of $h_\chi$, for both the character and the character height function.
One important application of the $\Sigma$-invariants is:
[@bieri10 Theorem 1.1]\[cit:bnsr\_fin\_props\] Let $G$ be a group of type ${\operatorname{F}}_m$ and $N$ a subgroup of $G$ containing $[G,G]$ (so $N$ is normal). Then $N$ is of type ${\operatorname{F}}_m$ if and only if for every $\chi\in{\operatorname{Hom}}(G,{\mathbb{R}})$ with $\chi(N)=0$ we have $[\chi]\in\Sigma^m(G)$.
For example, if $\chi \colon G\twoheadrightarrow{\mathbb{Z}}$ is a discrete character, then $\ker(\chi)$ is of type ${\operatorname{F}}_m$ if and only if $[\pm\chi]\in\Sigma^m(G)$.
The setup of Definition \[def:bnsr\] is particularly tractable in the situation where $Y$ is an affine cell complex and $\chi$ is affine on cells. Then discrete Morse theory enters the picture, and higher (essential) connectivity properties can be deduced from higher connectivity properties of ascending/descending links.
An *affine cell complex* $Y$ is the quotient of a disjoint union of euclidean polytopes modulo an equivalence relation that maps every polytope injectively into $Y$, with images called *cells*, such that such cells intersect in faces (see [@bridson99 Definition I.7.37]). In particular, every cell has an affine structure. The link ${\operatorname{lk}}_Y v$ of a vertex $v$ of $Y$ is the set of directions in $Y$ emanating out of $v$. The link is naturally a spherical simplicial complex, whose closed cells consist of directions pointing into closed cells of $Y$. If every cell is a cube of some dimension, we call $Y$ an affine cube complex.
The following is taken directly from [@witzel15]:
\[def:morse\] The most general kind of *Morse function* on $Y$ that we will be using is a map $(h,s) \colon Y \to {\mathbb{R}}\times {\mathbb{R}}$ such that both $h$ and $s$ are affine on cells. The codomain is ordered lexicographically, and the conditions for $(h,s)$ to be a Morse function are the following: the function $s$ takes only finitely many values on vertices of $Y$, and there is an $\varepsilon > 0$ such that every pair of adjacent vertices $v$ and $w$ either satisfy ${\lvert h(v) - h(w) \rvert} \ge \varepsilon$, or else $h(v) = h(w)$ and $s(v)\ne s(w)$.
Let us summarize some setup from [@witzel15]: We call $h$ the *height*, $s$ the *secondary height* and $(h,s)$ the *refined height*. Every cell has a unique vertex of maximal refined height and a unique vertex of minimal refined height. The *ascending star* ${\operatorname{st}}^{(h,s)\uparrow}_Y v$ of a vertex $v$ (with respect to $(h,s)$) is the subcomplex of ${\operatorname{st}}_Y v$ consisting of cells $\sigma$ such that $v$ is the vertex of minimal refined height in $\sigma$. The *ascending link* ${\operatorname{lk}}^{(h,s)\uparrow}_Y v$ of $v$ is the link of $v$ in ${\operatorname{st}}^{(h,s)\uparrow}_Y v$. The *descending star* and the *descending link* are defined analogously. Since $h$ and $s$ are affine, ascending and descending links are full subcomplexes. We denote by $Y^{p \le h \le q}$ the full subcomplex of $Y$ supported on vertices $v$ with $p \le h(v) \le q$.
With our definition of Morse function as above, we have the following Morse Lemma, which was proved in [@witzel15] (compare to [@bestvina97 Corollary 2.6]):
\[lem:morse\] Let $p,q,r \in {\mathbb{R}}\cup \{\pm \infty\}$ with $p \le q \le r$. If for every vertex $v \in Y^{q < h \le r}$ the descending link ${\operatorname{lk}}^{(h,s)\downarrow}_{Y^{p \le h}} v$ is $(k-1)$-connected then the pair $(Y^{p \le h \le r},Y^{p \le h \le q})$ is $k$-connected. If for every vertex $v \in Y^{p \le h < q}$ the ascending link ${\operatorname{lk}}^{(h,s)\uparrow}_{Y^{h \le r}} v$ is $(k-1)$-connected then the pair $(Y^{p \le h \le r},Y^{q \le h \le r})$ is $k$-connected.
For the sake of keeping things self-contained, we redo the proof from [@witzel15].
The “ascending” version is like the “descending” version with $(h,s)$ replaced by $-(h,s)$, so we only prove the descending version. Using induction (and compactness of spheres if $r = \infty$) we can assume that $r-q \le \varepsilon$, where $\varepsilon > 0$ is as in Definition \[def:morse\]. By compactness of spheres, it suffices to show that there exists a well order $\preceq$ on the vertices of $Y^{q < h \le r}$ such that the pair $$(S_{\preceq v},S_{\prec v}) {\mathrel{\mathop{:}}=}\left(Y^{p \le h \le q} \cup \bigcup_{w \preceq v} {\operatorname{st}}^{(h,s)\downarrow}_{Y^{p \le h}} w \text{, } Y^{p \le h \le q} \cup \bigcup_{w \prec v} {\operatorname{st}}^{(h,s)\downarrow}_{Y^{p \le h}} w\right)$$ is $k$-connected for every vertex $v\in Y^{q < h \le r}$. Let $\preceq$ be any well order satisfying $v\prec v'$ whenever $s(v)<s(v')$ (this exists since $s$ takes finitely many values on vertices). Note that $S_{\preceq v}$ is obtained from $S_{\prec v}$ by coning off $S_{\prec v} \cap \partial {\operatorname{st}}v$. We claim that this intersection equals the boundary $B$ of ${\operatorname{st}}^{(h,s)\downarrow} v$ in $Y_{p \le h}^{(h,s)\le (h,s)(v)}$, which is homeomorphic to ${\operatorname{lk}}^{(h,s)\downarrow}_{Y^{p \le h}} v$ and hence $(k-1)$-connected by assumption. The inclusion $S_{\prec v} \cap \partial {\operatorname{st}}v \subseteq B$ is evident. Since $S_{\prec v} \cap \partial {\operatorname{st}}v$ is a full subcomplex of $\partial {\operatorname{st}}v$, for the converse it suffices to verify that any vertex $w$ adjacent to $v$ with $(h,s)(w) < (h,s)(v)$ lies in $S_{\prec v}$. If $h(w) < h(v)$ then $h(w) \le h(v) - \varepsilon \le r - \varepsilon \le q$, so $w \in Y^{p \le h \le q}$. Otherwise $s(w) < s(v)$ and hence $w \prec v$.
In practice, the following form is all we will need.
\[cor:morse\] If $Y$ is $(m-1)$-connected and for every vertex $v \in Y^{h < q}$ the ascending link ${\operatorname{lk}}^{(h,s)\uparrow}_Y v$ is $(m-1)$-connected, then $Y^{q \le h}$ is $(m-1)$-connected.
This follows from the Morse Lemma using $p=-\infty$ and $r=\infty$.
The groups and characters {#sec:groups_and_chars}
=========================
Thompson’s group $F$ admits many generalizations. In this paper we will be concerned with a family of groups usually denoted $F_{n,\infty}$, which we abbreviate to $F_n$ ($2\le n\in{\mathbb{N}}$); the group $F_2$ is $F$. As a warning, when dealing with generalizations of Thompson groups, e.g., in [@brown87; @brin98], the notation $F_n$ often refers to a different group, in which $F_{n,\infty}$ sits with finite index (not to mention that $F_n$ also often denotes the free group of rank $n$). We will not be concerned with these though, so here **the notation $F_n$ will always refer to the group denoted $F_{n,\infty}$ in [@brown87; @brin98; @kochloukova12]**. In this section we give three viewpoints of $F_n$ and its characters. The three viewpoints of $F_n$ are: its standard infinite presentation, as a group of homeomorphisms of $[0,1]$, and as a group of $n$-ary tree pairs. The equivalence of these was proved in the original paper by Brown [@brown87 Section 4]. For all three ways of viewing $F_n$, we also discuss characters of $F_n$ from that viewpoint. The last one will be the most important, since it is the one we use later to compute the $\Sigma^m(F_n)$.
Presentation {#sec:presentation}
------------
The standard infinite presentation for $F_n$ ([@brown87 Proposition 4.8]) is $$F_n\cong{\langle x_i~(i\in{\mathbb{N}}_0)\mid x_j x_i = x_i x_{j+(n-1)} \text{ for all }i<j \rangle}\text{.}$$ It is easy to abelianize this presentation, and get that $F_n/[F_n,F_n] \cong {\mathbb{Z}}^n$. One basis for this is $\bar{x}_0,\dots,\bar{x}_{n-1}$. From this, one could get a basis for ${\operatorname{Hom}}(F_n,{\mathbb{R}}) \cong {\mathbb{R}}^n$ by taking the dual basis. This was one tool used in [@kochloukova12] to compute $\Sigma^2(F_n)$.
Piecewise linear homeomorphisms {#sec:homeos}
-------------------------------
A more hands-on basis for ${\operatorname{Hom}}(F_n,{\mathbb{R}})$ can be described by viewing $F_n$ as piecewise linear self homeomorphisms of $[0,1]$. We will not prove anything in this subsection, since the model for $F_n$ we will actually use comes in the next subsection; here we are just giving some intuition for $F_n$ and its characters. Each element $f\in F_n$ is an orientation preserving homeomorphism $f\colon[0,1] \to [0,1]$ that is piecewise linear with slopes powers of $n$, and whose finitely many points of non-differentiability lie in ${\mathbb{Z}}[1/n]$. Already this gives us two interesting characters, usually denoted $\chi_0$ and $\chi_1$. The character $\chi_0$ is the log base $n$ of the right derivative at $0$, and $\chi_1$ is the log base $n$ of the left derivative at $1$.
Any such $f\in F_n$ is determined by certain sets of *breakpoints* in the domain and range, as we now describe. Build a finite set $P \subseteq [0,1]$ by starting with the points $\{0,1\}$, and then do finitely many iterations of the following procedure:
> Pick two points $x$ and $x'$ already in $P$, with no points in between them yet in $P$, and then add to $P$ the $n-1$ new points $\frac{(n-i)x+ix'}{n}$ for $0<i<n$.
For example, after one iteration of this, $P$ consists of $\{0,1/n,2/n,\dots,(n-1)/n,1\}$. Call $P$ a *legal set of breakpoints*. If $Q$ is another legal set of breakpoints with $|P|=|Q|$, then we can define $f \colon [0,1] \to [0,1]$ by sending the points of $P$, in order, to the points of $Q$, and then extending affinely between breakpoints. By construction, slopes will be powers of $n$ and breakpoints will lie in ${\mathbb{Z}}[1/n]$. Moreover, every $f\in F_n$ arises in this way [@brown87 Proposition 4.4].
One can show that every element of ${\mathbb{Z}}[1/n] \cap [0,1]$ appears in some legal set of breakpoints. Moreover, while a point can appear in more than one legal set of breakpoints, and have a different “position” in different legal sets of breakpoints, the “position modulo $n-1$” is a well defined measurement. The equivalence classes induced by this measurement are in fact the $F_n$-orbits in ${\mathbb{Z}}[1/n] \cap (0,1)$. (Again, proofs are left to the reader.) For each $0\le i\le n-2$, let $O_i$ denote the $F_n$-orbit of points of ${\mathbb{Z}}[1/n] \cap (0,1)$ appearing in a legal set of breakpoints in a position congruent to $i$ modulo $n-1$.
Now we can define characters on $F_n$. For a point $x\in(0,1]$ define $LD|_x \colon F_n \to {\mathbb{Z}}$ to be the log base $n$ of the left derivative at $x$. Similarly for $x\in[0,1)$ let $RD|_x$ be the log base $n$ of the right derivative at $x$. These are not group homomorphisms. However, summing these over a complete $F_n$-orbit $O_i$ would define a homomorphism. To get these sums to be finite, we will actually sum up $LD|_x - RD|_x$, since then for a given $f$ this can be nonzero at only finitely many points. For $0\le i\le n-2$ define: $$\psi_i(f) {\mathrel{\mathop{:}}=}\sum\limits_{x\in O_i} LD|_x (f) - RD|_x (f) \text{.}$$ This is a group homomorphism $\psi_i \colon F_n \to {\mathbb{Z}}$. As a remark, the characters $-\chi_0$ and $\chi_1$ are also of this form, namely for $-\chi_0$ we sum over the orbit of $0$ (which is just $\{0\}$) and for $\chi_1$ we sum over the orbit $\{1\}$. (Technically this only makes sense if we declare $LD|_0=0$ and $RD|_1=0$.)
Note that $\sum_{i=0}^{n-2} \psi_i = \chi_0 - \chi_1$. However, one can check that $\chi_0,\psi_0,\dots,\psi_{n-3},\chi_1$ are linearly independent, and so form a basis of ${\operatorname{Hom}}(F_n,{\mathbb{R}})\cong {\mathbb{R}}^n$. In the next subsection we will redefine the $\psi_i$ using a different model for $F_n$, and in particular will prove all of these facts.
$n$-ary trees {#sec:trees}
-------------
This brings us to the descriptions of the $F_n$ and their characters that we will use for the rest of the paper, namely making use of $n$-ary trees.
An *$n$-ary tree* will always mean a finite connected tree with a single vertex of degree $n$ or $0$, its *root*, some number of degree $1$ vertices, the *leaves*, and all other vertices of degree $n+1$. The *trivial tree* ${\mathrm{I}}$ is the one where the root has degree $0$ (so there are no leaves or other vertices). The *$n$-caret* ${\Lambda}_n$ is the non-trivial $n$-ary tree in which every vertex is either the root or a leaf. Every $n$-ary tree can be obtained as a union of $n$-carets.
For an $n$-ary tree $T$, each leaf of $T$ has a unique reduced path to the root. The length of this path (i.e., its number of edges) defines the *depth* of that leaf. As a remark, the trivial tree is characterized as having a leaf of depth $0$, and the $n$-caret is characterized as having all its leaves of depth $1$.
For each $n$-ary tree $T$, say with $r$ leaves, we fix a planar embedding of $T$, and hence an order on the leaves. We label the leaves $0$ through $r-1$, left to right. The next definition is of various measurements that we will call *proto-characters* on $T$, which will later be used to define characters on elements of $F_n$.
\[def:proto\_chars\] Let $T$ be an $n$-ary tree with leaves labeled $0$ through $r-1$, left to right. Define $L(T)$ to be the depth of the $0$th leaf. Define $R(T)$ to be the depth of the $(r-1)$st leaf. For each $0\le j\le r-1$, define $d_j(T)$ to be the depth of the $j$th leaf, and then for each $0\le j\le r-2$ define $$\delta_j(T) {\mathrel{\mathop{:}}=}d_j(T) - d_{j+1}(T) \text{.}$$ This is the *$j$th change of depth* of $T$. For each $0\le i\le n-2$ define $$D_i(T) {\mathrel{\mathop{:}}=}\sum \{\delta_j(T) \mid 0\le j \le r-2 \text{, } j\equiv i \mod (n-1)\} \text{.}$$
As a quick (and trivial) example, if ${\Lambda}_n$ is the $n$-caret then $L({\Lambda}_n)=R({\Lambda}_n)=1$, and $D_i({\Lambda}_n)=0$ for all $i$, since $d_j({\Lambda}_n)=1$ for all $j$. A less trivial example is given in Figure \[fig:proto\].
(0,0) – (1,1) – (2,0) (1,1) – (1,0) (-1,-1) – (0,0) – (1,-1) (0,0) – (0,-1) (-1,-2) – (0,-1) – (1,-2) (0,-1) – (0,-2); (-1,-1) circle (1.5pt) (0,-2) circle (1.5pt) (1,-1) circle (1.5pt) (2,0) circle (1.5pt); (-1,-2) circle (1.5pt) (1,-2) circle (1.5pt) (1,0) circle (1.5pt); (-1,-2) circle (1.5pt) (1,-2) circle (1.5pt) (1,0) circle (1.5pt);
An *$n$-ary tree pair* $(T_-,T_+)$ consists of $n$-ary trees $T_-$ and $T_+$ such that $T_-$ and $T_+$ have the same number of leaves. Two $n$-ary tree pairs are *equivalent* if they can be transformed into each other via a sequence of reductions and expansions. An *expansion* amounts to adding an $n$-caret to the $k$th leaf of $T_-$ and one to the $k$th leaf of $T_+$, for some $k$. A *reduction* is the reverse of an expansion. We denote the equivalence class of $(T_-,T_+)$ by $[T_-,T_+]$.
These $[T_-,T_+]$ are the elements of $F_n$. The multiplication, say of $[T_-,T_+]$ and $[U_-,U_+]$, written $[T_-,T_+] \cdot [U_-,U_+]$, is defined as follows. First note that $T_+$ and $U_-$ admit an $n$-ary tree $S$ that contains them both, so using expansions we have $[T_-,T_+] = [\hat{T}_-,S]$ and $[U_-,U_+] = [S,\hat{U}_+]$ for some $\hat{T}_-$ and $\hat{U}_+$. Then we define $$[T_-,T_+] \cdot [U_-,U_+] {\mathrel{\mathop{:}}=}[\hat{T}_-,S] \cdot [S,\hat{U}_+] = [\hat{T}_-,\hat{U}_+] \text{.}$$ This multiplication is well defined, and it turns out the resulting structure is a group, namely $F_n$.
Having described elements of $F_n$ using the $n$-ary tree pair model, we now describe characters. We make use of the proto-characters from Definition \[def:proto\_chars\].
\[def:chars\] Let $f=[T,U]=(T,U)\in F_n$. Define $$\chi_0(f) {\mathrel{\mathop{:}}=}L(U) - L(T) \text{ and } \chi_1(f) {\mathrel{\mathop{:}}=}R(U) - R(T) \text{.}$$ For $0\le i\le n-2$ define $$\psi_i(f) {\mathrel{\mathop{:}}=}D_i(U) - D_i(T) \text{.}$$
\[lem:chars\_well\_def\] The functions $\chi_0$, $\chi_1$ and $\psi_i$ ($0\le i\le n-2$) are well defined group homomorphisms from $F_n$ to ${\mathbb{Z}}$.
For well definedness, we need to show that for $\chi\in\{\chi_0,\chi_1,\psi_i\}_{i=0}^{n-2}$, if $T'$ (respectively $U'$) is obtained from $T$ (respectively $U$) by adding an $n$-caret to the $k$th leaf, then $\chi(T',U')=\chi(T,U)$. If suffices to show that for $A\in \{L,R,D_i\}_{i=0}^{n-2}$, the value $A(T')-A(T)$ depends only on $i$, $k$ and $r$, where $r$ is the number of leaves of $T$. Since $U$ has the same number of leaves, this will show that $A(T')-A(T) = A(U')-A(U)$, and so $A(U')-A(T') = A(U)-A(T)$ and $\chi(T',U')=\chi(T,U)$. For $A=L,R$ this is clear: $L(T')-L(T)=1$ if $k=0$ and $L(T')-L(T)=0$ otherwise, and $R(T')-R(T)=1$ if $k=r-1$ and $R(T')-R(T)=0$ otherwise. Now let $A=D_i$. We then have the following:
1. If $0<k$ and $k-1\equiv_{n-1} i$, then $D_i(T')=D_i(T)-1$.
2. If $k<r-1$ and $k\equiv_{n-1} i$, then $D_i(T')=D_i(T)+1$.
3. Otherwise $D_i(T')-D_i(T) = 0$.
In particular, $D_i(T')-D_i(T)$ depends only on $i$, $k$ and $r$.
It is now easy to check that the $\chi$ are group homomorphisms. If we have two elements to multiply, represent them with a common tree and get $[T,U]\cdot [U,V] = [T,V]$; then for $A\in\{L,R,D_i\}$ we have $A(U) - A(T) + A(V) - A(U) = A(V) - A(T)$, so any $\chi\in\{\chi_0,\chi_1,\psi_i\}$ is a homomorphism.
As the proof showed, we now know how the measurements $L$, $R$ and $D_i$ change when an $n$-caret is added to the $k$th leaf of an $n$-ary tree. For example if $0<k$ and $k-1\equiv_{n-1} i$ then $D_i$ goes down by $-1$, and if $k<r-1$ and $k\equiv_{n-1} i$ then $D_i$ goes up by $1$. See Figure \[fig:proto\_change\] for an example.
(-0.5,0) – (4.5,0) – (2,2) – (-0.5,0) (0,0) – (0,-1) (1,0) – (1,-1) (2,0) – (2,-1) (3,0) – (3,-1) (4,0) – (4,-1); (0,-1) circle (1.5pt) (2,-1) circle (1.5pt) (4,-1) circle (1.5pt); (1,-1) circle (1.5pt) (3,-1) circle (1.5pt); (1,-1) circle (1.5pt) (3,-1) circle (1.5pt); at (2,0.75) [$T$]{};
(-0.5,0) – (4.5,0) – (2,2) – (-0.5,0) (0,0) – (0,-1) (1,0) – (1,-1) (2,0) – (2,-1) (3,0) – (3,-1) (4,0) – (4,-1); (0,-1) circle (1.5pt) (4,-1) circle (1.5pt); (1,-1) circle (1.5pt) (3,-1) circle (1.5pt); (1,-1) circle (1.5pt) (3,-1) circle (1.5pt); (1.5,-2) – (2,-1) – (2.5,-2) (2,-1) – (2,-2); (1.5,-2) circle (1.5pt) (2.5,-2) circle (1.5pt); (2,-2) circle (1.5pt); (2,-2) circle (1.5pt); at (2,0.75) [$T'$]{};
\[prop:char\_basis\] As elements of ${\operatorname{Hom}}(F_n,{\mathbb{R}})\cong {\mathbb{R}}^n$, the $n$ characters $$\chi_0,\psi_0,\dots,\psi_{n-3},\chi_1$$ are linearly independent, and hence form a basis. A dependence involving $\psi_{n-2}$ is that $\psi_0+\cdots+\psi_{n-2}=\chi_0 - \chi_1$.
For the second statement, just note that for any tree $T$, $D_0(T)+\cdots D_{n-2}(T) = L(T) - R(T)$.
We turn to the statement about linear independence. For $0\le k\le n-1$, let $T_k$ be the tree consisting of an $n$-caret with another $n$-caret on its $k$th leaf, so $T_k$ has leaves labeled $0$ through $2n-2$. It is straightforward to compute $L(T_0)=2$, $L(T_k)=1$ for $k>0$, $R(T_{n-1})=2$, $R(T_k)=1$ for $k<n-1$, and the following for the $D_i$ ($0\le i\le n-2$): $$\begin{aligned}
D_i(T_k) = \left\{\begin{array}{ll} -1 & \text{if } i = k-1 \\
1 & \text{if } i = k \\
0 & \text{else.}
\end{array}\right.
\end{aligned}$$ For $0\le i\le n-2$, we therefore have $$(D_i(T_0),D_i(T_1),\dots,D_i(T_{n-1})) = (0,\dots,0,1,-1,0,\dots,0)$$ with the $1$ at $D_i(T_i)$. We will also need to use trees $T_k'$, obtained by attaching the root of $T_k$ to the last leaf of an $n$-caret. For each $k$ we have $L(T_k')=1$, $R(T_k')=R(T_k)+1$ and $D_i(T_k')=D_i(T_k)$ for all $0\le i\le n-3$.
Consider the $n$ elements $[T_0,T_{n-1}],\dots,[T_{n-2},T_{n-1}],[T_0',T_{n-1}']$ of $F_n$. Our goal now is to hit them with the $n$ characters $\chi_0,\psi_0,\dots,\psi_{n-3},\chi_1$ to get an $n$-by-$n$ matrix, and then show that this matrix is non-singular. In particular this will prove that these $n$ characters are linearly independent. The $\chi_0$ row is $(-1,0,\dots,0)$ and the $\chi_1$ row is $(1,\dots,1)$. For $0\le i\le n-3$, $D_i(T_{n-1})=0$, so $\psi_i([T_k,T_{n-1}])=-D_i(T_k)$ for $0\le k\le n-2$, and similarly $\psi_i([T_0',T_{n-1}'])=-D_i(T_0')$. Hence we can compute the rows for $\psi_i$, using our previous computation of the $D_i(T_k)$. We get that the $\psi_0$ row is $(-1,1,0,\dots,0,-1)$, the $\psi_1$ row is $(0,-1,1,0,\dots,0)$, and so forth up to the $\psi_{n-3}$ row, which is $(0,\dots,0,-1,1,0)$. Arranging these rows into a matrix, we need to show non-singularity of the matrix: $$\begin{aligned}
\begin{pmatrix}
-1 & 0 & 0 & 0 & \dots & 0 & 0 & 0 \\
-1 & 1 & 0 & 0 & \dots & 0 & 0 & -1\\
0 & -1 & 1 & 0 & \dots & 0 & 0 & 0 \\
0 & 0 & -1 & 1 & \dots & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & \dots & 0 & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\
0 & 0 & 0 & 0 & \dots & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & \dots & -1 & 1 & 0 \\
1 & 1 & 1 & 1 & \dots & 1 & 1 & 1
\end{pmatrix}
\end{aligned}$$ This is visibly “almost” lower triangular; the second row (the $\psi_0$ row) is the only problem, if $n>2$ (note that if $n=2$ then the only rows are $\chi_0$ and $\chi_1$, and this matrix is lower triangular and non-singular). We hit this row with elementary row operations, namely if $r_i$ is the $i$th row we replace $r_2$ with $$r_2 + r_n - r_{n-1} - 2r_{n-2} - 3r_{n-3} - \cdots - (n-3)r_3 \text{.}$$ The new second row is $(0,n-1,0,\dots,0)$, and hence the matrix reduces to a lower triangular matrix whose determinant is readily computed to be $-(n-1)$. This matrix is therefore non-singular, and so the characters $\chi_0,\psi_0,\dots,\psi_{n-3},\chi_1$ are linearly independent elements of ${\operatorname{Hom}}(F_n,{\mathbb{R}})\cong {\mathbb{R}}^n$.
Clearly this proof would have been faster if, instead of $\psi_0$, we used the character $\psi_0 + \chi_1 - \psi_{n-3} - 2\psi_{n-4} - 3\psi_{n-5} - \cdots - (n-3)\psi_1$, but since computing the $\Sigma^m(F_n)$ will involve being able to tell whether our basis characters increase, decrease, or neither under certain moves, it will be advantageous to have basis characters with the easiest possible descriptions.
The $\psi_i$ here agree with the $\psi_i$ in Subsection \[sec:homeos\], provided the connection between the homeomorphism model and the $n$-ary tree pair model is made correctly. For $(T,U)$ we view $U$ as the “domain tree” and $T$ as the “range tree”. Each tree defines a subdivision of $[0,1]$ into as many subintervals as there are leaves. Then, the subdivision given by the domain tree is taken to the subdivision given by the range tree, defining a homeomorphism as described in Subsection \[sec:homeos\]. It is straightforward to check that the two definitions of $\psi_i$ agree.
Stein–Farley complexes {#sec:stein_farley}
======================
In this section we recall the Stein–Farley ${\operatorname{CAT}}(0)$ cube complex $X_n$ on which $F_n$ acts, and extend the characters $\chi \colon F_n \to {\mathbb{R}}$ to functions $\chi \colon X_n \to {\mathbb{R}}$. The complex $X_n$ was first constructed by Stein [@stein92] building off ideas of Brown, and shown to be ${\operatorname{CAT}}(0)$ by Farley [@farley03], who viewed $F_n$ as a *diagram group*, à la Guba and Sapir [@guba97]. To define $X_n$, we first expand from considering $n$-ary trees to considering $n$-ary forests. An *$n$-ary forest* is a disjoint union of finitely many $n$-ary trees. The roots and leaves of the trees are *roots* and *leaves* of the forest. We fix an order on the trees, and hence on the leaves. An *$n$-ary forest pair* $(E_-,E_+)$ consists of $n$-ary forests $E_-$ and $E_+$ such that $E_-$ and $E_+$ have the same number of leaves. We call the roots of $E_-$ *heads* and the roots of $E_+$ *feet* of the pair (the terminology comes from flipping $E_+$ upside down and identifying the leaves of $E_-$ and $E_+$).
Just like the tree case, we have a notion of equivalence. Two $n$-ary forest pairs are *equivalent* if they can be transformed into each other via a sequence of reductions or expansions. We denote the equivalence class of $(E_-,E_+)$ by $[E_-,E_+]$. Let ${\mathcal{P}}$ be the set of equivalence classes of $n$-ary forest pairs.
This set has two important pieces of structure. First, it is a groupoid. If $[E_-,E_+]$ has $k$ heads and $\ell$ feet, and $[D_-,D_+]$ has $\ell$ heads and $m$ feet, then we can define their product, written $[E_-,E_+] \cdot [D_-,D_+]$, which is an $n$-ary forest pair with $k$ heads and $m$ feet. Like in $F_n$, with $n$-ary tree pairs, to define the product we first note that $E_+$ and $D_-$ admit an $n$-ary forest $C$ that contains them both. Then applying expansions we can write $[E_-,E_+] = [\hat{E}_-,C]$ and $[D_-,D_+] = [C,\hat{D}_+]$ for some $\hat{E}_-$ and $\hat{D}_+$, and then define $$[E_-,E_+] \cdot [D_-,D_+] {\mathrel{\mathop{:}}=}[\hat{E}_-,C] \cdot [C,\hat{D}_+] = [\hat{E}_-,\hat{D}_+] \text{.}$$ For ${\mathcal{P}}$ to be a groupoid with this multiplication, we need identities and inverses. A forest in which all trees are trivial is called a *trivial forest*. The trivial forest with $\ell$ trees is denoted ${\operatorname{id}}_\ell$. We can view an $n$-ary forest $E$ as an $n$-ary forest pair via $E \mapsto [E,{\operatorname{id}}_\ell]$, where $\ell$ is the number of leaves of $E$. It is clear that for any element with $k$ heads and $\ell$ feet, $[{\operatorname{id}}_k,{\operatorname{id}}_k]$ is the left identity and $[{\operatorname{id}}_\ell,{\operatorname{id}}_\ell]$ is the right identity. We also have inverses, namely the (left and right) inverse of $[E_-,E_+]$ is $[E_+,E_-]$.
Since $F_n$ lives in ${\mathcal{P}}$ as the set of elements with one head and one foot, we have an action of $F_n$, by multiplication, on the subset ${\mathcal{P}}_1$ of elements with one head.
The second piece of structure on ${\mathcal{P}}$ is an order relation. The order is defined by: $[E_-,E_+] \le [D_-,D_+]$ whenever there is an $n$-ary forest $C$ such that $[E_-,E_+] \cdot C = [D_-,D_+]$. We informally refer to right multiplication by an $n$-ary forest pair of the form $[C,{\operatorname{id}}_\ell]$ as *splitting* the feet of $[E_-,E_+]$. Multiplying by $[{\operatorname{id}}_\ell,C]$ is called *merging*. This terminology comes from viewing $E_+$ upside down with its leaves attached to those of $E_-$, forming a “strand diagram” à la [@belk14]. It is straightforward to check that $\le$ is a partial order, so ${\mathcal{P}}$ is a poset. The subset ${\mathcal{P}}_1$ of elements with one head is a subposet.
The topological realization of the poset $({\mathcal{P}}_1,\le)$ is a contractible simplicial complex on which $F_n$ acts, and the *Stein–Farley complex* $X_n$ is a certain invariant subcomplex with a natural cubical structure. Given $n$-ary forest pairs $[E_-,E_+] \le [E_-,E_+] \cdot E$, write $[E_-,E_+] \preceq [E_-,E_+] \cdot E$ whenever $E$ is an *elementary $n$-ary forest*. This means that each $n$-ary tree of $E$ is either trivial or a single $n$-caret. Now $X_n$ is defined to be the subcomplex of $|{\mathcal{P}}_1|$ consisting of chains $x_0<\cdots<x_k$ with $x_i \preceq x_j$ for all $i\le j$. The cubical structure is given by intervals: given $x\preceq y$ ($x$ with $r$ feet), the interval $[x,y]{\mathrel{\mathop{:}}=}\{z\mid x\le z\le y\}$ is a Boolean lattice of dimension $r$, and so the simplices in $[x,y]$ form an $r$-cube. Note that $x\prec y$ are adjacent, i.e., share a $1$-cube, if and only if $y=x \cdot E$ for $E$ an elementary $n$-ary forest with just a single $n$-caret.
[@farley03] $X_n$ is a ${\operatorname{CAT}}(0)$ cube complex.
Every cube $\sigma$ has a unique vertex $x$ with fewest feet and a unique vertex $y$ with most feet. There is a unique elementary $n$-ary forest $E$ with $y=x\cdot E$, and the other vertices of $\sigma$ are obtained by multiplying $x$ by subforests of $E$. We use the following notation: suppose $x$ has $\ell$ feet and $E=(A_0,\dots,A_{\ell-1})$, where each $A_i$ is either ${\mathrm{I}}$ or ${\Lambda}_n$; here ${\mathrm{I}}$ is the trivial tree and ${\Lambda}_n$ is the tree with one $n$-caret. Let $\Phi$ be the set of subforests of $E$, written $\Phi {\mathrel{\mathop{:}}=}{\langle A_0,\ldots,A_{\ell-1} \rangle}$. Then the vertex set of $\sigma$ is precisely $x\Phi$.
If we center ourselves at a different vertex $z$ of $\sigma$, then we also have to allow merges. Say $z$ has $r > \ell$ feet. Then we can write $\sigma = z\Psi$ where $\Psi$ is of the form ${\langle A_0,\ldots,A_{\ell-1} \rangle}$, where each $A_i$ is either ${\mathrm{I}}$, ${\Lambda}_n$ or ${\mathrm{V}}_n$. Here ${\mathrm{V}}_n$ is the inverse of the tree with one $n$-caret (so an upside-down $n$-caret). The tuple $(A_0,\ldots,A_{\ell-1})$ can be thought of as an $n$-ary forest pair, with all the ${\Lambda}_n$ in the first forest and all the ${\mathrm{V}}_n$ in the second forest (and some ${\mathrm{I}}$s included if necessary). Then the set $\Psi$ is the set of all $n$-ary forest pairs that can be obtained by removing some of the carets. As before, the vertex set of $\sigma$ is $z\Psi$.
Note that the action of $F_n$ on $X_n$ is free, since the action on vertices is given by multiplication in a groupoid, and if an element stabilizes a cube $[x,y]$ then it fixes $x$.
Character height functions {#sec:char_height_fxns}
--------------------------
In Subsection \[sec:trees\], we defined characters on $F_n$ by first defining “proto-characters” on $n$-ary trees, and then viewing elements of $F_n$ as $n$-ary tree pairs. It is straightforward to extend these proto-characters to be defined on $n$-ary forests. To be precise, each leaf of an $n$-ary forest is connected to a unique root, which gives it a depth, so $n$-ary forests $E$ admit the measurements $L(E)$, $R(E)$, $\delta_j(E)$ and $D_i(E)$. Much like the proto-characters on $n$-ary trees induce the characters (group homomorphisms) $\chi_0$, $\chi_1$ and $\psi_i$ from $F_n$ to ${\mathbb{Z}}$, also the proto-characters on $n$-ary forests induce groupoid homomorphisms ${\mathcal{P}}\to {\mathbb{Z}}$ extending these characters.
In particular, the $\chi_i$ and $\psi_i$ can now be evaluated on vertices of $X_n$. Moreover, any character $\chi$ on $F_n$ can be written as a linear combination $$\label{eq:character_linear_combination}
\chi = a\chi_0 + c_0\psi_0 + \cdots + c_{n-3}\psi_{n-3} + b\chi_1$$ thanks to Proposition \[prop:char\_basis\]. Hence $\chi$ extends to arbitrary $n$-ary forest pairs by interpreting as a linear combination of the extended characters.
It will be important to know how our basis characters vary between adjacent vertices of $X_n$.
\[lem:vary\_chars\] Let $x$ be a vertex in $X_n$, say with feet numbered $0$ through $r-1$, left to right. Let ${\Lambda}_n(r,k)$ be the elementary $n$-ary forest with $r$ roots and a single non-trivial tree, namely an $n$-caret on the $k$th root. Let $y=x\cdot {\Lambda}_n(r,k)$. We have the following:
1. If $k=0$ then $\chi_0(y)=\chi_0(x)-1$.
2. If $k>0$ then $\chi_0(y)=\chi_0(x)$.
3. If $k<r-1$ then $\chi_1(y)=\chi_1(x)$.
4. If $k=r-1$ then $\chi_1(y)=\chi_1(x)-1$.
5. If $0<k$ and $k-1\equiv_{n-1} i$, then $\psi_i(y) = \psi_i(x) + 1$.
6. If $k<r-1$ and $k\equiv_{n-1} i$, then $\psi_i(y) = \psi_i(x) - 1$.
7. Otherwise $\psi_i(y) = \psi_i(x)$.
Let $\chi\in\{\chi_0,\chi_1,\psi_i\}_{i=0}^{n-3}$ be a basis character. Let $A\in\{L,R,D_i\}_{i=0}^{n-3}$ be the corresponding proto-character. Since $\chi$ is a groupoid morphism ${\mathcal{P}}\to {\mathbb{Z}}$, we have $$\chi(y) = \chi(x)+\chi([{\Lambda}_n(r,k),{\operatorname{id}}_{r+(n-1)}]) = \chi(x) - A({\Lambda}_n(r,k)) \text{.}$$ Hence, to check the cases in the statement, it suffices to check the following, all of which are readily verified:
1. If $k=0$ then $L({\Lambda}_n(r,k)) = 1$.
2. If $k>0$ then $L({\Lambda}_n(r,k)) = 0$.
3. If $k<r-1$ then $R({\Lambda}_n(r,k)) = 0$.
4. If $k=r-1$ then $R({\Lambda}_n(r,k)) = 1$.
5. If $0<k$ and $k-1\equiv_{n-1} i$, then $D_i({\Lambda}_n(r,k)) = -1$.
6. If $k<r-1$ and $k\equiv_{n-1} i$, then $D_i({\Lambda}_n(r,k)) = 1$.
7. Otherwise $D_i({\Lambda}(r,k)) = 0$.
Note that since we only consider $0\le i\le n-3$, if $y=x \cdot {\Lambda}_n(r,r-1)$ (i.e., if we get from $x$ to $y$ by splitting the last foot), then no $\psi_i$ changes, since $r-2\equiv_{n-1} n-2$.
So far we know that any character $\chi$ on $F_n$ can be extended to all the vertices of $X_n$. Now we extend it to the entire complex.
\[lem:affine\_extend\] Any character $\chi$ extends to an affine map $\chi \colon X_n \to {\mathbb{R}}$.
Before proving this, we reduce the problem using the following:
[@witzel15 Lemma 2.4]\[lem:affine\_on\_cubes\] Let $\varphi \colon \{0,1\}^r \to {\mathbb{R}}$ be a map that can be affinely extended to the $2$-faces of the cube $[0,1]^r$. Then $\varphi$ can be affinely extended to all of $[0,1]^r$.
This was proved in [@witzel15], and we repeat the proof here for the sake of being self contained. There is a unique affine function $\tilde{\varphi} \colon {\mathbb{R}}^r \to{\mathbb{R}}$ that agrees with $\varphi$ on the zero vector and the $r$ standard basis vectors. We claim that $\tilde{\varphi}$ agrees with $\varphi$ on all the other vertices of $[0,1]^r$ as well, and hence defines an affine extension of $\varphi$ to all of $[0,1]^r$. Let $v=(v_1,\dots,v_r)$ be a vertex with at least two entries equal to $1$ (and the others all $0$). Pick $i\ne j$ with $v_i=v_j=1$. For any $w$ obtained from $v$ by zeroing out $v_i$, $v_j$, or both, we have by induction that $\tilde{\varphi}(w)=\varphi(w)$. These three $w$ vertices, plus $v$, define a $2$-face of $[0,1]^r$. By assumption, $\varphi$ can be affinely extended to this $2$-face, and the value on $v$ is uniquely determined by the values on the other three vertices. Hence $\tilde{\varphi}(v)=\varphi(v)$.
Let $\square_2=v \Phi$ be a $2$-cube in $X_n$, say $\Phi={\langle A_0,\dots,A_{r-1} \rangle}$, with exactly two $A_i$ being ${\Lambda}_n$ and all others being ${\mathrm{I}}$. Thanks to Lemma \[lem:affine\_extend\], we just need to show that $\chi$ extends affinely to $\square_2$. Say $A_j=A_k={\Lambda}_n$ for $j<k$, and let $v_j=v{\langle {\mathrm{I}},\dots,A_j,\dots,{\mathrm{I}}\rangle}$, $v_k=v{\langle {\mathrm{I}},\dots,A_k,\dots,{\mathrm{I}}\rangle}$ and $v_{j,k}=v{\langle {\mathrm{I}},\dots,A_j,\dots,A_k,\dots,{\mathrm{I}}\rangle}$. Hence the vertices of $\square_2$ are $v$, $v_j$, $v_k$ and $v_{j,k}$. Now we just need to show that $\chi(v_j)-\chi(v) = \chi(v_{j,k})-\chi(v_k)$.
It suffices to do this for $\chi\in\{\chi_0,\chi_1,\psi_i\}_{i=0}^{n-3}$. It is clear that $\chi_0(v_j)-\chi_0(v) = \chi_0(v_{j,k})-\chi_0(v_k)$, namely they equal $-1$ if $j=0$ and equal $0$ otherwise, and similarly we always have $\chi_1(v_j)-\chi_1(v) = \chi_1(v_{j,k})-\chi_1(v_k) = 0$. Next consider $\psi_i$. By Lemma \[lem:vary\_chars\], we have that $\psi_i(v_j)-\psi_i(v)=1$ if and only if $0<j$ and $j-1\equiv_{n-1} i$, which also holds if and only if $\psi_i(v_{j,k})-\psi_i(v_k)=1$. Also, $\psi_i(v_j)-\psi_i(v)=-1$ if and only if $j\equiv_{n-1} i$ if and only if $\psi_i(v_{j,k})-\psi_i(v_k)=-1$ (since $j<k$, we know $j$ cannot be the highest index of a foot of either $v$ or $v_k$). The only other option is $\psi_i(v_j)-\psi_i(v) = \psi_i(v_{j,k})-\psi_i(v_k) = 0$.
These extended characters $\chi$ will be our height functions. Our secondary height will be given by the number of feet function $f$.
\[obs:feet\_affine\] There is a map $f \colon X_n \to {\mathbb{R}}$ that is affine on cubes and assigns to any vertex its number of feet. It is a Morse function.
That $f$ extends affinely is straightforward. When we say that $f$ is a Morse function, in the language of Definition \[def:morse\] this means that $(f,0)$ is a Morse function. This is true because adjacent vertices $v$ and $w$ satisfy ${\lvert f(v)-f(w) \rvert}=n-1$.
Let $X_n^{p\le f\le q}$ be the subcomplex of $X_n$ supported on vertices $v$ with $p\le f(v)\le q$.
\[prop:char\_morse\] Let $\chi$ be a character. The pair $(\chi,f)$ is a Morse function on $X_n^{p\le f\le q}$ for any $p\le q<\infty$.
We check the conditions required by Definition \[def:morse\]. We have extended $\chi$ and $f$ to affine functions in Lemma \[lem:affine\_extend\] and Observation \[obs:feet\_affine\]. By construction $f$ takes finitely many values on $X_n^{p\le f\le q}$. Write $\chi=a\chi_0+c_0\psi_0+\cdots+c_{n-3}\psi_{n-3}+b\chi_1$. Let $$\varepsilon{\mathrel{\mathop{:}}=}\min\{{\lvert d \rvert}\mid d=\alpha a + \beta b + \gamma_0 c_0 +\cdots+\gamma_{n-3} c_{n-3} \ne 0 \text{ for } \alpha,\beta,\gamma_i\in\{-1,0,1\}\}\text{.}$$ Since we only consider such $d$ that are non-zero, and there are finitely many, we have $0<\varepsilon$. For any pair of adjacent vertices $v$ and $w$, we know from Lemma \[lem:vary\_chars\] that for any basis character $\phi\in\{\chi_0,\chi_1,\psi_i\}_{i=0}^{n-3}$, we have $\phi(v) - \phi(w) \in \{-1,0,1\}$. Hence for any character $\chi$ we have $\chi(v) - \chi(w) = \alpha a + \beta b + \gamma_0 c_0 +\cdots+\gamma_{n-3} c_{n-3}$ for some $\alpha,\beta,\gamma_i\in\{-1,0,1\}$. In particular, either ${\lvert \chi(v) - \chi(w) \rvert} \ge \varepsilon$ or else $\chi(v)=\chi(w)$. The condition $f(v)\ne f(w)$ is always satisfied anyway for adjacent vertices, so we conclude that $(\chi,f)$ is a Morse function.
Links and matchings {#sec:links_matchings}
===================
We will use Morse theory to reduce the computation of $\Sigma^m(F_n)$ to questions about ascending links in $X_n$. In this section we discuss a useful model for links in $X_n$.
Let $\Delta$ be a simplicial complex, say of dimension $d$. Let $D\subseteq\{0,\dots,d\}$. A *$D$-matching* is a subset $\mu$ of $\Delta$, consisting of $k$-simplices for $k\in D$ such that any two simplices in $\mu$ are disjoint. If $D=\{k\}$ is a singleton we may write “$k$-matching” instead of “$\{k\}$-matching”. For example a $0$-matching is just any collection of $0$-simplices, and a $1$-matching is what is usually called a *matching* on the graph $\Delta^{(1)}$. For our purposes, we will be interested in certain $(n-1)$-dimensional complexes $\Delta=\Delta^n(r)$, defined below, and $D=\{0,n-1\}$, so $D$-matchings are collections of pairwise disjoint $0$-simplices and $(n-1)$-simplices. In general, the $D$-matchings of $\Delta$ form a simplicial complex, denoted ${\mathcal{M}}_D(\Delta)$, with face relation given by inclusion, called the *$D$-matching complex* of $\Delta$.
Define $\Delta^n(r)$ as follows. It is a simplicial complex on $r$ vertices, labeled $v_0$ through $v_{r-1}$, such that a collection of vertices spans a simplex precisely when ${\lvert i-j \rvert}<n$ for all vertices $v_i$ and $v_j$ in the collection. For example, $\Delta^1(r)$ is a discrete set of $r$ vertices, and $\Delta^2(r)$ is the linear graph on $r$ vertices. The complex $\Delta^3(9)$ is shown in Figure \[fig:Delta\^3(9)\]. To keep notation straight, we reiterate that $r$ is the number of vertices of $\Delta^n(r)$, and $n$ is the maximum number of vertices that may share a simplex.
(0,0) – (8,0) – (7,1) – (1,1); (0,0) – (1,1) – (2,0) – (3,1) – (4,0) – (5,1) – (6,0) – (7,1) – (8,0) (0,0) – (8,0) (1,1) – (7,1); (0,0) circle (1.5pt) (2,0) circle (1.5pt) (4,0) circle (1.5pt) (6,0) circle (1.5pt) (8,0) circle (1.5pt); (1,1) circle (1.5pt) (3,1) circle (1.5pt) (5,1) circle (1.5pt) (7,1) circle (1.5pt); (1,1) circle (1.5pt) (3,1) circle (1.5pt) (5,1) circle (1.5pt) (7,1) circle (1.5pt); at (0,-.3) [$0$]{}; at (1,1.3) [$1$]{}; at (2,-.3) [$2$]{}; at (3,1.3) [$3$]{}; at (4,-.3) [$4$]{}; at (5,1.3) [$5$]{}; at (6,-.3) [$6$]{}; at (7,1.3) [$7$]{}; at (8,-.3) [$8$]{};
For any $0\le i\le j\le r-1$ with $j-i<n$, let $e_{[i,j]}$ denote the $(j-i)$-simplex $\{i,i+1,\dots,j\}$, so $\{e_{[i,j]}\}$ is a $0$-simplex in the $(j-i)$-matching complex. When a matching $\{e\}$ consists of a single simplex $e$, we will usually abuse notation and just write $e$ for the matching. For example $e_{[i,j]}$ now represents both a $(j-i)$ simplex in $\Delta^n(r)$ and a $0$-simplex in ${\mathcal{M}}_{j-i}(\Delta^n(r))$, and $v_k$ represents both a $0$-simplex in $\Delta^n(r)$ and a $0$-simplex in ${\mathcal{M}}_0(\Delta^n(r))$.
\[lem:top\_matching\_conn\] For $n,r\in{\mathbb{N}}$, ${\mathcal{M}}_{n-1}(\Delta^n(r))$ is $(\lfloor\frac{r-n}{2n-1}\rfloor-1)$-connected.
Note that $n$ is fixed. We induct on $r$. The base case is that ${\mathcal{M}}_{n-1}(\Delta^n(r))$ is non-empty when $n\le r$, which is true. Now assume that $3n-1\le r$. In this case, for any $(n-1)$-matching $\mu$ in ${\mathcal{M}}_{n-1}(\Delta^n(r))$, either $e_{[i,i+(n-1)]}\in \mu$ for some $0\le i\le n-1$, or else every $0$-simplex of $\mu$ is an $(n-1)$-simplex of $\Delta^n(r)$ that is disjoint from $e_{[0,n-1]}$. In particular, ${\mathcal{M}}_{n-1}(\Delta^n(r))$ is covered by the contractible subcomplexes $S_i{\mathrel{\mathop{:}}=}{\operatorname{st}}(e_{[i,i+(n-1)]})$ for $0\le i\le n-1$. The $S_i$ all contain the matching $e_{[r-n,r-1]}$, since $3n-1\le r$ implies $2n-2<r-n$, so the nerve of the covering is contractible (a simplex). Any intersection $S_{i_1}\cap\cdots \cap S_{i_t}$ for $t>1$ is isomorphic to a matching complex of the form ${\mathcal{M}}_{n-1}(\Delta^n(r'))$ for $r'\ge r-(2n-1)$. By induction this is $(\lfloor\frac{r-(2n-1)-n}{2n-1}\rfloor-1)$-connected, and hence $(\lfloor\frac{r-n}{2n-1}\rfloor-2)$-connected. The result now follows from the Nerve Lemma [@bjoerner94 Lemma 1.2].
For example, ${\mathcal{M}}_2(\Delta^3(9))$ is connected, which is clear from Figure \[fig:Delta\^3(9)\].
There is an analogy between $\{0,n-1\}$-matchings on $\Delta^n(r)$ and points in the link of a vertex $x\in X_n$ with $r$ feet. That is, each $0$-matching is a single vertex of $\Delta^n(r)$, so corresponds to splitting a foot of $x$ into $n$ new feet, and each $(n-1)$-matching is a collection of $n$ sequential vertices of $\Delta^n(r)$, so corresponds to merging $n$ sequential feet of $x$ into one new foot. We make this rigorous in the next lemma.
Let $x$ be a vertex of $X_n$ with $r$ feet. The cofaces of $x$ are the cells $\sigma = x\Psi$, for every $\Psi$ such that $x\Psi$ makes sense. If $\Psi={\langle A_0,\dots,A_{\ell-1} \rangle}$ for $A_i\in\{{\mathrm{I}},{\Lambda}_n,{\mathrm{V}}_n\}$ ($0\le i\le \ell-1$), then the rule is that $r$ must equal the number of $A_i$ that are ${\mathrm{I}}_n$ or ${\Lambda}_n$, plus $n$ times the number that are ${\mathrm{V}}_n$.
\[lem:vertex\_link\] If a vertex $x\in X_n$ has $r$ feet then ${\operatorname{lk}}x \cong {\mathcal{M}}_{\{0,n-1\}}(\Delta^n(r))$.
Define a map $g\colon {\operatorname{lk}}x \to {\mathcal{M}}_{\{0,n-1\}}(\Delta^n(r))$ as follows. For a coface $x\Psi$ with $\Psi={\langle A_0,\dots,A_{\ell-1} \rangle}$, $g$ sends $x\Psi$ to a $\{0,n-1\}$-matching of $\Delta^n(r)$ where each ${\Lambda}_n$ is a $0$-simplex in $\Delta^n(r)$ and each ${\mathrm{V}}_n$ is an $(n-1)$-simplex in $\Delta^n(r)$. More precisely, for each $0\le i\le \ell-1$, let $m_i$ be the number of $0\le j<i$ such that $A_j={\mathrm{V}}_n$, and then $$g(x\Psi) {\mathrel{\mathop{:}}=}\{v_{k+(n-1)m_k},e_{[\ell+(n-1)m_\ell,\ell+(n-1)m_\ell+(n-1)]}\mid A_k={\Lambda}_n \text{ and } A_\ell={\mathrm{V}}_n\} \text{.}$$ For example, $g(x{\langle {\mathrm{I}},{\mathrm{V}}_n,{\Lambda}_n \rangle})=\{e_{[1,n]},v_{n+1}\}$. It is straightforward to check that $g$ is a simplicial isomorphism.
See Figure \[fig:lk\_model\] for an example of the correspondence ${\operatorname{lk}}x \cong {\mathcal{M}}_{\{0,n-1\}}(\Delta^n(r))$.
(-0.5,0) – (4.5,0) – (2,2) – (-0.5,0) (0,0) – (0,-1) (1,0) – (1,-1) (2,0) – (2,-1) (3,0) – (3,-1) (4,0) – (4,-1); (0,-1) circle (1.5pt) (2,-1) circle (1.5pt) (4,-1) circle (1.5pt); (1,-1) circle (1.5pt) (3,-1) circle (1.5pt); (1,-1) circle (1.5pt) (3,-1) circle (1.5pt); (1,-1) – (2,-2) – (3,-1) (2,-2) – (2,-1); (3.5,-2) – (4,-1) – (4.5,-2) (4,-1) – (4,-2); at (2,0.75) [$x$]{}; at (5.5,0) [$\mapsto$]{};
(0,0) – (4,0) – (3,1) – (1,1); (1,1) – (2,0) – (3,1); (0,0) – (1,1) – (2,0) – (3,1) – (4,0) (0,0) – (4,0) (1,1) – (3,1); (0,0) circle (1.5pt) (2,0) circle (1.5pt); (4,0) circle (2.5pt); (1,1) circle (1.5pt) (3,1) circle (1.5pt); (1,1) circle (1.5pt) (3,1) circle (1.5pt);
For a vertex $x \in X_n$ recall that $f(x)$ denotes its number of feet. The function $f$ extends to an affine Morse function on $X_n$ (Observation \[obs:feet\_affine\]). Viewing ${\operatorname{lk}}x$ as ${\mathcal{M}}_{\{0,n-1\}}(\Delta^n(r))$ for $x$ with $f(x)=r$, the ascending link of $x$ with respect to $f$ is ${\mathcal{M}}_0(\Delta^n(r))$ and the descending link is ${\mathcal{M}}_{n-1}(\Delta^n(r))$.
\[cor:f\_lks\] For $x$ a vertex with $f(x)=r$, ${\operatorname{lk}}^{f\uparrow}_{X_n} x$ is contractible and ${\operatorname{lk}}^{f\downarrow}_{X_n} x$ is $(\lfloor\frac{r-n}{2n-1}\rfloor-1)$-connected.
We have ${\operatorname{lk}}^{f\uparrow}_{X_n} x \cong {\mathcal{M}}_0(\Delta^n(r))$, which is an $(r-1)$-simplex, hence contractible. We have ${\operatorname{lk}}^{f\downarrow}_{X_n} x \cong {\mathcal{M}}_{n-1}(\Delta^n(r))$, which is $(\lfloor\frac{r-n}{2n-1}\rfloor-1)$-connected by Lemma \[lem:top\_matching\_conn\].
In Section \[sec:computations\] we will need a subcomplex of the form $X_n^{p \le f \le q}$ that is $(m-1)$-connected. It will be convenient to have one of the form $X_n^{p \le f \le pn^2}$.
\[lem:sublevel\_conn\] For any $p\ge m$ the complex $X_n^{p \le f \le pn^2}$ is $(m-1)$-connected.
We first claim that $X_n^{f \le pn^2}$ is $(m-1)$-connected. By the Morse Lemma (specifically Corollary \[cor:morse\]) it suffices to show that for any vertex $x$ with $f(x)>pn^2$, the $f$-descending link ${\operatorname{lk}}^{f\downarrow}_{X_n} x$ is $(m-1)$-connected. Setting $r=f(x)$, we know from Corollary \[cor:f\_lks\] that the $f$-descending link is $(\lfloor\frac{r-n}{2n-1}\rfloor-1)$-connected. Since $r\ge pn^2+1 \ge mn^2+1$, this is $(\lfloor\frac{mn^2-n+1}{2n-1}\rfloor-1)$-connected. To see that $mn^2-n+1 \ge m(2n-1)$ (which now suffices), we note that the roots of the polynomial $mx^2+(-2m-1)x+(m+1)$ are $1$ and $1+\frac{1}{m}$.
Now we pass from $X_n^{1 \le f \le pn^2}$ to $X_n^{p \le f \le pn^2}$. In fact these are homotopy equivalent, since ascending links of vertices with respect to $f$ are contractible (Corollary \[cor:f\_lks\]), and for a vertex with fewer than $p$ feet, the entire ascending link is contained in $X_n^{1 \le f \le pn^2}$.
\[obs:cocompact\] For $p,q\in{\mathbb{N}}$, the action of $F_n$ on $X_n^{p \le f \le q}$ is cocompact.
For each $r$, $F_n$ acts transitively on vertices with $r$ feet. The result is thus immediate since $X_n$ is locally compact.
In particular, we now have highly connected spaces on which our groups act freely and cocompactly, which is part of the setup for Definition \[def:bnsr\]. To compute the $\Sigma$-invariants using Morse theory, we will use our knowledge of how characters vary between adjacent vertices (Lemma \[lem:vary\_chars\]). Since we are modeling vertex links by $\{0,n-1\}$-matching complexes on $\Delta^n(r)$, we need to translate Lemma \[lem:vary\_chars\] into the language of $\{0,n-1\}$-matchings.
Let $\chi$ be a character of $F_n$, extended to $X_n$ as in Lemma \[lem:affine\_extend\]. Let $x\in X_n$ be a vertex with $r=f(x)$ feet, so ${\operatorname{lk}}x \cong {\mathcal{M}}_{\{0,n-1\}}(\Delta^n(r))$. Under this isomorphism, call a vertex of ${\mathcal{M}}_{\{0,n-1\}}(\Delta^n(r))$ *$\chi$-ascending* if the corresponding vertex $y$ in ${\operatorname{lk}}x$ has $\chi(y)>\chi(x)$. Analogously define *$\chi$-descending* and *$\chi$-preserving*. Say a simplex $\mu$ in ${\mathcal{M}}_{\{0,n-1\}}(\Delta^n(r))$ is $\chi$-ascending/descending/preserving if all its vertices are.
\[obs:asc\_lks\_matchings\] Let $(\chi,f) \colon X_n^{p\le f\le q} \to {\mathbb{R}}\times {\mathbb{R}}$ be a Morse function as in Proposition \[prop:char\_morse\]. Let $x$ be a vertex in $X_n^{p\le f\le q}$ with $r=f(x)$ feet. Then the $(\chi,f)$-ascending link of $x$ in $X_n$ is isomorphic to the full subcomplex of ${\mathcal{M}}_{\{0,n-1\}}(\Delta^n(r))$ supported on those $0$-simplices $v_k$ and $e_{[k,k+(n-1)]}$ such that $v_k$ is either $\chi$-ascending or $\chi$-preserving, and $e_{[k,k+(n-1)]}$ is $\chi$-ascending. The $(\chi,f)$-ascending link of $x$ in $X_n^{p\le f\le q}$ is then obtained by removing any $\{0,n-1\}$-matchings $\mu$ such that $r+(n-1)\mu_0>q$ or $r-(n-1)\mu_{n-1}<p$, where $\mu_i$ is the number of vertices of $\mu$ that are $i$-matchings.
To increase $(\chi,f)$, we must either increase $\chi$ or else preserve $\chi$ and increase $f$. The $v_k$ correspond to vertices in ${\operatorname{lk}}x$ with $r+(n-1)$ feet, and the $e_{[k,k+(n-1)]}$ to vertices in ${\operatorname{lk}}x$ with $r-(n-1)$ feet. Hence the first claim follows. For the second claim, just note that $\mu$ corresponds to a simplex in ${\operatorname{lk}}x$, and hence to a cube in $X_n$ containing $x$, and $r+(n-1)\mu_0$ is the maximum number of feet of a vertex in that cube; similarly $r-(n-1)\mu_{n-1}$ is the minimum number of feet of a vertex in that cube.
\[cor:char\_match\] If $k=0$ then $v_k$ is $\chi_0$-descending and $e_{[k,k+(n-1)]}$ is $\chi_0$-ascending. Otherwise they are both $\chi_0$-preserving. If $k=r-1$ then $v_k$ is $\chi_1$-descending and $e_{[k-(n-1),k]}$ is $\chi_1$-ascending. Otherwise they are both $\chi_1$-preserving. If $0<k$ and $k-1\equiv_{n-1} i$, then $v_k$ is $\psi_i$-ascending and $e_{[k,k+(n-1)]}$ is $\psi_i$-descending. If $k<r-1$ and $k\equiv_{n-1} i$, then $v_k$ is $\psi_i$-descending and $e_{[k-(n-1),k]}$ is $\psi_i$-ascending. Anything not covered by these cases is $\psi_i$-preserving. In all of these cases, “ascending” entails an increase by $+1$ and “descending” entails a decrease by $-1$.
Translating to ${\operatorname{lk}}x$, $v_k$ corresponds to $x \cdot [{\Lambda}_n(r,k),{\operatorname{id}}_{r+(n-1)}]$, $e_{[k,k+(n-1)]}$ corresponds to $x \cdot [{\operatorname{id}}_r,{\Lambda}_n(r-(n-1),k)]$ and $e_{[k-(n-1),k]}$ corresponds to $x \cdot [{\operatorname{id}}_r,{\Lambda}_n(r-(n-1),k-(n-1))]$. Hence Lemma \[lem:vary\_chars\] implies all of these facts.
Note that in particular if $r-1>0$ then $v_{r-1}$ is $\psi_i$-preserving for all $0\le i\le n-3$, since $r-2\equiv_{n-1} n-2$.
Some examples of $\psi_i$-ascending, descending or preserving $0$-simplices in ${\mathcal{M}}_{\{0,2\}}(\Delta_3(5))$, as governed by Corollary \[cor:char\_match\], are shown in Figure \[fig:char\_match\].
(0,0) – (4,0) – (3,1) – (1,1); (0,0) – (1,1) – (2,0) – (3,1) – (4,0) (0,0) – (2,0) – (4,0) (1,1) – (3,1); (0,0) circle (2.5pt); (2,0) circle (1.5pt) (4,0) circle (1.5pt); (1,1) circle (1.5pt) (3,1) circle (1.5pt); (1,1) circle (1.5pt) (3,1) circle (1.5pt); at (0,-.4) [$v_0$]{};
(0,0) – (4,0) – (3,1) – (1,1); (0,0) – (1,1) – (2,0) – (3,1) – (4,0) (0,0) – (2,0) – (4,0) (1,1) – (3,1); (0,0) circle (1.5pt) (2,0) circle (1.5pt) (4,0) circle (1.5pt); (1,1) circle (2.5pt) (3,1) circle (1.5pt); (1,1) circle (2.5pt); (3,1) circle (1.5pt); at (1,1.3) [$v_1$]{};
(0,0) – (4,0) – (3,1) – (1,1); (1,1) – (2,0) – (3,1); (0,0) – (1,1) – (2,0) – (3,1) – (4,0) (0,0) – (2,0) – (4,0) (1,1) – (3,1); (0,0) circle (1.5pt) (2,0) circle (1.5pt) (4,0) circle (1.5pt); (1,1) circle (1.5pt) (3,1) circle (1.5pt); (1,1) circle (1.5pt) (3,1) circle (1.5pt); at (2,1.3) [$e_{[1,3]}$]{};
(0,0) – (4,0) – (3,1) – (1,1); (2,0) – (3,1) – (4,0); (0,0) – (1,1) – (2,0) – (3,1) – (4,0) (0,0) – (2,0) – (4,0) (1,1) – (3,1); (0,0) circle (1.5pt) (2,0) circle (1.5pt) (4,0) circle (1.5pt); (1,1) circle (1.5pt) (3,1) circle (1.5pt); (1,1) circle (1.5pt) (3,1) circle (1.5pt); at (3,-.4) [$e_{[2,4]}$]{};
Proof of theorem A {#sec:computations}
==================
In this section we prove Theorem A, that $\Sigma^m(F_n)=\Sigma^2(F_n)$ for all $n,m\ge 2$. The forward inclusion always holds, so the work to do is the reverse inclusion. Throughout this section, $\chi$ is a character of $F_n$ with $[\chi]\in\Sigma^2(F_n)$.
For the first three lemmas, we will make use of a certain ascending HNN-extension of $F_n$. (We should mention that there is nothing novel here, and the reduction done over the course of these three lemmas was already contained in the work of Kochloukova [@kochloukova12].) Let $F_n(1)$ be the subgroup of $F_n$ generated by the $x_i$ for $i>0$ (see Subsection \[sec:presentation\]). It is well known that $F_n=F_n(1)*_{x_0}$ and $F_n(1)\cong F_n$.
\[lem:poles\] If $\chi=-\chi_i$ for $i=0,1$ then $[\chi]\in\Sigma^\infty(F_n)$.
By symmetry it suffices to do the $i=0$ case. We know $F_n=F_n(1)*_{x_0}$ and that $F_n(1)\cong F_n$ is of type ${\operatorname{F}}_\infty$. Also, $-\chi_0(F_n(1))=0$ and $-\chi_0(x_0)=1$, so the result follows from [@bieri10 Theorem 2.1].
Now suppose $\chi=a\chi_0+b\chi_1$. Since $[\chi]\in\Sigma^2(F_n)$, we know from [@kochloukova12 Proposition 9,Theorem 10] that $a<0$ or $b<0$. This could also be deduced using the action of $F_n$ on $X_n$, following the proof of the $n=2$ case in [@witzel15]. This would take many pages of technical details though, so we content ourselves with just citing Kochloukova to say that we know $a<0$ or $b<0$.
\[lem:equator\] If $\chi=a\chi_0+b\chi_1$ with $a<0$ or $b<0$ then $[\chi]\in\Sigma^\infty(F_n)$.
By symmetry we can assume $b<0$. Since $\chi_0(F_n(1))=0$, we have that $\chi|_{F_n(1)}$ is equivalent to $-\chi_1$ when restricted to $F_n(1)$. Now, $F_n(1)\cong F_n$ is of type $F_\infty$ and $F_n=F_n(1)*_{x_0}$, so by [@bieri10 Theorem 2.3] and Lemma \[lem:poles\] $[\chi]\in\Sigma^\infty(F_n)$.
Now we can assume $n>2$ and $\chi$ has non-zero $\psi_i$ component for some $i$.
\[lem:push\_to\_hemispheres\] Assume that we already know every non-trivial character of the form $\chi'=\sum_{i=0}^{n-3} c_i \psi_i$ has $[\chi']\in\Sigma^\infty(F_n)$. Then for any $\chi=a\chi_0 + \sum_{i=0}^{n-3} c_i \psi_i + b\chi_1$ with $c_i\ne 0$ for at least one $i$, we have $[\chi]\in\Sigma^\infty(F_n)$.
Note that such a $\chi$ restricted to $F_n(1)$ is still non-trivial. As in the previous proof, we can restrict to $F_n(1)$ and ensure that without loss of generality $a=0$. If $b\ne 0$ then appealing to symmetry, we can rather assume $a\ne 0$ but $b=0$. Now by the first sentence we can reduce to the case $a=b=0$. But this is exactly the case already handled in the assumption.
This brings us to the final case, where we assume that $\chi$ is a linear combination of the $\psi_i$ for $0\le i\le n-3$, and show that $[\chi]\in\Sigma^\infty(F_n)$. This is where Kochloukova’s approach in [@kochloukova12] became difficult to extend beyond the $\Sigma^2$ case, and where our setup from the previous sections will prove to be able to handle all the $\Sigma^m$.
Let $m\in{\mathbb{N}}$ and set $p{\mathrel{\mathop{:}}=}4m+5$. Let $Y{\mathrel{\mathop{:}}=}X_n^{p\le f\le pn^2}$. Then $Y$ is $(m-1)$-connected (Lemma \[lem:sublevel\_conn\]), and $F_n$ acts freely and cocompactly on $Y$ (Observation \[obs:cocompact\]), so we have the requisite setup of Definition \[def:bnsr\]. (Of course $Y$ would have already been $(m-1)$-connected just using $p=m$, but having $p=4m+5$ will be important in the proof of Proposition \[prop:asc\_lk\_conn\].) According to Definition \[def:bnsr\], we need to show that $(Y^{t\le\chi})_{t\in{\mathbb{R}}}$ is essentially $(m-1)$-connected and then we will have $[\chi]\in\Sigma^m(F_n)$. In fact we will show that every $Y^{t\le\chi}$ is $(m-1)$-connected.
The proof that $Y^{t\le\chi}$ is $(m-1)$-connected will be quite technical, so we first sketch the proof here, to serve as an outline for what follows.
Thanks to Morse theory, it suffices to show that all $(\chi,f)$-ascending links of vertices $x$ are $(m-1)$-connected. Since we are working in $Y$, we know the number of feet of $x$ lies between $p$ and $pn^2$. We consider the cases $p\le f(x)\le pn$ and $pn \le f(x) \le pn^2$ separately. In the first case, even if we split every foot of $x$, we remain in $Y$, so all splittings are “legal”. In particular if there is some ascending splitting move that is joinable in $X_n$ to every other ascending move, then these joins can even take place in $Y$, and the ascending link is a (contractible) cone. It turns out that the move where we split the rightmost foot serves as such a cone point. Now consider the second case, $pn \le f(x) \le pn^2$. Here there may be splitting moves that push us out of $Y$, but every merging move keeps us inside $Y$. It is too much to hope for to find an ascending merging move joinable to every other ascending move. However, we do find an ascending merging move consisting of a “large” simplex $\sigma_q$ such that every ascending vertex is joinable to “almost all” of $\sigma_q$. We prove in Lemma \[lem:popular\_simplex\] that this is sufficient to get high connectivity.
Now we begin the technicalities. First we need a lemma that is a useful tool for proving higher connectivity of certain complexes. Heuristically, if there is a simplex $\sigma$ such that every vertex is joinable to “most” of $\sigma$, then we can conclude higher connectivity properties. The case when the complex is finite and flag was proved by Belk and Forrest, and written down by Belk and Matucci in [@belk15]. Here we show that the requirement of being finite can easily be relaxed. We also replace the requirement of being flag with something weaker, and rephrase the condition from [@belk15] so that in the flag case it is the same. This is a necessary modification, since the complexes we will apply this lemma to in the proof of Proposition \[prop:asc\_lk\_conn\] are not flag.
Let $\Delta$ be a simplicial complex. Two simplices $\rho_1$ and $\rho_2$ are *joinable* to each other if they lie in a common simplex. For a fixed simplex $\sigma$ in $\Delta$, we will call $\Delta$ *flag with respect to $\sigma$* if whenever $\rho$ is a simplex and $\sigma'$ is a face of $\sigma$ such that every vertex of $\rho$ is joinable to every vertex of $\sigma'$, already $\rho$ is joinable to $\sigma'$. For example if $\Delta$ is flag with respect to every simplex, then it is flag.
\[lem:popular\_simplex\] Let $\Delta$ be a simplicial complex, and let $k\in{\mathbb{N}}$. Suppose there exists an $\ell$-simplex $\sigma$ such that $\Delta$ is flag with respect to $\sigma$, and for every vertex $v$ in $\Delta$, $v$ is joinable to some $(\ell-k)$-face of $\sigma$. Then $\Delta$ is $(\lfloor\frac{\ell}{k}\rfloor-1)$-connected.
First note that our hypotheses on $\Delta$ are preserved under passing to any full subcomplex $\Delta'$ containing $\sigma$. Indeed, joinability of two simplices is preserved under passing to any full subcomplex containing them, so $\Delta'$ is still flag with respect to $\sigma$ and every vertex of $\Delta'$ is still joinable to some $(\ell-k)$-face of $\sigma$. In particular, without loss of generality $\Delta$ is finite. Indeed, any homotopy sphere lies in some full subcomplex $\Delta'$ of $\Delta$ that contains $\sigma$ and is finite, and the complex $\Delta'$ still satisfies our hypotheses since it is full and contains $\sigma$. If the sphere is nullhomotopic in $\Delta'$ then it certainly is nullhomotopic in $\Delta$, so from now on we may indeed assume $\Delta$ is finite.
We induct on the number $V$ of vertices of $\Delta$. If $V=\ell+1$ then $\Delta=\sigma$ is contractible. Now suppose $V>\ell+1$, so we can choose a vertex $v\in \Delta \setminus \sigma$. The subcomplex obtained from $\Delta$ by removing $v$ along its link $L{\mathrel{\mathop{:}}=}{\operatorname{lk}}v$ has fewer vertices than $\Delta$, is full, and contains $\sigma$, so by the first paragraph and by induction it is $(\lfloor\frac{\ell}{k}\rfloor-1)$-connected. It now suffices to show that $L$ is $(\lfloor\frac{\ell}{k}\rfloor-2)$-connected.
Let $\tau {\mathrel{\mathop{:}}=}\sigma\cap {\operatorname{st}}v$, so $\tau$ also equals $\sigma\cap L$. Since $\Delta$ is flag with respect to $\sigma$, $\tau$ is a face of $\sigma$. Say $\tau$ is an $(\ell-k')$-simplex, which since $v$ is joinable to an $(\ell-k)$-face of $\sigma$ tells us that $k'\le k$. Now let $w$ be a vertex in $L$. By similar reasoning we know that $\tau_w {\mathrel{\mathop{:}}=}\sigma \cap {\operatorname{st}}w$ is an $(\ell-k'_w)$-simplex for $k'_w\le k$. Intersecting the two faces $\tau$ and $\tau_w$ of $\sigma$ thus yields a face $\omega_w$ that is an $(\ell-k'-k'')$-simplex for $k''\le k'_w\le k$. Since $v$ and $w$ are joinable to $\omega_w$, and $\Delta$ is flag with respect to $\sigma$, the edge connecting $v$ and $w$ is also joinable to $\omega_w$. In particular $\omega_w$ is joinable to $w$ in $L$. We have shown that there is an $(\ell-k')$-simplex, $\tau$, in $L$ such that every vertex $w$ of $L$ is joinable in $L$ to an $(\ell-k'-k)$-face of $\tau$, namely any $(\ell-k'-k)$-face of $\omega_w$. We also claim that $L$ is flag with respect to $\tau$. Indeed, if $\rho$ is a simplex in $L$ and $\tau'$ is a face of $\tau$ such that every vertex of $\rho$ is joinable to every vertex of $\tau'$, then $\rho*v$ is a simplex in $\Delta$ all of whose vertices are joinable in $\Delta$ to all the vertices of $\tau'$, so $\rho*v$ is joinable in $\Delta$ to $\tau'$ and indeed $\rho$ is joinable in $L={\operatorname{lk}}v$ to $\tau'$. Now we can apply the induction hypothesis to $L$, and conclude that $L$ is $(\lfloor\frac{\ell-k'}{k}\rfloor-1)$-connected, and hence $(\lfloor\frac{\ell}{k}\rfloor-2)$-connected.
As a trivial example (which works for any $\Delta$), if there exists a simplex $\sigma$ such that every vertex is joinable to some vertex of $\sigma$, so we can use $k=\ell$, then $\Delta$ is $0$-connected. To tie Lemma \[lem:popular\_simplex\] to the version in [@belk15], note that if $\Delta$ is flag, then $v$ being joinable to an $(\ell-k)$-face of $\sigma$ is equivalent to $v$ being joinable to all but at most $k$ vertices of $\sigma$.
We return to the complex $Y=X_n^{p\le f\le pn^2}$ (recall $p=4m+5$) and the problem of showing that every $Y^{t\le\chi}$ is $(m-1)$-connected. Consider $$h{\mathrel{\mathop{:}}=}(\chi,f) \colon Y \to {\mathbb{R}}\times {\mathbb{R}}\text{,}$$ ordered lexicographically. This is a Morse function by Proposition \[prop:char\_morse\], so by the Morse Lemma \[lem:morse\] (specifically Corollary \[cor:morse\]), it suffices to show the following:
\[prop:asc\_lk\_conn\] Let $x$ be a vertex in $Y$. Then the $h$-ascending link ${\operatorname{lk}}^{h\uparrow}_Y x$ is $(m-1)$-connected.
Let $r{\mathrel{\mathop{:}}=}f(x)$. We view ${\operatorname{lk}}x$ as ${\mathcal{M}}_{\{0,n-1\}}(\Delta^n(r))$, so the $h$-ascending link is as described in Observation \[obs:asc\_lks\_matchings\]. We will split the problem into two cases: when $r$ is “small” and when $r$ is “big”. First suppose $p\le r\le pn$. In this case, for any $\{0,n-1\}$-matching $\mu$, with $\mu_0$ the number of $0$-simplices in $\mu$ that are $0$-matchings (so $\mu_0 \le pn$), we have $r+(n-1)\mu_0 \le pn + (n-1)pn = pn^2$. In particular the addition of $0$-matchings to a $\{0,n-1\}$-matching will never push us out of $Y$.
We know from Corollary \[cor:char\_match\] that $v_{r-1}$ is $\psi_i$-preserving for all $0\le i\le n-3$, and hence $\chi$-preserving. If $e_{[r-n,r-1]}$ represents a vertex of ${\operatorname{lk}}_Y x$, i.e., if $p\le r-(n-1)$, then $e_{[r-n,r-1]}$ is $\chi$-preserving since $r-1>0$. Hence $v_{r-1}$ is $h$-ascending and $e_{[r-n,r-1]}$ is not, by Observation \[obs:asc\_lks\_matchings\]. But $e_{[r-n,r-1]}$ is the only $0$-simplex of ${\mathcal{M}}_{\{0,n-1\}}(\Delta^n(r))$ not joinable to $v_{r-1}$, so ${\operatorname{lk}}^{h\uparrow}_Y x$ is contractible, via the conical contraction $\mu \le \mu\cup\{v_{r-1}\} \ge \{v_{r-1}\}$.
Now suppose $pn \le r\le pn^2$. In this case, for any $\{0,n-1\}$-matching $\mu$, with $\mu_{n-1}$ the number of $0$-simplices in $\mu$ that are $(n-1)$-matchings (so $n\mu_{n-1} \le r$), we claim that $r-(n-1)\mu_{n-1} \ge p$. Indeed, if $\mu_{n-1} \ge p$, then $r-(n-1)\mu_{n-1} \ge \mu_{n-1} \ge p$, and if $\mu_{n-1}<p$ then $r-(n-1)\mu_{n-1}> pn - (n-1)p =p$. In analogy to the previous case, this means that the addition of $(n-1)$-matchings to a $\{0,n-1\}$-matching will never push us out of $Y$.
For $0\le q\le n-2$, let $\sigma_q$ be the $((s/2)-1)$-simplex $$\sigma_q {\mathrel{\mathop{:}}=}\{e_{[q+(n-1),q+2(n-1)]},e_{[q+3(n-1),q+4(n-1)]}, \dots, e_{[q+(s-1)(n-1),q+s(n-1)]}\} \text{,}$$ where $s\in 2{\mathbb{N}}$ is as large as possible such that $q+s(n-1)<r-1$; see Figure \[fig:sigma\_0\] for an example. Since $r\ge pn \ge 9n$, such an $s$ certainly exists. By maximality of $s$, we must have $q+(s+2)(n-1) \ge r-1$, and since $r \ge pn$ and $q\le n-2$, we then have $s\ge \lfloor\frac{(p-3)n+3}{n-1}\rfloor$. By definition $p=4m+5$, and it is straightforward to check that this bound gives us the bound $s\ge 4m+2$.
We now want to cleverly choose $q$ so that $\sigma_q$ is $\chi$-ascending, and hence $h$-ascending. Recall that $\chi=c_0\psi_0+\cdots+c_{n-3}\psi_{n-3}$, and now also set $c_{n-2}{\mathrel{\mathop{:}}=}0$. Let $0\le q\le n-2$ be any value such that, with subscripts considered mod $(n-1)$, we have $c_{q-1}<c_q$. Since the $c_i$ cannot all be zero, such a $q$ exists. For this choice of $q$, and any $1\le t\le s-1$, we claim that $e_{[q+t(n-1),q+(t+1)(n-1)]}$ is $\chi$-ascending, which will then imply that $\sigma_q \in {\operatorname{lk}}^{h\uparrow}_Y x$. By Corollary \[cor:char\_match\], since $0<q+(n-1)$ and $q+s(n-1)<r-1$, we know that $e_{[q+t(n-1),q+(t+1)(n-1)]}$ is $\psi_{q-1}$-descending (subscript taken mod $(n-1)$), $\psi_q$-ascending, and $\psi_i$-preserving for all other $0\le i\le n-2$. Then since $c_{q-1}<c_q$, Corollary \[cor:char\_match\] tells us that $e_{[q+t(n-1),q+(t+1)(n-1)]}$ is indeed $\chi$-ascending, and so $h$-ascending, namely it increases $\chi$ by $c_q-c_{q-1}>0$.
With this $h$-ascending $((s/2)-1)$-simplex $\sigma_q$ in hand, we want to apply Lemma \[lem:popular\_simplex\] to ${\operatorname{lk}}^{h\uparrow}_Y x$. Note that ${\operatorname{lk}}^{h\uparrow}_{X_n} x$ is flag, but ${\operatorname{lk}}^{h\uparrow}_Y x$ might not be, since filling in missing simplices might require pushing $f$ above $pn^2$. However, we claim ${\operatorname{lk}}^{h\uparrow}_Y x$ is flag with respect to $\sigma_q$. Indeed, if $\rho$ is a simplex and $\sigma'_q$ is a face of $\sigma_q$ such that every vertex of $\rho$ is joinable to every vertex of $\sigma_q'$, then since $X_n$ is flag we can consider the simplex $\rho*\sigma'_q$ in $X_n$, and since $\sigma_q$ consists only of $(n-1)$-matchings, $f$ achieves its maximum on $\rho*\sigma'_q$ already on $\rho$. Hence if $\rho$ came from $Y$, then $\rho*\sigma'_q$ is also in $Y$, and so ${\operatorname{lk}}^{h\uparrow}_Y x$ is flag with respect to $\sigma_q$. Now we want to show that every vertex of ${\operatorname{lk}}^{h\uparrow}_Y x$ is joinable to “most” of $\sigma_q$. Let $\mu$ be any $0$-simplex in ${\mathcal{M}}_{\{0,n-1\}}(\Delta^n(r))$. We claim that $\mu$ is joinable to all but at most two vertices of $\sigma_q$. Indeed, if $\mu=\{v_j\}$ then $\mu$ fails to be joinable to $\{e_{[k,k+(n-1)]}\}$ if and only if $k\le j\le k+(n-1)$, and there is at most one such $\{e_{[k,k+(n-1)]}\}$ in $\sigma_q$ with this property. Similarly if $\mu=\{e_{[j,j+(n-1)]}\}$ then $\mu$ fails to be joinable to $\{e_{[k,k+(n-1)]}\}$ if and only if $k\le j\le k+(n-1)$ or $k\le j+(n-1)\le k+(n-1)$, and there are at most two such $\{e_{[k,k+(n-1)]}\}$ in $\sigma_q$ with this property. We now know that for any $0$-simplex $\mu$ in ${\operatorname{lk}}^{h\uparrow}_Y x$, $\mu$ is joinable in ${\operatorname{lk}}^{h\uparrow}_Y x$ to an $((s/2)-3)$-face of $\sigma_q$. By Lemma \[lem:popular\_simplex\], we conclude that ${\operatorname{lk}}^{h\uparrow}_Y x$ is $(\lfloor\frac{(s/2)-1}{2}\rfloor-1)$-connected. Recall that $s\ge 4m+2$, and so ${\operatorname{lk}}^{h\uparrow}_Y x$ is $(m-1)$-connected.
(0,0) – (14,0) – (13,1) – (1,1); (2,0) – (3,1) – (4,0) (6,0) – (7,1) – (8,0) (10,0) – (11,1) – (12,0); (0,0) – (14,0) – (13,1) – (1,1) – (0,0) (1,1) – (2,0) – (3,1) – (4,0) – (5,1) – (6,0) – (7,1) – (8,0) – (9,1) – (10,0) – (11,1) – (12,0) – (13,1); (0,0) circle (1.5pt) (2,0) circle (1.5pt) (4,0) circle (1.5pt) (6,0) circle (1.5pt) (8,0) circle (1.5pt) (10,0) circle (1.5pt) (12,0) circle (1.5pt) (14,0) circle (1.5pt); (1,1) circle (1.5pt) (3,1) circle (1.5pt) (5,1) circle (1.5pt) (7,1) circle (1.5pt) (9,1) circle (1.5pt) (11,1) circle (1.5pt) (13,1) circle (1.5pt); (1,1) circle (1.5pt) (3,1) circle (1.5pt) (5,1) circle (1.5pt) (7,1) circle (1.5pt) (9,1) circle (1.5pt) (11,1) circle (1.5pt) (13,1) circle (1.5pt); at (3,-.3) [$e_{[2,4]}$]{}; at (7,-.3) [$e_{[6,8]}$]{}; at (11,-.3) [$e_{[10,12]}$]{};
We summarize this section by writing down the proof of Theorem A.
Let $\chi=a\chi_0+c_0\psi_0+\cdots+c_{n-3}\psi_{n-3}+b\chi_1$ with $[\chi]\in\Sigma^2(F_n)$. If all the $c_i$ are zero then since $[\chi]\in\Sigma^2(F_n)$ we know either $a<0$ or $b<0$, and so $[\chi]\in\Sigma^\infty(F_n)$ by Lemma \[lem:equator\]. Now suppose the $c_i$ are not all zero. By Lemma \[lem:push\_to\_hemispheres\], without loss of generality $a=b=0$. Then by Proposition \[prop:asc\_lk\_conn\] and Corollary \[cor:morse\], $Y^{t\le\chi}$ is $(m-1)$-connected for all $t\in{\mathbb{R}}$. Hence by Definition \[def:bnsr\] we conclude that $[\chi]\in\Sigma^\infty(F_n)$.
Houghton groups {#sec:houghton}
===============
Let $(H_n)_{n\in{\mathbb{N}}}$ be the family of Houghton groups, introduced in [@houghton78]. An element $\eta$ of $H_n$ is an automorphism of $\{1,\dots,n\}\times{\mathbb{N}}$ such that for each $1\le i\le n$ there exists $m_i\in{\mathbb{Z}}$ and $N_i\in{\mathbb{N}}$ such that for all $x\ge N_i$ we have $(i,x)\eta=(i,x+m_i)$. That is, $\eta$ “eventually acts as translations” on each ray $\{i\}\times{\mathbb{N}}$. We have that $H_n$ is of type ${\operatorname{F}}_{n-1}$ but not of type ${\operatorname{F}}_n$ [@brown87 Theorem 5.1].
It is known that ${\operatorname{Hom}}(H_n,{\mathbb{R}})$ is generated by characters $\chi_1,\dots,\chi_n$, given by $\chi_i(\eta)=m_i$ for each $i$ (with $m_i$ as above). Since $\eta$ is an automorphism, $\sum m_i = 0$ for any $\eta$, and hence $\chi_1+\cdots+\chi_n=0$ as characters. In fact for $n\ge 2$, $\chi_1,\dots,\chi_{n-1}$ form a basis of ${\operatorname{Hom}}(H_n,{\mathbb{R}})\cong{\mathbb{R}}^{n-1}$. Since $H_n$ is of type ${\operatorname{F}}_{n-1}$, one can ask about $\Sigma^m(H_n)$ for $m\le n-1$. Bieri and Strebel \[unpublished\], and independently Brown [@brown87bns Proposition 8.3], proved that for $n\ge2$ the complement of $\Sigma^1(H_n)$ is $\{[-\chi_i]\}_{i=1}^n$. Note that when $n=2$, $S(H_2)=\{[\chi_1],[-\chi_1]\}$ and $\chi_1=-\chi_2$ so in fact $\Sigma^1(H_2)=\emptyset$.
In this section we prove:
\[thrm:houghton\_pos\] Let $n\in{\mathbb{N}}$ and let $\chi = a_1\chi_1 + \cdots + a_n \chi_n$ be a non-trivial character, i.e., the $a_i$ are not all equal. Up to symmetry, we can assume $a_1\le \cdots \le a_n$. Let $1\le m(\chi)\le n-1$ be maximal such that $a_{m(\chi)}\ne a_n$. Then $[\chi]\in\Sigma^{m(\chi)-1}(H_n)$.
For example, $[\chi_n]\in \Sigma^{n-2}(H_n)$. Note that since $\chi_1+\cdots+\chi_n=0$, without loss of generality $a_{m(\chi)+1}=a_n=0$. With this convention we would for example not write $\chi_n$ but rather $-\chi_1-\cdots-\chi_{n-1}$. Also note that the only $\chi$ with $m(\chi)=1$ are those equivalent to $-\chi_i$, so we recover the fact that the $[-\chi_i]$ are the only things in the complement of $\Sigma^1(H_n)$.
This leaves open the question of whether $[\chi]\not\in\Sigma^{m(\chi)}(H_n)$ always holds, which we expect should be true.
\[conj:houghton\_neg\] With the setup of Theorem \[thrm:houghton\_pos\], moreover $[\chi]\not\in\Sigma^{m(\chi)}(H_n)$.
This conjecture holds for low values of $m(\chi)$. When $m(\chi)=1$, without loss of generality $\chi=-\chi_1$, and $[-\chi_1]\not\in\Sigma^1(H_n)$ as mentioned above. When $m(\chi)=2$ (so $n\ge3$), without loss of generality $[\chi]$ lies in the convex hull in $S(H_n)$ of $[-\chi_1]$ and $[-\chi_2]$. Since these are not in $\Sigma^1(H_n)$, [@kochloukova02 Theorem A1] tells us that $[\chi]$ is not in $\Sigma^2(H_n)$. In general, Conjecture \[conj:houghton\_neg\] is equivalent to conjecturing that the complement of $\Sigma^m(H_n)$ is the union of all convex hulls of all $\le m$-tuples from the complement of $\Sigma^1(H_n)$; for metabelian groups this is conjectured to always hold, and is called the $\Sigma^m$-Conjecture (see, e.g., [@bieri10 Section 1.3]).
We will prove Theorem \[thrm:houghton\_pos\] by inspecting the proper action of $H_n$ on a proper ${\operatorname{CAT}}(0)$ cube complex $X_n$. Our reference for $X_n$ is [@lee12] (this is a preprint including the author’s PhD thesis results). This cube complex was also remarked upon by Brown in [@brown87], though not explicitly constructed. We will not prove everything in the setup here, but will sometimes just cite [@lee12]. The vertices of $X_n$ are elements of the monoid $M$ of injective maps $\{1,\dots,n\}\times{\mathbb{N}}\hookrightarrow \{1,\dots,n\}\times{\mathbb{N}}$ that are eventually translations. In particular $H_n$ sits in $X_n$ as a discrete set of vertices, namely those maps $\phi$ that are bijective. To describe the higher dimensional cells of $X_n$, we need to discuss $M$ in a bit more detail.
There are $n$ elements of $M$ of particular interest, namely for each $1\le i\le n$ we have a map $$t_i \colon \{1,\dots,n\}\times{\mathbb{N}}\to \{1,\dots,n\}\times{\mathbb{N}}$$ given by sending $(j,x)$ to itself if $j\ne i$ and $(i,x)$ to $(i,x+1)$ for all $x\in{\mathbb{N}}$. It is clear that for any $\phi\in M$, there exists a product of $t_i$s, say $\tau$, such that $\tau \circ \phi$ is a product of $t_i$s. Here our maps act on the right, so this composition means first do $\tau$, then do $\phi$. Heuristically, $\phi$ acts as translations outside of some finite region $S\subseteq \{1,\dots,n\}\times{\mathbb{N}}$, so just choose $\tau$ such that the range of $\tau$ lies outside $S$.
Back to defining the higher cells of $X_n$, we now declare that two vertices $\phi,\psi$ share an edge whenever $\phi = t_i \circ \psi$ or $\psi = t_i \circ \phi$ for some $1\le i\le n$. Already we have that $X_n$ is connected, thanks to the discussion in the previous paragraph. Now for $2\le k\le n$, we declare that we have a $k$-cube supported on every set of vertices of the following form: start with a vertex $\phi$, let $K$ be a subset of $\{1,\dots,n\}$ with $|K|=k$, and look at the set of $2^k$ vertices $$\left\{\left(\prod_{i\in J} t_i\right)\circ\phi \mid J\subseteq K\right\} \text{.}$$ Since the $t_i$ all commute, we do not need to specify an order in which to compose them. These vertices span a $k$-cube in $X_n$. For example, when $n\ge2$ any vertex $\phi$ lies in the square $\{\phi,t_1\circ\phi,t_2\circ\phi, t_1\circ t_2\circ \phi\}$.
It is known that $X_n$ is a ${\operatorname{CAT}}(0)$ cube complex [@lee12]. The group $H_n$ acts on the vertices of $X_n$ via $(\phi)\eta {\mathrel{\mathop{:}}=}\phi \circ \eta$, and this extends to an action of $H_n$ on $X_n$. There is an $H_n$-invariant height function $f$ (called $h$ in [@lee12]) on the vertices of $X_n$, namely if $\phi\in M$ and $F(\phi) {\mathrel{\mathop{:}}=}(\{1,\dots,n\}\times{\mathbb{N}}) \setminus \operatorname{image}(\phi)$, so $F(\phi)$ is finite, then $$f(\phi) {\mathrel{\mathop{:}}=}|F(\phi)| \text{.}$$ Note that $f(\phi)=0$ if and only if $\phi$ is bijective, i.e., $\phi\in H_n$. It is clear that $f$ is $H_n$-invariant. Also note that for any cube $\sigma$ in $X_n$, there is a unique vertex of $\sigma$ with minimal $f$-value, and so any cube stabilizer is contained in a vertex stabilizer. Vertex stabilizers are finite, since if $\phi \circ \eta = \phi$ then $\eta$ must fix all points outside $F(\phi)$.
In summary, $H_n$ acts properly on the $n$-dimensional proper ${\operatorname{CAT}}(0)$ cube complex $X_n$. The action is not cocompact, since it is $f$-invariant and $f$ takes infinitely many values, but it is cocompact on $f$-sublevel sets:
\[lem:houghton\_cocpt\] The action of $H_n$ on any $X_n^{p\le f\le q}$ is cocompact.
Since $X_n$ is locally compact, we just need to see that $H_n$ is transitive on vertices with the same $f$ value. Let $\phi$ and $\psi$ be vertices with $f(\phi)=f(\psi)$. Let $\alpha\in S_\infty\le H_n$ be any bijection taking $F(\phi)$ bijectively to $F(\psi)$ (since $f(\phi)=f(\psi)$ such a $\alpha$ exists). Define $\eta\in H_n$ via: $$(x)\eta {\mathrel{\mathop{:}}=}\left\{\begin{array}{ll}
(y)\psi & \text{if } x=(y)\phi \\
(x)\alpha & \text{if } x\in F(\phi) \text{.}
\end{array}\right.$$ Now $\phi \circ \eta = \psi$ by definition, and $\eta$ clearly eventually acts by translations, so we just need to show $\eta$ is bijective. Note that $\eta$ takes $\operatorname{image}(\phi)$ bijectively to $\operatorname{image}(\psi)$, and also takes $F(\phi)$ bijectively to $F(\psi)$, so indeed $\eta$ is bijective.
Extending $f$ to a Morse function on $X_n$ (technically the Morse function is $(f,0)$, if we use our definition of Morse function in Definition \[def:morse\]), to figure out the higher connectivity properties of the $X_n^{p\le f\le q}$ it suffices to look at $f$-descending links of vertices.
[@lee12 Lemma 3.52]\[cit:houghton\_f\_desc\_lk\_conn\] Let $\phi$ be a vertex in $X_n$. If $f(\phi)\ge 2n-1$ then the descending link of $\phi$ is $(n-2)$-connected.
In particular, Corollary \[cor:morse\] says that $X_n^{f\le q}$ is $(n-2)$ connected for $q\ge 2n-2$. Setting $Y{\mathrel{\mathop{:}}=}X_n^{f\le 3n-3}$, we have the whole setup of Definition \[def:bnsr\], namely $H_n$ acts properly and cocompactly on the $(n-2)$-connected complex $Y$ (it will become clear later why we use $3n-3$ instead of $2n-2$). Hence to understand $\Sigma^m(H_n)$, we can inspect filtrations of the form $(Y^{t\le \chi})_{t\in{\mathbb{R}}}$ for $\chi\in{\operatorname{Hom}}(H_n,{\mathbb{R}})$.
We have to explain what $\chi$ means as a function $Y \to {\mathbb{R}}$. The generating characters $\chi_i$ of $H_n$ measure the length of the eventual translations that every element of $H_n$ must have. Of course vertices of $X_n$ are also functions on $\{1,\dots,n\}\times{\mathbb{N}}$ that act as eventual translations, so the $\chi_i$ are all naturally defined on vertices of $X_n$, and extend affinely to $X_n$. Note that whereas $\chi_1+\cdots+\chi_n=0$ as characters on $H_n$, now more generally $\chi_1+\cdots+\chi_n=f$ as functions on $X_n$.
Let $\chi=a_1\chi_1+\cdots a_n\chi_n$ be a non-trivial character of $H_n$, so the $a_i$ are not all equal. Up to symmetry assume $a_1\le\cdots\le a_n$. Choose $1\le m(\chi)\le n-1$ maximal with $a_{m(\chi)}\ne a_n$. Since $\chi_1+\cdots+\chi_n=0$ as a character of $H_n$, without loss of generality $a_{m(\chi)+1}=a_n=0$. For instance, instead of $\chi_n$, we consider the equivalent character $-\chi_1-\cdots-\chi_{n-1}$. In general now the first $m(\chi)$ many coefficients of $\chi$ are negative, and all the coefficients from the $(m(\chi)+1)$st one on are zero.
Consider the function $h{\mathrel{\mathop{:}}=}(\chi,f)\colon Y \to {\mathbb{R}}$ with $(\chi,f)$ ordered lexicographically. This is a Morse function, à la Definition \[def:morse\], for reasons similar to those in the proof of Proposition \[prop:char\_morse\] for $F_n$. The key property is that the basis characters vary by $0$, $1$, or $-1$ between adjacent vertices. We now claim that all $h$-ascending links of vertices in $Y$ are $(m(\chi)-2)$-connected, and then Theorem \[thrm:houghton\_pos\] will follow from Corollary \[cor:morse\].
\[lem:houghton\_asc\_lk\_model\] Let $\phi$ be a vertex in $Y$. An adjacent vertex $\psi$ is in the $h$-ascending link of $\phi$ if and only if either
1. $\psi=t_i \circ \phi$ for some $m(\chi)+1\le i\le n$, or else
2. $\phi=t_i \circ \psi$ for some $1\le i\le m(\chi)$.
That $\psi$ is in the link of $\phi$ means there is $1\le i\le n$ such that $\psi=t_i \circ \phi$ or $\phi=t_i \circ \psi$. In the former case, $\psi$ has higher $\chi_i$ and $f$ values than $\phi$ and equal $\chi_j$ values (for $j\ne i$), and in the latter case $\psi$ has lower $\chi_i$ and $f$ values than $\phi$ and equal $\chi_j$ values (for $j\ne i$). Hence in the former case $\psi$ in the $h$-ascending link of $\phi$ if and only if $m(\chi)+1\le i\le n$, since then $\chi$ does not change but $f$ goes up, and in the latter case $\psi$ in the $h$-ascending link of $\phi$ if and only if $1\le i\le m(\chi)$, since then $\chi$ goes up.
Note that if $\phi=t_i \circ \psi$ and $\psi'=t_j \circ \phi$ ($i\ne j$) then $\psi$ and $\psi'$ share an edge in ${\operatorname{lk}}\phi$. In particular ${\operatorname{lk}}^{h\uparrow}_Y \phi$ is a join, of its intersection with ${\operatorname{lk}}^{f\uparrow}_Y \phi$ and its intersection with ${\operatorname{lk}}^{f\downarrow}_Y \phi$. Call the former the *ascending up-link* and the latter the *ascending down-link*. The two cases in Lemma \[lem:houghton\_asc\_lk\_model\] are thus complete descriptions of the vertices in, respectively, the ascending up-link and ascending down-link.
\[prop:houghton\_asc\_lk\_conn\] Let $\phi$ be a vertex in $Y$. Then ${\operatorname{lk}}^{h\uparrow}_Y \phi$ is $(m(\chi)-2)$-connected.
We know $0\le f(\phi)\le 3n-3$. First suppose $0\le f(\phi)\le 2n+m(\chi)-3$. The subscripts $i$ for which $t_i \circ \phi$ is ascending are those satisfying $m(\chi)+1\le i\le n$, so there are $n-m(\chi)$ of them, and since $(2n+m(\chi)-3) + (n-m(\chi)) = 3n-3$ we have in this case that the entire $f$-ascending link of $\phi$ in $X_n$ is contained in $Y$. This tells us that the ascending up-link of $\phi$ consists of the $(n-m(\chi)-1)$-simplex $\{t_{m(\chi)+1},\dots,t_n\}$, which is contractible, and hence ${\operatorname{lk}}^{h\uparrow}_Y \phi$ is contractible.
Now suppose $2n+m(\chi)-2 \le f(\phi)\le 3n-3$, so $Y$ does not contain the entire ascending up-link of $\phi$ in $X_n$, but rather only its $(3n-f(\phi)-4)$-skeleton. This is the $(3n-f(\phi)-4)$-skeleton of an $(n-m(\chi)-1)$-simplex, so it is $(3n-f(\phi)-5)$-connected. Since $f(\phi)\ge 2n+m(\chi)-2 \ge 2n-1$ though, in this case we have that the entire ascending down-link of $\phi$ in $X_n$ is contained in $Y$. Lemma \[lem:houghton\_asc\_lk\_model\] tells us that this $h$-ascending down-link is isomorphic to the $f$-descending link in $X_{m(\chi)}$ of a vertex with $f$ value equal to $f(\phi)$. Since $f(\phi)\ge 2n+m(\chi)-2 \ge 2m(\chi)-1$, this is $(m(\chi)-2)$-connected by Citation \[cit:houghton\_f\_desc\_lk\_conn\]. In this case, taking the join, we see that ${\operatorname{lk}}^{h\uparrow}_Y \phi$ is $((3n-f(\phi)-4)+(m(\chi)-1))$-connected, and hence $(3n-f(\phi)+m(\chi)-5)$-connected. The result now follows since $f(\phi)\le 3n-3$.
The superlevel sets $Y^{t\le \chi}$ are all $(m(\chi)-2)$-connected by Corollary \[cor:morse\] and Proposition \[prop:houghton\_asc\_lk\_conn\], so by Definition \[def:bnsr\], $[\chi]\in\Sigma^{m(\chi)-1}(H_n)$.
As for negative properties, i.e., Conjecture \[conj:houghton\_neg\], it is difficult in general to tell using Morse theory that a filtration is *not* essentially $(m-1)$-connected. Even if we know the ascending link of a vertex is not $(m-1)$-connected, we do not know whether gluing in that vertex served to kill a pre-existing $(m-1)$-sphere, or served to create a new $m$-sphere. For example if a vertex’s ascending link is two points, we do not know whether gluing in that vertex connects up two previous disconnected components, or creates a loop. This is basically what makes it so difficult to prove that character classes lie in $S(G)\setminus\Sigma^m(G)$; for example even when $G$ is metabelian this problem remains open in general.
As a remark, to show that $(Y^{t\le \chi})_{t\in{\mathbb{R}}}$ is not essentially $(m(\chi)-1)$-connected, it suffices to prove that $Y^{0\le\chi}$ is not $(m(\chi)-1)$-connected, by tricks for negative properties discussed in [@witzel15]. Also, thanks to how we have realized $\chi$ as a linear combination of the $\chi_i$ using non-positive coefficients, for any vertex $x\in X_n^{0\le\chi}$ the whole $f$-descending link of $x$ lies in $X_n^{0\le\chi}$. Hence $Y^{0\le\chi}$ is $(m(\chi)-1)$-connected if and only if $X_n^{0\le\chi}$ is. This reduces Conjecture \[conj:houghton\_neg\] to proving that $X_n^{0\le\chi}$ is not $(m(\chi)-1)$-connected, but this is still a hard problem when $m(\chi)>2$, beyond the scope of our present techniques.
As a final remark, one typical trick for deducing negative properties is finding a retract onto a more manageable quotient with negative properties. However, for $H_n$, every proper quotient $Q$ has $\Sigma^1(Q)=S(Q)$, and so it seems very unlikely this trick could work. To see this fact about such $Q$, we note that for $n\ge3$, $[H_n,H_n]=S_\infty$, the infinite symmetric group, and the second derived subgroup $H_n^{(2)}$ is $A_\infty$, the infinite alternating group. One can check that every non-trivial normal subgroup of $H_n$ contains $A_\infty$, so any $Q$ as above is a quotient of $H_n/A_\infty$. But every kernel of a character on $H_n$ becomes finitely generated when taken mod $A_\infty$, so indeed $\Sigma^1(Q)=S(Q)$.
[BLV[Ž]{}94]{}
M. Bestvina and N. Brady, *Morse theory and finiteness properties of groups*, Invent. Math. **129** (1997), no. 3, 445–470.
M. G. Brin and F. Guzm[á]{}n, *Automorphisms of generalized [T]{}hompson groups*, J. Algebra **203** (1998), no. 1, 285–348.
K.-U. Bux and C. Gonzalez, *The [B]{}estvina-[B]{}rady construction revisited: geometric computation of [$\Sigma$]{}-invariants for right-angled [A]{}rtin groups*, J. London Math. Soc. (2) **60** (1999), no. 3, 793–801.
R. Bieri and R. Geoghegan, *Sigma invariants of direct products of groups*, Groups Geom. Dyn. **4** (2010), no. 2, 251–261.
R. Bieri, R. Geoghegan, and D. H. Kochloukova, *The [S]{}igma invariants of [T]{}hompson’s group [$F$]{}*, Groups Geom. Dyn. **4** (2010), no. 2, 263–273.
M. R. Bridson and A. Haefliger, *Metric spaces of non-positive curvature*, Die Grundlehren der Mathematischen Wissenschaften, vol. 319, Springer, 1999.
A. Bj[ö]{}rner, L. Lov[á]{}sz, S. T. Vre[ć]{}ica, and R. T. [Ž]{}ivaljevi[ć]{}, *Chessboard complexes and matching complexes*, J. London Math. Soc. (2) **49** (1994), no. 1, 25–39.
J. Belk and F. Matucci, *Conjugacy and dynamics in [T]{}hompson’s groups*, Geom. Dedicata **169** (2014), 239–261.
[to3em]{}, *R[ö]{}ver’s simple group is of type $\textrm{F}_\infty$*, Preprint. arXiv:1312.2282, 2015.
R. Bieri, W. D. Neumann, and R. Strebel, *A geometric invariant of discrete groups*, Invent. Math. **90** (1987), no. 3, 451–477.
R. Bieri and B. Renz, *Valuations on free resolutions and higher geometric invariants of groups*, Comment. Math. Helv. **63** (1988), no. 3, 464–497.
K. S. Brown, *Finiteness properties of groups*, Proceedings of the [N]{}orthwestern conference on cohomology of groups ([E]{}vanston, [I]{}ll., 1985), vol. 44, 1987, pp. 45–75.
[to3em]{}, *Trees, valuations, and the [B]{}ieri-[N]{}eumann-[S]{}trebel invariant*, Invent. Math. **90** (1987), no. 3, 479–504.
K.-U. Bux, *Finiteness properties of soluble arithmetic groups over global function fields*, Geom. Topol. **8** (2004), 611–644 (electronic).
D. S. Farley, *Finiteness and [$\rm CAT(0)$]{} properties of diagram groups*, Topology **42** (2003), no. 5, 1065–1082.
V. Guba and M. Sapir, *Diagram groups*, Mem. Amer. Math. Soc. **130** (1997), no. 620, viii+117.
G. Higman, *Finitely presented infinite simple groups*, Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, 1974, Notes on Pure Mathematics, No. 8 (1974).
C. H. Houghton, *The first cohomology of a group with permutation module coefficients*, Arch. Math. (Basel) **31** (1978/79), no. 3, 254–258.
D. H. Kochloukova, *The [$\Sigma^m$]{}-conjecture for a class of metabelian groups*, Groups [S]{}t. [A]{}ndrews 1997 in [B]{}ath, [II]{}, London Math. Soc. Lecture Note Ser., vol. 261, Cambridge Univ. Press, Cambridge, 1999, pp. 492–502.
[to3em]{}, *Subgroups of constructible nilpotent-by-abelian groups and a generalization of a result of [B]{}ieri, [N]{}eumann and [S]{}trebel*, J. Group Theory **5** (2002), no. 2, 219–231.
[to3em]{}, *On the [$\Sigma^2$]{}-invariants of the generalised [R]{}. [T]{}hompson groups of type [$F$]{}*, J. Algebra **371** (2012), 430–456.
[to3em]{}, *Rank and deficiency gradients of generalized [T]{}hompson groups of type [$F$]{}*, Bull. Lond. Math. Soc. **46** (2014), no. 5, 1050–1062.
S. R. Lee, *Geometry of [H]{}oughton’s groups*, arXiv:1212.0257, 2012.
H. Meinert, *The homological invariants for metabelian groups of finite [P]{}rüfer rank: a proof of the [$\Sigma^m$]{}-conjecture*, Proc. London Math. Soc. (3) **72** (1996), no. 2, 385–424.
[to3em]{}, *Actions on [$2$]{}-complexes and the homotopical invariant [$\Sigma^2$]{} of a group*, J. Pure Appl. Algebra **119** (1997), no. 3, 297–317.
J. Meier, H. Meinert, and L. VanWyk, *Higher generation subgroup sets and the [$\Sigma$]{}-invariants of graph groups*, Comment. Math. Helv. **73** (1998), no. 1, 22–44.
[to3em]{}, *On the [$\Sigma$]{}-invariants of [A]{}rtin groups*, Topology Appl. **110** (2001), no. 1, 71–81, Geometric topology and geometric group theory (Milwaukee, WI, 1997).
M. Stein, *Groups of piecewise linear homeomorphisms*, Trans. Amer. Math. Soc. **332** (1992), no. 2, 477–514.
S. Witzel and M. C. B. Zaremsky, *The [$\Sigma$]{}-invariants of [T]{}hompson’s group [$F$]{}, via [M]{}orse theory*, Submitted. arXiv:1501.06682, 2015.
[^1]: Meaning that for all $t\in{\mathbb{R}}$ there exists $s\le t$ such that the inclusion $Y^{t\le h_\chi} \to Y^{s\le h_\chi}$ induces the trivial map in $\pi_k$ for all $k\le m-1$.
[^2]: A subcomplex is *full* if as soon as it contains a simplex’s vertices, it also contains the simplex.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'With a rapidly increasing number of devices connected to the internet, big data has been applied to various domains of human life. Nevertheless, it has also opened new venues for breaching users’ privacy. Hence it is highly required to develop techniques that enable data owners to privatize their data while keeping it useful for intended applications. Existing methods, however, do not offer enough flexibility for controlling the utility-privacy trade-off and may incur unfavorable results when privacy requirements are high. To tackle these drawbacks, we propose a compressive-privacy based method, namely RUCA (Ratio Utility and Cost Analysis), which can not only maximize performance for a privacy-insensitive classification task but also minimize the ability of any classifier to infer private information from the data. Experimental results on Census and Human Activity Recognition data sets demonstrate that RUCA significantly outperforms existing privacy preserving data projection techniques for a wide range of privacy pricings.'
address: |
Princeton University\
Department of Electrical Engineering\
Princeton, NJ, 08544, USA
bibliography:
- 'refs.bib'
nocite: '[@*]'
title: Ratio Utility and Cost Analysis for Privacy Preserving Subspace Projection
---
Compressive privacy, Subspace methods, Projection matrix, Principal/Discriminant component analysis
Introduction {#sec:intro}
============
With our daily activities moving online, vast amounts of personal information are being collected, stored and shared across the internet, often without the data owner’s knowledge. Even when the data owners trust data keepers such as Internet Service Providers and Statistics Bureaus to keep their personal information private, the data are often needed to be analyzed and released for Statistics, Commercial and Research purposes. This raises obvious concerns about the privacy of data contributors, as not only are the data vulnerable to inadvertent leakages, but also to malicious inference by other parties. Thus privacy-protection methods should be employed that allow data collectors and owners to control the types of information that can be inferred from their data.
Consider a scenario where mobile users upload their sensor readings to the cloud, which in turn trains a classifier that allows smartphones to identify their users from sensor readings in the background as in [@Shi]. This approach takes advantage of the vast storage and computation resources of the cloud. However, without proper processing the same data can be used to infer sensitive information about users, such as location, context and activities performed [@Christin]. This is especially alarming given the fact that private information about users may not only be inferred by the cloud but possibly by other users as well through classifiers, which may include training samples in them [@Lin].
A number of approaches based on data projection and/or noise addition have been proposed to preserve the statistics of the data for machine learning applications, while making privacy-sensitive information unavailable. Additive noise based randomization was proposed in [@RAgrawal], but was shown to be susceptible to reconstruction attacks using spectral properties of random noise and data [@HKargupta]. Liu et al. [@KunLiu] proposed projection of the data to a lower dimensional space via a Random Projection Matrix. Later, a more suitable system was proposed in [@BinLiu] for collaborative-learning, where the cloud trains a classifier with data from multiple users. Each user randomly generates a hidden Projection Matrix and adds variable levels of noise to projected samples before sending them to the cloud.
In [@SYKungDCA; @SYKungDUCA], Kung presented a supervised version of Principle Component Analysis (PCA) called Discriminant Component Analysis (DCA) in order to project the data into a lower dimensional space that maximizes the discriminant power as in Fisher Discriminant Analysis [@SYKungKernelBook]. The recent work of Diamantaras and Kung [@Kostas] inspired by this approach introduced another criterion called Multiclass Discriminant Ratio (MDR), and projects the data based on a pair of desirable and undesirable classification tasks.
Dimension reduction through data projection removes both application-relevant and privacy-sensitive information from the data. DCA and MDR attempt to remove as little application-relevant information as possible by optimizing the projection subspace for the intended classification task. Yet they do not offer any flexibility for finding a favorable trade-off between utility and privacy.
To address these problems, we propose a methodology called RUCA (Ratio Utility and Cost Analysis), which forms a bridge between DCA (utility driven projection) and MDR (privacy emphasized projection) and allows data owners to select a compromise between them. RUCA can be considered as a generalization of DCA and MDR, and it can also be extended to multiple privacy-sensitive classifications. Experimental results on Census and Human Activity Recognition data sets show that our methodology can provide better classification accuracies for the desired task while outperforming state-of-the-art privacy preserving data projection methods in terms of accuracies obtained from privacy-sensitive classifications.
Our methodology for privacy preservation is described in Section \[sec:method\], and it is formulated as the problem of maximizing separability of projected data for a desired classification task, while minimizing separability for undesirable classifications. We then present Generalized Eigenvalue Decomposition as a method for finding the optimal Projection Matrix that achieves this task. Our methods are tested on real data with possible utility and privacy classifications in Section \[sec:eval\] and are compared with other projection based privacy protection methods. Finally, we conclude in Section \[sec:conc\].
Methodology {#sec:method}
===========
Problem Statement
-----------------
For simplicity, we shall assume that there is a single privacy-sensitive classification on the data, though it is straightforward to generalize this to the case where there are multiple privacy-sensitive classifications. We assume that the data of our concern is fully represented by a set of $N$ $M$-dimensional vectors $\{\x_1,\x_2,\cdots,\x_N\}$. For the desired classification, which we name utility classification, we have a set of labels $y_i$ associated with the vectors $\x_i$. For an undesirable classification, which we name privacy classification, we have a set of labels $s_i$ associated with the vectors $\x_i$. There are two or more classes for each classification task, i.e. $y_i \in 1,\cdots,L$, $s_i \in 1,\cdots,P$, where $L$ and $P$ are the numbers of utility and privacy classes, respectively.
Let $\W$ be an $M \times K$ projection matrix where $K<M$ and $\z_i=\W^T\x_i$ denote the projection of a vector $\x_i$ to a $K$-dimensional subspace. Let $\X$ denote the $M \times N$ matrix whose columns correspond to the data entries $\x_i$ and $\Z$ denote the $K \times N$ matrix whose columns correspond to the projected entries $\z_i$. Given $\X$, our problem is to find a matrix $\W$ such that given the projected data matrix $\Z=\W^T\X$:
1. A classifier can achieve similar performance on the task of finding the labels $\{y_1,y_2,\cdots,y_N\}$, compared to the case where the full data matrix $\X$ is given.
2. Conversely, any classifier achieves poor performance, ideally as poor as random guessing, on the task of finding the set of labels $\{s_1,s_2,\cdots,s_N\}$.
Projection Method
-----------------
{width="6cm"}
\(a) Census
{width="6cm"}
\(b) HAR (Activity)
{width="6cm"}
\(c) HAR (Identity)
To achieve the task outlined above, we need to select a subspace such that separability between classes based on the utility labels $y_i$ is maximized, while separability between classes based on privacy labels $s_i$ is minimized. For utility driven dimension reduction, given the subspace dimension $K$, DCA [@SYKungDCA] involves searching for the projection matrix $\W_{DCA} \in \R^{M \times K}$: $$\W_{DCA} = \underset{\W:\W^T[\Sbar+\rho \I]\W=\I}{\text{arg\ max}}\operatorname{tr}(\W^T \SBu \W)
\label{eq:WDCA}$$ where $\operatorname{tr}(\cdot)$ is the trace operator and $\rho \I$ is a small regularization term added for numerical stability. $\Sbar$ is the center adjusted scatter matrix: $$\Sbar = \Xbar\Xbar^T=\sum_{i=1}^{N}[\x_i-\bmu][\x_i-\bmu]^T$$ where $\bmu$ denotes the mean of the samples $\{\x_i\}_{i = 1}^{N}$. $\Sbar$ is divided into two additive parts: $$\Sbar = \SBu+\SWu
\label{eq:Sbar}$$ where $\SBu$ and $\SWu$ are utility between-class and within-class scatter matrices, respectively. These are defined as $$\begin{aligned}
\SBu &= \sum_{c=1}^L N^u_c[\bmu-\bmu^u_c][\bmu-\bmu^u_c]^T \\
\SWu &= \sum_{c=1}^L \sum_{y_i=c} [\x_i-\bmu^u_c][\x_i-\bmu^u_c]^T\end{aligned}$$ where $\bmu^u_c$ is the mean and $N^u_c$ is the number of samples in utility class $c$, respectively. Privacy between-class scatter matrix $\SBp$ can be defined similarly: $$\SBp = \sum_{c=1}^P N^p_c[\bmu-\bmu^p_c][\bmu-\bmu^p_c]^T$$ where $\bmu^p_c$ is the mean and $N^p_c$ is the number of samples in privacy class $c$, respectively.
Optimal solution to the problem given in Equation \[eq:WDCA\] remains the same when $\Sbar$ is replaced with $\SWu$ due to the relationship given in Equation \[eq:Sbar\]. Even though Equation \[eq:WDCA\] applies more restrictive orthonormality constraints to the columns of the projection matrix $\W$, the subspace spanned by these columns constitutes an optimal solution for *Multiclass Discriminant Analysis* (MDA) criterion [@DudaPatternBook] (with an additional regularization term $\rho \I$): $$MDA=\frac{\det(\W^T \SBu \W)}{\det(\W^T (\SWu+\rho \I) \W)}$$ where $\det(\cdot)$ is the determinant operator.
In addition, an optimal solution to both of these problems can be derived from the first $K$ principal generalized eigenvectors of the matrix pencil $(\SBu,\Sbar+\rho \I)$ [@GHGolub].
*Multiclass Discriminant Ratio* (MDR) is a natural extension to MDA criterion for the case where there are two conflicting goals: To maximize separability for a utility classification problem and to minimize separability for a privacy classification problem [@Kostas]. It is defined as: $$MDR = \frac{\det(\W^T \SBu \W)}{\det(\W^T (\SBp+\rho \I) \W)}$$
Analogous to DCA and MDA, an optimal solution to MDR can be derived from the first $K$ principal generalized eigenvectors of the matrix pencil $(\SBu,\SBp+ \rho \I)$. Thus DCA and MDR, barring an orthonormality constraint on the columns of the projection matrix, are very similar and can both be solved via Generalized Eigenvalue Decomposition.
We shall add additional parameters to DCA to obtain a compromise between DCA and MDR, which we will call *Ratio Utility and Cost Analysis* (RUCA): $$\W_{RUCA} = \underset{\W:\W^T[\mathbf{S}_{RUCA}+\rho \I]\W=\I}{\text{arg\ max}}\operatorname{tr}(\W^T \SBu \W) \\$$ where $\mathbf{S}_{RUCA}$ is a privacy-regularized scatter matrix: $$\mathbf{S}_{RUCA}=\Sbar+\rho_{p} \SBp$$ where $\rho_p$ is a privacy parameter different from $\rho$.
Note that when $\rho_p$ is 0, this projection method becomes DCA and when $\rho_p$ is very large, it becomes MDR as the term $\rho_p \SBp$ dominates. By varying $\rho_p$, it is possible to establish a more favorable trade-off between utility and privacy than MDR. Additionally, RUCA can be generalized to multiple privacy classifications by including multiple between-class scatter matrices in the regularization: $$\mathbf{S}_{RUCA}=\Sbar+\sum_{i}^{} {\rho_{p}}_i {\SBp}_i$$
Finally, an optimal solution to RUCA can be derived from the first $K$ principal generalized eigenvectors of the matrix pencil $(\SBu,\mathbf{S}_{RUCA}+\rho \I)$. In other words, columns of the projection matrix $\W$ correspond to $K$ largest eigenvalues $\lambda_i$ satisfying the following relationship: $$\SBu \w_i = \lambda_i (\mathbf{S}_{RUCA}+\rho \I)\w_i
\label{eq:eigen}$$
In all the subspace optimization techniques described above, the left hand side of the characteristic equation remains the same as in Equation \[eq:eigen\]. Due to the fact that rank of $\SBu$ is at most $L-1$, there are at most $L-1$ non-zero eigenvalues associated with the generalized eigenvalue decompositions. In practice, another small regularization term $\rho' \I$ may be added to $\SBu$ to make it full rank, which will allow users to rank the columns of $\W$ in cases where $K \geq L$. As columns corresponding to eigenvalues ranking $L$ or lower don’t normally contribute to our criteria, they are expected to have little contribution to the effectiveness of utility classification.
Experimental Results {#sec:eval}
====================
Data Sets
---------
[|l|Y|Y|Y|Y|]{} Projection Method & Income & Marital Status & Gender\
Random Projection [@KunLiu] & 81.07 & 40.22 & 53.63\
MDR [@Kostas] & 81.93 & 36.64 & 54.40\
RUCA ($\rho_p=16$) & 82.99 & **35.87** & 51.21\
RUCA ($\rho_p=8$) & 83.30 & 37.05 & 51.17\
RUCA ($\rho_p=4$) & 84.07 & 40.40 & 51.31\
RUCA ($\rho_p=2$) & 84.97 & 45.80 & 51.61\
RUCA ($\rho_p=1$) & 85.67 & 50.66 & 52.07\
DCA [@SYKungDCA] & **86.24** & 58.41 & 52.49\
PCA & 81.06 & 38.20 & 55.59\
Full-Dimensional & 86.91 & 81.78 & 75.63\
\[table:Census\]
We have tested our approach with multiple applications on Census (Adult) and Human Activity Recognition (HAR) [@HAR] data sets, both of which are available at UCI Machine Learning Repository [@UCI]. For the Census data set we used Income as the utility classification where we try to classify an individual as with high- or low-income, parallel with the original purpose of the data set. Privacy classifications were chosen as Marital Status and Gender, both of which were given as categorical features in the original data. We grouped ‘Married-civ-spouse’, ‘Married-spouse-absent’ and ‘Married-AF-spouse’ into a single category called ‘Married’. ‘Divorced’, ‘Separated’ and ‘Widowed’ were grouped into a single category called ‘Used to be Married’. We left the ‘Never Married’ category as is.
We first removed the samples with missing features in the data set and randomly sampled the rest of the training and testing sets (separately) in order to create two sets in which all privacy classes have equal number of samples, i.e. numbers of males and females were equal in our training and testing sets, and so were the number of samples categorized as ’Married’, ’Never Married’ and ’Used to be Married’. All categorical features were turned to numerical ones via binary encoding, as we determined it to yield higher classification accuracies than one-hot encoding with this data. After these operations we had 10086 samples remaining in the training set and 4962 samples remaining in the testing set with 29 features.
In HAR data set, we had Activity and Identity as labels available to us, either of which can be utility or privacy based on the application. Therefore we tested for both cases. Activity had 6 types of labels: ‘Walking’, ‘Walking Upstairs’, ‘Walking Downstairs’, ‘Sitting’, ‘Standing’ and ‘Laying’. Identity, on the other hand, had 21 types of labels based on the individuals who contributed to the data.
Training and testing sets of the HAR data set consist of samples contributed by two disjoint sets of users. Therefore we extracted testing sets for Identity classification by randomly picking samples from the original training set. When Activity classification was chosen as utility, we tested Activity classification accuracy on the original testing set and Identity classification accuracy on the extracted testing set. The numbers of training, privacy testing and utility testing samples were 4011, 1890 and 2947, respectively, with 561 features.
When Identity classification was chosen as utility, we tested both Identity and Activity classification accuracies on the same testing set, which was extracted from the original training set. The numbers of training and testing samples were 4026 and 1890 respectively with 561 features. As with the Census data, we kept the number of samples in all privacy classes equal in all sets.
Results
-------
[|l|Y|Y|Y|Y|]{} Projection Method & Activity & Identity\
Random Projection [@KunLiu] & 56.14 & **14.73**\
MDR [@Kostas] & 89.68 & 22.53\
RUCA ($\rho_p=1000$) & 90.99 & 20.93\
RUCA ($\rho_p=100$) & 91.34 & 21.20\
RUCA ($\rho_p=10$) & 92.24 & 21.98\
RUCA ($\rho_p=1$) & **92.37** & 22.98\
DCA [@SYKungDCA] & 92.07 & 23.95\
PCA & 79.61 & 30.20\
Full-Dimensional & 93.01 & 62.64\
\[table:HARa\]
All our experiments were performed using RBF SVM on the original and projected data. Training and testing sets were separated as described in the last section before the experiments commenced. With the Census data set we performed 50 iterations at which we randomly picked 10% of the training samples. At each iteration and with each projection method, a 5-fold cross-validation grid search was performed to find the best parameters for training utility and privacy classifiers.
With the HAR data set we performed 50 iterations at which we randomly picked 25% of the training samples. Once again, optimal parameters for SVM-RBF were determined via 5-fold cross-validation at each iteration. PCA and Random Projection were also included in our experiments for comprehensiveness.
In order to compare RUCA’s performance with other projection methods, we adopt a simple performance criterion: $$Performance = Acc_{U}+\beta(1-Acc_{P})$$ where $Acc_U$ and $Acc_P$ denote the utility and privacy classification accuracies, respectively, and $\beta$ denotes the *Privacy Pricing*. Higher $\beta$ indicates that higher emphasis is placed on privacy, while $\beta=0$ indicates that all the emphasis is placed on utility.
Figure \[fig:ROC\] displays the utility-privacy trade-off curves obtained by progressively adding more components with each projection method. We stopped adding components as they started contributing predominantly to privacy classification. To obtain the results provided in Tables \[table:Census\], \[table:HARa\] and \[table:HARb\], we picked $K=1$, $K=5$ and $K=20$, respectively, because we had $L=2$, $L=6$ and $L=21$ for Income, Activity and Identity classification problems, respectively.
The curves in Figure \[fig:ROC\](a) demonstrate a trade-off between utility and privacy as the privacy parameter $\rho_p$ is increased. Even RUCA with a low privacy parameter achieves higher privacy levels than possible with DCA. RUCA with $\rho_p=1$ outperforms PCA and DCA when $\beta \geq 0.067$, whereas RUCA with $\rho_p=4$ outperforms MDR and all remaining methods for all privacy pricings. Based on the trade-off curves in Figures \[fig:ROC\](b) and \[fig:ROC\](c), RUCA outperforms both DCA and MDR on HAR data for all privacy pricings. Furthermore, RUCA outperforms all other methods in (b) for all privacy pricings and all other methods in (c) when $\rho_p \geq 0.226$. PCA and Random Projection, on the other hand, are seen to under-perform in all plots when the privacy pricing is high.
By comparing the curves in (b) and (c), it becomes apparent that Identity classification when Activity is private is much harder than Activity classification when Identity is private on HAR data. Steepness of the drops in (b) suggests that more utility performance can be obtained by sacrificing relatively little privacy, which is not the case in (c).
Results with $K=1$ for the Census data set are given in Table \[table:Census\]. Clearly, DCA alone reduces gender classification accuracy close to random guessing (50%) by sacrificing less than 1% (absolute) utility classification accuracy. Accordingly for this application, a nonzero privacy parameter $\rho_{p}$ was only applied to the between-class scatter matrix of Marital Status classification and privacy parameter was kept at 0 for Gender classification. The table demonstrates a clear utility-privacy trade-off as $\rho_p$ is increased, similar to Figure \[fig:ROC\](a). RUCA outperforms DCA when $\beta \geq 0.073$ and all other methods for all privacy pricings. Results indicate that a small privacy parameter $\rho_p$ provides significantly better privacy while sacrificing little utility classification performance, whereas with a large $\rho_p$ it is possible to get better utility classification performance for the same privacy classification performance as other methods.
Tables \[table:HARa\] and \[table:HARb\] show similar results for HAR data set when Activity classification and Identity classification are chosen as utility, respectively. Utility performance doesn’t immediately drop, though privacy classification accuracies decrease as $\rho_p$ is increased. Here RUCA outperforms all other methods for all privacy pricings, except for Random Projection as seen in Table \[table:HARa\]. Although Random Projection provides better privacy for HAR data set when Activity classification is chosen as utility, it only outperforms RUCA when $\beta \geq 5.621$, i.e. when much higher emphasis is placed on privacy.
[|l|Y|Y|Y|Y|]{} Projection Method & Identity & Activity\
Random Projection [@KunLiu] & 38.47 & 81.72\
MDR [@Kostas] & 52.57 & 73.46\
RUCA ($\rho_p=1000$) & 59.03 & **69.81**\
RUCA ($\rho_p=100$) & 59.05 & 69.84\
RUCA ($\rho_p=10$) & **59.06** & 70.21\
RUCA ($\rho_p=1$) & 58.91 & 74.70\
DCA [@SYKungDCA] & 58.52 & 79.41\
PCA & 50.07 & 90.37\
Full-Dimensional & 64.85 & 95.77\
\[table:HARb\]
Conclusion {#sec:conc}
==========
We have presented a novel subspace projection method that allows data offered by users in a collaborative learning environment to be used for the intended purpose, with minimal loss of private information. We formulated a new criterion called Ratio Utility and Cost Analysis, which combines utility driven DCA with privacy emphasized MDR. Our method allows users to define multiple undesirable classifications on their data and achieve better utility for a given level of privacy. Using publicly available Census (Adult) and Human Activity Recognition data sets, we have demonstrated that our approach can provide better classification performance for the intended task for an equally low privacy classification performance when compared with state-of-the-art methods. Future work will include the extension of RUCA to privacy preserving non-linear projections, as well as an optimization method for the privacy parameters.
| {
"pile_set_name": "ArXiv"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.